Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models

Non-Bac ktrac ki ng Sp ectrum of Degree-Corrected Sto c hastic Blo c k Mo dels Lennart Gulik ers ∗ , Marc Lelarge † , Lauren t Massouli ´ e ‡ August 27, 2018 Abstract Motiv ated by comm u nity detection, w e characterise the sp ectrum of the non-backtrac k ing matrix B in the Degree-Co rrected Stochastic Bloc k Mo del. Sp eciﬁcally , w e consider a random graph on n vertices partitioned into tw o asymp- totically equal-sized clusters. The vertices ha ve i.i.d. w eights { φ u } n u =1 with second moment Φ (2) . The intra-cluster connection probability for v ertices u and v is φ u φ v n a and the inter-cluster connection probability is φ u φ v n b . W e sho w that with high p robabilit y , the follo wing holds: The leading eigen v alue of the non-bac k trac k ing matrix B is asymptotic to ρ = a + b 2 Φ (2) . The second eigen v alue is asymptotic to µ 2 = a − b 2 Φ (2) when µ 2 2 > ρ , but asymptotically b ound ed by √ ρ when µ 2 2 ≤ ρ . All the remaining eigenv alues are asymptotically b ounded by √ ρ . As a result, a clustering positively-correlated with the true comm unities can b e obtained based on the second eigen vector of B in the regime where µ 2 2 > ρ. In a previous work w e obtained that detection is impossible when µ 2 2 < ρ, mean- ing th at t here o ccurs a phase-transition in the sparse regime of the Degree-Corrected Sto chas tic Block Model. As a corollary , w e obtain that D egree-Corrected Erd˝ os-R´ enyi graphs asymptotically satisfy the graph Riemann hyp othesis, a quasi-Ramanujan prop erty . A by-pro duct of our proof is a weak law of large n umb ers for local-functionals on Degree-Corrected Sto chastic Block Mo dels, which could b e of in d ep endent interest. 1 In tro duction The non-backtrac king matrix B of a graph G = ( V , E ) is indexed by the set of its orien ted edges ~ E = { ( u, v ) : { u, v } ∈ E } . F or e = ( e 1 , e 2 ) , f = ( f 1 , f 2 ) ∈ ~ E , B is deﬁned as B ef = 1 e 2 = f 1 1 e 1 6 = f 2 . This matrix w as introduced by Hashimoto [ 10 ] in 1989. W e study the spectru m of B when G is a random graph generated according to the Degree-Corrected Stochastic Blo ck Mod el (DC-SBM) [ 11 ]. W e c h aracterise its leading eigen va lues and corresp onding eigenv ectors when t h e num b er of vertices in G tend s to inﬁ nity . Our motiv ation stems from comm unity detection problems: exp erimen t s in [ 14 ] show that the sp ectral metho d based on the non-backtrac k ing matrix seems ∗ Microsoft Researc h - INRIA Joint Cen tre and ´ Ecole Normale Sup´ erieure, F rance. E-mail: lennart.gulike r s@inria.f r † INRIA Paris and ´ Ecole Normale Sup´ erieure, F rance. E- mail: marc.lelarge@ens.fr ‡ Microsoft Research - INRIA Join t Centre, F rance. E-mai l: laurent.massoulie@inria.fr 1 to work well on real datasets. W e test the robustness of this metho d and show in particular that, ab ov e a certain threshold, the second eigenv ector of B is correlated with th e underlying comm u nities. The DC-SBM [ 11 ] is an extension of t h e or dinary Stochastic Block Model (SBM) [ 8 ]. The latter mod el has as a drawbac k that vertices in the same communit y are stochastically indistinguishable and it th erefore fails to accurately describe netw orks with high h eterogeneit y . Compare t his to ﬁ t ting a straigh t line on intrinsicall y curved data, whic h is do omed to miss imp ortant information. The DC-S BM is a more realistic mod el: it allo ws for very general degree -sequences. The sp ecial case of t he DC-SBM under consideration h ere is deﬁned as follow s: It is a random graph on n ve rt ices partitioned in to tw o asymptotically equal-sized clusters. The vertices hav e b oun ded i.i.d. weigh ts { φ u } n u =1 with second moment Φ (2) . The intra-cluster connection probability for vertices u and v is φ u φ v n a and the in ter-cluster connection probability is φ u φ v n b , for tw o constants a, b > 0 . Note t hat those graphs are thus sparse, which is a challenging regime for community detection. I ndeed, in the or dinary SBM (obtained by putting φ 1 = . . . = φ n = 1), an instance of the graph migh t not contain en ough informatio n to distinguish b etw een the tw o clusters if the diﬀerence b etw een a and b is small. M ore precisely , reconstruction is impossible when ( a − b ) 2 ≤ 2( a + b ) [ 18 ]. Interesti ngly , positively-correlated recon- struction can b e obtained b y thresholding th e second-eigen vector of B [ 2 ] immediately above the threshold (i.e., ( a − b ) 2 > 2( a + b )). The SBM thus has a ph ase-transition in its sparse regime. Does the DC-SBM exhib it a simi lar beh aviour? W e show ed in an earlier w ork [ 6 ] that detection is imp ossible when ( a − b ) 2 Φ (2) ≤ 2( a + b ). In our current work we analyse the regime where ( a − b ) 2 Φ (2) > 2( a + b ). W e answer the follo wing questions: is detection p ossible in this regime and if so, can w e use again the non-bac kt racking matrix or do w e need to mo dify it? A priori this is uncle ar, b e c ause an algorithm solely b ase d on B c annot use any i nformation on the weights as input. Our main result sho ws that the sp ectral method b ased on the non-backtrac k ing matrix (thus the same metho d as in [ 2 ]) successfully detects communities in th e regime ( a − b ) 2 Φ (2) > 2( a + b ). Surprisingly , no mo diﬁ cation of th e matrix, nor information about the w eights is needed (compare this to th e adjacency matrix, which n eed s to b e adapted to the degree-corrected setting [ 7 ]), whic h shows th e robustness of the metho d. Moreov er as in the stand ard SBM, th e algorithm is optimal in the sense that it wo rk s all t he w a y do wn to th e d etectabilit y - threshold. Informally , w e ha ve the follo wing results: With high probability , the leading eigen- v alue of the non-backtrac k ing matrix B is asymptotic to ρ = a + b 2 Φ (2) . The second eigen va lue is asymptotic to µ 2 = a − b 2 Φ (2) when µ 2 2 > ρ , bu t asymptotically bound ed by √ ρ when µ 2 2 ≤ ρ . All the remaining eigenv alues are asymp totically b oun d ed by √ ρ . F urther, a clustering p ositiv ely- correlated wi th th e true communities can b e obtained based on the second eigen vector of B in the regime where µ 2 2 > ρ (i.e., precisel y when ( a − b ) 2 Φ (2) > 2( a + b )). A side-result is that Degree-Correc ted Erd˝ os-R´ enyi graphs asymptotically satisf y the graph Riemann hypothesis, a q uasi-Ramanujan prop erty . In our proof w e derive and use a wea k law of large num b ers for local-functionals on Degree-Corrected Sto chastic Block Mo dels, which could b e of in d ep endent interest. 1.1 Comm unit y detect ion backgroun d In th is pap er we are interested in communit y d etection: The problem of clustering vertices in a gra ph into groups of ”similar” n od es. In particular, th e graphs h ere are generated according to the DC-S BM and the goal is to retrieve the spin (or group- membership) of the no d es b ased on a single observ ation of the DC-SBM. 2 When the av erage degree of a vertex grows suﬃciently fast with th e size of the net- w ork (i.e., the av erage degree is Ω(log ( n ))), we sp eak ab out dense netw orks. Comm unity- detection is then wel l understo od and we consider instead sparse graphs where the a verage degree is boun ded by a constan t. This setting is more realis tic as most real netw orks are sparse, b ut is at the same time more challe n ging. Indeed, traditional metho d s based on the Adjacency or Lapla cian matrix w orking well in the dense cas e break down when employ ed in the sparse case. In the sparse regime, with high probability , at least a p ositiv e fraction of th e no des is isolated. Consequently , one cann ot hope to ﬁnd the community-mem b ership of al l vertices . W e th erefore address here the problem of ﬁnding a clustering that is p ositively correlated with the true comm un ity-structure. In [ 3 ] it wa s ﬁrst conjectured that a d etectabilit y p hase transition exists in the or di nary SBM: When ( a − b ) 2 > 2( a + b ), the b elief propagation algorithm wo u ld succeed in ﬁnd ing such a p ositiv ely correla t ed clustering. Conv ersely , due t o a lack of information, detection would b e imp ossible when ( a − b ) 2 ≤ 2( a + b ). In [ 18 ], imp ossibility of reconstruction when ( a − b ) 2 ≤ 2( a + b ) is sho wn for the SBM. This pap er builds further on a tree-reconstruction problem in [ 4 ]. The authors of [ 14 ] conjectured that detection using th e second eigen vector of B w ould succeed all the wa y do wn to the co njectured detectability th reshold. Tw o v ari- ants of this so-called sp ectral redemption conjecture w ere prov en b efore the w ork in [ 2 ] app eared: In [ 16 ] it is sho wn th at detection based on the second eigenv ector of a matrix counting self-a voiding path s in th e graph le ad s to consistent recov ery when  a − b 2  2 > a + b 2 . Indep endently , in [ 17 ], the authors prov e th e p ositive side of the conjecture by using a constructin g based on counting non-backtrac king paths in graph s generated according to the SBM. More recently , in [ 2 ] the sp ectral redemption conjecture is p ro ved. This work more- o ver determines the limits of comm u nity detection b ased on the non- backtrac king sp ec- trum in the p resence of an arbitrary number of comm unities. Here w e extend the w ork in [ 2 ] to the more general setting of the DC-SBM. 1.2 Quasi R aman ujan prop erty F ollo wing the deﬁnition introduced in [ 15 ], a k -regular graph is Ramanujan if its second largest absolute eigenv alue is no larger than 2 √ k − 1. In [ 9 ], a graph is said to satisfy the graph Riemann h yp othesis if B has no eigen v alues λ such that | λ | ∈ ( √ ρ B , ρ B ), where ρ B is the Perron-F roben ius eigenv alue of B . The graph Riemann hyp othesis can b e seen as a generalization of th e Ramanujan property , b ecause a regular graph satisﬁes the graph Riemann h yp oth esis if and only if it has the Ramanuja n property [ 9 , 19 ]. Now , put a = b = 1 to obtain a D egree-Corrected Erd˝ os-R´ enyi graph where vertices u and v are connected by an edge with probabilit y φ u φ v n . Our results imply that, with high probabilit y , ρ B = Φ (2) + o (1), while all other eigen v alues are in absolute v alue smaller than √ Φ (2) + o (1). Consequently , th ese Degree-Corrected Erd˝ os-R´ enyi graphs asymptotically satisfy the graph Riemann hypothesis. 1.3 Outline and main diﬀerences with ordinary SBM W e f ollo w th e same general approac h as in [ 2 ]. W e focus primarily on th e d iﬀerences and complications here: w e often omit or shorten the proof of a statement if it may b e prov en in a very similar w ay . 3 In Section 2 w e deﬁne the DC-SBM and state the assumptions we make. This is then follo we d by Theorem 2.1 on the sp ectrum of B and its consequ en ces for community detection, Theorem 2.2 . In Section 3 , we give the necessary bac k groun d on non-backtrac king matrices. F ur- ther, w e give an extension of th e Bauer-Fike Theorem, that ﬁrst appeared in [ 2 ]. In Section 4 w e give the pro of of Theorem 2.1 . It builds on Propositions 4.1 and 4.2 . Their proofs are deferre d to later sections. In Section 5 we consider tw o-type branching pro cess where t h e oﬀsp rin g distribution is go verned by a Po isson mixture to capture the weigh ts of the ver tices. W e associate tw o martingales to this pro cess and ex tend limiting results by Kesten and S tigum [ 12 , 13 ]. Ho eﬀding’s inequ alit y pla ys an important role here t o pro ve concentrations results for the w eigh ts. F urther, w e deﬁne a cross-generational fun ctional on t hese branching pro cesses that is correlated with the spin of the ro ot. In Section 6 we state a coupling b etw een local neigh b ourho o ds and the branc hing process with w eights in Section 5 . W e established this coupling in an earlier w ork [ 6 ], it is technically more invol ved th an the ordinary coupling on graphs with unit weig ht. It is crucial that the weigh ts in the graph and the branching pro cess are perfectly coupled. W e further establish a gro wth condition on the lo cal neigh b ourh oo ds, using a stochastic domination argument that is more inv olved than its analogue in unw eighed graphs. In S ection 7 w e deﬁn e lo cal functionals th at map graphs, together with their spins and weigh ts to the real numbers. W e establish, using Efron-Stein’s inequ alit y , a weak la w of large numbers for t h ose functionals, which could b e of indep endent interest. Part of the w ork here is again hidden in the coupling from [ 6 ]. In Section 8 w e apply th ose local functionals to establish Prop osition 4.1 . In S ection 9 w e decomp ose p ow ers of the matrix B as a sum of prod ucts. This tec h- nique app eared ﬁrst in [ 16 ] for matrices counting self-a voiding path s and w as elab orated in [ 2 ]. T o bound the n orm of the individual matrices occurring in the decomposition, w e use th e t race metho d initiated in [ 5 ]. In d oing so, w e need to b oun d the exp ectation of pro ducts of higher moments of the w eigh t s ove r certain paths. This is a signiﬁcan t complication with respect to the ordinary SBM, see Section 9.2 for a comparison. In Section 10 we pro ve that p ositively correlated clustering is p ossible based on the second eigenv ector of B , i.e., Theorem 2.2 . W e use the symmetry present in th e tw o-comm u nities setting here, whic h gets in general b roken in mo dels with more than tw o comm un ities. Detailed pro ofs of the statements in Sections 5 , 6 , 7 , 9 and 10 can b e found in App endices A - E . In eac h section we give a d etailed comparison with the ordinary S BM. 2 Main Results W e deﬁne our model more precisely and state the tw o main theorems. W e consider random graphs on n no des V = { 1 , . . . , n } d rawn according to the Degree-Corrected Sto chastic Blo ck Mod el [ 11 ]. The vertices are partitio ned into t wo clusters of sizes n + and n − by giving each vertex v a spin σ ( v ) from { + , −} . The vertices hav e i.i.d. we igh t s { φ u } n u =1 go verned by some la w ν with supp ort in [ φ min , φ max ] , where 0 < φ min ≤ φ max < ∞ are constants. W e den ote the second moment of the w eights by Φ (2) . An edge is drawn b etw een nod es u and v with probabilit y φ u φ v n a when u and v ha ve the same spin and with probability φ u φ v n b otherwise. The mo del parameters a and b are constant. W e assume that for some constant γ ∈ (0 , 1], n ± = n 2 + O ( n 1 − γ ) , (2.1) i.e., the comm u nities ha ve nearly equal size. 4 The ordinary SBM on tw o or more communities was ﬁrst in tro duced in [ 8 ], which is a generalization of Erd˝ os-R´ enyi graphs. The Degree-Corrected SBM appeared ﬁrst in [ 11 ]. General inhomogeneous random graphs are considered in [ 1 ]. Note that we retrieve the tw o-communities ordinary SBM by giving all nodes unit w eight. Local n eighbourho od s in the sparse graphs under consideration are tree-like with high p robabilit y . In [ 6 ] we sho wed that these trees are distributed according to a P oisson-mixtu re tw o-type branc h in g process, d etailed in Section 5 b elow. W e denote the mean progen y matrix of the branching p rocess by M = Φ (2) 2  a b b a  . (2.2) W e introduce the orthonormal v ectors g 1 = 1 √ 2  1 1  , and g 2 = 1 √ 2  1 − 1  , (2.3) together with the scalars ρ = µ 1 = a + b 2 Φ (2) , and µ 2 = a − b 2 Φ (2) . (2.4) Then, g k ( k = 1 , 2) are the left-eigenv ectors of M asso ciated t o eigen v alues µ k : g ∗ k M = µ k g ∗ k , k = 1 , 2 . (2.5) Note that ρ an d µ 2 are also asymptotically eigen v alues of the ex p ected adjacency matrix conditioned on the w eights. Indeed, if A denotes the adjacency matrix, and if ψ 1 and ψ 2 are the vectors deﬁned for u ∈ V by ψ 1 ( u ) = 1 √ 2 φ u and ψ 2 ( u ) = 1 √ 2 σ u φ u , then E [ A | φ 1 , . . . , φ n ] = a + b n ψ 1 ψ ∗ 1 + a − b n ψ 2 ψ ∗ 2 − a 1 n diag { φ 2 u } . Put b ψ i = ψ i k ψ i k 2 . Then, by the law of large num b ers, for i = 1 , 2,    E [ A | φ i , . . . , φ n ] b ψ i − µ i b ψ i    2 → 0 , in probabilit y , as n tends to ∞ . Finally , w e deﬁn e for k ∈ { 1 , 2 } , χ k ( e ) = g k ( σ ( e 2 )) φ e 2 , for e ∈ ~ E . (2.6) W e show that the c andidate eigenv e ctors ζ k = B ℓ B ∗ ℓ ˇ χ k k B ℓ B ∗ ℓ ˇ χ k k (2.7) are th en , for ℓ ∼ log ( n ), asymptotically aligned with the ﬁrst tw o eigen vectors of B . Note the w eight in ( 2.6 ), whic h is not present in the ordinary SBM. Theorem 2.1 (Degree-Co rrected Extension of Theorem 4 in [ 2 ]) . L et G b e dr awn ac- c or di ng to the DC-SBM such that assumption ( 2.1 ) holds. Assume that ℓ = C min log( n ) , with C min > 0 a smal l c onstant deﬁne d in ( 2.9 ) . If µ 2 2 > ρ , then, with high pr ob abili ty, t he eigenva lues λ i of B satisfy | λ 1 − ρ | = o (1) , | λ 2 − µ 2 | = o (1) , and, for i ≥ 3 , | λ i | ≤ √ ρ + o (1) . 5 F urther, if , for k ∈ { 1 , 2 } , ξ k is a normalize d eigenve ctor asso ciate d to λ k , then ξ k is asymptotic al ly aligne d with ζ k . The ve ctors ξ 1 and ξ 2 ar e asymptotic al ly ortho gonal. If ρ > 1 , and µ 2 2 ≤ ρ , t hen, wi th hi gh pr ob ability, the eigenvalues λ i of B satisfy | λ 1 − ρ | = o (1) , and, for i ≥ 2 , | λ i | ≤ √ ρ + o (1) . F urther, ξ 1 is asymptotic al ly aligne d with ζ 1 . Note that µ 2 2 > ρ implies ρ > 1, so that we consider the DC-SBM precisely in the regime where a gian t comp onent emerges, see [ 1 ]. In Theorem 2.2 w e sho w th at p ositiv ely correlated clustering is possible based on the second eigen vector of B when ab ove the feasibilit y threshold. More precisely , let b σ = { b σ ( v ) } v ∈ V b e estimators for the spins of the vertices. F ollo wing [ 3 ], we sa y th at b σ has p ositive ov erlap with the true spin conﬁguration σ = { σ ( v ) } v ∈ V if for some δ > 0, with h igh probabilit y , min p 1 n n X v = 1 1 b σ ( v )= p ◦ σ ( v ) > 1 2 + δ, where p runs o ver the identit y mapping on { + , −} and t he p ermutation that swaps + and − . Theorem 2.2 (Degree-Corrected Extension of Theorem 5 in [ 2 ]) . L et G b e dr awn ac c or di ng to the DC-SBM such th at ass um ption ( 2.1 ) holds and such that µ 2 2 > ρ . L et ξ 2 b e the se c ond normalize d eigenv e ctor of B . Then, ther e exists a deterministic thr eshold τ ∈ R , such that the fol lowing pr o c e dur e yields asymptotic al ly p ositive overlap: Put for vertex v ∈ V i ts estimator b σ ( v ) = + if P e : e 2 = v ξ 2 ( e ) > τ √ n and put b σ ( v ) = − otherwise. 2.1 Notation W e sa y that a sequ ence ( E n ) n of events happ ens with high probability (w.h.p.) if lim n →∞ P ( E n ) = 1. W e denote by k · k both the euclidean norm for v ectors and the op erator norm of matrices. I.e., for vec tors x = ( x 1 , . . . , x m ), and a matrix A , k x k = p P m u =1 x 2 u , and k A k = sup x, k x k =1 k Ax k . Belo w w e use that the neigh b ourh o o ds with a radius n o larger than C coupling log ρ ( n ) can b e coupled w.h.p. to certain branching pro cesses, where C coupling :=  1 3 − 1 9 log(4 /e )  ∧  1 80 ∧ γ 4  log ρ (2( a + b ) φ 2 max ) . (2.8) W e put, C min = 1 10 C coupling (2.9) and consider often neigh b ourho o ds of radius C min log ρ ( n ). W e denote the k -th momen t of the weigh t distribution ν by Φ ( k ) . I .e., E  φ k 1  = Φ ( k ) . The non-b ac k tracking property for orien ted edges e, f ∈ ~ E is d en oted by e → f , i.e., e 2 = f 1 and f 2 6 = e 1 . In proofs, we often use the symbols c 1 , c 2 , . . . for suitably chosen constants . 3 Preliminaries 3.1 Bac kground on non-bac ktr ac king matrix W e rep eat here t he most important observ ations made in [ 2 ]. 6 Firstly , for any k ≥ 1, B k ef counts the n umb er of non-backtrac king p aths b etw een orien ted edges e and f . A non-backtracking path is deﬁned as an oriented path b etw een tw o oriented edges such that n o ed ge is th e inv erse of its preceding edge, i.e., the path makes no backtrac k. Another imp ort observ ation is that ( B ∗ ) ef = B f e = B e − 1 f − 1 , where for oriented edge e = ( e 1 , e 2 ), w e set e − 1 = ( e 2 , e 1 ). If w e introduce the swap notation, for x ∈ R ~ E , ˇ x e = x e − 1 , e ∈ ~ E , then for any x, y ∈ R ~ E , and integ er k ≥ 0, h y , B k x i = h B k ˇ y , ˇ x i . Denote by P th e m atrix on R ~ E × ~ E , deﬁ ned on orien ted edges e, f as P ef = 1 f = e − 1 . Then, P x = ˇ x , P ∗ = P and P − 1 = P . F urther, ( B k P ) ∗ = P ( B ∗ ) k = B k P, so that w e can write the symmetric matrix B k P in diagonal form: Let ( σ k,j ) j b e eigen va lues of B k P ordered in decreasing order of absolute val u e, and let ( x k,j ) j b e the correspondin g orthonormal eigenvec tors. Then, B k = ( B k P ) P = X j σ k,j x k,j x ∗ k,j P = X j σ k,j x k,j ˇ x ∗ k,j = X j s k,j x k,j y ∗ k,j , (3.1) where s k,j = | σ k,j | and y k,j = sign( σ k,j ) ˇ x k,j . Since P is an orthogonal matrix, ( ˇ x k,j ) j form an orthonormal base for R ~ E and the term furthest on the right of ( 3.1 ) is th us the sp ectral v alue decomp osition of B k . Now , if B is irreducible and if ξ denotes the normali zed Pe rron eigen vector of B with eigen v alue λ 1 ( B ) > 0, we hav e λ 1 ( B ) = lim k →∞ ( σ k, 1 ) 1 /k , and lim k →∞ k x k, 1 − ξ k = 0 . In [ 2 ], the Bauer-Fik e Theorem is extended to pro ve the sp ectral claims w e make here. 3.2 Extension of Bauer-Fik e Theorem T ailored to our need s, we u se the follow ing p rop osition from [ 2 ]: Proposition 3.1 (Sp ecial case of Proposition 8 in [ 2 ]) . L et ℓ = C log ρ n , with C > 0 . L et A ∈ M n ( R ) , such that f or some ve ctors x 1 = x ℓ, 1 , y 1 = y ℓ, 1 , x 2 = x ℓ, 2 , y 2 = y ℓ, 2 ∈ R , some matrix R ℓ ∈ M n ( R ) , and some non-zer o c onstants ρ > µ 2 with µ 2 2 > ρ , A ℓ = ρ ℓ x 1 y ∗ 1 + µ ℓ 2 x 2 y ∗ 2 + R ℓ . (3.2) Assume ther e exist c 0 , c 1 > 0 such that for al l i ∈ { 1 , 2 } , h y i , x i i ≥ c 0 , k x i kk y i k ≤ c 1 . Assume further that h x 1 , y 2 i = h x 2 , y 1 i = h x 1 , x 2 i = h y 1 , y 2 i = 0 and for some c > 0 k R ℓ k < ρ ℓ/ 2 log c ( n ) . L et ( λ i ) 1 ≤ i ≤ n , b e the eigenvalues of A with | λ n | ≤ . . . ≤ | λ 1 | . T hen, | λ 1 − ρ | = o (1) , | λ 2 − µ 2 | = o (1) , and, for i ≥ 3 , | λ i | ≤ √ ρ + o (1) . F urther, ther e exist unit ei genve ctors ψ 1 , ψ 2 of A wi th eigenvalues λ 1 , r esp e ctively λ 2 such t hat || ψ i − x i k x i k || = o (1) . 7 Pr oo f . This is a sp ecial case of Proposition 8 in [ 2 ]. In the notation of the latter, we hav e ℓ ′ = ℓ − 2, θ 1 = ρ , θ 2 = µ 2 , θ = µ 2 , γ ≥ a + b | a − b | > 1. F urther c 0 ( c 0 γ k − c 1 ) + 4 c 1 ∧ c 2 0 2( ℓ ∨ ℓ ′ ) c 1 ∼ 1 log ρ n , and t hus k R ℓ k ≤ log c ( n )  √ ρ | µ 2 |  ℓ | µ 2 | ℓ = o (1) 1 log ρ n | θ | ℓ . T o pro ve the case µ 2 2 > ρ of Theorem 2.1 , w e thus need to ﬁnd candidate v ectors x 1 , x 2 , y 1 and y 2 that meet the cond itions in Proposition 3.1 and further verify that the remainder R ℓ has small norm. Note that the last condition is true whenever k B ℓ x k ≤ ρ ℓ/ 2 log c ( n ) for al l normalized x in span { y 1 , y 2 } ⊥ . T o address the case µ 2 2 ≤ ρ of Theorem 2.1 , we app eal to Proposition 7 in [ 2 ], whic h is very similar in spirit to Proposition 3.1 . 4 Pro of of Theorem 2.1 4.1 The case µ 2 2 > ρ . W e start with the case µ 2 2 > ρ . W e decomp ose, for some v ectors x 1 , y 1 , x 2 and y 2 and matrix R ℓ , B ℓ = ρ ℓ x 1 y ∗ 1 + µ ℓ 2 x 2 y ∗ 2 + R ℓ , and we show that the assumptions of Prop osition 3.1 are met. Let ℓ be as in Theorem 2.1 and recall χ k and ζ k from ( 2.6 ) and ( 2.7 ). F or ease of notation, we introduce for k ∈ { 1 , 2 } , ϕ k = B ℓ χ k k B ℓ χ k k , and θ k = k B ℓ ˇ ϕ k k . (4.1) Then, ζ k = B ℓ ˇ ϕ k θ k . T o prov e the main theorem, w e need th e follo wing tw o prop ositions. The pro ofs are deferred to Section 8 and 9.1 . The material in Section 8 builds on ingredients from Sections 6 - 7 , where we assume that µ 2 2 > ρ , un less stated otherwise. Proposition 4.1 (Degree-Corr ected Extension of Prop osition 19 in [ 2 ]) . Assume that µ 2 2 > ρ . L et ℓ = C log ρ n wi th 0 < C < C min . F or some b, c > 0 , with high pr ob abil ity, (i) b | µ ℓ k | ≤ θ k ≤ c | µ ℓ k | if k ∈ { 1 , 2 } , (ii) sign ( µ ℓ k ) h ζ k , ˇ ϕ k i ≥ b if k ∈ { 1 , 2 } , (iii) |h ϕ 1 , ϕ 2 i| ≤ (log n ) 3 n C − ( γ 2 ∧ 1 40 ) , (iv) |h ζ j , ˇ ϕ k i| ≤ (log n ) 3 n 3 2 C − ( γ 2 ∧ 1 40 ) if k 6 = j ∈ { 1 , 2 } . (v) |h ζ 1 , ζ 2 i| ≤ (log n ) 8 n 2 C − ( γ 2 ∧ 1 40 ) . Put H = sp an { ˇ ϕ 1 , ˇ ϕ 2 } , then Proposition 4.2 (Degree-Corrected Ext ension of Prop osition 20 in [ 2 ]) . L et ℓ = C log ρ n wi th 0 < C < C min . F or some c > 0 , with high pr ob ability, sup x ∈ H ⊥ , k x k =1 k B ℓ x k ≤ (log n ) c ρ ℓ/ 2 . (4.2) 8 Put ¯ ϕ 1 = ˇ ϕ 1 , and ¯ ϕ 2 = ˇ ϕ 2 −h ˇ ϕ 1 , ˇ ϕ 2 i ˇ ϕ 1 || ˇ ϕ 2 −h ˇ ϕ 1 , ˇ ϕ 2 i ˇ ϕ 1 || , then ¯ ϕ 1 and ¯ ϕ 2 are orthonormal and || ¯ ϕ 2 − ˇ ϕ 2 || = o ( ρ − ℓ/ 2 ) , due to Proposition 4.1 (iii). Let ¯ ζ 1 b e t he normalized orthogonal pro jection of ζ 1 on span { ¯ ϕ 2 } ⊥ . Similarly , let ¯ ζ 2 b e the normalized orthogonal pro jection of ζ 2 on span { ¯ ζ 1 , ¯ ϕ 1 } ⊥ . Then h ¯ ζ 1 , ¯ ζ 2 i = 0 and for i = 1 , 2, || ¯ ζ i − ζ i || = o ( ρ − ℓ/ 2 ) , as follo ws from Proposition 4.1 ( iv ) and ( v ). W e set D = θ 1 ¯ ζ 1 ¯ ϕ ∗ 1 + θ 2 ¯ ζ 2 ¯ ϕ ∗ 2 = ρ ℓ  θ 1 ρ ℓ ¯ ζ 1  ¯ ϕ ∗ 1 + µ ℓ 2  θ 2 µ ℓ 2 ¯ ζ 2  ¯ ϕ ∗ 2 . Note that, k B ℓ ¯ ϕ 1 k = θ 1 = O ( ρ ℓ ) , and k B ℓ ¯ ϕ 2 k = k B ℓ ((1 + o (1)) ˇ ϕ 2 + o (1) ¯ ϕ 1 ) k = O ( ρ ℓ ) . As a consequence, from Proposition 4.2 , k B ℓ k = O ( ρ ℓ ) . Since D ¯ ϕ i = B ℓ ˇ ϕ i + θ i ( ¯ ζ i − ζ i ) , k B ℓ ¯ ϕ i − D ¯ ϕ i k ≤ k B ℓ kk ¯ ϕ i − ˇ ϕ i k + θ i k ¯ ζ i − ζ i k = O  ρ ℓ/ 2  . Let P b e the orthogonal p ro jection on H = span { ¯ ϕ 1 , ¯ ϕ 2 } = span { ˇ ϕ 1 , ˇ ϕ 2 } , then k B ℓ P − D k = O  ρ ℓ/ 2  . Put R ℓ = B ℓ − D . W rite for y ∈ R ~ E with unit norm, y = h + h ⊥ , with h ∈ H and h ⊥ ∈ H ⊥ , then k R ℓ y k = k B ℓ h ⊥ + ( B ℓ − D ) h k ≤ sup x ∈ H ⊥ , k x k =1 k B ℓ x k + k B ℓ P − D k = O  log c ( n ) ρ ℓ/ 2  , (4.3) as follo ws f rom Prop osition 4.2 . W e ﬁnish by app lying Proposition 3.1 with x 1 = θ 1 ρ ℓ ¯ ζ 1 , y 1 = ¯ ϕ 1 , x 2 = θ 2 µ ℓ 2 ¯ ζ 2 , and, y 2 = ¯ ϕ 2 . 4.2 The case µ 2 2 ≤ ρ . In case µ 2 2 ≤ ρ , Prop osition 4.1 ( i ) and ( ii ) continue to hold for k = 1. F urth er, Proposition 4.1 ( iii ) as w ell as Proposition 4.2 conti nue to hold. W e need how ever the follo wing bound for k = 2: Proposition 4. 3. Assume that µ 2 2 ≤ ρ . L et ℓ = C log ρ n with 0 < C < C min . F or some c > 0 , with high pr ob ability, θ 2 ≤ (log n ) c ρ ℓ/ 2 . Using t h is prop osition and || ¯ ϕ 2 − ˇ ϕ 2 || = o ( ρ − ℓ/ 2 ) , w e get k B ℓ ¯ ϕ 2 k ≤ (log n ) c +1 ρ ℓ/ 2 . It remains to apply Proposition 7 from [ 2 ]. 5 P oisson-mixture t w o-t yp e branching pro cesses The p roofs of th e statements in this section are deferred to Ap p endix A . 9 5.1 A theorem of Kesten and Stigum W e consider the follo wing branching process starting with a single particle, the ro ot o , having spin σ o ∈ { + , −} and wei ght φ o ∈ [ φ min , φ max ] (whic h we often take ran- dom). The ro ot i s replaced in generatio n 1 by P oi  a 2 Φ (1) φ o  particles of spin σ o and P oi  b 2 Φ (1) φ o  particles of spin − σ o . F urth er, the we ights of those particles are i.i.d. distributed foll owing law ν ∗ , the si ze- biased v ersion of ν , deﬁned for x ∈ [ φ min , φ max ] by ν ∗ ([0 , x ]) = 1 Φ (1) Z x φ min y d ν ( y ) . (5.1) F or generation t ≥ 1, a particle with spin σ and w eight φ ∗ is replaced in the n ext generation by P oi  a 2 Φ (1) φ ∗  particles with the same spin and Po i  b 2 Φ (1) φ ∗  particles of the opp osite sign. Again, the w eights of the particles in generation t + 1 follow in an i.i.d. fashion the law ν ∗ . The oﬀspring-size of an ind ividual is thus a P oisson- mixture . W e use the n otation Z t =  Z t (+) Z t ( − )  for the population at generation t ≥ 1, where Z t ( ± ) is the number of t yp e ± particles in generation t . W e let ( F t ) t ≥ 1 denote the natural ﬁltration associated to ( Z t ) t ≥ 1 . W e associate tw o matrices to the branching p rocess, namely M deﬁned in ( 2.2 ), and, for a ro ot w ith w eigh t φ o , M φ o = Φ (1) φ o Φ (2) M . (5.2) Then, M is the tr ansition matrix for generations t ≥ 1 and later: E [ Z t +1 | Z t ] = M Z t , for all t ≥ 1 , (5.3) and M φ o describes th e transition from th e ro ot to the ﬁrst generation: E [ Z 1 | Z 0 , φ o ] = M φ o Z 0 , (5.4) where, by assumption Z 0 =  1 σ o =+ 1 σ o = −  . Note th at the diﬀerence betw een t he ro ot and later generations stems from t he fact that the root’s w eigh t is deterministic in the conditional exp ectation, whereas the weigh t of a particle in any later generation has exp ectation Φ (2) Φ (1) . Recall from ( 2.5 ) that g k ( k = 1 , 2) are the left-eigenv ectors of M asso ciated to eigen va lues µ k : g ∗ k M = µ k g ∗ k , k = 1 , 2 . (5.5) Note that M φ o has th e same left-eigenv ectors as M , while the corresp onding eigenv alues are given by µ k,φ o = Φ (1) φ o Φ (2) µ k , k = 1 , 2 . (5.6) Theorem 5.1 sh o ws that a Kesten-Stigum theorem applies to th e ”classical” branch- ing pro cess obtained after restricting the above p ro cess to generations 1 and later. Corollar y 5.2 , th en, joins this classical branching pro cess to the transition from the root to generation 1. W e further consider the vector Ψ t = (Ψ t (+) , Ψ t ( − )), con taining sums of the weigh ts, Ψ t ( ± ) = X u ∈ Y t 1 σ u = ± φ u , (5.7) where Y t is the set of p articles at distance t from the ro ot, and where φ u and σ u denote the w eight resp ectively spin of a particle u . N ote that Ψ t = Z t in case of un it weig hts. 10 The martingale Theorem 5.3 is not presen t in [ 2 ]. W e need it to b ound the vari an ce of the cross-generational functional deﬁned in Section 5.3 . Theorem 5.1 (Degree-Corrected Extension of Theorem 21 in [ 2 ]) . Assume that µ 2 2 > ρ . Put F t = { Z s } s ≤ t . F or any k = 1 , 2 ,  X k ( t ) := h g k , Z t i µ t − 1 k − h g k , Z 1 i  t ≥ 1 , is an F t -martingale c onver ging a.s. and i n L 2 such that for some C > 0 and al l t ≥ 1 , E [ X k ( t )] = 0 and E  X 2 k ( t ) | Z 1  ≤ C k Z 1 k 1 . Corollary 5.2. Assume that µ 2 2 > ρ . F or k = 1 , 2 , with the weight φ o = ψ o of the r oo t ﬁxe d, the se quenc e of r andom variables ( Y k,ψ o ( t )) t ≥ 1 =  h g k ,Z t i µ t − 1 k µ k,ψ o  t ≥ 1 c onver ges almost surely and in L 2 to a r andom variable Y k,ψ o ( ∞ ) wi th E [ Y k,ψ o ( ∞ ) | σ o ] = g k ( σ o ) . F urther, t he L 2 -c onver genc e takes plac e uniformly over al l ψ o . Theorem 5.3. Assume tha t µ 2 2 > ρ . Put G t = { Ψ s } s ≤ t . F or any k = 1 , 2 ,  X k ( t ) := h g k , Ψ t i µ t − 1 k − h g k , Ψ 1 i  t ≥ 1 , is an G t -martingale c onver ging a.s. and in L 2 such t hat for some C > 0 and al l t ≥ 1 , E [ X k ( t )] = 0 and E  X 2 k ( t ) | Z 1  ≤ C k Z 1 k 1 . 5.2 Quan titative v ersion of the Kesten-Stigum theorem W e now quantify th e growth of the p opulation size. The latter is deﬁned as S t = k Z t k 1 , t ≥ 0 , i.e., the num b er of individuals in generation t ≥ 0. Given S t , for t ≥ 1 we ha ve S t +1 = Poi S t X l =1 X ( l ) t ! , (5.8) where  X ( l ) t  l are i.i.d. copies of a + b 2 Φ (1) φ ∗ , where φ ∗ follo ws law ν ∗ . Note that in the or di nary Stochastic Blo ck Model (i.e., when all vertices hav e unit w eight), the argument of the P oisson random v ariables in ( 5.8 ) is deterministic, contrary to the general case under consideration here. Using ( 5.3 ) recursively in conjunction with ( 5.4 ), it follo ws that E [ S t | φ o ] = Φ (1) φ o Φ (2) ρ t , ∀ t ≥ 1 . In the follo wing lemma w e sho w that deviations from this av erage are small. In fact, there exists a constan t C suc h that for each t ≥ 0, S t is asymptotically stochastica lly dominated by an Exp onential random v ariable with mean C ρ t . An imp ortant ingredi- ent in th e pro of b elow is Hoeﬀdin g’s inequality , whic h we use to derive a concentration result for the parameter of t he P oisson v ariable in ( 5.8 ). Lemma 5.4 (Degree-Corrected Extension of Lemma 23 in [ 2 ]) . Assume S 0 = 1 . Ther e exist c, c ′ > 0 such that f or al l s ≥ 0 , P  ∀ k ≥ 1 , S k ≤ sρ k  ≥ 1 − c ′ e − cs . F rom Theorem 5.1 and Corollary 5.2 , we know that the d iﬀerent comp onents (ex- pressed in t h e basis of eigen vectors of M ) grow ex p onentiall y with rate ρ , resp ectively µ 2 . W e now qu antif y the error. R ecall Ψ t from ( 5.7 ). 11 5.2.1 The case µ 2 2 > ρ . Theorem 5.5 (Degree-Corrected Extension of Theorem 24 in [ 2 ]) . Assume that µ 2 2 > ρ . L et β > 0 , Z 0 = δ x and φ o = ψ o b e ﬁxe d. Ther e exists C = C ( x, β ) > 0 such that with pr ob ability at le ast 1 − n − β , for al l k ∈ { 1 , 2 } , al l 0 ≤ s < t ≤ C min log( n ) , with 0 ≤ s < t , |h g k , Z s i − µ s − t k h g k , Z t i| ≤ C ( s + 1) ρ s/ 2 (log n ) 3 / 2 , and, |h g k , Ψ s i − µ s − t k h g k , Ψ t i| ≤ C ρ s/ 2 (log n ) 5 / 2 . 5.2.2 The case µ 2 2 ≤ ρ . Theorem 5.6. Assume that µ 2 2 ≤ ρ . L et β > 0 , Z 0 = δ x and φ o = ψ o b e ﬁxe d. Ther e exists C = C ( x, β ) > 0 suc h that with pr ob abili ty at le ast 1 − n − β , for al l t ≥ 1 , |h g 2 , Ψ t i| ≤ C t 2 ρ t/ 2 (log n ) 2 , and, E  |h g 2 , Ψ t i| 2  ≤ C t 3 ρ t . 5.3 B ℓ B ∗ ℓ ˇ χ k on trees: a cross generation functional Recall our cla im that B ℓ B ∗ ℓ ˇ χ k are asymptotically aligned with the eigenv ectors of B . In the DC-SBM, the local-neighbourho o d of a vertex has with high probability a tree- lik e structure describ ed by the branching process ab ove. In th is section w e analyse B ℓ B ∗ ℓ ˇ χ k on trees. T o this end we deﬁne a cross-generational functional slightl y diﬀerent from its ana- logue in [ 2 ] due to the presence of w eigh t s: Q k,ℓ = X ( u 0 ,...,u 2 ℓ +1 ) ∈P 2 ℓ +1 g k ( σ ( u 2 ℓ +1 )) φ u 2 ℓ +1 , (5.9) where P 2 ℓ +1 is the set of paths ( u 0 , . . . , u 2 ℓ +1 ) ( of length 2 ℓ + 1) in the tree starting from u 0 = o with b oth ( u 0 , . . . , u ℓ ) and ( u ℓ , . . . , u 2 ℓ +1 ) non-backtrac king and u ℓ − 1 = u ℓ +1 . Note that these paths thus mak e a back-track exactly once at step ℓ + 1. Explicitly , w e hav e Q 1 ,ℓ = X ( u 0 ,...,u 2 ℓ +1 ) ∈P 2 ℓ +1 1 √ 2 φ u 2 ℓ +1 , (5.10) and, Q 2 ,ℓ = X ( u 0 ,...,u 2 ℓ +1 ) ∈P 2 ℓ +1 1 √ 2 σ ( u 2 ℓ +1 ) φ u 2 ℓ +1 . (5.11) Consider a tree T ′ and a leaf e 1 on it that has unique neigh b our, sa y , o . Then, if e is the oriented edges from e 1 to o and if B T ′ denotes the non-backtrac king matrix deﬁned on T ′ ,  B ℓ T ′ B ∗ ℓ T ′ ˇ χ k  ( e ) = Q k,ℓ + g k ( σ ( e 1 )) φ e 1 k Z ℓ k 1 , (5.12) where Q k,ℓ and Z ℓ are deﬁ n ed on th e tree T with ro ot o obtained after remo ving vertex e 1 from T ′ . In th e seq uel we analyse Q k,ℓ on t he branc h ing p rocess deﬁn ed abov e, starting with a single particle, the root o . Let V indicate the p articles of the random tree. Denote the spin of a particle v ∈ V by σ v ∈ { + , −} and its wei ght by φ v ∈ S . F or t ≥ 0, let Y v t denote the set of particles, includi ng their spins and weigh ts, of generation t from v in t he subtree of particles with common an cestor v ∈ V . Let 12 Z v t = ( Z v, + t , Z v, − t ) denote the num b er of ± vertices in generation t ; i.e., Z v, ± t = P u ∈ Y v t 1 σ ( u )= ± . Finally , let Ψ v t = (Ψ v, + t , Ψ v, − t ) , with Ψ v, ± t = P u ∈ Y v t 1 σ ( u )= ± φ u . W e rewrite Q k,ℓ into a more managea ble form: First observ e that ev ery path in P 2 ℓ +1 , after reac h in g u ℓ +1 , clim bs back to a dep th t from which it then ag ain mov es dow n the tree (that is , in the direction aw ay from th e root). Let us call th e vertex at leve l t (to which the path climbs bac k before descending again) u . Then, (if t 6 = 0) there are t wo children of u , say v and w suc h that w lies on the p ath b etw een u and u ℓ +1 and v is in betw een u and u 2 ℓ +1 . F or such ﬁ xed v and w in Y u 1 , only the children u 2 ℓ +1 ∈ Y v t determine the con trib u tion of a path to ( 5.9 ), regardless of the c hoice of u ℓ +1 ∈ Y w ℓ − t − 1 . Hence, for such ﬁxed u an d v, w ∈ Y u 1 and u 2 ℓ +1 , there are | Y w ℓ − t − 1 | = S w ℓ − t − 1 paths giving the same con tribu t ion to ( 5.9 ): Q k,ℓ = ℓ − 1 X t =0 X u ∈ Y o t L u k,ℓ , (5.13 ) where, for | u | = t ≥ 0, L u k,ℓ = X w ∈ Y u 1 S w ℓ − t − 1   X v ∈ Y u 1 \{ w } h g k , Ψ v t i   . (5.14) The follo wing theorem is an extension of Theorem 25 in [ 2 ]. The imp ortant observ a- tion is that, again, for Z 0 = δ τ ﬁxed,  Q 2 ,ℓ /µ 2 ℓ 2  ℓ conv erges to a random v ariable with mean a constant times τ , that is, the spin of th e root. Its proof uses b oth martingale theorems stated abov e. W e use the second martingale statement, which is not present in the ordinary SBM, to b ound the v ariance of Q k,ℓ : Theorem 5.7 (Degree-Corrected Extension of Theorem 25 in [ 2 ]) . Assume that µ 2 2 > ρ . L et Z 0 = δ x and φ o = ψ o b e ﬁxe d. F or k ∈ { 1 , 2 } ,  Q k,ℓ /µ 2 ℓ k  ℓ c onver ges in L 2 as ℓ tends to inﬁnity to a r andom variable wi th me an Φ (3) Φ (2) ρ µ 2 k − ρ µ k,ψ o g k ( x ) . F urther, the L 2 -c onver genc e takes plac e uniformly for al l ψ o . 5.3.1 The case µ 2 2 ≤ ρ . Theorem 5.8. Assume that µ 2 2 ≤ ρ . L et Z 0 = δ x and φ o = ψ o b e ﬁxe d. Ther e exists a c onstant C such that E  Q 2 2 ,ℓ  ≤ C ρ 2 ℓ ℓ 5 . 5.4 Orthogonalit y: D ecor relation in branch ing pro cess Again, as in [ 2 ], Q 1 ,ℓ and Q 2 ,ℓ are uncorrelated when deﬁned on t h e branching process above. The pro of presented here is simpler than the corresponding one in [ 2 ] and u ses that for the tw o communities-case, Q 1 ,ℓ and Q 2 ,ℓ are explicitly know n . The orthogonalit y of the candidate eigen vectors (i.e., ( iii ) − ( v ) in Prop osition 4.1 ) follo ws from t h is fact, see Prop osition 7.3 ( ii ) , ( iii ) and Proposition 7.4 ( ii ) b elow. Theorem 5.9 (Degree-Corrected Extension of 28 in [ 2 ]) . Assume that the spin σ o of the r o ot is dr awn uniformly fr om { + , −} . Then f or any ℓ ≥ 0 , E [ Q 1 ,ℓ Q 2 ,ℓ |T ] = 0 . 6 Coupling of lo cal neigh b ourho o d The p roofs of th e statements in this section are deferred to Ap p endix B . 13 6.1 Coupling Here we establish the connection b etw een neighbourh oo ds in the DC-S BM and the branching pro cess in Section 5 . W e established th is coupling in an earlier pap er [ 6 ] using an exploration pro cess that we rep eat below. Compared to the ordinary SBM, vertices are no w weigh ted, so that t w o facts need to be veriﬁed: At eac h step of the exploration p ro cess, unexplored vertices hav e a weigh t dra wn from a distribution close in total v ariation distance to ν . D etected v ertices on their turn fol low a la w close to ν ∗ . W e distinguish betw een t wo diﬀerent concepts of neighbourho od : the classica l neigh- b ourho od that is rooted at a v ertex and another neighbourho o d that starts with an edge. F or the latter, we need th e foll owing concept of oriente d d istance ~ d , whic h for e, f ∈ ~ E ( V ) is deﬁned as ~ d ( e, f ) = min γ ℓ ( γ ) where th e minimum is taken ov er all self-a voiding paths γ = ( γ 0 , γ 1 , · · · , γ ℓ +1 ) in G such that ( γ 0 , γ 1 ) = e , ( γ ℓ , γ ℓ +1 ) = f and for all 1 ≤ k ≤ ℓ + 1, { γ k , γ k +1 } ∈ E . and where f or such a path γ , ℓ ( γ ) = ℓ . N ote th at ~ d ( e, f ) = ~ d ( f − 1 , e − 1 ), i.e., ~ d is not symmetric. W e in tro du ce the vector Y t ( e ) = ( Y t ( e )( i )) i ∈{ + , −} where, for i ∈ { + , − } , Y t ( e )( i ) =    n f ∈ ~ E : ~ d ( e, f ) = t, σ ( f 2 ) = i o    , (6.1) w e den ote th e number of vertices at orien ted d istance t from e by S t ( e ) = k Y t ( e ) k 1 =    n f ∈ ~ E : ~ d ( e, f ) = t o    , and we deﬁne vector Ψ t ( e ) = (Ψ t ( e )( i )) i ∈{ + , −} where, for i ∈ { + , − } , Ψ t ( e )( i ) = X f ∈ ~ E : ~ d ( e,f )= t 1 σ ( f 2 )= i φ f 2 . (6.2 ) W e denote the classical neighbourho o d of radius r rooted at vertex v by ( G, v ) r and the neigh b ourho o d around oriented edge e = ( e 1 , e 2 ) b y ( G, e ) r . With the deﬁnitions above, w e then ha ve, ( G, e ) r = ( G ′ , e 2 ) r , where G ′ is the graph G with edge { e 1 , e 2 } remo ved. In particular, S t ( e ) = S ′ t ( e 2 ) , where S ′ t is S t deﬁned on G ′ . The tw o branc h ing pro cesses that describ e the neighbourh oo ds are almost identica l, the only diﬀerence lies in t he weigh t of the root: I n the classical branc hing processes, the w eight is dra wn according to distribution ν . In the b ranching process starting at an ed ge oriented tow ards, sa y , o , th e ro ot o h as wei ght go verned by ν ∗ . See Prop osition 6.1 b elo w. As a corollary we obtain an analogue of Theorem 5.5 for local neigh b ourho od s: the compon ents of Ψ t ( e ) gro w exp onential ly , see Coroll ary 6.3 . W e boun d the gro wth of S t in Lemma 6.4 . W e use a coupling argument to sho w that the w eights of the unexp lored vertices and selected vertices are sto chastica lly dominated by v ariables fol lowing law ν , resp ectively ν ∗ . This argument is not needed in the ordinary SBM. F ollo wing [ 17 ], we need to verif y that certain problematic stru ctures, namely tangles , are excluded with high probability . W e sa y that a graph H is tangle-fre e if all its ℓ − neighbourho o d s con tain at most one cy cle. If there is at least on e ℓ − neigh b ourho o d in H that contains more than one cycle, w e call H tangled. Note t h at in the sequ el w e shall often suppress the dep en dence on ℓ and simply call a graph tangle-free or tangled; the ℓ dep endence is then t acitly assumed. 14 F ollo wing standard arguments we establish in Lemma 6.5 that the graph is with high probability log ( n )-tangle free. W e prepare b y recalling the exploration process in [ 6 ] starting at a vertex: At time m = 0, c ho ose a v ertex ρ in V ( G ), where G is an instan t of t h e DC-SBM. Initially , it is the only active vertex: A (0) = { ρ } . All other v ertices are neu tral at start: U (0) = V ( G ) \ { ρ } . N o vertex has b een explored yet: E (0) = ∅ . At each time m ≥ 0 w e arbitrarily pick an active vertex u in A ( m ) t hat has shortest distance to ρ , and explore all its edges in { uv : v ∈ U ( m ) } : if uv ∈ E ( G ) for v ∈ U ( m ), then w e set v activ e in step m + 1, otherwise it remains neutral. At the end of step m , we designate u to be explorated. Thus, E ( m + 1) = E ( m ) ∪ { u } , A ( m + 1) = ( A ( m ) \ { u } ) ∪ ( N ( u ) ∩ U ( m )) , and, U ( m + 1) = U ( m ) \ N ( u ) . Proposition 6.1 (Degree-Corrected Ext ension of Prop osition 31 in [ 2 ]) . L et ℓ = C log ρ ( n ) , with C < C c oupling . L et ρ ∈ V and e = ( e 1 , e 2 ) ∈ ~ E . L et ( T , o ) b e the br anching pr o c ess wi th r o ot o deﬁne d in Se ction 5 , wher e the r o ot has spin σ ( v ) and weight governe d by ν . Simi larly, L et ( T ′ , o ) b e that same br anching pr o c ess, when the r oo t has spin σ ( e 2 ) and weight governe d by ν ∗ . Then, the total variation distanc e b e- twe en t he law of ( G, v ) ℓ and ( T , o ) ℓ go es to zer o as 1 − n − ( γ 2 ∧ 1 40 ) . The same is true for the di ﬀer enc e b etwe en the law of ( G, e ) ℓ and ( T ′ , o ) . Remark 6.2. Note that with the eve nt ( G, v ) ℓ = ( T , o ) ℓ , we me an t hat t he gr aph and tr e e ar e e qual, including their spins and weights . Se e [ 6 ] for mor e details. Corollary 6.3 (Degree-Corrected Extension of Corollary 32 in [ 2 ]) . Assume µ 2 2 > ρ . L et ℓ = C log ρ n with 0 < C < C c oupling . F or e ∈ ~ E ( V ) , we deﬁne the event E ( e ) th at for al l 0 ≤ t < ℓ and k ∈ { 1 , 2 } : |h g k , Ψ t ( e ) i − µ t − ℓ k h g k , Ψ ℓ ( e ) i| ≤ (log n ) 3 ρ t/ 2 . Then, with hi gh pr ob abili ty, the numb er of e dges e ∈ ~ E suc h that E ( e ) do es not hold is at most log( n ) n 1 − ( γ 2 ∧ 1 40 ) . Lemma 6.4 (Degree-Corrected Extension of Lemma 29 in [ 2 ]) . Ther e exist c, c ′ > 0 such t hat for al l s ≥ 0 and for any w ∈ [ n ] ∪ ~ E ( V ) , P  ∀ t ≥ 0 : S t ( w ) ≤ s ¯ ρ t n  ≥ 1 − ce − c ′ s . Conse quently, for any p ≥ 1 , ther e exists c ′′ > 0 such that E  max v ∈ [ n ] ,t ≥ 0  S t ( v ) ¯ ρ t n  p  ≤ c ′′ (log n ) p . Lemma 6. 5 (Degree-Corrected Extension of Lemma 30 in [ 2 ]) . L et ℓ = C log ρ ( n ) , with 0 < C < C c oupling . Then, w.h.p. , at m ost ρ 2 ℓ log( n ) vertic es have a cycle in their ℓ - neighb ourho o d. F urther, w. h.p., the gr aph is ℓ - tangle- f r e e. 6.2 Geometric gr owth Here w e sho w that for k ∈ { 1 , 2 } , h B ℓ χ k , δ e i gro ws nearly geometrically in t with rate µ k . Corollary 6.7 t hen establishes a b ound for r ≤ ℓ on sup h B ℓ χ k ,x i =0 , k x k =1 kh B r χ k , x ik crucial for the norm b ou n ds in Section 9 . Proposition 6.6 (Degree-Corrected Extension of Prop osition 33 in [ 2 ]) . Assume µ 2 2 > ρ . L et ℓ = C log ρ ( n ) , with 0 < C < C c oupling ∧  1 2 −  γ 4 ∧ 1 80  = C c oupling . F or e ∈ ~ E ( V ) , let ~ E ℓ b e the set of oriente d e dges such that either ( G, e 2 ) ℓ is not a tr e e or the event E ( e ) (deﬁne d in Cor ol lary 6.3 ) do es not hold. Then, w.h.p. for k ∈ { 1 , 2 } : 15 (i) | ~ E ℓ | ≪ (log n ) 2 n 1 − γ 2 ∧ 1 40 , (ii) for al l e ∈ ~ E \ ~ E ℓ , 0 ≤ r ≤ ℓ , |h B r χ k , δ e i − µ r − ℓ k h B ℓ χ k , δ e i| ≤ (log n ) 4 ρ r / 2 , (iii) for al l e ∈ ~ E ℓ , 0 ≤ r ≤ ℓ , |h B r χ k , δ e i| ≤ (log n ) 2 ρ r . Corollary 6.7 (Degree-Corrected Ex tension of Corollary 34 in [ 2 ]) . L et ℓ = C log ρ ( n ) , with 0 < C < C c oupling ∧  1 − γ 2 ∧ 1 40  ∧  γ 4 ∧ 1 80  = C c oupling . W. h.p. for any 0 ≤ r ≤ ℓ − 1 and k ∈ { 1 , 2 } : sup h B ℓ χ k ,x i =0 , k x k =1 kh B r χ k , x ik ≤ (log n ) 5 n 1 / 2 ρ r / 2 . 7 A w eak la w of large num b ers for lo cal func- tionals on the DC-SB M The p roofs of th e statements in this section are deferred to Ap p endix C . Here w e sho w that a w eak la w of large num b ers applies for lo cal functionals deﬁned on weighte d c olour e d ran d om graphs generated according to th e D C-SBM. By a weighte d c olour e d graph we mean a graph G = ( V , E ) together with maps σ : V → { + , − } and φ : V → [ φ min , φ max ]. F or v ∈ V , w e iden t ify σ ( v ) as the spin of v and φ ( v ) as its weig ht. W e denote by G ∗ the set of r o ote d weighte d c olour e d graphs. W e denote an element of G ∗ by ( G, o ): G = ( V , E ) is then a weigh ted coloured gra p h and o ∈ V is some distinguished vertex. A function τ : G ∗ → R is sai d to b e ℓ − lo cal if τ ( G, o ) dep ends only on ( G, o ) ℓ . T o derive the claimed w eak la w when G is dra wn according to the DC-S BM, we prepare with a v ariance b oun d for P n v = 1 τ ( G, v ), see Proposition 7.1 . The bou n d follo ws from the la w of total v ariance, V ar n X v = 1 τ ( G, v ) ! = E " V ar n X v = 1 τ ( G, v )      φ 1 , . . . , φ n !# + V ar E " n X v = 1 τ ( G, v )      φ 1 , . . . , φ n #! , together with an application of Efron-Stein’s inequality to b oth terms on the right. Note that E  P n v = 1 τ ( G, v )   φ 1 , . . . , φ n  is a constant in the or di nary SBM, whereas here it needs a careful analysis. The sample av erage 1 n P n v = 1 τ ( G, v ) concentrates then around E [ τ ( T , o )], where ( T , o ) is the branching pro cess from Section 5 , with root o ha v ing spin drawn uniformly from { + , − } and weigh t go verned by ν , see Prop osition 7.2 . The coupling, and in particular the matc h ing of the w eights, pla ys an imp ortant role in its pro of. In the next section we app ly the latter proposition to some sp eciﬁc functionals. Proposition 7.1 (D egree-Corrected Extension of Proposition 35 in [ 2 ]) . L et G b e dr awn ac c or ding to the DC-SBM. Ther e exists c > 0 such that if τ , ϕ : G ∗ → R ar e ℓ -lo c al, | τ ( G, o ) | ≤ ϕ ( G, o ) and ϕ is non-de cr e asing by the addition of e dges, th en V ar n X v = 1 τ ( G, v ) ! ≤ cnρ 2 ℓ  E  max v ∈ [ n ] ϕ 4 ( G, v )  1 / 2 . 16 Proposition 7.2 (D egree-Corrected Extension of Proposition 36 in [ 2 ]) . L et G b e dr awn ac c or ding to the D C -SBM. L et ( T , o ) b e the br anching pr o c ess fr om Se ction 5 , with r o ot o having spin dr awn uni formly f r om { + , −} and weight governe d by ν . L et ℓ = C log ρ ( n ) , with C < C c oupling . T her e exists c > 0 such that i f τ , ϕ : G ∗ → R ar e ℓ -lo c al, | τ ( G, o ) | ≤ ϕ ( G, o ) and ϕ is non-de cr e asing by the addition of e dges, th en E "      1 n n X v = 1 τ ( G, v ) − E [ τ ( T , o )]      # ≤ c 2 n − ( γ 2 ∧ 1 40 ) E  max v ∈ [ n ] ϕ 4 ( G, v )  1 / 4 ∨ E  ϕ 2 ( T , o )  1 / 2 ! + O ( n − γ ) (7.1) 7.1 Application with some sp eciﬁc lo cal functionals Here we consider h B ℓ χ 1 , B ℓ χ 2 i , h B 2 ℓ χ k , B ℓ χ j i , and h B ℓ B ∗ ℓ χ 1 , B ℓ B ∗ ℓ χ 2 i , q uantities occurring in Prop osition 4.1 . Explicitly , B ℓ χ k ( e ) = P f B ℓ ef g k ( σ ( f 2 )) φ f 2 , where w e recall th at B ℓ ef is the num b er of non-backtrac kin g w alks from e to f . Now, if t h e orien t ed ℓ − n eigh b ourho od of e is a tr e e , then B ℓ χ k ( e ) = h g k , Ψ ℓ ( e ) i . With this intuition in mind , we analyse likewi se expressions in Proposition 7.3 b elo w. Inspired by ( 5.12 ), which exp resses B ℓ B ∗ ℓ χ k on tr e es in t erms of the operator Q k,ℓ , w e extend the latter to an operator d eﬁned on general graphs. First, for e ∈ ~ E ( V ) and t ≥ 0, set Y t ( e ) = { f ∈ ~ E : ~ d ( e, f ) = t } . Then, for k ∈ { 1 , 2 } , we set P k,ℓ ( e ) = ℓ − 1 X t =0 X f ∈ Y t ( e ) L k ( f ) , (7.2) with L k ( f ) = X ( g,h ) ∈ Y 1 ( f ) \Y t ( e ); g 6 = h h g k , ˜ Ψ t ( g ) i ˜ S ℓ − t − 1 ( h ) , where ˜ Ψ t ( g ), ˜ S ℓ − t − 1 ( h ) = k ˜ Y ℓ − t − 1 ( h ) k 1 are the v ariables Ψ t ( g ), respectively S ℓ − t − 1 ( h ), deﬁned on the graph G where all edges in ( G, e 2 ) t hav e b een remov ed. N ote that, if ( G, e ) 2 ℓ is a tree, then ˜ Ψ s ( g ) = Ψ s ( g ) for s ≤ 2 ℓ − t . Compare P k,ℓ to Q k,ℓ in ( 5.9 ) and L k ( f ) to L k,ℓ in ( 5.14 ). Finally , deﬁne S k,ℓ ( e ) = S ℓ ( e ) g k ( σ ( e 1 )) φ e 1 . ( 7.3) W e then hav e an ext ension of ( 5.12 ), when ( G, e 2 ) 2 ℓ is a tree: B ℓ B ∗ ℓ ˇ χ k ( e ) = P k,ℓ ( e ) + S k,ℓ ( e ) . (7.4) W e analyse ( 7.4 ) in Proposition 7.4 b elow. 7.1.1 The case µ 2 2 > ρ . Proposition 7.3 (Degree-Corr ected Extension of Prop osition 37 in [ 2 ]) . Assume that µ 2 2 > ρ . L et ℓ = C log ρ n wi th 0 < C < C c oupling . (i) F or any k ∈ { 1 , 2 } , ther e exists c ′ k > 0 such that, in pr ob ability, 1 n X e ∈ ~ E h g k , Ψ ℓ ( e ) i 2 µ 2 ℓ k → c ′ k . (ii) F or any k ∈ { 1 , 2 } , ther e exists c ′′ k > 0 such that, in pr ob abil i ty, 1 n X e ∈ ~ E h g k , Y ℓ ( e ) i 2 µ 2 ℓ k → c ′′ k . 17 (iii) E         1 n X e ∈ ~ E h g 1 , Ψ ℓ ( e ) ih g 2 , Ψ ℓ ( e ) i         ≤ (log n ) 3 n 2 C − ( γ 2 ∧ 1 40 ) + n − γ . (iv) F or any k 6 = j ∈ { 1 , 2 } , E         1 n X e ∈ ~ E h g k , Ψ 2 ℓ ( e ) ih g j , Ψ ℓ ( e ) i         ≤ (log n ) 3 n 3 C − ( γ 2 ∧ 1 40 ) + n − γ . (v) F or any k ∈ { 1 , 2 } , i n pr ob ability 1 n X e ∈ ~ E h g k , Ψ 2 ℓ ( e ) ih g k , Ψ ℓ ( e ) i µ 3 ℓ k → c ′′′ k . Proposition 7.4 (Degree-Corr ected Extension of Prop osition 38 in [ 2 ]) . Assume that µ 2 2 > ρ . L et ℓ = C log ρ n wi th C < C c oupling . (i) F or any k ∈ { 1 , 2 } , ther e exists c ′′′′ k > 0 such that in pr ob ability 1 n X e ∈ ~ E P 2 k,ℓ ( e ) µ 4 ℓ k → c ′′′′ k . (ii) E         1 n X e ∈ ~ E ( P 1 ,ℓ ( e ) + S 1 ,ℓ ( e ))( P 2 ,ℓ ( e ) + S 2 ,ℓ ( e ))         ≤ (log n ) 8 n 4 C − ( γ 2 ∧ 1 40 ) 7.1.2 The case µ 2 2 ≤ ρ . Most of the above claims con tinue to hold if µ 2 2 ≤ ρ . W e treat the excep t ions here. Proposition 7.5. Assume that µ 2 2 ≤ ρ . L et ℓ = C log ρ n wi th 0 < C < C c oupling . Ther e exists some c > 0 , such th at w.h.p., 1 n X e ∈ ~ E h g 2 , Ψ ℓ ( e ) i 2 ρ ℓ ≥ c. Proposition 7.6. Assume that µ 2 2 ≤ ρ . L et ℓ = C log ρ n wi th C < C c oupling . Ther e exists c > 0 suc h that w.h.p., 1 n X e ∈ ~ E P 2 2 ,ℓ ( e ) ρ 2 ℓ log 5 ( n ) ≤ c. 8 Pro of op Prop ositions 4.1 and 4.3 W e introduce for k ∈ { 1 , 2 } the vector N k,ℓ , deﬁned on e ∈ ~ E as N k,ℓ ( e ) = h g k , Ψ ℓ ( e ) i . If ( G, e 2 ) ℓ is a t ree, then N k,ℓ ( e ) = h B ℓ χ k , δ e i , and we hav e a simila r expression for B ℓ B ∗ ℓ ˇ χ k in ( 7.4 ). No w, at most ρ 2 ℓ log( n ) v ertices hav e a cycle in their ℓ -neighbourho od (see Lemma 6.5 ). Therefore: 18 Lemma 8.1 (Degree-Corrected Extension of Lemma 39 i n [ 2 ]) . L et ℓ = C log ρ n wi th 0 < C < C min . T hen, w.h. p. k B ℓ χ k − N k,ℓ k = O  (log n ) 5 / 2 ρ 2 ℓ  = o  ρ ℓ/ 2 √ n  , k B ℓ B ∗ ℓ ˇ χ k − P k,ℓ − S k,ℓ k = O ((log n ) 4 ρ 4 ℓ ) and k B ℓ B ∗ ℓ ˇ χ k − P k,ℓ k = O ( ρ ℓ √ n ) . Pr oo f . The pro of of Lemma 39 in [ 2 ] can b e easily adapted t o the current setting. The k ey idea is pointed out a b ov e. It thus remains to boun d | ( B ℓ χ k − N k,ℓ )( e ) | and | ( B ℓ B ∗ ℓ ˇ χ k − P k,ℓ )( e ) | on edges e for which ( G, e 2 ) ℓ is not a tree. F or this, use that with h igh p robabilit y the graph is 2 ℓ -tangle free so that there are at most tw o non- backtrac king paths b etw een e and any edge at d istance ℓ . W e can thus in our calculations replace B ℓ χ k by N k,ℓ and B ℓ B ∗ ℓ ˇ χ k by P k,ℓ . F rom Propositions 7.3 and 7.4 , Proposition 4.1 then follo ws: Pr oo f of Pr op osition 4.1 . This pro of follo ws the corresp onding pro of in [ 2 ]. W e giv e th e key observ ations: ( i ) F rom Prop osition 7.3 ( i ), k N k,ℓ k ∼ √ nµ ℓ k and fro m Proposition 7.4 ( i ), k P k,ℓ k ∼ √ nµ 2 ℓ k . ( ii ) F rom Proposition 7.3 ( v ), |h N k,ℓ , N k, 2 ℓ i| ∼ nµ 3 ℓ k . ( iii ) F rom Prop osition 7. 3 ( iii ), |h N 1 ,ℓ , N 2 ,ℓ i| ∼ (log n ) 3 n 3 C − ( γ 2 ∧ 1 40 ) . ( iv ) F rom Proposition 7.3 ( iv ), |h N k, 2 ℓ , N j,ℓ i| ∼ (log n ) 3 n 4 C − ( γ 2 ∧ 1 40 ) . ( v ) F rom Prop osition 7.4 ( ii ), |h P 1 ,ℓ + S 1 ,ℓ , P 2 ,ℓ + S 2 ,ℓ i| ∼ (log n ) 8 n 5 C − ( γ 2 ∧ 1 40 ) . Proposition 4.3 follow s similarly from th e case µ 2 2 ≤ ρ treated in Section 7.1 : Pr oo f of Pr op osition 4.3 . This follo ws from Prop ositions 7.5 and 7.6 in conjunction with Lemma 8.1 . 9 Norm of non-bac ktrac king matrices The p roofs of th e statements in this section are deferred to Ap p endix D . In this section the pro duct ov er an empty set is deﬁned to b e one. It is conv enient to extend matri x B and vector χ k to th e set of directed edges on the c omplete graph, ~ E K ( V ) = { ( u, v ) : u 6 = v ∈ V } : F or e, f ∈ ~ E K ( V ), B ef is then extended to B ef = A e A f 1 e 2 = f 1 1 e 1 6 = f 2 , (9.1) where A is the adjacency matrix. F or eac h e ∈ ~ E K ( V ) we set χ k ( e ) = g k ( σ ( e 2 )) φ e 2 . F or integer k ≥ 1, e, f ∈ ~ E K ( V ), w e let Γ k ef b e the set of non-backtrac king w alks γ = ( γ 0 , . . . , γ k ) of length k fro m ( γ 0 , γ 1 ) = e to ( γ k − 1 , γ k ) = f on t he c omplete graph with vertex set V . By induction it follow s that ( B k ) ef = X γ ∈ Γ k +1 ef k Y s =0 A γ s γ s +1 . (9.2) Indeed, note that Q k s =0 A γ s γ s +1 is one when γ is a path in G and zero otherwise. T o eac h w alk γ = ( γ 0 , . . . , γ k ), w e associate the graph G ( γ ) = ( V ( γ ) , E ( γ )), with the set of vertices V ( γ ) = { γ i , 0 ≤ i ≤ k } and t h e set of edges E ( γ ) = {{ γ i , γ i +1 } , 0 ≤ i ≤ k − 1 } . F rom Lemma 6.5 , the graphs follow in g th e D C-SBM are tangle-free with high prob- abilit y . H ence, it makes sense to consider the subset F k +1 ef ⊂ Γ k +1 ef of tangle-free 19 non-backtrac king wa lk s on th e c omplete graph. I ndeed, if G is t angle-free, we n eed only consider the tangle-free paths in th e summation ( 9.2 ): ( B ( k ) ) ef = X γ ∈ F k +1 ef k Y s =0 A γ s γ s +1 , (9.3) and B k = B ( k ) for 1 ≤ k ≤ ℓ . Deﬁne for u 6 = v the c entr e d random v ariable A uv = A uv − φ u φ v n W σ u σ v , (9.4) where W =  a b b a  . Compare this to the SBM without degree -corrections in Section 10 . 1 of [ 2 ]: φ u = 1 for all u in the latter model. Using A w e shall attempt to center B k when the underlying graph G is tan gle-free through considering ∆ ( k ) ef = X γ ∈ F k +1 ef k Y s =0 A γ s γ s +1 . (9.5) F urther, w e set ∆ (0) ef = 1 e = f A e and B (0) ef = 1 e = f A e . (9.6) T o decompose ( 9.3 ), follo wing a decomp osition that app eared ﬁrst in [ 16 ], we u se ℓ Y s =0 x s = ℓ Y s =0 y s + ℓ X t =0 t − 1 Y s =0 y s ( x t − y t ) ℓ Y s = t +1 x s , with x s = A γ s γ s +1 and y s = A γ s γ s +1 on a path γ ∈ F k +1 ef : ℓ Y s =0 A γ s γ s +1 = ℓ Y s =0 A γ s γ s +1 + ℓ X t =0 t − 1 Y s =0 A γ s γ s +1  φ γ t φ γ t +1 n W σ γ t σ γ t +1  ℓ Y s = t +1 A γ s γ s +1 . Summing ov er all γ ∈ F ℓ +1 ef then giv es B ( ℓ ) ef = X γ ∈ F ℓ +1 ef ℓ Y s =0 A γ s γ s +1 + ℓ X t =0 X γ ∈ F ℓ +1 ef t − 1 Y s =0 A γ s γ s +1  φ γ t φ γ t +1 n W σ γ t σ γ t +1  ℓ Y s = t +1 A γ s γ s +1 = ∆ ( ℓ ) ef + ℓ X t =0 X γ ∈ F ℓ +1 ef t − 1 Y s =0 A γ s γ s +1  φ γ t φ γ t +1 n W σ γ t σ γ t +1  ℓ Y s = t +1 A γ s γ s +1 . (9.7) Consider the t wo p rod ucts in t h e su m mation ov er F ℓ +1 ef on the right of ( 9.7 ): W e can, for 1 ≤ t ≤ ℓ − 1, replace the summation o ver F ℓ +1 ef by summing o ver all pairs γ ′ = ( γ 0 , . . . , γ t ) ∈ F t eg and γ ′′ = ( γ t +1 , . . . , γ ℓ +1 ) ∈ F ℓ − t g ′ f for some g, g ′ ∈ ~ E ( V ) suc h that there exists a non-b ac k tracking path with one intermediate edge, on the c omplete graph, b etw een oriented ed ges g and g ′ (w e denote this prop erty by g 2 → g ′ ). How ever caution is needed, as this summation also includ es tang l e d p aths, namely those in the 20 sets { F ℓ +1 t,ef } ℓ t =0 . Where, for 1 ≤ t ≤ ℓ − 1, F ℓ +1 t,ef is deﬁ n ed as the collection of all tangle d paths γ = ( γ 0 , . . . , γ ℓ +1 ) = ( γ ′ , γ ′′ ) ∈ Γ ℓ +1 ef with γ ′ and γ ′′ as abov e. F or t = 0, F ℓ +1 0 ,ef consists of all non- backtrac king tangle d paths ( γ ′ , γ ′′ ) with γ ′ = ( e 1 ) and γ ′′ ∈ F ℓ g ′ f for any g ′ such that g ′ 1 = e 2 . F or t = ℓ , F ℓ +1 ℓ,ef is the set of non-backtrac king tangle d paths ( γ ′ , γ ′′ ) suc h that γ ′′ = ( f 2 ) and γ ′ ∈ F ℓ eg for some g ∈ ~ E ( V ) with g 2 = f 1 . W e rewrite ( 9.7 ) as B ( ℓ ) = ∆ ( ℓ ) + 1 n K B ( ℓ − 1) + 1 n ℓ − 1 X t =1 ∆ ( t − 1) K (2) B ( ℓ − t − 1) + 1 n ∆ ( ℓ − 1) b K − 1 n ℓ X t =0 R ( ℓ ) t , (9.8) where for e, f ∈ E K , K ef = 1 e → f φ e 1 φ e 2 W σ ( e 1 ) σ ( e 2 ) , ( 9.9) the weighte d non-backtrac k ing matrix on the c omplete graph (recall that e → f repre- sents the non-backtrac k ing p rop erty), b K ef = 1 e → f φ f 1 φ f 2 W σ ( f 1 ) σ ( f 2 ) , ( 9.10) K (2) ef = 1 e 2 → f φ e 2 φ f 1 W σ ( e 2 ) σ ( f 1 ) , (9.11) and where ( R ( ℓ ) t ) ef = X γ ∈ F ℓ +1 t,ef t − 1 Y s =0 A γ s γ s +1 φ γ t φ γ t +1 W σ ( γ t ) σ ( γ t +1 ) ℓ Y s = t +1 A γ s γ s +1 . (9.12) Indeed, ℓ − 1 X t =1 ∆ ( t − 1) K (2) B ( ℓ − t − 1) ! ef = ℓ − 1 X t =1 X g,g ′ ∆ ( t − 1) eg K (2) gg ′ B ( ℓ − t − 1) g ′ f = ℓ − 1 X t =1 X g,g ′ X γ ′ ∈ F t eg X γ ′′ ∈ F ℓ − t g ′ f t − 1 Y s =0 A γ ′ s γ ′ s +1 1 g 2 → g ′ φ γ ′ t φ γ ′′ 0 · W σ ( γ ′ t ) σ ( γ ′′ 0 ) ℓ − t − 1 Y s =0 A γ ′′ s γ ′′ s +1 , (9.13)  K B ( ℓ − 1)  ef = X g X γ ′′ ∈ F ℓ gf 1 e → g φ e 1 φ e 2 W σ ( e 1 ) σ ( e 2 ) A e 2 ,g 2 ℓ − 2 Y s =1 A γ ′′ s γ ′′ s +1 A f 1 f 2 , (9.14) and,  ∆ ( ℓ − 1) b K  ef = X g X γ ′ ∈ F ℓ eg A e 1 e 2 ℓ − 2 Y s =1 A γ ′ s γ ′ s +1 A g 1 f 1 1 g → f φ f 1 φ f 2 W σ ( f 1 ) σ ( f 2 ) (9.15) that is ex actly the splitting describ ed j ust b elow ( 9.7 ), where we also p ointed out the need to comp ensate for tangle d p aths occuring in ( 9.13 ), which is precisely the role of R ( ℓ ) t in ( 9.8 ). T o boun d ( 9.8 ), w e introdu ce W = 2 Φ (2) ( ρχ 1 ˇ χ ∗ 1 + µ 2 χ 2 ˇ χ ∗ 2 ) =  φ e 2 φ f 1 W σ ( e 2 ) σ ( f 1 )  ef , (9.16) 21 and, L = K (2) − W . (9.17) Note the presence of wei ghts in ( 9.16 ), hen ce our choice for the cand idate eigenv ectors. F urther, w e set for 1 ≤ t ≤ ℓ − 1, S ( ℓ ) t = ∆ ( t − 1) LB ( ℓ − t − 1) . ( 9.18) W e then hav e: Proposition 9.1 (Degree-Corrected Extension of Prop osition 13 in [ 2 ]) . If G is tangle- fr e e and x ∈ C ~ E ( V ) with norm smal ler than one, we have k B ℓ x k ≤ k ∆ ( ℓ ) k + 1 n k K B ( ℓ − 1) k + 1 n X j =1 , 2 2 µ j Φ (2) ℓ − 1 X t =1 k ∆ ( t − 1) χ j k||h ˇ χ j , B ℓ − t − 1 x i|| + 1 n ℓ − 1 X t =1 k S ( ℓ ) t k + φ 2 max ( a ∨ b ) k ∆ ( ℓ − 1) k + 1 n ℓ X t =0 k R ( ℓ ) t k . Pr oo f . Due to the tangle-freeness, B ℓ = B ( ℓ ) . F urth er K (2) = L + W and || K || ≤ φ 2 max ( a ∨ b ) n . In app endix D w e prov e the follow ing b ounds on the m atrices in Prop osition 9.1 : Proposition 9.2 (Degree-Corrected Ext ension of Prop osition 14 in [ 2 ]) . L et ℓ = C log ρ n wi th C < 1 . With hi gh pr ob abili ty, the fol l owing norm b ounds hold for al l k , 0 ≤ k ≤ ℓ , and i = 1 , 2 : k ∆ ( k ) k ≤ (log n ) 10 ρ k/ 2 , (9.19) k ∆ ( k ) χ i k ≤ (log n ) 5 ρ k/ 2 √ n, (9.20) k R ( ℓ ) k k ≤ (log n ) 25 ρ ℓ − k/ 2 , (9.21) k K B ( k ) k ≤ √ n (log n ) 10 ρ k , (9.22) and the fol lowing b ound holds for al l k , 1 ≤ k ≤ ℓ − 1 : k S ( ℓ ) k k ≤ √ n (log n ) 20 ρ ℓ − k/ 2 . (9.23) 9.1 Pro of of P rop osition 4.2 F rom Prop ositions 9.1 and 9.2 , the geometric growth in Corolla ry 6.7 together with the tangle-freeness due to Lemma 6.5 , the proof of Prop osition 4.2 follo ws: Let j ∈ { 1 , 2 } . If, for some vector x , h ˇ ϕ j , x i = 0, then h B ℓ χ j , ˇ x i = 0. Therefore, using Corolla ry 6.7 , sup k x k =1 , h ˇ ϕ j ,x i =0 h ˇ χ j , B ℓ − t − 1 x i = sup k x k =1 , h B ℓ χ j , ˇ x i =0 h B ℓ − t − 1 χ j , ˇ x i = sup k ˇ x k =1 , h B ℓ χ j , ˇ x i =0 h B ℓ − t − 1 χ j , ˇ x i ≤ log 2 ( n ) n 1 / 2 ρ ℓ − t − 1 2 . (9.24) With high probabilit y , the graph is ℓ − tangle free ( L emma 6.5 ). Thus, in voking Propositions 9.1 and 9.2 , with h igh probabilit y , sup x ∈ H ⊥ , k x k =1 k B ℓ x k ≤ log 10 ( n ) ρ ℓ 2 + n − 1 / 2 log 10 ( n ) ρ ℓ − 1 + c 1 log 8 ( n ) ρ ℓ 2 + n − 1 / 2 log 21 ( n ) ρ ℓ + c 2 log 10 ( n ) ρ ℓ 2 + n − 1 log 26 ( n ) ρ ℓ ≤ log c ( n ) ρ ℓ 2 , (9.25) 22 since C < 1. 9.2 Comparison with t he Sto c hastic Blo c k Mo del in [ 2 ] Putting φ u = 1 for all u , we retrieve exactly the same b ounds as i n the Stochastic Blo ck Mo del , that is equations (30) − (34) in [ 2 ]. Belo w we use the trace method and therefore path counti ng com b inatorial argu- ments to establish Prop osition 9.2 . In particular, we b ound the exp ectation of exp res- sions of the form E " 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s # , (9.26) for certain paths γ = ( γ 1 , . . . , γ 2 m ) with γ i = ( γ i, 0 , · · · , γ i,k ) ∈ V k +1 , where A is deﬁn ed in ( 9.4 ). In b ou n ding ( 9.26 ) the foll owing term o ccurs: Y u ∈ V ( γ ) Φ ( d u ) , where ( d u ) u are the degrees of the v ertices in a sp eciﬁc tree (or forest) spanning the path γ . See, for instance, ( D.4 ) and ( D.17 ) b elo w. Here lies a ma jor complication with respect to the Stochastic Block Model: those t erms are n ot present in the latter mo del. In ( D.8 ) and ( D.19 ) we ﬁnd | V ( γ ) | Y u =1 Φ ( d u ) ≤ C P u : d u > 2 ( d u − 2) 2  Φ (2)  | V ( γ ) |− n C , where C 2 > 1 is some constan t and where n C ≥ 1 is the number of comp onents on the path γ . T o compare this term with p ow ers of Φ (2) (whic h are present in pow ers of ρ = a + b 2 Φ (2) ), we bound P u : d u > 2 ( d u − 2), see in particular Lemma D.2 and Lemma D.5 . 10 Detection: Pro of of Theorem 2.2 The p roofs of th e statements in this section are deferred to Ap p endix E . W e need th e follow ing special case of a lemma in [ 2 ]: Lemma 10.1 (Special case of Lemma 40 in [ 2 ]) . Assume that t her e exists a function F : V → { 0 , 1 } such that in pr ob ability, for any i ∈ { + , − } , lim n →∞ 1 n n X v = 1 1 σ ( v )= i F ( v ) = f ( i ) 2 , wher e f : { + , −} → [0 , 1] is such that f (+) > f ( − ) . Then, assigning to e ach vertex a lab el b σ ( v ) = + i f F ( v ) = 1 and b σ ( v ) = − if F ( v ) = 0 , yields asymptotic al ly p ositive overlap with the true spins. Recall the eigen vector ξ 2 from Theorem 2.1 . Bel ow we use the function F : v 7→ 1 P e : e 2 = v ξ 2 ( e ) > τ √ n or F : v 7→ 1 P e : e 2 = v ξ 2 ( e ) ≤ τ √ n for some ﬁx ed parameter τ . W e verify also that ξ 2 is aligned with P 2 ,ℓ . I t is therefore useful to introdu ce the vector I ℓ , deﬁned elemen t - wise by I ℓ ( v ) = X e ∈ ~ E : e 2 = v P 2 ,ℓ ( e ) , (10.1) for v ∈ V . 23 F urther, put b c = a + b 2 (Φ (1) ) 2 Φ (3) Φ (2) ρ µ 2 2 − ρ µ 2 The follow ing lemma sho ws that I ℓ is correlated with the spins: Lemma 10.2 (Degree-Corrected Extension of Lemma 41 in [ 2 ]) . L et ℓ = C log ρ n with C < C c oupling and i ∈ { + , −} . Ther e exists a r andom variable Y i such that E [ Y i ] = 0 , E [ | Y i | ] < ∞ and for any c ontinuity p oint t of the distribution of Y i , in L 2 , 1 n n X v = 1 1 σ ( v )= i 1 I ℓ ( v ) µ − 2 ℓ 2 − b cg 2 ( i ) ≥ t → 1 2 P ( Y i ≥ t ) . Recall from Theorem 2.1 that the eigenv ector ξ 2 is asymptotically aligne d with B ℓ B ∗ ℓ ˇ χ 2 k B ℓ B ∗ ℓ ˇ χ 2 k , (10.2) where ℓ ∼ log ρ ( n ). Hence, for some unknown sign ω , the vector ξ ′ 2 = ω ξ 2 is asymptoti- cally close t o ( 10.2 ). F rom Lemma 8.1 we kn o w that B ℓ B ∗ ℓ ˇ χ 2 and P 2 ,ℓ are asymptoti- cally close. Consequ ently , prop erly renormalizing ξ ′ 2 will make it asymptotically close to P 2 ,ℓ , so t hat w e can replace P 2 ,ℓ in ( 10.1 ) by ξ ′ 2 . That is, we set for v ∈ V , I ( v ) = X e : e 2 = v s √ nξ ′ 2 ( e ) , with s = p c ′′′′ 2 the limit in Prop osition 7.4 . Then, I and I ℓ /µ 2 ℓ 2 are close, whic h leads to the foll owing lemma: Lemma 10.3 (Degree-Corrected Exten sion of Lemma 42 in [ 2 ]) . L et i ∈ { + , − } and b Y i b e as in L emma 10. 2 . F or any c ontinuity p oi nt t of the distr i bution of b Y i , in L 2 , 1 n n X v = 1 1 σ ( v )= i 1 I ( v ) − b cg 2 ( i ) ≥ t → 1 2 P  b Y i ≥ t  . Put for i ∈ { + , −} , X i = b Y i + b cg 2 ( i ) = b Y i + 1 √ 2 b ci . Then, for all t ∈ R that are contin uity p oints of the distribution of X i , the follo wing conv ergence h olds in probabilit y 1 n n X v = 1 1 σ ( v )= i 1 I ( v ) >t → 1 2 P ( X i > t ) . Since E [ X + ] > 0, the argument b elow (90) in [ 2 ] establishes the ex istence of a con tinuity p oint t 0 ∈ R such that P ( X + > t 0 ) > P ( X − > t 0 ). F urther, w e note that X + is in distribut ion equal t o − X − , a fact th at we use below. W e are no w in a position to apply Lemma 10.1 and thereby ﬁnishing the pro of of Theorem 2.2 : If ω = 1, then we d eﬁne F , for v ∈ V , by F ( v ) = 1 P e : e 2 = v ξ 2 ( e ) > t 0 s √ n = 1 I ( v ) >t 0 . Then, lim n →∞ 1 n n X v = 1 1 σ ( v )=+ F ( v ) = 1 2 P ( X + > t 0 ) =: f (+) 2 , and, lim n →∞ 1 n n X v = 1 1 σ ( v )= − F ( v ) = 1 2 P ( X − > t 0 ) =: f ( − ) 2 , 24 so that f (+) > f ( − ) and Lemma 10.1 applies. If, ho wev er, ω = − 1, then w e d eﬁne F , for v ∈ V , by F ( v ) = 1 P e : e 2 = v ξ 2 ( e ) ≤ t 0 s √ n = 1 − I ( v ) ≤ t 0 . Then, th is time, lim n →∞ 1 n n X v = 1 1 σ ( v )=+ F ( v ) = lim n →∞ 1 n n X v = 1 1 σ ( v )=+ 1 I ( v ) > − t 0 = 1 2 P ( X + > − t 0 ) =: f (+) 2 , since − t 0 is a continuit y point of X + , whic h follow s from th e fact that X + is in distri- bution equal to − X − and t 0 is a continuit y p oint of X − . Similarly , lim n →∞ 1 n n X v = 1 1 σ ( v )= − F ( v ) = 1 2 P ( X − > − t 0 ) =: f ( − ) 2 . Now , f (+) = P ( X + > − t 0 ) = 1 − P ( X − > t 0 ) > 1 − P ( X + > t 0 ) = P ( X − > − t 0 ) = f ( − ) , exactly the setting of Lemma 10.1 . A Pro ofs of Section 5 Pr oo f of T he or em 5.1 . F or 1 ≤ q < t , we h a ve Z t − M t − s Z s = t − 1 X u = s M t − u − 1 ( Z u +1 − M Z u ) , consequently , as g ∗ k M = µ k g ∗ k , h g k , Z t i µ t − 1 k = h g k , Z q i µ q − 1 k + t − 1 X u = q h g k , Z u +1 − M Z u i µ u k , (A.1) compare to (55) in [ 2 ]. Hence, ( X k ( t )) t ≥ 1 is an F t -martingale with mean 0. W e shall inv oke D o ob’s martingale conv ergence theorem to prov e th e assertion. That is, we shall sho w that for some C > 0 and all t ≥ 1, E  X 2 k ( t ) | Z 1  ≤ C k Z 1 k 1 . Let, for i, j ∈ { + , −} , Z s +1 ( i, j ) denote the num b er of type i ind iv iduals in gener- ation s + 1 which descend from from a type j particle in the s - th generation. Then, E  k Z s +1 − M Z s k 2 2 | Z s  = X i,j ∈{ + , −} E  ( Z s +1 ( i, j ) − M ij Z s ( j )) 2 | Z s ( j )  . (A.2) W e calculate ﬁrst, for some integer z ≥ 0, E  ( Z s +1 ( i, j ) − M ij Z s ( j )) 2 | Z s ( j ) = z  = E " z X l =1 ( Y l ( i, j ) − M ij ) ! 2      Z s ( j ) = z # = z X l =1 E  ( Y l ( i, j ) − M ij ) 2  , (A.3) 25 where ( Y l ( i, j )) z l =1 are i.i.d. copies of Poi  1 i = j a +1 i 6 = j b 2 Φ (1) φ ∗  , where φ ∗ follo ws the biased la w ν ∗ . Put c 1 = max i,j ∈{ + , −} E  ( Y l ( i, j ) − M ij ) 2  < ∞ . Then, plugging ( A.3 ) into ( A.2 ), w e obtain E  k Z s +1 − M Z s k 2 2 | Z s  ≤ 2 c 1 k Z s k 1 . Consequently , E  k Z s +1 − M Z s k 2 2 | Z 1  = E  E  k Z s +1 − M Z s k 2 2 | Z s  | Z 1  ≤ 2 c 1 E [ k Z s k 1 | Z 1 ] = 2 c 1 ρ s − 1 k Z 1 k 1 . (A.4) Com binin g the ab ov e with ( A.1 ) for q = 1, w e obtain E  X 2 k ( t ) | Z 1  = t − 1 X s =1 E  h g k , ( Z s +1 − M Z s ) i 2 | Z 1  µ 2 s k ≤ k g k k 2 2 t − 1 X s =1 E  k Z s +1 − M Z s k 2 2 | Z 1  µ 2 s k ≤ 2 c 1 k g k k 2 2 ∞ X s =1 ρ s − 1 µ 2 s k k Z 1 k 1 . (A.5) The assertion now foll ows u p on noting that C := 2 c 1 max k ∈{ + , −} k g k k 2 2 ∞ X s =1 ρ s − 1 µ 2 s k < ∞ , since ρ < µ 2 k . Pr oo f of C or ol l ary 5.2 . F rom Theorem 5.1 we know that there exists a random v ariable X k ( ∞ ) suc h that X k ( t ) := h g k , Z t i µ t − 1 k − h g k , Z 1 i a.s. → X k ( ∞ ) , as t → ∞ . Now, h g k , Z 1 i = µ k,ψ o h g k , Z 0 i + h g k , Z 1 − M ψ o Z 0 i . W e com bine this with the deﬁ nition of X k ( t ) to obtain h g k , Z t i µ k,ψ o µ t − 1 k = h g k , Z 0 i + h g k , Z 1 − M ψ o Z 0 i µ k,ψ o + X k ( t ) µ k,ψ o , where the right hand side is seen to conv erge in b oth senses to the random v ariable Y k,ψ o ( ∞ ) = h g k , Z 0 i + h g k , Z 1 − M ψ o Z 0 i µ k,ψ o + X k ( ∞ ) µ k,ψ o . Indeed,     h g k , Z t i µ k,ψ o µ t − 1 k − Y k,ψ o ( ∞ )     ≤ 1 µ k,φ min | X k ( t ) − X k ( ∞ ) | , for all ψ o . The un iform conv ergence follo ws, since E  | X k ( t ) − X k ( ∞ ) | 2   φ 0 = ψ o  = ∞ X z =0 E  | X k ( t ) − X k ( ∞ ) | 2   k Z 1 k = z  P ( k Z 1 k = z | φ 0 = ψ o ) ≤ e a + b 2 Φ (1) ( φ max − φ min ) E  | X k ( t ) − X k ( ∞ ) | 2 | φ 0 = φ max  (A.6) 26 Pr oo f of T he or em 5.3 . F or 1 ≤ q < t , we h a ve again h g k , Ψ t i µ t − 1 k = h g k , Ψ q i µ q − 1 k + t − 1 X u = q h g k , Ψ u +1 − M Ψ u i µ u k . (A.7) Since E [Ψ u +1 | Ψ u ] = M Ψ u , ( X k ( t )) t ≥ 1 is an G t -martingale with mean 0. W e show again that for some C > 0 and all t ≥ 1, E  X 2 k ( t ) | Z 1  ≤ C k Z 1 k 1 . Let, for i, j ∈ { + , − } , Ψ s +1 ( i, j ) denote t he sum o ver the weigh ts of typ e i in d ivid- uals in generation s + 1 which descend from a type j particle in the s -th generation. Then, E  k Ψ s +1 − M Ψ s k 2 2 | Z s  = X i,j ∈{ + , −} E  (Ψ s +1 ( i, j ) − M ij Ψ s ( j )) 2 | Z s ( j )  . (A.8) W e calculate ﬁrst, for some integer z ≥ 0, E  (Ψ s +1 ( i, j ) − M ij Ψ s ( j )) 2 | Z s ( j ) = z  = E     z X l =1   Y l ( i,j ) X l ′ =1 φ i ll ′ − M ij φ j l     2       Z s ( j ) = z   (A.9) where φ i ll ′ and φ j l are all independ ent and gov erned by the biased la w ν ∗ , and where ( Y l ( i, j )) z l =1 are i.i.d. copies of P oi  1 i = j a +1 i 6 = j b 2 Φ (1) φ ∗  , with φ ∗ go verned by ν ∗ . Thus the summands indexed by l are indep en d ent. W e hav e E   Y l ( i,j ) X l ′ =1 φ i ll ′ − M ij φ j l       Z s ( j )   = 1 i = j a + 1 i 6 = j b 2 Φ (1) Φ (2) Φ (1) Φ (2) Φ (1) − M ij Φ (2) Φ (1) = 0 Therefore, E  (Ψ s +1 ( i, j ) − M ij Ψ s ( j )) 2 | Z s ( j ) = z  = z X l =1 E     Y l ( i,j ) X l ′ =1 φ i ll ′ − M ij φ j l   2   . (A.10) Put c 1 = max i,j ∈{ + , −} E   P Y l ( i,j ) l ′ =1 φ i ll ′ − M ij φ j l  2  < ∞ . Then, plugging ( A.10 ) into ( A.8 ), w e obtain E  k Ψ s +1 − M Ψ s k 2 2 | Z s  ≤ 2 c 1 k Z s k 1 . Pr oo f of L emma 5.4 . F or k ≥ 1, put ǫ k = ρ − k/ 2 √ k and f k = k Y ℓ =1 (1 + ǫ ℓ ) . Due to con vergence of ( f k ) k , there ex ist constan t s c 0 , c 1 > 0 such that for all k ≥ 1, c 0 ≤ f k ≤ c 1 and ǫ k ≤ c 1 , (A.11) exactly as (57) in [ 2 ]. Recall the law of S k +1 from ( 5.8 ). W e shall ﬁ rstly derive a concentra t ion re- sult for P S k l =1 X ( l ) k , by using Ho eﬀding’s inequality . N ote that by deﬁ nition X ( l ) k ∈ 27 a + b 2 Φ (1) [ φ min , φ max ]. Put γ = ( a + b 2 Φ (1) ) 2 ( φ max − φ min ) 2 , th en Ho eﬀding’s equality reads P      n X l =1 X ( l ) k − nρ      ≥ t ! ≤ 2 exp  − 2 t 2 nγ  . Hence, in particular, P         sf k ρ k X l =1 X ( l ) k − sf k ρ k ρ       ≥ sf k ρ k ρ ǫ k +1 2   ≤ 2 exp  − f k ρ ( k + 1) 2 γ s  ≤ 2 exp ( − c 2 s ) , (A.12) for some c 2 > 0, du e to ( A.11 ). W e use the last result to obtain P  S k +1 > sf k +1 ρ k +1 | S k ≤ sf k ρ k  ≤ P   P oi   sf k ρ k X l =1 X ( l ) k   > sf k +1 ρ k +1   ≤ P  P oi  sf k ρ k +1  1 + ǫ k +1 2  > sf k +1 ρ k +1   1 − 2e − c 2 s  + 2e − c 2 s . (A.13) W e b ound sf k +1 ρ k +1 = sf k ρ k +1  1 + ǫ k +1 2  1 + ǫ k +1 1 + ǫ k +1 2 ≥ sf k ρ k +1  1 + ǫ k +1 2  (1 + c 3 ǫ k +1 ) , where c 3 = 1 2 1 1+max l ǫ l / 2 > 0. Combining t he last estimate with ( A.13 ) and the inequal- it y P (Poi ( λ ) ≥ λs ) ≤ e − λI ( s ) , where I : x 7→  x log x − x + 1 if x > 0; ∞ if x ≤ 0 , (A.14) entai ls that P  S k +1 > sf k +1 ρ k +1 | S k ≤ sf k ρ k  ≤ exp  − sf k ρ k +1  1 + ǫ k +1 2  I (1 + c 3 ǫ k +1 )  +2e − c 2 s . It remains to b oun d I (1 + c 3 ǫ k ) from b elo w. But, due t o the form of I , there exists a θ > 0 such that for x ∈ [0 , c 3 max k ǫ k ], I (1 + x ) ≥ θ x 2 . Consequently P  S k +1 > sf k +1 ρ k +1 | S k ≤ sf k ρ k  ≤ 3e − c 4 sk , for some constan t c 4 > 0. Hence, P  ∃ k : S k > sc 1 ρ k  ≤ ∞ X k =1 3 e − c 4 sk = 3 1 − e − c 4 s e − c 4 s , from whic h the statement follow s. Pr oo f of T he or em 5.5 . W e claim t h at there exist constan ts c, c ′ > 0 such that for an y s ≥ 0 P  k Z t +1 − M Z t k 2 > s k Z t k 1 / 2 1   F t  ≤ c ′ e − c ( s ∧ s 2 ) . (A.15) T o prov e ( A.15 ), w e shall employ Ho eﬀding’s in eq uality to establish a concentration result for λ + = Φ (1) 2   a Z + t X i =1 Φ + i + b Z − t X i =1 Φ − i   , (A.16) 28 and, λ − = Φ (1) 2   b Z + t X i =1 Φ + i + a Z − t X i =1 Φ − i   (A.17) around their resp ective means y + = E ∗  λ +  and y − = E ∗  λ −  , where (Φ ± i ) i are i.i.d. random v ariables with law ν ∗ , and where E ∗ [ · ] = E [ ·| Z t ] . This in conjunction with the classical tail b ound for Y d = Poi( λ ): P ( | Y − λ | > λs ) ≤ 2 e − λδ ( s ) , (A.18) where δ : x 7→ I (1 − x ) ∧ I ( 1 + x ), with I deﬁned in ( A.14 ), shall allo w us to prov e concentrati on of  Z + t +1 Z − t +1  =  P oi  λ +  P oi  λ −   around E ∗  Z + t +1 Z − t +1  =  y + y −  = M Z t . Let t + , t − > 0. Then, Ho eﬀding’s inequality gives P ∗         Z ± t X i =1 Φ ± i − Z ± t Φ (2) Φ (1)       ≥ t ±   ≤ 2 exp  − 2( t ± ) 2 Z ± t γ  , (A.19) where γ = ( φ min − φ max ) 2 , and where P ∗ ( · ) = P ( ·| Z t ) . Hence, P ∗  | λ + − y + | ≤ Φ (1) 2  at + + bt −   ≥ P ∗         Z + t X i =1 Φ + i − Z + t Φ (2) Φ (1)       ≤ t + ,       Z − t X i =1 Φ − i − Z − t Φ (2) Φ (1)       ≤ t −   ≥  1 − 2 ex p  − 2( t + ) 2 Z + t γ   1 − 2 ex p  − 2( t − ) 2 Z − t γ  . (A.20) Plugging t + = s √ y + √ 3Φ (1) a and t − = s √ y − √ 3Φ (1) b into the last equation leads to P ∗  | λ + − y + | ≤ s 2 k y k 1 / 2 1  ≥  1 − 2 ex p  − 4 / 3 (Φ (1) ) 2 a 2 γ y + Z + t s 2   1 − 2 ex p  − 4 / 3 (Φ (1) ) 2 a 2 γ y − Z − t s 2  ≥  1 − 2 e − c 0 s 2  2 ≥ 1 − 4 e − c 0 s 2 , (A.21) for some constan t c 0 > 0, since y ± Z ± t is boun ded aw ay from zero by some constant. W e use the last inequality to obtain P ∗  Z + t +1 − y + > s k y k 1 / 2 1  ≤ P ∗  P oi  y + + s 2 k y k 1 / 2 1  −  y + + s 2 k y k 1 / 2 1  > s 2 k y k 1 / 2 1  + 4 e − c 0 s 2 . (A.22) W e contin ue by inv oking ( A.18 ), P ∗  P oi  y + + s 2 k y k 1 / 2 1  −  y + + s 2 k y k 1 / 2 1  > s 2 k y k 1 / 2 1  ≤ 2 exp − ( y + + s 2 k y k 1 / 2 1 ) δ s 2 k y k 1 / 2 1 y + + s 2 k y k 1 / 2 1 !! . 29 W e note the existence of a θ > 0 suc h t h at for all x ∈ [0 , 1], δ ( x ) ≥ θx 2 , so t hat ( y + + s 2 k y k 1 / 2 1 ) δ s 2 k y k 1 / 2 1 y + + s 2 k y k 1 / 2 1 ! ≥ θ s 2 4 k y k 1 y + + s 2 k y k 1 / 2 1 ≥ c 2 ( s 2 ∧ s ) , for some constan t c 2 > 0, b ecause y + + s 2 k y k 1 / 2 1 ≤ max { 2 y + , s k y k 1 / 2 1 } . Similarly , to boun d P ∗  Z + t +1 − y + ≤ − s k y k 1 / 2 1  from abov e, we need t o estimate P ∗  P oi  y + − s 2 k y k 1 / 2 1  −  y + − s 2 k y k 1 / 2 1  ≤ − s 2 k y k 1 / 2 1  ≤ 2 exp − ( y + − s 2 k y k 1 / 2 1 ) δ s 2 k y k 1 / 2 1 y + − s 2 k y k 1 / 2 1 !! , (A.23) when y + > s 2 k y k 1 / 2 1 (if y + < s 2 k y k 1 / 2 1 , then Z + t +1 − y + > − s 2 k y k 1 / 2 1 , so t hat P ∗  Z + t +1 − y + ≤ − s k y k 1 / 2 1  = 0). W e distinguish b etw een t wo cases: Firstly , when y + − s 2 k y k 1 / 2 1 > s 2 k y k 1 / 2 1 , we h av e ( y + − s 2 k y k 1 / 2 1 ) δ s 2 k y k 1 / 2 1 y + − s 2 k y k 1 / 2 1 ! ≥ θ s 2 4 k y k 1 y + − s 2 k y k 1 / 2 1 ≥ θ k y k 1 y + s 2 4 ≥ c 3 s 2 , (A.24) for some constan t c 3 , due to ou r observ ation ab ov e. Secondly , in case y + − s 2 k y k 1 / 2 1 < s 2 k y k 1 / 2 1 , we use t h e existence of a θ ′ > 0 such that for all x ≥ 1, δ ( x ) ≥ θ ′ x : ( y + − s 2 k y k 1 / 2 1 ) δ s 2 k y k 1 / 2 1 y + − s 2 k y k 1 / 2 1 ! ≥ θ ′ k y k 1 / 2 1 2 s ≥ c 4 s, (A.25) for some constan t c 4 > 0. Com binin g ( A.22 ) - ( A.25 ), leads to P  | Z + t +1 − y + | > s k y k 1 / 2 1  ≤ 2  e − c 2 ( s 2 ∧ s ) + e − c 4 s + e − c 3 s 2  + 8 e − c 0 s 2 ≤ c 5 e − c 6 ( s 2 ∧ s ) . (A.26) An identical b ound h olds (after p ossibly redeﬁning the val u es of c 5 and c 6 ) for | Z − t +1 − y − | . Finally , noting that k y k 1 = ρ k Z t k 2 , w e hav e P  k Z t +1 − M Z t k 2 > s k Z t k 1 / 2 1   F t  ≤ P  | Z + t +1 − y + | ≥ s √ 2 k Z t k 1 / 2 1   F t  + P  | Z − t +1 − y − | ≥ s √ 2 k Z t k 1 / 2 1   F t  ≤ c ′ e − c ( s 2 ∧ s ) , (A.27) that is exactly claim ( A.15 ). W e are n ow in a p osition t o d erive a similar b ound as (59) in [ 2 ]: P  ∀ t ≥ 1 : k Z t +1 − M Z t k 2 ≤ u ( t + 1) log n k Z t k 1 / 2 1  ≥ 1 − c ′ X t ≥ 1 e − cut log n ≥ 1 − C ′ n − C u . (A.28) Recalling ( A.1 ), w e hav e, for s ≥ 1, |h g k , Z s i − µ s − t k h g k , Z t i| ≤ µ s − 1 k k g k k 2 t − 1 X u = s k Z u +1 − M Z u k 2 µ u k · 30 F rom Equation ( A.28 ) w e kn ow that, for all u ≥ 1, k Z u +1 − M Z u k 2 ≤ c 9 (log n )( u + 1) k Z u k 1 / 2 1 , (A.29) where c 9 is so large that A.29 holds with probability 1 − n − β . F urther, k Z h k 1 itself is b ounded by Lemma 5.4 : k Z h k 1 ≤ c 10 (log n ) ρ h , (A.30) also with probabilit y at least 1 − n − β . With the same p robabilit y , for k ∈ { 1 , 2 } , |h g k , Z s i − µ s − t k h g k , Z t i| ≤ c 11 (log n ) 3 / 2 µ s − 1 k t − 1 X u = s ( u + 1) √ ρ µ k u ≤ c 12 (log n ) 3 / 2 ( s + 1) ρ s/ 2 . (A.31) The p roof th e last claim, write h g k , Ψ s i − µ s − t k h g k , Ψ t i = Φ (2) Φ (1)  h g k , Z s i − µ s − t k h g k , Z t i  + ǫ s − µ s − t k ǫ t , (A.32) where, for s ≥ 1, ǫ s = g k (+)  Ψ s (+) − Z + s Φ (2) Φ (1)  + g k ( − )  Ψ s ( − ) − Z − s Φ (2) Φ (1)  . W e b ound ǫ t using ( A.19 ), P  ∀ t ≥ 1 : ǫ t ≤ t log n k Z t k 1 / 2 1  ≥ 1 − c 13 X t ≥ 1 e − c 14 t 2 log 2 n ≥ 1 − C ′ n − C u . ( A.33) So that, with probabilit y 1 − n − β , | ǫ s − µ s − t k ǫ t | ≤ c 15 log 5 / 2 ( n )  ρ s/ 2 + | µ k | s − t ρ t/ 2  ≤ c 16 log 5 / 2 ( n ) ρ s/ 2 , since | µ k | > ρ 1 / 2 . Pr oo f of T he or em 5.6 . W e h av e, k Ψ u +1 − M Ψ u k 2 ≤ Φ (2) Φ (1) k Z u +1 − M Z u k 2 + k Ψ u +1 − Φ (2) Φ (1) Z u +1 k 2 + k M  Ψ u − Φ (2) Φ (1) Z u  k 2 . (A.34) W e use ( A.19 ), to obtain that for any β > 0 (similar to ( A.28 )) P  ∀ t ≥ 1 : k Ψ t − Φ (2) Φ (1) Z t k 2 ≤ t log n k Z t k 1 / 2 1  ≥ 1 − n − β . (A.35) Com bing ( A.34 ), ( A.35 ) and ( A.28 ), giv es that with p robabilit y 1 − n − β , for all u ≥ 1, k Ψ u +1 − M Ψ u k 2 ≤ c 2 u log n k Z u k 1 / 2 1 . (A.36) W e can no w apply the arg u ment at th e end of Theorem 24 in [ 2 ]. The second claim follo ws by u sing th e last part of the proof of Theorem 24 in [ 2 ], where the v ariable U needs to b e rep laced by U = sup t ≥ 1 k Ψ t +1 − M Ψ t k 2 t k Z t k 1 / 2 1 . It is important here t hat E  U 4  = O (1), which is ensured by ( A.36 ). 31 Pr oo f of T he or em 5.7 . W e start by calculating the exp ectation and v ariance of P u ∈ Y o t L u k,ℓ conditional on F t (deﬁned in Theorem 5.1 ). W e use this to sho w t h at, as ℓ → ∞ , uni- formly for all ψ o , ¯ Q k,ℓ µ 2 ℓ k → Φ (3) Φ (2) ρ µ 2 k − ρ µ k,ψ o Y k,ψ o ( ∞ ) , (A.37) almost surely and in L 2 , where Y k,ψ o ( ∞ ) is deﬁned in Corol lary 5.2 , and where ¯ Q k,ℓ = ℓ − 1 X t =0 E F t X u ∈ Y o t L u k,ℓ . The latter is reminiscent of Q k,ℓ = ℓ − 1 X t =0 X u ∈ Y o t L u k,ℓ , and we show that ¯ Q k,ℓ and Q k,ℓ are in fact close in L 2 -distance: k ¯ Q k,ℓ − Q k,ℓ k = o ( µ 2 ℓ k ) . Consider for t ≥ 0 and ℓ ≥ t + 2, E F t ,Y o t ,Y u 1 L u k,ℓ = X ( v,w ) ∈ Y u 1 ,v 6 = w E F t ,Y o t ,Y u 1 S w ℓ − t − 1 E F t ,Y o t ,Y u 1 h g k , Ψ v t i = X ( v,w ) ∈ Y u 1 ,v 6 = w ρ w ρ ℓ − t − 2 φ v µ t k h g k , Z v 0 i (A.38) where ρ w = a + b 2 Φ (1) φ w , with φ w a random v ariable that follo ws la w ν ∗ . The second equality in ( A.38 ) follo ws after calculating E [Ψ v t | Y v 0 ] = Φ (2) Φ (1) E [ Z v t | Y v 0 ] = Φ (2) Φ (1) Φ (1) φ v Φ (2) M t Z v 0 = φ v M t Z v 0 , where the factor Φ (1) φ v Φ (2) accounts for the fact th at the ”parental” vertex v has determin- istic typ e φ v (and transitions are th us given by M φ v = Φ (1) φ v Φ (2) M ), w h ereas vertices in the later generations ha ve i.i.d. weigh ts (for which M is the t ransition matrix). Now, E F t ,Y o t L u k,ℓ = E F t ,Y o t E F t ,Y o t ,Y u 1 L u k,ℓ = E F t ,Y o t X ( v,w ) ∈ Y u 1 ,v 6 = w ρ w ρ ℓ − t − 2 φ v µ t k h g k , Z v 0 i = ρ ℓ − t − 2 µ t k E F t ,Y o t | Y u 1 | ( | Y u 1 | − 1) E F t ,Y o t ρ ∗ E F t ,Y o t φ ∗ h g k ,  1 σ ∗ =+ 1 σ ∗ = −  i , (A.39) where φ ∗ has la w ν ∗ , ρ ∗ is an i.i.d. cop y of a + b 2 Φ (1) φ ∗ and σ ∗ = σ u with probabilit y a a + b , and σ ∗ = − σ u with probabilit y b a + b (further, ρ ∗ , φ ∗ and σ ∗ are indep endent). W e th us ha ve E F t ,Y o t L u k,ℓ = ρ ℓ − t − 2 µ t k · ρ 2 u · ρ · Φ (2) Φ (1) · ( g k (1) c ( σ u , +) + g k (2) c ( σ u , − )) , (A.40) where ρ u = a + b 2 Φ (1) φ u (with φ u the w eight of u ) and for ( x, y ) ∈ { + , −} × { + , −} , c ( x, y ) = a a + b if x = y and c ( x, y ) = b a + b otherwise. Now , as g k is an eigen vector of M with eigen va lu e µ k , w e hav e ( g k (1) c ( σ u , +) + g k (2) c ( σ u , − )) = 2 a + b µ k Φ (2) h g k , Z u 0 i = µ k ρ h g k , Z u 0 i . 32 T ogether with ( A.39 ) this gives E F t ,Y 0 t L u k,ℓ = ρ ℓ − t − 2 µ t +1 k ρ 2 u Φ (2) Φ (1) h g k , Z u 0 i . (A.41) Summing ov er u ∈ Y o t using the last equation y ields E F t X u ∈ Y o t L u k,ℓ = E F t X u ∈ Y o t E F t ,Y 0 t L u k,ℓ = ρ ℓ − t − 2 µ t +1 k Φ (2) Φ (1) E F t X u ∈ Y o t ρ 2 u h g k , Z u 0 i = ρ ℓ − t − 2 µ t +1 k h g k , Z t i  a + b 2  2 Φ (2) ·  ψ o Φ (2) if t = 0; Φ (3) if t > 0. (A.42) W e lea ve it to the reader to verify that the same inequalit y holds for l = t + 1. W e con tinue by b ounding the v ariance of L u k,ℓ : V ar F t L u k,ℓ ≤ E F t ( L u k,ℓ ) 2 = E F t X ( v,w ) ∈ Y u 1 ,v 6 = w X ( v ′ ,w ′ ) ∈ Y u 1 ,v ′ 6 = w ′ S w ℓ − t − 1 S w ′ ℓ − t − 1 h g k , Ψ v t ih g k , Ψ v ′ t i ≤ E F t | Y u 1 | 2 E ∞ S 2 ℓ − t − 1 E ∞ h g k , Ψ v t i 2 , (A.43) where E ∞ [ · ] = max τ ′ ∈{ + , −} E [ ·| φ o = φ max , σ o = τ ′ ]. Now, E F t | Y u 1 | 2 ≤ c 0 . F rom Lemma 5.4 , we know that S k d ≤ Exp  c 1 ρ k  , hence E ∞ S 2 ℓ − t − 1 ≤ 2 c 2 1  ρ ℓ − t − 1  2 . T o b ound E ∞ h g k , Ψ v t i 2 , recall from Theorem 5.3 that E "  h φ k , Ψ t i µ t − 1 k − h g k , Ψ 1 i  2      Z 1 # ≤ C 2 k Z 1 k 1 . Consequently , as E [ k Z 1 k 1 ] is b ounded, E ∞ h g k , Ψ v t i 2 ≤ c 3 µ 2 t k . Returning t o ( A.43 ), w e hav e V ar F t X u ∈ Y o t L u k,ℓ ≤ c 4 µ 2 t k ρ 2( ℓ − t ) S t . (A.44) W e hav e ¯ Q k,ℓ = ℓ − 1 X t =0 E F t X u ∈ Y o t L u k,ℓ = ρ ℓ µ k h g k , Z 0 i ψ o + ℓ − 1 X t =1 ρ ℓ − t µ t +1 k h g k , Z t i Φ (3) Φ (2) = ρ ℓ µ k h g k , Z 0 i ψ o + Φ (3) Φ (2) ℓ − 1 X t =1 ρ ℓ − t µ 2 t k µ k,ψ o Y k,ψ o ( t ) , (A.45) where Y k,ψ o ( t ) is deﬁned in Corol lary 5.2 . W e consider ¯ Q k,ℓ µ 2 ℓ k = o (1) + Φ (3) Φ (2) ℓ − 1 X t =1  µ 2 k ρ  t − ℓ µ k,ψ o Y k,ψ o ( t ) , (A.46) 33 and verify our claim ( A.37 ). T o do so, split for arbitrary ﬁxe d ǫ > 0, ℓ − 1 X t =1 r t − ℓ Y k ( t ) = T ǫ − 1 X t =1 r t − ℓ Y k ( t ) + ℓ − 1 X t = T ǫ r t − ℓ Y k ( t ) , where r = µ 2 k ρ , Y k is shorthand notation for Y k,ψ o , and where T ǫ = min { t : ∀ s ≥ t , | Y k ( ∞ ) − Y k ( s ) | ≤ ǫ } . Then, T ǫ − 1 X t =1 r t − ℓ Y k ( t ) ≤ | sup t Y k ( t ) | r − ℓ r T ǫ T ǫ a.s. → 0 , as ℓ → ∞ , since ( Y k ( t )) t is con vergen t (uniformly in ψ o ) and hen ce b ounded . F urther, ℓ − 1 X t = T ǫ r t − ℓ Y k ( t ) = ℓ − T ǫ X u =1 ( Y k ( ∞ ) + O ( ǫ )) a.s. → ∞ X u =1 r − u ( Y k ( ∞ ) + O ( ǫ ) ) = 1 r − 1 ( Y k ( ∞ ) + O ( ǫ )) , (A.47) where the limit is taken for ℓ → ∞ . Since ǫ > 0 w as arbitrary , ( A.37 ) follo ws. L 2 -conv ergence follo ws from [ 2 ] (this con vergence t akes place un iformly for al l ψ o due to Theorem 5.1 ). F urther, that k ¯ Q k,ℓ − Q k,ℓ k = o ( µ 2 ℓ k ) ca n be established by follo wing the pro of in [ 2 ]. Indeed, from the latter pro of w e k now that, for some constan t c 6 indep endent of ψ o , k Q k,ℓ − ¯ Q k,ℓ k 2 ≤ ℓ − 1 X t =0          V ar F t   X u ∈ Y o t L u k,ℓ     1 / 2        2 ≤ c 5 ℓ X t =0 µ t k ρ ℓ − t k √ S t k 2 ≤ c 6 µ ℓ k ρ ℓ/ 2 , (A.48) due to the v ariance bou n d in ( A.44 ) and L emma 5.4 . Finally , com binin g th e u niform boun ds ( A.37 ) and ( A.48 ), en tails that     Q k,ℓ µ 2 ℓ k − Φ (3) Φ (2) ρ µ 2 k − ρ µ k,ψ o Y k,ψ o ( ∞ )     2 → 0 , uniformly for all ψ o . Pr oo f of T he or em 5.8 . U sing ( A.43 ) and Theorem 5.6 , w e have V ar F t L u k,ℓ ≤ c 1 ρ 2( ℓ − t ) t 3 ρ t . Plugging this b ound , together with ( A.42 ) here, into (66) in [ 2 ] establishes the clai m. Pr oo f of T he or em 5.9 . R ecall the explicit expressions for Q 1 ,ℓ and Q 2 ,ℓ from ( 5.10 ) , respectively ( 5.11 ). No w, conditional on T and the weigh ts (denoted by T φ ), P 2 ℓ +1 is deterministic, hence E [ Q 1 ,ℓ Q 2 ,ℓ |T , T φ ] = Q 1 ,ℓ X ( u 0 ,...,u 2 ℓ +1 ) ∈P 2 ℓ +1 φ u 2 ℓ +1 E [ σ ( u 2 ℓ +1 ) |T ] = 0 , 34 b ecause, E [ σ ( u ) |T , σ o ] =  a − b a + b  | u | σ o , for a ve rt ex u at d istance | u | from the ro ot, by construction of the branching pro cess. B Pro ofs of Section 6 Pr oo f of Pr op osition 6.1 . The second statement follo ws from the ﬁrst after recalling that ( G, e ) ℓ = ( G ′ , e 2 ) ℓ , where G ′ is th e graph G with edge { e 1 , e 2 } remov ed. S ince e ∈ ~ E , e 2 then has a biased weigh t go verned by ν ∗ . In [ 6 ], w e established a coupling betw een th e branc hing p rocess and the DC-SBM where the sp in s are dra wn unif ormly from { + , −} , with error probabilit y n − 1 2 log(4 /e ) . Thus, we are done if we couple the n eigh b ourho o ds in the latter graph to the DC- SBM with deterministic spins under consideration here. Now , with probability at least 1 − e − Ω( n − 1 / 2 ) w e can couple the graphs such that at most c 1 n 3 4 ∨ (1 − γ ) hav e un equal spins (call the corresp onding set of vertices S ) a nd al l w eights are equal. F urther, we ma y assume that the subgraph s obtained after remo v ing S are identi cal. The ℓ -neighbourho o ds in b oth graphs are exactly th e same if they are b oth disjoin t with S . Conditional on | S | and | G ℓ | , this happ ens with p robabilit y at least 1 − c 2 | G ℓ || S | n . F rom [ 6 ], w e k n ow that with probabilit y 1 − n − log(4 /e ) , | G ℓ | < n 1 8 ∧ γ 2 . Thus, cond itional on the b ounds for | S | and | G ℓ | , the n eigh b ourho o ds are the same with p robabilit y at least 1 − c 3 n − ( 1 8 ∨ γ 2 ) . All together, P (( G, v ) ℓ = ( T , o ) ℓ ) ≥ 1 − c 4 n − ( 1 8 ∧ γ 2 ) ∧ ( 1 2 log(4 /e ) ) . Pr oo f of C or ol l ary 6.3 . This p roof follo ws the pro of of Corollary 32 in [ 2 ]. Indeed (although with a slightly diﬀeren t probabilit y) the graph n eigh b ourho od ( Y t ( e )) 0 ≤ t ≤ ℓ and branc hing pro cess ( Z t ) 0 ≤ t ≤ ℓ coincide aga in, and moreov er, th e w eights are equal in b oth processes. Pr oo f of L emma 6.4 . As observed in [ 2 ], the second statemen t follo ws from the ﬁrst. Adapting our pap er [ 6 ], at step m in the exploration pro cess, the w eigh t s of the vertices in U ( m ) are indep en dent, and those with spin τ ha ve w eight go verned by ν ( m ) τ , where d ν ( m ) τ ( ψ ) = g τ ( ψ ) R φ max φ min g τ ( ψ ′ )d ν ( ψ ′ ) d ν ( ψ ) , where g τ ( · ) = Q m i =1  1 − κ ( x i ,τ · ) n  , with x u = σ u φ u the typ es of t he already explored vertices and κ ( x, y ) = | xy | (1 { xy > 0 } a + 1 { xy < 0 } b ). W e claim that v ariables follow ing ν ( m ) τ are stochastical ly dominated by v ariables go verned by ν . In deed, use that for any non-decreasing f , h : R → R and any random v ariable X we h ave E [ f ( X ) h ( X )] ≥ E [ f ( X )] E [ h ( X )]. Then, for ψ ≥ 0, ν ( m ) τ ([0 , ψ ]) = E [ − g τ ( φ ) · − 1 φ ≤ ψ ] E [ g τ ( φ )] ≥ E [ g τ ( φ )] E [1 φ ≤ ψ ] E [ g τ ( φ )] = ν ([0 , ψ ]) , with φ ∼ ν . Secondly , we claim that th e weigh t of a vertex when it is just d iscov ered is stochasti- cally dominated by va riables go verned by ν ∗ . T o prov e this, let m ≥ 0 and assume the claim t o hold for all l ≤ m . Consider vertex v ex plored in step m + 1 (itself discov ered in step, say , l ≤ m ) with weig ht φ ∗ ( l ) v . Its children are sel ected from the set U ( m ) in whic h they ha ve indep end ent w eights ( φ ( m ) u ) u ∈U ( m ) all sto chastically dominated by ν . W e compare this to a setting S where a particle with weigh t φ ∗ ∼ ν ∗ has its c hildren selected follo wing the same rules from a reservoir of |U ( m ) | particles with spins as in U ( m ) and i.i.d. w eights ( φ u ) u ∈U ( m ) ∼ ν . Du e t o th e assumed stochastic domination, 35 there exists a coupling of the exploratio n pro cess and the setting S , such t h at p oint- wise φ ∗ ( l ) v ≤ φ ∗ and φ ( m ) u ≤ φ u for a ll u . T o decide whether u ∈ U ( m ) is sel ected as a chil d , we can dra w u niformly from [0 , 1] a n u mber U u and include u in the exploration process exactly when (1 σ u = σ v a +1 σ u = − σ v b ) φ ∗ ( l ) v φ ( m ) u n ≥ U u and in the setting S exactly when (1 σ u = σ v a +1 σ u = − σ v b ) φ ∗ φ u n ≥ U u . Sin ce b y assumption φ ∗ φ u ≥ φ ∗ ( l ) v φ ( m ) u , for each u , w e conclu d e that the n ewly selected particles are also sto chastical ly dominated. Denote the vertices in S t by 1 , . . . , S t and their weigh ts b y ( b φ ∗ v ) v ∈ S t . W e shall use the same strategy as in Lemma 5.4 to b ound S t +1 = S t X v = 1 b D ∗ v , where b D ∗ v is the oﬀspring-size of v . In particular, to u se large deviation theory as in ( A.12 ), we shall calculate for θ ≥ 0, E h e θ P S t v =1 b D ∗ v    S t i . Caution is n eed ed here as the v ariables ( b D ∗ v ) v ∈ S t are not ind ep endent. Let F m b e the sigma-alg eb ra generated b y the exp loration process upto step m (includ ed). If v ertex v is explored in step m + 1, then, b D ∗ v = X u ∈U ( m ) Ber (1 σ u = σ v a + 1 σ u = − σ v b ) b φ ∗ v φ ( m ) u n ! , where we recall th at conditioned on F m , φ ( m ) u is sto chastical ly dominated by ν and b φ ∗ v by ν ∗ . Hence, using that 1 + y ≤ e y for all y ∈ R , E h e θ b D ∗ v    F m , b φ ∗ v i ≤ E " Y u 1 + b φ ∗ v φ ( m ) u n (1 σ u = σ v a + 1 σ u = − σ v b )( e θ − 1) !      F m , b φ ∗ v # ≤ 1 + a b φ ∗ v Φ (1) n ( e θ − 1) ! n σ v 1 + b b φ ∗ v Φ (1) n ( e θ − 1) ! n − σ v ≤ e r n b φ ∗ v Φ (1) ( e θ − 1) , (B.1) where r n = max { n + a + n − b n , n − a + n + b n } . Thus, if φ ∗ has la w ν ∗ , E h e θ b D ∗ v    F m i ≤ E h e r n φ ∗ Φ (1) ( e θ − 1) i , (B.2) since for t ≥ 0, E  e tX  ≤ E  e tY  if X d ≤ Y . Iterating ( B.2 ), we obtain E h e θ P S t v =1 b D ∗ v    S t i ≤  E h e r n φ ∗ Φ (1) ( e θ − 1) i S t = E h e r n P S t v =1 φ ∗ v Φ (1) ( e θ − 1)    S t i , where { φ ∗ v } v are i.i.d. with la w ν ∗ . Thus, we h ave E h e θ P S t v =1 b D ∗ v i ≤ E  e θ Poi  P S t v =1 r n φ ∗ v Φ (1)   , compare this to ( 5.8 ): the characteri stic function of P S t v = 1 b D ∗ v is dominated by t he chara cteristic function of the P oisson-mixtu re in ( 5.8 ) if w e replace a + b 2 with r n . Hence w e can rep eat the pro of of Lemma 5.4 , with ρ n := r n Φ (2) instead of ρ . Pr oo f of L emma 6.5 . Fix a v ertex v . Let m ≥ 0 be the smallest in t eger suc h that all vertices within distance R of v hav e b een rev ealed at step m of the exploration process. Now , th e exploration pro cess construct s a spanning tree T m for G R ( v ). How ever, edges b etw een vertices in ∂ G r ( r ≤ ℓ ) are not insp ected, and neither is it veriﬁed wheth er tw o 36 vertices in ∂ G r share a common n eigh b our in ∂ G r +1 ( r ≤ R − 1). The n u mber of th ose uninsp ected edges is b ounded by | G r | 2 . Hence, among them at most Bin( | G r | 2 , c 1 n ) are actually present in G r . Thus, using t wice Marko v’s inequality in conjunction with Lemma 6.4 , for some c 2 > 0, P ( G r ( v ) is not a tree) ≤ E  | G r | 2  c 1 n ≤ c 3 ρ 2 ℓ n , and, P X v 1 G r ( v ) is not a tree ≥ ρ 2 ℓ log( n ) ! ≤ c 4 log( n ) . F or the other claim, if the graph is t angled, th en there is a ve rt ex such that among its un insp ected edges in th e exploration pro cess at step m , at least tw o are in fact present. No w, P  Bin  | G r | 2 , c 1 n  ≥ 2  ≤  c 1 n  4 E  | G r | 4  ≤ c 5 ρ 4 ℓ n 4 . A union b ound o ver all v ertices then gives P ( G tangled) ≤ c 6 ρ 4 ℓ n 3 = o (1) . Pr oo f of Pr op osition 6.6 . ( i ) follow s from Lemma 6.4 and Corollary 6.3 . T o prov e ( ii ), recall th at B r ~ e ~ g is the number of non-backtrac king paths of length r (i.e., containing r + 1 edges) betw een ~ e and ~ g . F urt h er, if G r ( e 2 ) is a tree, th en there is exactly one path betw een e and any edge g on th e tree. Hence h B r χ k , δ e i = h g k , Ψ r ( e ) i . An app eal to Corollary 6.3 then establishes ( ii ). F urther, ( iii ) follo ws from the fact that G is ℓ -tangle-free with high probability , so that there are at most t wo non- b acktrac king walks of length r betw een any edges ~ e and ~ f . Th us, |h B r χ k , δ e i| ≤ 2 k g k k ∞ φ max S t ( e ) ≤ log 2 ( n ) ρ r , with p robabilit y at least 1 − e − Ω( n ) , due to L emma 6.4 . Pr oo f of C or ol l ary 6.7 . W e start with the cas e µ 2 2 > ρ . Using that h B ℓ χ k , x i = 0 and Proposition 6.6 ( iii), w e write, |h B r χ k , x i| = | X e ∈ ~ E ℓ x e h B r χ k , δ e i + X e / ∈ ~ E ℓ x e h B r χ k , δ e i − µ r − ℓ k   X e ∈ ~ E ℓ x e h B ℓ χ k , δ e i + X e / ∈ ~ E ℓ x e h B ℓ χ k , δ e i   | ≤ (log n ) 2 ρ r q | ~ E ℓ | + X e / ∈ ~ E ℓ | x e |kh B r χ k , δ e i − µ r − ℓ k h B ℓ χ k , δ e ik + µ r − ℓ k log( n ) 2 ρ ℓ q | ~ E ℓ | . (B.3) Now , | µ k | > 1 and for e / ∈ ~ E ℓ , bou n d ( ii ) in Prop osition 6.6 applies, so that w.h.p. |h B r χ k , x i| ≤ 2 ρ ℓ (log n ) 2 q | ~ E ℓ | + ρ r / 2 (log n ) 4 p | E | ≤ ρ ℓ (log n ) 3 n 1 2 − γ 4 ∧ 1 80 + ρ r / 2 (log n ) 9 2 n 1 2 ≤ ρ r / 2 (log n ) 5 n 1 / 2 , (B.4) 37 since ρ ℓ = n C ≪ n γ 4 ∧ 1 80 . In case µ 2 2 ≤ ρ , redeﬁne ~ E ℓ as the set of oriented edges suc h that ( G, e 2 ) ℓ is not a tree or |h g 1 , Ψ t ( e ) i − ρ t − ℓ h g 1 , Ψ ℓ ( e ) i| > (log n ) 4 ρ t/ 2 or |h g 2 , Ψ t ( e ) i| > (log n ) 4 ρ t/ 2 . Note that | ~ E ℓ | can no w by b oun ded with the same arguments as in the pro of of Corol lary 6.3 . W rite h B r χ k , x i = P e ∈ ~ E ℓ x e h B r χ k , δ e i + P e / ∈ ~ E ℓ x e h B r χ k , δ e i , T o boun d the sum o ver E ℓ , use Cauc hy-Sch wartz inequality and Proposition 6.6 ( iii ), which also h olds if µ 2 2 ≤ ρ . F or the sec ond sum, use that, if e / ∈ ~ E ℓ , then |h B r χ k , δ e i| ≤ ( log n ) 4 ρ r / 2 , as follo ws from Theorem 5.6 and the coupling result for lo cal n eighbourho od s. C Pro ofs of Section 7 Pr oo f of Pr op osition 7.1 . W e start by using the law of total v ariance for Y = P n v = 1 τ ( G, v ): V ar ( Y ) = E [V ar ( Y | φ 1 , . . . , φ n )] + V ar ( E [ Y | φ 1 , . . . , φ n ]) , and shall apply Efron-Stein’s inequality on b oth terms. Deﬁne th e function h for ( ψ 1 , . . . , ψ n ) ∈ [ φ min , φ max ] n as h ( ψ 1 , . . . , ψ n ) = E [ Y | φ 1 = ψ 1 , . . . , φ n = ψ n ] . W e n eed to b ound | h ( ψ 1 , . . . ψ k − 1 , ψ k , ψ k +1 , . . . , ψ n ) − h ( ψ 1 , . . . ψ k − 1 , ψ ′ k , ψ k +1 , . . . , ψ n ) | 2 for arbitrary ψ ′ k ∈ [ φ min , φ max ]. Den ote by G ψ 1 ,...,ψ k ,...,ψ n the random graph G , conditional on φ 1 = ψ 1 , . . . , φ n = ψ n . Assume without loss of generalit y that ψ k ≥ ψ ′ k . Then, there exists a coupling of G ψ 1 ,...,ψ k ,...,ψ n and G ψ 1 ,...,ψ ′ k ,...,ψ n such that G ψ 1 ,...,ψ ′ k ,...,ψ n is a su b graph of G ψ 1 ,...,ψ k ,...,ψ n obtained after removing some edges betw een k and its neighbours in the latter graph. F or this coupling, | τ ( G ψ 1 ,...,ψ k ,...,ψ n , u ) − τ ( G ψ 1 ,...,ψ ′ k ,...,ψ n , u ) | is nonzero only if u ∈ V ( G ψ 1 ,...,ψ k ,...,ψ n , k ) ℓ , and it is b ounded by max v ϕ ( G ψ 1 ,...,ψ k ,...,ψ n , v ) + max v ϕ ( G ψ 1 ,...,ψ ′ k ,...,ψ n , v ). Consequently , | h ( ψ 1 , . . . ψ k − 1 , ψ k , ψ k +1 , . . . , ψ n ) − h ( ψ 1 , . . . ψ k − 1 , ψ ′ k , ψ k +1 , . . . , ψ n ) | 2 ≤ E h | V ( G ψ 1 ,...,ψ k ,...,ψ n , k ) ℓ |  max v ϕ ( G ψ 1 ,...,ψ k ,...,ψ n , v ) + max v ϕ ( G ψ 1 ,...,ψ ′ k ,...,ψ n , v i 2 ≤ E  | V ( G k, ∞ , k ) | 2 | φ 1 = ψ 1 , . . . , φ k − 1 = ψ k − 1 , φ k +1 = ψ k +1 , . . . , φ n = ψ n  · 3 E h max v ϕ 2 ( G, v )    φ 1 = ψ 1 , . . . , φ k = ψ k , . . . , φ n = ψ n i + E  | V ( G k, ∞ , k ) | 2 | φ 1 = ψ 1 , . . . , φ k − 1 = ψ k − 1 , φ k +1 = ψ k +1 , . . . , φ n = ψ n  · 3 E h max v ϕ 2 ( G, v )    φ 1 = ψ 1 , . . . , φ k = ψ ′ k , . . . , φ n = ψ n i (C.1) where G k, ∞ is the random graph G conditioned on φ k = φ max , and where w e used H¨ older’s inequality and th e fact that ( x + y ) 2 ≤ 3( x 2 + y 2 ) for an y x, y ∈ R . H ence, using again H¨ older’s inequalit y , Efron-Stein’s inequalit y becomes V ar ( E [ Y | φ 1 , . . . , φ n ]) ≤ 1 2 n X k =1 E  | h ( φ 1 , . . . , φ k , . . . , φ n ) − h ( φ 1 , . . . φ ′ k , . . . , φ n ) | 2  ≤ 3 n X k =1 q E [ | V ( G k, ∞ , k ) | 4 ℓ ] r E h max v ϕ 4 ( G, v ) i , (C.2) where ( φ ′ k ) k is an i.i.d. cop y of ( φ k ) k . Now , du e t o 6.4 , E  | V ( G k, ∞ , k ) | 4 ℓ  ≤ c 1 2 ρ 4 ℓ . Thus, V ar ( E [ Y | φ 1 , . . . , φ n ]) ≤ c 2 nρ 2 ℓ r E h max v ϕ 4 ( G, v ) i . T o boun d V ar ( Y | φ 1 = ψ 1 , . . . , φ n = ψ n ) w e u se again Efron-Stein’s inequ alit y . D e- ﬁne for 1 ≤ k ≤ n , X k = { 1 ≤ v ≤ k : { v, k } ∈ E } , where E is the edge set of G . 38 Then, conditioned on the weigh ts ( φ u = ψ u ), { X k } k are indep endent. Let { X ′ k } k b e an indep end ent copy of { X k } k and deﬁ n e G k as the graph on vertex set V with edge set ∪ v 6 = k X v ∪ X ′ k . Thus, conditional on the w eights, G k equals G excep t for the edges in { 1 ≤ v ≤ k } w h ich are redrawn indep endently . Now , for some function F ψ 1 ,...,ψ u , n X v = 1 τ ( G, v ) = F ψ 1 ,...,ψ n ( X 1 , . . . , X k , . . . , X n ) , and hence, n X v = 1 τ ( G k , v ) = F ψ 1 ,...,ψ n ( X 1 , . . . , X ′ k , . . . , X n ) . Proceeding as ab ov e, we obt ain V ar ( Y | φ 1 = ψ 1 , . . . , φ n = ψ n ) ≤ 1 2 n X k =1 E  | F ψ 1 ,...,ψ n ( X 1 , . . . , X k , . . . , X n ) − F ψ 1 ,...,ψ n ( X 1 , . . . X ′ k , . . . , X n ) | 2  ≤ 1 2 n X k =1 q E [ | V ( G, k ) | 4 ℓ ∩ | V ( G k , k ) | 4 ℓ ] s E   max v ϕ ( G, v ) + max v ϕ ( G k , v )  4  ≤ c 3 nρ 2 ℓ r E h max v ϕ 4 ( G, v ) i . (C.3) Pr oo f of Pr op osition 7.2 . W e recall that th e coupling b etw een neighbourh oo ds and branching pro cesses is such that, in case of success, the weig hts are equal in b oth processes. Therefore, as in the pro of of Proposition 36 in [ 2 ], w e obtain E " 1 n n X v = 1 τ ( G, v ) # = E [ τ ( T , o )] + ǫ ( n ) , where ǫ ( n ) = O ( n − γ ) + c 1 n − ( γ 2 ∧ 1 40 ) s E  max v ∈ [ n ] ϕ 2 ( G, v )  ∨ E [ ϕ 2 ( T , o )] . This error stems from the probability for the coupling to fail. Hence, E "      1 n n X v = 1 τ ( G, v ) − E [ τ ( T , o )]      # ≤ v u u t V ar 1 n n X v = 1 τ ( G, v ) ! + ǫ ( n ) . An app eal to Proposition 7.1 then ﬁnishes t he pro of. Pr oo f of Pr op osition 7.3 . W e give the key steps used to prov e Proposition 37 in [ 2 ] together with the m ain diﬀerences in th e cu rrent setting. F or ( i ), consider the branching process deﬁn ed in Section 5 , whic h we denote again by Z t ( ± ). W e denote the associated random rooted tree by ( T , o ). Put τ ( G, v ) = P e ∈ ~ E ,e 1 = v h g k , Ψ ℓ ( e ) i 2 µ 2 ℓ k . Then, 1 n P v τ ( G, v ) = 1 n P e ∈ ~ E h g k , Ψ ℓ ( e ) i 2 µ 2 ℓ k and τ ( G, v ) ≤ ϕ ( G, v ) := φ 2 max S 2 ℓ ( v ) ρ ℓ . It follo ws from Lemma 6.4 that E  max v ∈ [ n ] ϕ 4 ( G, v )  = O  (log n ) 8 ρ 4 ℓ  . 39 W e hav e τ ( T , o ) = P v ∈ Z o 1 h g k , Ψ v ℓ i 2 µ 2 ℓ k . Theorem 5.3 sa y s that  h g k , Ψ t i µ t − 1 k  t ≥ 1 conv erges in L 2 and so does it conditional on || Z o 1 || = 1. Hence, E [ τ ( T , o )] conv erges. An app eal to Proposition 7.2 in conjunction with the triangle ineq ualit y then es- tablishes that 1 n P v τ ( G, v ) con verges t o a constant, sa y c ′ k . Statement ( ii ) follo ws simi larly . The statements ( iii ) − ( v ) follo w after p rop erly choosing lo cal functionals. W e further use that E [ φ u φ v g 1 ( σ u ) g 2 ( σ v ) |T ] = E  φ u φ v 1 2 σ v |T  = 0 , for any tw o nod es u, v . F urther, on the branching pro cess, E [ h g k , Ψ 2 ℓ ih g j , Ψ ℓ i| Ψ ℓ ] = h g j , Ψ ℓ ih g k , M ℓ Ψ ℓ i = µ ℓ k h g k , Ψ ℓ ih g j , Ψ ℓ i . Pr oo f of Pr op osition 7.4 . S tarting with ( i ), w e deﬁn e the local fun ct ion τ as τ ( G, v ) = P e ∈ ~ E ,e 1 = v P 2 k,ℓ ( e ) µ − 4 ℓ k , for a ro oted graph ( G, v ). Let M ( v ) = max 0 ≤ t ≤ ℓ max u ∈ ( G,v ) t max s ≤ 2 ℓ − t ( S s ( u ) /ρ s ) . By monotonicit y , the statement of Lemma 6.4 holds also for ˜ S ℓ − t − 1 ( h ) and ˜ S t ( g ). W e use this fact to b ound p o wers of M ( v ) in the follo wing calcula tion: τ ( G, v ) ≤ ρ − 2 ℓ X e ∈ ~ E ,e 1 = v   ℓ − 1 X t =0 X f ∈ Y t ( e ) k g k k ∞ φ max ˜ S t +1 ( f ) ˜ S ℓ − t ( f )   2 ≤ c 1 ρ − 2 ℓ X e ∈ ~ E ,e 1 = v   ℓ − 1 X t =0 X f ∈ Y t ( e ) M 2 ( v ) ρ t +1 ρ ℓ − t   2 = c 1  M 2 ( v ) ρ  2 X e ∈ ~ E ,e 1 = v ℓ − 1 X t =0 S t ( e ) ! 2 = c 2  M 2 ( v ) ρ  2 X e ∈ ~ E ,e 1 = v  M ( v ) ρ ℓ  2 ≤ c 2 M 7 ( v ) ρ 2 ℓ . W e put ϕ ( G, v ) = c 2 M 7 ( v ) ρ 2 ℓ . Then, E  max v ϕ ( G, v ) 4  = O ((log n ) 28 ρ 8 ℓ ) , and the same b ound holds for ϕ ( T , o ). F rom Prop osition 7.2 , w e then know that E         1 n X e ∈ ~ E P 2 k,ℓ ( e ) µ 4 ℓ k − E [ τ ( T , o )]         ≤ c 3 n − ( γ 2 ∧ 1 40 ) (log n ) 7 ρ 2 ℓ , (C.4) where τ ( T , o ) = 1 µ 4 ℓ k X v ∈ Y o 1 P 2 k,ℓ ( o → v ) = 1 µ 4 ℓ k X v ∈ Y o 1  Q v k,ℓ  2 , (C.5) where Q v k,ℓ is equal to Q k,ℓ deﬁned on the subtree of all v ertices with common ancestor v . W e need to show that the exp ectation of τ ( T , o ) conv erges for ℓ → ∞ . Conditional on σ o , and | Y o 1 | , { Q v k,ℓ } v ∈ Y o 1 are indep en dent copies of Q k,ℓ deﬁned on the b ranching process in Section 5 where the ro ot has spin σ o with p rob ab ility a a + b and random w eight go verned by the biased law ν ∗ . The uniform L 2 conv ergence in Theorem 5.7 establishes the claim. 40 W e now prov e ( ii ). Put τ ( G, v ) = P e ∈ ~ E ,e 1 = v ( P 1 ,ℓ ( e ) + S 1 ,ℓ ( e ))( P 2 ,ℓ ( e ) + S 2 ,ℓ ( e )) . W e claim that E [ τ ( T , o )] = 0. Consider τ ( T , o ) = P v ∈ Z o 1 ( P 1 ,ℓ ( o → v ) + S 1 ,ℓ ( o → v ))( P 2 ,ℓ ( o → v ) + S 2 ,ℓ ( o → v )). Firstly , for k ∈ { 1 , 2 } , P k,ℓ ( o → v ) = Q v k,ℓ . Now, it follo ws from Theorem 5.9 , that E  Q v 1 ,ℓ Q v 2 ,ℓ  = 0 , since σ v is dra wn uniformly from { + , −} . Secondly , S 1 ,ℓ ( o → v ) S 2 ,ℓ ( o → v ) = 1 2 φ 2 o σ o S 2 ℓ ( o → v ) has also zero ex p ectation. Thirdly , Q v 1 ,ℓ S 2 ,ℓ ( o → v ) = 1 2 X ( u 0 ,...,u 2 ℓ +1 ) ∈P v 2 ℓ +1 φ u 2 ℓ +1 φ o σ o S ℓ ( o → v ) , (C.6) where P v 2 ℓ +1 is P 2 ℓ +1 (from ( 5.9 )) deﬁn ed on the subtree of all v ertices with common ancestor v . The exp ectation of Q v 1 ,ℓ S 2 ,ℓ ( o → v ) is thus zero since σ o is indep endent of all other terms in ( C.6 ). Lastly , Q v 2 ,ℓ = P ( u 1 ,...,u 2 ℓ +1 ) ∈P v 2 ℓ +1 σ u 2 ℓ +1 , is seen to hav e zero exp ectation. Those four statements com bin ed establish E [ τ ( T , o )] = 0. As above , w e calculate E  max v ϕ ( G, v ) 4  = O ((log n ) 28 ρ 16 ℓ ) . Pr oo f of Pr op osition 7.5 . Put τ as in Prop osition 7.3 ( i ), then τ ( T , o ) = P v ∈ Z o 1 h g 2 , Ψ v ℓ i 2 µ 2 ℓ 2 . Now , E  h g 2 , Ψ v ℓ i 2  = E  h g 2 , Φ (2) Φ (1) Z v ℓ i 2  + E  h g 2 , Ψ v ℓ − Φ (2) Φ (1) Z v ℓ i 2  ≥  Φ (2) Φ (1)  2 E  h g 2 , Z v ℓ i 2  . Now , Theorem 2 . 4 in [ 12 ] sa y s that for some random va riable X with strictly positive v ariance, weakly , h g 2 ,Z v ℓ i ρ ℓ/ 2 → X , as ℓ → ∞ . Because of the wea k conve rgence, we hav e for any θ > 0, E   h g 2 ,Z v ℓ i ρ ℓ/ 2  2 ∧ θ  → E  X 2 ∧ θ  , as ℓ → ∞ . Now by Leb esque’s dominated conv ergence theorem, E  X 2 ∧ θ  → E  X 2  > 0 , as θ → ∞ . Pr oo f of Pr op osition 7.6 . U se τ from Proposition 7.4 ( i ), together with th e b ound E  Q 2 2 ,ℓ  ≤ C ρ 2 ℓ ℓ 5 from Theorem 5.8 . D Pro of of Prop osition 9.2 D.1 Bound on k ∆ ( k ) k W e set m =  log n 13 log(log n )  . W e b ound the norm of k ∆ ( k ) k by using the trace metho d. F ollow ing (36) in [ 2 ] (which remains true for the DC-SBM), w e obtain k ∆ ( k − 1) k 2 m ≤ X γ ∈ W k,m 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s , (D.1) where W k,m is the coll ection con taining all sequences of paths γ = ( γ 1 , . . . , γ 2 m ) suc h that for all i : • γ i = ( γ i, 0 , · · · , γ i,k ) ∈ V k +1 is a non-backtrac king tangle-free path of length k , and, • ( γ i,k − 1 , γ i,k ) = ( γ i +1 , 1 , γ i +1 , 0 ), 41 where w e put γ 0 = γ 2 m . Recall the notation G ( γ ) = ( V ( γ ) , E ( γ )). F urther introdu ce the notation E φ ( · ) = E [ ·| φ 1 , . . . , φ n ]. W e b ound, for a giv en γ ∈ W k,m , E φ 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s ! = Y e ∈ E ( γ ) E φ  A p ( γ ) e 1 e 2 e 1 e 2  , (D.2) where for e ∈ E ( γ ), p ( γ ) e 1 e 2 denotes t he n umber of times the edge e is tra versed on the w alk γ . In ( D.2 ) we used that A is symmetric and th at, cond itional on the wei ghts, edges are indep endently present. Note that for an y edge uw , and in t eger p , E φ A p uw ≤ φ u φ w W σ ( u ) σ ( w ) n . Belo w in Lemma D.2 , we construct a spann ing tree T ( γ ) = ( V ( γ ) , E T ( γ )) of γ . In particular, for the e − ( v − 1) edges not present in T , we hav e φ u φ w W σ ( u ) σ ( w ) n ≤ c 1 n , with c 1 = φ 2 max ( a ∨ b ) . Putting th is in to ( D.2 ), w e get Y e ∈ E ( γ ) E φ  A p ( γ ) e 1 e 2 e 1 e 2  ≤ ( c 1 /n ) e − v +1 Y e ∈ E T ( γ ) φ e 1 φ e 2 W σ ( e 1 ) σ ( e 2 ) n = ( c 1 /n ) e − v +1 Y u ∈ V ( γ ) φ d u u Y e ∈ E T ( γ ) W σ ( e 1 ) σ ( e 2 ) n , (D.3) where d u is the d egree of u in the sp anning tree. Consequently , E   Y e ∈ E ( γ ) A p ( γ ) e 1 e 2 e 1 e 2   ≤ ( c/n ) e − v +1 Y u ∈ V ( γ ) Φ ( d u ) Y e ∈ E T ( γ ) W σ ( e 1 ) σ ( e 2 ) n . (D.4) Let τ : [ v ( γ )] 7→ V ( γ ) be the bijectio n describing the order the vertic es are v isited for the ﬁrst time. I.e., for 1 ≤ u ≤ v ( γ ) − 1, τ ( u ) is seen for the ﬁ rst time, before τ ( u + 1). W e shall sa y that a p ath γ c is canonical if V ( γ c ) = [ v ( γ c )] and the vertices are ﬁrst visited in the order 1 , . . . , v ( γ c ). With every path γ there corresp onds (th rough the bijection τ ) a canonical path γ c . Consequ ently , if W k,m ( v , e ) denotes the set of canonical paths in W k,m with v v ertices and e edges, and I γ c the set of all injections from [ v ( γ c )] to [ n ], E   X γ ∈ W k,m 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s   ≤ km +1 X v = 3 km X e = v − 1 X γ c ∈W k,m ( v,e ) X τ ∈ I γ c E   Y e ∈ E ( γ c ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 )   , (D.5) b ecause any non-backtrac k in g path has at least 3 vertice s, and v − 1 ≤ e ≤ k m , si n ce ( D.2 ) is non-zero only if each edge is travers ed at least twice. W e now b ound the term P τ ∈ I γ c E  Q e ∈ E ( γ c ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 )  in ( D.5 ). Using ( D.4 ), w e hav e, X τ ∈ I γ c E   Y e ∈ E ( γ c ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 )   ≤ ( c 1 /n ) e − v +1 v ( γ c ) Y u =1 Φ ( d u ) X τ ∈ I γ c Y e ∈ E T ( γ c ) W σ ( τ ( e 1 )) σ ( τ ( e 2 )) n . (D.6) Our ob jective is to compare Q v ( γ c ) u =1 Φ ( d u ) P τ ∈ I γ c Q e ∈ E T ( γ c ) W σ ( τ ( e 1 )) σ ( τ ( e 2 )) n with nρ ( v − 1) . W e start by analysing the term con taining th e spins: 42 Lemma D.1. F or any c anonic al p ath γ c ∈ W k,m , X τ ∈ I γ c Y e ∈ E T ( γ c ) W σ ( τ ( e 1 )) σ ( τ ( e 2 )) n ≤ (1 + o (1)) n  a + b 2  v − 1 . (D.7) Pr oo f . Let l b e any leaf on t he tree with unique neighbour g . Then, writing τ u = τ ( u ) for u ∈ { 1 , . . . , v } , X τ ∈ I γ c Y e ∈ E T ( γ c ) W σ ( τ ( e 1 )) σ ( τ ( e 2 )) n ≤ n X τ 1 =1 · · · n X τ v =1 Y e ∈ E T ( γ c ) W σ ( τ e 1 ) σ ( τ e 2 ) n . Keeping τ g ﬁxed, n X τ l =1 Y e ∈ E T ( γ c ) W σ ( τ e 1 ) σ ( τ e 2 ) n = Y e ∈ E T ( γ c ) \{ g ,l } W σ ( τ e 1 ) σ ( τ e 2 ) n n X τ l =1 W σ ( τ g ) σ ( τ l ) n = Y e ∈ E T ( γ c ) \{ g ,l } W σ ( τ e 1 ) σ ( τ e 2 ) n  a + b 2 + O ( n − γ )  , due to assumption ( 2.1 ). Rep eating inductively this proced ure (by remo ving leav es from the tree) prove s the assertion. It remains to b oun d Q v ( γ c ) u =1 Φ ( d u ) . T o do so, we note that, since the w eights are assumed to be b ou n ded, Φ ( d u ) ≤ C d u − 2 2 Φ (2)  Φ (1)  d u − 2 if d u ≥ 2 , with C 2 = φ max Φ (1) > 1. Consequently , v ( γ c ) Y u =1 Φ ( d u ) ≤ C P u : d u > 2 ( d u − 2) 2 Y u : d u > 2 Φ (2)  Φ (1)  d u − 2 Y u : d u ≤ 2 Φ ( d u ) ≤ C P u : d u > 2 ( d u − 2) 2  Φ (2)  1 2 P v u =1 d u = C P u : d u > 2 ( d u − 2) 2  Φ (2)  v − 1 , (D.8) where w e used that by Jensen’s inequality  Φ (1)  2 ≤ Φ (2) . Now , th e sum P u : d u > 2 ( d u − 2) is small for a tree spanning a path in W k,m : Lemma D. 2. F or any γ ∈ W k,m , wi th v vertic es and e e dges, ther e exists a tr e e sp anning γ with de gr e es ( d u ) v u =1 such t hat: X u : d u > 2 ( d u − 2) ≤ e − ( v − 1) + 2 m. (D.9) Pr oo f . W e construct a sp anning tree, while t ra versing γ . W e den ote by p ( t ) the graph constructed at step t ≥ 0. Pu t p (0) = { γ 1 , 0 , ∅ } and r = s = 0 (t h e mea n ing of these tw o c ounters b ecomes clear in th e algorithm b elow). Consider edge f tra versed in step t + 1 of the walk: If f or ˇ f has already been trav ersed, then con t inue with step t + 2. Otherwise, if b oth f and ˇ f ha ve not yet b een trav ersed, distinguish b etw een the follo wing cases : 43 1. f 1 is a leaf of p ( t ) and a) p ( t ) contai n s a cycle, then if f 2 / ∈ p ( t ), pu t p ( t + 1) = p ( t ) ∪ f , otherwise, if f 2 ∈ p ( t ), put p ( t + 1) = p ( t ); b) p ( t ) does not con t ain a cycle, then p ut p ( t + 1) = p ( t ) ∪ f . If f 2 ∈ p ( t ), th en put e c = f ; 2. f 1 is not a leaf of p ( t ) and a) p ( t ) con tains a cycle, then pu t p ( t + 1) = ( p ( t ) \ e c ) ∪ f . If f 2 ∈ p ( t ), put e c = f . Increase the v alue of r with one. b) p ( t ) d oes not con t ain a cycle, t hen put p ( t + 1) = p ( t ) ∪ f . If f 2 ∈ p ( t ), put e c = f . Otherwise, if f 2 / ∈ p ( t ), increase the val u e of s with one. Once the path is completely trave rsed, remov e e c to obtain a spanning tree. Note that at eac h stage of the construction, the graph contains at most on e cycle and in this case, remo ving e c will mak e th e graph in to a tree. F urther, cases 1 .a and 1 .b do not contribute t o P u : d u > 2 ( d u − 2), since the lea ve in p ( t ) b ecomes a vertex with degree at mos t 2 in p ( t + 1). A cycle formed in step t + 1 will temp or arily increase th e degree of the v ertex that is merg ed by the leaf , ho wev er this edge e c will later b e remove d. In case 2 .a , the degree of ve rtex f 1 increases with one, how ever, at t h e same time an edge is remo ved. The num b er of times 2 .a happ ens, r , is thus bou n ded by the n u mber of times an edge is remov ed : r ≤ e − ( v − 1). In case 2 .b , w e need only to consider the case where no cy cle is formed. But, b efore arriving at such a vertex considered in 2 .b , the path must hav e made a backtrac k. Hence s ≤ 2 m . (In fact, betw een tw o subsequent o ccurrences of ev ent 2, the w alk should at le ast either make a backtrac k or ’get bac k to the tree’ by forming a cycle: giving the same b ound for s + r ) . All together, X u : d u > 2 ( d u − 2) ≤ r + s ≤ e − ( v − 1) + 2 m. Finally , w e recall the b ound on the cardinalit y of W k,m from [ 2 ]: Lemma D .3 ( Lemma 17 in [ 2 ]) . L et W k,m ( v , e ) b e the set of c anonic al p aths wi th v ( γ ) = v and e ( γ ) = e . We have |W k,m ( v , e ) | ≤ k 2 m (2 km ) 6 m ( e − v +1) . (D.10) Hence, com bining ( D.1 ), ( D.5 ) - ( D.10 ), E h k ∆ ( k − 1) k 2 m i ≤ km +1 X v = 3 km X e = v − 1 |W k,m ( v , e ) |  c n  e − ( v − 1) nC e − ( v − 1)+2 m ρ v − 1 ≤ nc m 5 ρ km km +1 X v = 3 km X e = v − 1 ℓ 2 m  c 7 (2 ℓm ) 6 m n  e − ( v − 1) ≤ nc m 5 ρ km ℓ 2 m ℓm ∞ X s =0  c 7 (2 ℓm ) 6 m n  s ≤ n ( c 8 log n ) m log 2 nρ km ≤ ( c 9 log n ) 16 m ρ km , (D.11) where we used the bound on m , in particular to deriv e con vergence of th e series, and the fact that n 1 /m = o (log n ) 14 . W e ﬁnish by using Mark ov’s inequalit y . 44 D.2 Bound on k ∆ ( k ) χ i k W e p oint out the diﬀerences with b ou n d (31) in [ 2 ]: H ere, w e hav e E h k ∆ ( k − 1) χ i k 2 i = E   X e,f,g ∆ ( k − 1) ef ∆ ( k − 1) eg ξ i ( f ) ξ i ( g )   ≤ φ 2 max E   X e,f,g ∆ ( k − 1) ef ∆ ( k − 1) eg   ≤ φ 2 max X γ ∈ W ′′ k, 1 E " 2 Y i =1 k Y s =1 A γ i,s − 1 ,γ i,s # , (D.12) where W ′′ k, 1 is deﬁned in [ 2 ]. I n the latter pap er it is also sho wn that th e same boun d , Lemma D.3 holds for the cardinality of W ′′ k, 1 ). Hence, using th e p enultimate line of ( D.11 ) with m = 1, gi ves E h k ∆ ( k − 1) χ i k 2 i ≤ c 1 n log 3 ( n ) ρ k . D.3 Bound on k R ( ℓ ) k k Put m =  log n 25 log(log n )  . W e apply th e same strategy as ab ov e: for 0 ≤ k ≤ ℓ − 1, we have the b ound k R ( ℓ − 1) k k 2 m ≤ tr n R ( ℓ − 1) k R ( ℓ − 1) k ∗  m o = X γ ∈ T ′ ℓ,m,k 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s φ γ i,k φ γ i,k +1 W σ ( γ i,k ) σ ( γ i,k +1 ) ℓ Y s = k +2 A γ i,s − 1 γ i,s ≤ c m 1 X γ ∈ T ′ ℓ,m,k 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s ℓ Y s = k +2 A γ i,s − 1 γ i,s , (D.13) where c 1 = φ 4 max ( a ∨ b ) 2 , and where T ′ ℓ,m,k is the collection containing all sequences of paths γ = ( γ 1 , . . . , γ 2 m ) suc h that • for all i : γ i = ( γ 1 i , γ 2 i ), where γ 1 i = ( γ i, 0 , · · · , γ i,k ) and γ 2 i = ( γ i,k +1 , · · · , γ i,ℓ ) are non-backtrac king tangle-free; • for all od d i : ( γ i, 0 , γ i, 1 ) = ( γ i − 1 , 0 , γ i − 1 , 1 ) and ( γ i,ℓ − 1 , γ i,ℓ ) = ( γ i +1 ,ℓ − 1 , γ i +1 ,ℓ ), with th e conv ention that γ 0 = γ 2 m . T o calculate the ex p ectation of k R ( ℓ − 1) k k 2 m , w e note that E " 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s ℓ Y s = k +2 A γ i,s − 1 γ i,s # is non-zero only if (for i ﬁx ed ) each edge { γ i,s − 1 , γ i,s } for 1 ≤ s ≤ k app ears more than once in the 2( ℓ − 1) m p airs {{ γ j,s − 1 , γ j,s }} j =2 m j =1 ,s 6 = k + 1 . Hen ce, E h k R ( ℓ − 1) k k 2 m i ≤ c m 1 X γ ∈ T ℓ,m,k E " 2 m Y i =1 k Y s =1 A γ i,s − 1 γ i,s ℓ Y s = k +2 A γ i,s − 1 γ i,s # , ( D.14) 45 where T ℓ,m,k = { γ ∈ T ′ ℓ,m,k | v ( γ ) ≤ e ( γ ) ≤ km + 2 m ( ℓ − 1 − k ) } . (D.15) Similarly as in establishing the boun d on k ∆ ( k ) k , w e say that a path γ c is c anonic al if V ( γ c ) = [ v ( γ c )] and the vertices are ﬁ rst visited in ord er. W e denote by T ℓ,m,k ( v , e ) the set of canonical paths in T ℓ,m,k with v vertices and e ed ges. Then: E h k R ( ℓ − 1) k k 2 m i ≤ c m 1 m (2 ℓ − 2 − k ) X v = 1 m (2 ℓ − 2 − k ) X e = v X γ c ∈T ℓ,m,k ( v,e ) X τ ∈ I γ c E   Y e ∈ E ( γ c ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 )   , (D.16) where I γ c is deﬁned as ab ove, p ( γ c ) e 1 e 2 is the num b er of t imes edge { e 1 , e 2 } occurs in {{ γ j,s − 1 , γ j,s }} s = k,j = 2 m s =1 ,j =1 and p ( γ c ) e 1 e 2 denotes t he number of times ed ge { e 1 , e 2 } o ccurs in the remainder of the collection of edges, {{ γ j,s − 1 , γ j,s }} s = ℓ,j =2 m s = k +2 ,j = 1 . Now , again, E  A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 )  ≤ φ τ ( e 1 ) φ τ ( e 2 ) W σ ( τ ( e 1 )) σ ( τ ( e 2 )) n . Belo w we constru ct a sp an n ing forest F = ( V ( γ ) , E F ( γ )) of γ (i.e., F is the disjoin t union of trees, eac h spanning another comp onent of G ( γ )). Let n C ≤ m denote the num b er of components of G ( γ ). Then, E   Y e ∈ E ( γ ) A p ( γ ) e 1 e 2 e 1 e 2 A p ( γ ) e 1 e 2 e 1 e 2   ≤ ( c/n ) e − ( v − n C ) Y u ∈ V ( γ ) Φ ( d u ) Y e ∈ E F ( γ ) W σ ( e 1 ) σ ( e 2 ) n , ( D.17) with d u the degree of v ertex u in the for est F , compare to ( D.4 ). Now , th is time, Lemma D.4. F or any c anonic al p ath γ c ∈ T ℓ,m,k ( v , e ) , X τ ∈ I γ c Y e ∈ E F ( γ c ) W σ ( τ ( e 1 )) σ ( τ ( e 2 )) n ≤ (1 + o (1)) n n C  a + b 2  v − n C . (D.18) Pr oo f . Apply Lemma D.1 subsequently to the diﬀerent comp onents of F . F urther, applying ( D.8 ) to diﬀerent comp onents in F giv es v ( γ ) Y u =1 Φ ( d u ) ≤ C P u : d u > 2 ( d u − 2) 2  Φ (2)  v − n C . (D.19) T ogether, X τ ∈ I γ c E   Y e ∈ E ( γ c ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 )   ≤ ( c/n ) e − v C P u : d u > 2 ( d u − 2) 2 ρ v − n C . (D.20) Again, we b ound P u : d u > 2 ( d u − 2): Lemma D.5. F or any γ ∈ T ℓ,m,k , with v vertic es and e e dges, ther e exists a for est sp anning γ with de gr e es ( d u ) v u =1 such t hat: X u : d u > 2 ( d u − 2) ≤ 18 m + e − ( v − n C ) . (D.21) 46 Pr oo f . As in Lemma D.2 , we construct th e spanning forest, while trav ersing γ . A gain p ( t ) denotes the graph constru cted at step t ≥ 0, with p (0) = { γ 1 , 0 , ∅ } . F urth er, w e introduce three counters: r = s = q = 0, together with e c = ∅ (b elo w, e c is either equal to ∅ or it is an edge suc h that p ( t ) contains one cy cle, but p ( t ) \ e C is a forest). At any step t , we let C 1 , . . . , C #comp onents b e the comp onents of p ( t ). Consider step t + 1 of th e walk: if the step consists in jumping to a vertex w , then put p ( t + 1) = ( p ( t ) \ e C ) ∪ { w } . Else, if t h e step consists in trav ersing an edge f = f 1 f 2 , th en: If f or ˇ f h as already b een trav ersed, con tinue with step t + 2. Otherwise, if b oth f and ˇ f h a ve not yet been tra versed, distinguish b etw een the follo wing cases: 1. f 1 is a lea ve or an isolated vertex of co mp onent C i of p ( t ) and a. C i does not con tain a cycle, then p ut p ( t + 1) = p ( t ) ∪ f . F urther, d istinguish b etw een the follo wing cases: i) f 2 / ∈ p ( t ); ii) f 2 ∈ C i , then put e c = f ; iii) f 2 ∈ C j 6 = i , then increase the val u e of s with one. b. C i conta ins a cycle, t hen distinguish b etw een t h e foll owing cases: i) f 2 / ∈ p ( t ), then put p ( t + 1) = p ( t ) ∪ f ; ii) f 2 ∈ C i , then put p ( t + 1) = p ( t ); iii) f 2 ∈ C j 6 = i , then put p ( t + 1) = p ( t ) ∪ f and increase the v alue of s with one. 2. f 1 in comp onent C i has degree at least 2 in p ( t ), then distinguish b etw een the follo wing cases: a. C i does not con tain a cycle, then p ut p ( t + 1) = p ( t ) ∪ f . F urther, d istinguish b etw een the follo wing cases: i) f 2 / ∈ p ( t ), then increase the va lue of q with one; ii) f 2 ∈ C i , then put e c = f ; iii) f 2 ∈ C j 6 = i , then increase the val u e of s with tw o. b. C i conta ins a cycle, then p ut p ( t + 1) = ( p ( t ) \ e c ) ∪ f . F urther, distinguish b etw een the follo wing cases: i) f 2 / ∈ p ( t ), then increase the va lue of r with one; ii) f 2 ∈ C i , then put e c = f ; iii) f 2 ∈ C j 6 = i , then increase the val u e of s with tw o. Once the path is completely trave rsed, remov e e c to obtain a spanning tree. The only cases th at contribute to P u : d u > 2 ( d u − 2) are 1 .a.iii, 1 .b.iii, 2 .a.i, 2 .a.iii, 2 .b.i and 2 .b.iii . Now , s counts the contribution of 1 .a.iii, 1 .b.iii, 2 .a.iii and 2 .b.iii . But, in all those 4 cases, t wo comp onents are merged, h ence s ≤ 6 #merges ≤ 12 m . By deﬁn ition of the event 2 .b.i , r is an upp er b oun d for the number of edges that are remov ed: r ≤ e − ( v − n C ). T o b oun d q (which counts t he occu rrence of 2 .a.i ), note that b etw een t w o subsequ ent occurren ces of the even t 2 .a.i , the wa lk makes at least one of the follo wing: a bac k trac k , a jump or a merge. Hence q ≤ 2 m + 2 m + 2 m = 6 m . Adding t he b ounds for r , q and s establishes ( D.21 ). Returning to ( D.20 ), w e get, since n C ≤ 2 m : X τ ∈ I γ c E   Y e ∈ E ( γ c ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 ) A p ( γ c ) e 1 e 2 τ ( e 1 ) τ ( e 2 )   ≤ ( c 1 /n ) e − v C 18 m + e − v + n C 2 ρ v − n C ≤  c 3 n  e − v c m 4 ρ v − n C . (D.22) 47 Putting this into ( D.16 ), we obtain E h k R ( ℓ − 1) k k 2 m i ≤ c m 5 m (2 ℓ − 2 − k ) X v = 1 m (2 ℓ − 2 − k ) X e = v X γ c ∈T ℓ,m,k ( v,e )  c 3 n  e − v ρ m (2 ℓ − k ) . (D.23) Now the cardinality of T ℓ,m,k ( v , e ) is b ounded in the follo wing lemma : Lemma D.6 (Lemma 18 in [ 2 ]) . L et T ℓ,m,k ( v , e ) b e the set of c anonic al p aths in T ℓ,m,k with v ( γ ) = v and e ( γ ) = e . W e have |T ℓ,m,k ( v , e ) | ≤ (4 ℓm ) 12 m ( e − v +1)+8 m . Hence, E h k R ( ℓ − 1) k k 2 m i ≤ c m 5 ρ m (2 ℓ − k ) m (2 ℓ − 2 − k ) X v = 1 m (2 ℓ − 2 − k ) X e = v (4 ℓm ) 12 m ( e − v +1)+8 m  c 3 n  e − v ≤ ρ m (2 ℓ − k ) c m 5 (4 ℓm ) 20 m m (2 ℓ − 2 − k ) X v = 1 ∞ X s =0  c 3 (4 ℓm ) 12 m n  s ≤ ρ m (2 ℓ − k ) c m 5 (4 ℓm ) 20 m 2 ℓm · O (1) ≤ ρ m (2 ℓ − k ) ( c 5 log( n )) 42 m . (D.24) W e used that, due to our choice of m , (4 ℓm ) 12 m ≤ n 24 / 25 . W e use ( D.24 ) together with Marko v ’s inequalit y : P  k R ( ℓ ) k k > (log( n )) 25 ρ ℓ − k/ 2  ≤ E h k R ( ℓ ) k k 2 m i (log( n )) 50 m ρ m (2 ℓ − k ) ≤ ( c 6 log( n )) − 8 m → 0 . (D.25) D.4 Bound k K B ( k ) k Put m =  log n 13 log(log n )  . (D.26) W e hav e, with the conven tion that e 2 m +1 = e 1 , k K B ( k − 2) k 2 m ≤ tr {  K B ( k − 2) K B ( k − 2) ∗  m } = X e 1 ,...,e 2 m m Y i =1 ( K B ( k − 2) ) e 2 i − 1 ,e 2 i ( K B ( k − 2) ) e 2 i +1 ,e 2 i . (D.27) Now ,  K B ( k − 2)  ef = X g K eg B ( k − 2) gf = X g 1 e → g φ e 1 φ e 2 W σ ( e 1 ) σ ( e 2 ) X γ ∈ F k − 1 gf k − 2 Y s =0 A γ s γ s +1 ≤ c 1 X g 1 e → g X γ ∈ F k − 1 gf k − 2 Y s =0 A γ s γ s +1 . (D.28) 48 Hence, k K B ( k − 2) k 2 m ≤ c m 2 X e 1 ,...,e 2 m m Y i =1    X g 1 e 2 i − 1 → g X γ ∈ F k − 1 ge 2 i k − 2 Y s =0 A γ s γ s +1       X g 1 e 2 i +1 → g X γ ∈ F k − 1 ge 2 i k − 2 Y s =0 A γ s γ s +1    = c m 2 X γ ∈ W k,m m Y i =1 k Y s =2 A γ 2 i − 1 ,s − 1 γ 2 i − 1 ,s k − 1 Y s =1 A γ 2 i,s − 1 γ 2 i,s , (D.29) where W k,m is the collection con taining all sequences of paths γ = ( γ 1 , . . . , γ 2 m ) with γ i = ( γ i, 0 , · · · , γ i,k ) ∈ V k +1 is non-backtrac king such that • for all i : ( γ i,k − 1 , γ i,k ) = ( γ i +1 , 1 , γ i +1 , 0 ), • for all o dd i : ( γ i, 1 , · · · , γ i,k ) is tangle-free, • for all even i : ( γ i, 0 , · · · , γ i,k − 1 ) is tangle-free, with th e conv ention that γ 2 m +1 = γ 1 . Recall the deﬁnition of W k,m and note that W k,m ⊂ W k,m . Fix γ ∈ W k,m \ W k,m and let S γ b e the set of all ˆ γ ∈ W k,m \ W k,m such that for all o dd i : ( ˆ γ i, 1 , · · · , ˆ γ i,k ) = ( γ i, 1 , · · · , γ i,k ) and for all even i : ( ˆ γ i, 0 , · · · , ˆ γ i,k − 1 ) = ( γ i, 0 , · · · , γ i,k − 1 ). Then | S γ | ≤ k m . Indeed, if for odd i , ˆ γ i is not tangle-free then n ecessarily ˆ γ i, 0 ∈ { ˆ γ i, 1 , . . . , ˆ γ i,k } , i.e., ˆ γ i, 0 can b e chosen in at most k diﬀerent wa ys. A simila r argument w orks in case i is ev en. Now , there alwa ys exists γ ∈ W k,m such that for all odd i : ( γ i, 1 , · · · , γ i,k ) = ( γ i, 1 , · · · , γ i,k ) and for all even i : ( γ i, 0 , · · · , γ i,k − 1 ) = ( γ i, 0 , · · · , γ i,k − 1 ) . As a consequ ence of these t wo observa t ions, we ha ve k K B ( k − 2) k 2 m ≤ c m 2 (1 + k m ) X γ ∈ W k,m m Y i =1 k Y s =2 A γ 2 i − 1 ,s − 1 γ 2 i − 1 ,s k − 1 Y s =1 A γ 2 i,s − 1 γ 2 i,s . (D.30) T o pro ceed follo wing the metho d used to b ound ∆ ( k ) , note that th e pro duct in ( D.30 ) is t aken o ver a path, consisting of 2 m n on - backtrac king tangle-free subpaths of length k − 1, t hat makes at most 2 m backtrac ks. Hence Lemma’s D.1 and D.2 may b e adapted to the curren t setting (for instance the righ t hand side of ( D.9 ) becomes e − ( v − m − 1) + 2 m ), entailing E h k K B ( k − 2) k 2 m i ≤ c m 2 (1 + k m ) 2 km +1 X v = 3 2 km X e = v − 1 |W k,m |  c 3 n  e − ( v − 1) − m c e − ( v − m − 1)+2 m 4 nρ v − 1 ≤ c m 5 (1 + k m ) n m +1 2 km +1 X v = 3 2 km X e = v − 1 |W k,m |  c 6 n  e − ( v − 1) ρ v − 1 ≤ c m 7 (1 + k m ) n m +1 ρ 2 km ℓ 2 m ℓm ∞ X s =0  c 6 (2 ℓm ) 6 m n  s ≤ c m 8 ( ℓm ) 2 ℓ 3 m n m +1 ρ 2 km ≤ ( c 9 log n ) 19 m n m ρ 2 km , (D.31) where we u sed our choice for m several times. An app eal to Mark ov’s inequalit y ﬁn ishes the pro of. 49 D.5 Bound on k S ( l ) k k This pro of follo ws almost line-to-line the pro of u sed in [ 2 ] to establish b ound (34) there. W e restrict ourselv es here to the diﬀerences: Observe that L ef = 0 unless e 2 → f does not h old, that is e = f , e → f , f − 1 → e or e → f − 1 , in which cases L ef = − φ e 2 φ f 1 W σ ( e 2 ) σ ( f 1 ) . Hence, we hav e the decomp osition L = − I ∗ − K ∗ , where ( I ∗ ) ef = 1 e = f φ e 1 φ e 2 W σ ( e 1 ) σ ( e 2 ) , and where ( K ∗ ) ef = φ e 2 φ f 1 W σ ( e 2 ) σ ( f 1 ) if e → f , f − 1 → e or e → f − 1 and ( K ∗ ) ef = 0 otherwise. Thus k S ( ℓ ) k k ≤ φ 2 max ( a ∨ b )  k ∆ ( k − 1) kk B ( ℓ − k − 1) k + k ∆ ( ℓ − 1) K ′ kk B ( ℓ − k − 1) k  , where K ′ is deﬁned in [ 2 ]. The rest of the pro of f ollo ws after applying t h e arguments used in [ 2 ] and follo wing the pro cedure set out ab o ve to obtain the b ound on K B ( k ) . E Pro ofs of Section 10 Pr oo f of L emma 10.1 . S ince b σ ( v ) = + if and only if F ( v ) = 1, it follow s that 1 n n X v = 1 1 σ ( v )=+ 1 b σ ( v )= σ ( v ) = 1 n n X v = 1 1 σ ( v )=+ F ( v ) → f (+) 2 , and 1 n n X v = 1 1 σ ( v )= − 1 b σ ( v )= σ ( v ) = 1 n n X v = 1 1 σ ( v )= − (1 − F ( v )) → 1 − f ( − ) 2 . Consequently , 1 n n X v = 1 1 b σ ( v )= σ ( v ) → 1 + f (+) − f ( − ) 2 > 1 2 , b ecause f ( + ) > f ( − ) by assumption. Pr oo f of L emma 10.2 . W e use Prop osition 7.2 with τ ( G, v ) = 1 σ ( v )= i 1 I ℓ ( v ) µ − 2 ℓ 2 − b cg 2 ( i ) ≥ t . Denote b y ( T , o ) the branching pro cess deﬁned in Section 5 where the ro ot h as spin σ o uniformly d ra wn from { + , −} . Den ote th e number of oﬀspring of the ro ot by D and let Q ℓ ( v ) b e eq ual to Q 2 ,ℓ deﬁned on the tree T v obtained after removing the subtree attac h ed to v from T . Then, τ ( T , o ) = 1 σ o = i 1 J ℓ µ − 2 ℓ 2 − b cg 2 ( i ) ≥ t , where J ℓ = D X v = 1 Q ℓ ( v ) = ( D − 1) Q 2 ,ℓ − L o 2 ,ℓ , (E.1) with L o 2 ,ℓ deﬁned in ( 5.14 ). W e need to calculate lim ℓ →∞ E [ τ ( T , o )]. T o this end, we ﬁrst sho w t h at, conditional on σ o = i , J ℓ µ 2 ℓ 2 − b cg 2 ( i ) conv erges in p robabilit y to some c enter e d rand om v ariable b Y i . 50 W e ﬁrst calculate E i [ J ℓ | φ o ], where E i [ · ] = E [ ·| σ o = i ]. Put r o = a + b 2 Φ (1) φ o , then E i [ J ℓ | φ o ] = ∞ X n =0 E i [ J ℓ | D = n, φ o ] P ( D = n | φ o ) = ∞ X n =0 n E i [ Q 2 ,ℓ | D = n − 1 , φ o ] r n o e − r o n ! = r o ∞ X n =1 E i [ Q 2 ,ℓ | D = n − 1 , φ o ] r n − 1 o e − r o ( n − 1)! = r o E i [ Q 2 ,ℓ | φ o ] . (E.2) Recall from Theorem 5.7 , that Q 2 ,ℓ µ 2 ℓ 2 conv erges in L 2 to some ran d om v ariable X , with mean ... Therefore, E      Q 2 ,ℓ µ 2 ℓ 2 − X         φ 0 = ψ o  = ∞ X z =0 E      Q 2 ,ℓ µ 2 ℓ 2 − X         k Z 1 k = z  P ( k Z 1 k = z | φ 0 = ψ o ) ≤ e a + b 2 Φ (1) ( φ max − φ min ) E      Q 2 ,ℓ µ 2 ℓ 2 − X         φ 0 = φ max  (E.3) Recall from Theorem 5.7 that, uniformly for all ψ o , E i  Q 2 ,ℓ µ 2 ℓ 2     φ o = ψ o  → Φ (3) Φ (2) ρ µ 2 2 − ρ µ 2 ,ψ o g 2 ( i ) as n → ∞ . H en ce, sup n,ψ o E i h Q 2 ,ℓ µ 2 ℓ 2    φ o = ψ o i < ∞ , so that we can apply Lebesque’s dominated conv ergence theorem: E i [ J 2 ,ℓ ] µ 2 ℓ 2 = E i  r o E i  Q 2 ,ℓ µ 2 ℓ 2     φ o  → b cg 2 ( i ) , (E.4) as n → ∞ . W e now com bine the right h and side of ( E.1 ), ( E.4 ), and Theorem 5.7 (and in particular ( A.43 ) whic h implies that L o 2 ,ℓ /µ 2 ℓ 2 → 0 as n → ∞ ) to establish the claim that, cond itional on σ o = i , J ℓ µ 2 ℓ 2 − b cg k ( i ) conv erges in probability to some c enter e d random v ariable b Y i . In particular, conditional on σ o = i , J ℓ µ 2 ℓ 2 − b cg 2 ( i ) conv erges in distribution to b Y i . So that, for t as in th e statemen t, E [ τ ( T , o )] = 1 2 P  J ℓ µ 2 ℓ 2 − b cg 2 ( i ) ≥ t     σ o = i  → 1 2 P  b Y i ≥ t  , as n → ∞ . Finally , n oting that the error term in Prop osition 7.2 is O  n − ( γ 2 ∧ 1 40 )  = o (1) ﬁnishes the proof. Pr oo f of L emma 10.3 . This foll o ws after repeating t he pro of in [ 2 ] in conjunction with Lemma 10.2 established here. References [1] B. Bollob´ as, S. Janson, and O. Riordan. The phase transitio n in inhomogeneous random graphs. R andom Struct. Algorithms , 31(1):3–122, Aug. 2007. 51 [2] C. Bordena ve, M. Lelarge, and L. Massouli ´ e. Non- backtrac king sp ectrum of ran- dom graphs: communit y detection and n on-regular raman u jan graphs. a rXiv pr eprint 1501.06087 , 2015. [3] A. Decelle, F. Krzak ala, C. Mo ore, and L. Zdeb orov´ a. 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