ERGMs on block models

ER GMs on blo c k mo dels E. Magnanini ∗ F ebruary 19, 2026 Abstract W e extend the classical edge-triangle Exp onen tial Random Graph Mo del (ER GM) to an inhomogeneous setting in which v ertices carry types deter- mined b y an underlying partition. This leads to a blo c k-structured ERGM where interaction parameters dep end on v ertex types. W e establish a large deviation principle for the asso ciated sequence of measures and deriv e the corresp onding v ariational form ula for the limiting free energy . In the ferro- magnetic regime, where the parameters go verning triangle densities are non- negativ e, w e reduce the v ariational problem to a scalar optimization problem, thereb y identifying the natural block counterpart of the replica symmetric regime. Under additional restrictions on the parameters, comparable to the classical Dobrushin uniqueness region, w e prov e uniqueness of the maximizer and deriv e a law of large num b ers for the edge densit y . Keyw ords: Edge-triangle mo del, free energy , blo c k mo dels, Euler-Lagrange equations, limit theorems. AMS Sub ject Classification 2010: 05C80; 60F15; 82B05. Con ten ts 1 In tro duction 2 1.1 Our con tribution . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Preliminaries 4 2.1 The mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The space of colored graphons . . . . . . . . . . . . . . . . . . . 6 2.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ∗ W eierstrass Institute for Applied Analysis and Sto c hastics, Anton-Wilhelm-Amo-Straße 39,, 10117 Berlin, German y . 1 2 3 Pro ofs 14 3.1 LDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . . 21 3.3 Scalar problem for the free energy . . . . . . . . . . . . . . . . 25 3.4 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5 SLLN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A Other helpful lemmas 32 A.1 Erd˝ os-R ´ en yi LDP . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.2 Con tinuit y of cell-restricted homomorphism densities . . . . . 33 A.3 Alternativ e deriv ation of the free energy . . . . . . . . . . . . . 37 1 In tro duction So cial netw orks, lik e man y biological and tec hnological net works, are kno wn to exhibit a high degree of clustering , also referred to as tr ansitivity . Infor- mally , if tw o no des share a common neigh b or, they are more lik ely to b e link ed to each other as well. A cen tral mo delling challenge is to relate such lo c al fe atur es , captured by the frequency of small subgraphs and therefore eas- ily measurable, to the glob al structur e of the net work. Among probabilistic mo dels for so cial net w orks (see e.g.[ 26 ]), Exp onential R andom Gr aph Mo dels (ER GMs) ha v e a particularly transparen t statistical justification. More pre- cisely , if one prescrib es the empirical v alues of finitely many net w ork statistics (suc h as the num b er of edges, triangles, or other small subgraphs), then among all probability distributions on graphs consistent with these constraints, the en tropy-maximizing distribution is a Gibbs measure [ 20 ]. F o cusing on un- constrained, undirected ERGMs, a substantial b ody of rigorous probabilistic results has b een developed. These include the deriv ation of the limiting free energy and the analysis of its phase diagram, as well as more general results on the asymptotic structure and t ypical b eha vior of such graphs (see, e.g., [ 8 , 7 , 25 , 31 , 2 , 13 ]). A complemen tary line of researc h in v estigates fluctu- ation phenomena and limit theorems for subgraph densities, obtained both through statistical mechanics techniques (see, e.g., [ 4 , 22 , 3 , 23 , 24 ]) and via Stein’s metho d, the latter also providing quan titative normal approximations [ 15 , 27 , 30 ]. Classical ERGMs are typically homo gene ous , in the sense that their law is inv ariant under relab ellings of the v ertex set. Real netw orks, how ever, often exhibit structural heterogeneity: v ertices carry attributes (or types) that significan tly influence link formation. In this w ork we study inhomo gene ous ER GMs in which heterogeneit y is in tro duced through an underlying partition of the v ertex set into finitely many blo c ks. Each v ertex is assigned a type (or color) according to the block to which it b elongs. Th us, in the underlying reference measure, the probabilit y to create an edge b et ween tw o vertices dep ends on their t yp es. Models of this kind w ere originally introduced in the ph ysics literature [ 17 ] to describe comm unity organization, where t yp es may 3 represen t p olitical orientation, scien tific field, thematic similarit y of w ebpages, or other attributes. F rom a probabilistic p oin t of view, inhomogeneous ER GMs can b e seen as exp onen tial tilts of dense inhomogeneous Erd˝ os-R ´ en yi graphs, in the same wa y that classical ERGMs arise as tilts of the homogeneous Erd˝ os-R ´ en yi mo del. V ertex types enter the Hamiltonian through the interaction parameters that w eight the v arious subgraph densities, dep ending on the t yp es of the vertices in volv ed in those subgraphs. F or instance, one ma y assign a higher w eigh t to triangles formed by v ertices of the same t yp e than to mixed-type configu- rations, thereby reflecting an underlying notion of proximit y , such as shared in terests or geographical closeness. Among recent developmen ts on blo c k-structured graph mo dels, we men- tion the deriv ation of a Large Deviation Principle (LPD) in [ 18 , 6 ], together with adv ances in the closely related theory of pr ob ability gr aphons [ 1 , 12 ], whic h extend the classical graphon framew ork to settings where edges carry additional decorations. 1.1 Our contribution The main nov elt y of this pap er is to provide a step further to w ards the under- standing of inhomogeneous ERGMs. W e show that the v ariational principle a v ailable for the free energy for homogeneous ER GMs can b e transferred to this richer setting, yielding a well-posed v ariational problem under minimal structural assumptions on the underlying partition. This leads to an explicit c haracterization of the maximizers and to a natural blo c k-structured coun ter- part of the r eplic a symmetric r e gime . Outside this regime, we exp ect effects asso ciated with symmetry br e aking and phase transitions, whose analysis we defer to future w ork. The tec hniques of the pro ofs rely on classical to ols, suit- ably adapted to the blo c k structure of the mo del. In what follows, we briefly list our main results. 1. In Thm. 2.12 , w e prov e a large deviation principle for the asso ciated se- quence of measures. F rom the corresp onding rate function, w e deriv e the v ariational formula for the free energy; an alternative direct deriv ation is pro vided in App endix A.3 . 2. In Theorem 2.13 , we derive the Euler-Lagrange equations that charac- terize an y optimizer of the v ariational form ula, without imp osing restric- tions on the parameter regime. 3. In Thm. 2.14 , assuming nonnegativit y of parameters tuning the triangle densities, we reduce the v ariational problem for the free energy to a scalar optimization problem, leading to the fixed-point system ( 2.24 ). In Thm. 2.15 , w e identify a restricted parameter regime (comparable to the so-called Dobrushin uniqueness region) under which this system admits a unique solution. Com bining these results, w e obtain our uniqueness statemen t in Cor. 2.18 . 4 4. Finally , relying on the uniqueness result, w e establish a strong la w of large n um b ers for the edge density , extending [ 4 , Thm. 3.10]. 1.2 Structure The rest of the pap er is organized as follows. 1. In Sec. 2 we in tro duce the mo del, the terminology , and our main as- sumption on the partition (see Assumption 2.1 ). 2. Subsec. 2.2 contains the main definitions, including the notion of c ol- or e d gr aphons (and their asso ciated space), as well as Def. 2.7 of the equiv alence relation. 3. In Subsec. 2.3 we state our main results, while Sec. 3 is devoted to their pro ofs. 4. Finally , App endix A collects auxiliary lemmas and provides a direct deriv ation of the free energy , as an alternativ e to applying the Legendre transform to the rate function. 2 Preliminaries 2.1 The mo del Let n ∈ N . W e denote by G n the set of all simple undirected graphs on the v ertex set [ n ] := { 1 , . . . , n } . F or eac h G ∈ G n , let X = ( X ij ) 1 ≤ i,j ≤ n b e its adjacency matrix, i.e. the n × n symmetric matrix with entries in { 0 , 1 } defined b y X ij = ( 1 , if { i, j } is an edge of G, i  = j 0 , otherwise . W e denote by A n the set of all suc h adjacency matrices. Bey ond the under- lying graph, we endow the v ertex set with a finite coloring. This is ac hiev ed b y fixing a partition of the v ertex set [ n ] in to k (nonempt y) classes, which represen t the differen t colors. F ormally , we consider a collection B ( k ) ≡ B ( k ) n =  B ( n ) 1 , . . . , B ( n ) k  (2.1) of disjoint subsets of [ n ] whose union is [ n ]. The in terpretation is that t wo v ertices i, j ∈ [ n ] ha ve the same color if and only if they b elong to the same elemen t ( blo ck ) B ( n ) a , a ∈ [ k ] of the partition. In what follo ws, w e consider only partitions B ( k ) n satisfying the following assumption (whic h ma y be view ed as the analogue of [ 5 , Eq. 2.2]). 5 Assumption 2.1 (Partitions) . Fix k ∈ N and a ve ctor of blo ck lengths b := ( b 1 , . . . , b k ) with b i > 0 and P k i =1 b i = 1 . L et B ( k ) = ( B 1 , . . . , B k ) b e a p artition of [0 , 1] into c onse cutive intervals with L eb esgue me asur e λ ( B i ) = b i . F or e ach n ≥ 1 c onsider a p artition B ( k ) = ( B ( n ) 1 , . . . , B ( n ) k ) of [ n ] into c on- se cutive intervals with c ar dinalities w ( n ) i := | B ( n ) i | satisfying k X i =1 w ( n ) i = n and w ( n ) i n − → b i as n → ∞ , i = 1 , . . . , k . (2.2) Equivalently, B ( k ) may b e viewe d as the limiting c oloring of the unit inter- val [0 , 1] . W e denote b y G ( B ) n := { ( G, B ( k ) ) : G ∈ G n } the set of colored graphs. In view of the bijection b et ween the sets A n and G n , we ma y regard at the Hamiltonian of the e dge-triangle mo del on blo cks as a function from A ( B ) n := { ( X , B ( k ) ) : X ∈ A n } on to R , defined by H ( B ) n ; α , h ( X ) := 1 n X i,j,ℓ ∈ [ k ] α ij ℓ X u ∈ B i , v ∈ B j , w ∈ B ℓ u 1 , the p artition function factorizes as Z ( B ) n ; h = Y 1 ≤ i 0 such that δ ≤ g ( x, y ) ≤ 1 − δ for a.e. ( x, y ) ∈ [0 , 1] 2 . (2.16) 2. F or a.e. ( x, y ) ∈ [0 , 1] 2 , g ( x, y ) = exp  h B ( x, y ) + ( T ( B ) α g )( x, y )  1 + exp  h B ( x, y ) + ( T ( B ) α g )( x, y )  , (2.17) wher e T ( B ) α is the (nonline ar) triangle op er ator  T ( B ) α g  ( x, y ) := Z 1 0  ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z )  dz , (2.18) with ∆ α ; B ( x, y , z ) := k X i,j,ℓ =1 α ij ℓ 1 B i ( x ) 1 B j ( y ) 1 B ℓ ( z ) , (2.19) and h B ( x, y ) := k X i,j =1 h ij 1 B i ( x ) 1 B j ( y ) . (2.20) Theorem 2.14 (Scalar problem) . Supp ose that α ≥ 0 . Define the set of admissible symmetric matric es C sym := n C = ( c ij ) i,j ∈ [ k ] ∈ [0 , 1] k × k : c ij = c j i ∀ i, j ∈ [ k ] o , (2.21) and for C ∈ C sym let g C b e the c orr esp onding B –blo ck gr aphon g C ( x, y ) := k X i,j =1 c ij 1 B i ( x ) 1 B j ( y ) , ( x, y ) ∈ [0 , 1] 2 . (2.22) Then: 1. Every maximizer e g ⋆ ( B ) ∈ f W ( k ) B of the variational pr oblem in ( 2.15 ) ad- mits a r epr esentative of the form ( g C , B ( k ) ) . In p articular, the variational pr oblem r e duc es to a finite-dimensional one: sup C ∈ C sym    1 6 k X i,j,ℓ =1 α ij ℓ t ( B ) ij ℓ ( H 2 , g C ) + 1 2 k X i,j =1 h ij t ( B ) ij ( H 1 , g C ) − 1 2 I ( B ) ( g C )    . (2.23) 2. Ther e exists at le ast one maximizer C ⋆ ≡ C ⋆ ( α , h ) ∈ C sym of ( 2.23 ) . 3. Any maximizer C ⋆ = ( c ⋆ ij ) i,j ∈ [ k ] satisfies the Euler–L agr ange fixe d-p oint system: for al l i, j ∈ [ k ] , c ⋆ ij = exp  h ij + P k ℓ =1 b ℓ c ⋆ iℓ c ⋆ j ℓ α ij ℓ  1 + exp  h ij + P k ℓ =1 b ℓ c ⋆ iℓ c ⋆ j ℓ α ij ℓ )  (2.24) (se e Fig. 2.3 for an il lustr ation of ( g C ⋆ , B k ) ). 12 x y b 1 b 2 b 3 b 1 b 2 b 3 Figure 2.3: Illustration of a maximizer ( g C ⋆ , B k ) of ( 2.15 ), where C ⋆ maximizes the scalar problem ( 2.23 ) and satisfies the system in ( 2.24 ). Theorem 2.15 (Contraction map) . L et k ∈ N , h = ( h ij ) i,j ∈ [ k ] ∈ R k × k , and α = ( α ij ℓ ) i,j,ℓ ∈ [ k ] ∈ R k × k × k . Define the lo gistic function σ : R → (0 , 1) by σ ( x ) := e x 1 + e x . We c onsider the map S ( B ) α ; h : [0 , 1] k × k → (0 , 1) k × k define d c omp onentwise, for C = ( c ij ) i,j ∈ [ k ] ∈ C sym , by  S ( B ) α ; h ( C )  ij := σ h ij + k X ℓ =1 b ℓ c iℓ c j ℓ α ij ℓ ! , i, j ∈ [ k ] . (2.25) Then S ( B ) α ; h is glob al ly Lipschitz on [0 , 1] k × k with r esp e ct to ∥ · ∥ ∞ , with Lipschitz c onstant L := 1 2 ∥ α ∥ ∞ , (2.26) wher e ∥ α ∥ ∞ := max i,j,ℓ ∈ [ k ] | α ij ℓ | . If L < 1 (i.e., if ∥ α ∥ ∞ < 2 ), then S ( B ) α ; h is a c ontr action on ([0 , 1] k × k , ∥ · ∥ ∞ ) and ther efor e admits a unique fixe d p oint C ⋆ ≡ C ⋆ ( α , h ) . Before mo ving to the pro ofs, some remarks are in order. Remark 2.16. When α ≡ α ∈ R , the c ondition ∥ α ∥ ∞ < 2 r e duc es to the sc alar c onstr aint | α | < 2 , which c oincides with the r ange of validity of [ 8 , Thm. 6.2] for the standar d e dge-triangle mo del. The r e gion | α | < 2 is some- times r eferr e d to as Dobrushin ’s uniqueness r e gion. 13 Remark 2.17. The r ole of Thms. 2.13 – 2.14 c an b e summarize d as fol lows. The Euler–L agr ange e quations char acterize maximizers of the variational pr ob- lem thr ough a p ointwise nonline ar e quation and imply that any maximizer takes values strictly b etwe en 0 and 1 . Under the additional assumption α ≥ 0 , the pr oblem admits a maximizer that is B –blo ck c onstant, so that the infinite- dimensional variational pr oblem r e duc es to a finite-dimensional one in terms of matric es C = ( c ij ) i,j ∈ [ k ] . The fixe d-p oint map S ( B ) α ; h is intr o duc e d in Thm. 2.15 to analyze the c orr esp ond- ing Euler–L agr ange system. F or sufficiently smal l α , this map is a c ontr action on the c omplete sp ac e [0 , 1] k × k and ther efor e admits at most one fixe d p oint. Com bining these results, one can readily deduce the following corollary . Corollary 2.18 (Uniqueness of the maximizer) . Assume that 0 ≤ ∥ α ∥ ∞ < 2 . Then the variational pr oblem ( 2.15 ) admits a unique maximizer e g ⋆ ( B ) ∈ f W ( k ) B . Mor e over, any r epr esentative of e g ⋆ ( B ) has the form g C ⋆ as in ( 2.22 ) , wher e C ⋆ ≡ C ⋆ ( α , h ) ∈ C sym is the unique solution of the finite-dimensional Euler– L agr ange system ( 2.24 ) . Ha ving established a uniqueness result, it is natural to turn to the question of whether a law of large n umbers holds for subgraph densities. W e presen t our results for the edge densit y; how ever, w e claim that the same argumen t applies to an y subgraph density . In the results b elow, we keep the dep endence on n inside the partition explicit, in order to a void any confusion with the measure in Eq. ( 2.29 ) (arising from Rem. 2.20 ). Let E n denote the random n um b er of edges of a ERG sampled according to the probabilit y measure P ( B n ) n ; α , h . The follo wing holds. Theorem 2.19 (Exponential con v ergence for E n ) . Supp ose 0 ≤ ∥ α ∥ ∞ < 2 . Then, 2 E n n 2 exp − − − → k X i,j =1 b i b j c ⋆ ij , w.r.t. P ( B n ) n ; α , h , as n → + ∞ , (2.27) wher e, for e ach i, j ∈ [ k ] , c ⋆ i,j solves ( 2.24 ) . Remark 2.20 (Kolmogorov extension) . F or n ∈ N , r e c al l that A n = { 0 , 1 } ( n 2 ) is the set of adjac ency matric es of simple gr aphs on [ n ] = { 1 , . . . , n } . F or m > n , denote by π m → n : A m → A n the c anonic al pr oje ction obtaine d by r estricting a c onfigur ation to the induc e d sub gr aph on [ n ] . Assume the pr oje ctive c onsistency c ondition P ( B n +1 ) n +1; α , h ◦ π − 1 n +1 → n = P ( B n ) n ; α , h for al l n ∈ N , (2.28) that is, under P ( B n +1 ) n +1; α , h the induc e d sub gr aph on [ n ] has law P ( B n ) n ; α , h . Then, by the Kolmo gor ov extension the or em, ther e exists a unique pr ob ability me a- sur e P ( B ) α , h on the infinite pr o duct sp ac e  { 0 , 1 } ( N 2 ) , F  whose mar ginals on 14 A n c oincide with P ( B n ) n ; α , h for al l n . Her e F denotes the Bor el σ -algebr a and B = ( B i ) i ∈ [ k ] is a deterministic p artition of N such that B ( i ) n = B i ∩ [ n ] for e ach i ∈ [ k ] (wher e B ( i ) n ar e the elements of the p artition given in Assumption 2.1 ). Remark 2.21. In view of R em. 2.20 and Assumption 2.1 , the p artition B of N is such that for al l i ∈ [ k ] , | B i ∩ [ n ] | n → b i as n → ∞ , wher e b i = λ ( B i ) . Finally , Thm. 2.22 together with Rem. 2.20 leads to the following. Theorem 2.22 (SLLN for E n ) . Supp ose 0 ≤ ∥ α ∥ ∞ < 2 and that the finite- size p artition B n satisfies ( 2.28 ) . Then, 2 E n n 2 a . s . − − − → k X i,j =1 b i b j c ⋆ ij , w.r.t. P ( B ) α , h , as n → + ∞ , (2.29) wher e, for e ach i, j ∈ [ k ] , c ⋆ i,j solves ( 2.24 ) , and P ( B ) α , h is the infinite-volume Gibbs me asur e c onstructe d in R em. 2.20 . 3 Pro ofs 3.1 LDP Pr o of of Thm. 2.12 (LDP). The idea of the pro of is to represent P ( B n ) n ; α , h as a tilted measure with resp ect to the Erd˝ os–R ´ enyi reference measure. T o this end, w e first derive an alternative representation of the Hamiltonian H ( B n ) n ; α , h ( X ) = 1 n X i,j,ℓ ∈ [ k ] α ij ℓ X u ∈ B i , v ∈ B j , w ∈ B ℓ u 0. Let φ : [0 , 1] 2 → R b e b ounded, measurable and symmetric, and set g u ( x, y ) := g ( x, y ) + uφ ( x, y ), u ∈ R . Then, g u is as w ell a bounded, symmetric function from [0 , 1] 2 to R . Throughout the pro of, w e consider | u | sufficiently small, so that g u ∈ W . Then, b y maximalit y w e ha v e F ( B ) α , h ( g u ) ≤ F ( B ) α , h ( g ), hence d du F ( B ) α , h ( g u )     u =0 = 0 . (3.39) W e compute the deriv ative in ( 3.39 ), starting from the en trop y term. Since I is C 1 on (0 , 1), u is assumed to b e sufficiently small, and g stays in [ δ, 1 − δ ], Leibniz integral rule yields − 1 2 d du     u =0 Z [0 , 1] 2 I ( g u ( x, y )) dx dy = − 1 2 Z [0 , 1] 2 I ′ ( g ( x, y )) φ ( x, y ) dx dy = − 1 2 Z [0 , 1] 2 φ ( x, y ) ln g ( x, y ) 1 − g ( x, y ) dx dy . (3.40) F or the edge term we hav e d du     u =0 1 2 Z [0 , 1] 2 h B ( x, y ) g u ( x, y ) dx dy ! = 1 2 Z [0 , 1] 2 h B ( x, y ) φ ( x, y ) dx dy . (3.41) Finally , for the triangle term in ( 3.36 ), we define Ξ α ; B ( g ) := Z [0 , 1] 3 ∆ α ; B ( x, y , z ) g ( x, y ) g ( y , z ) g ( z , x ) dx dy dz . F or eac h ( x, y , z ) ∈ [0 , 1] 3 w e ha ve g u ( x, y ) g u ( y , z ) g u ( z , x ) =  g ( x, y ) + uφ ( x, y )  g ( y , z ) + uφ ( y , z )  ×  g ( z , x ) + uφ ( z , x )  (3.42) (again, since u is taken sufficiently small and all the functions inv olved are b ounded, differentiation can b e interc hanged with integration). Ex- panding the pro duct in ( 3.42 ), and isolating the terms of order u and u 2 , w e get g u ( x, y ) g u ( y , z ) g u ( z , x ) = g ( x, y ) g ( y , z ) g ( z , x ) + u  φ ( x, y ) g ( y , z ) g ( z , x ) + g ( x, y ) φ ( y , z ) g ( z , x ) + g ( x, y ) g ( y , z ) φ ( z , x )  + u 2 R u ( x, y , z ) , 23 where R u ( x, y , z ) denotes a remainder term collecting all con tributions of order at least tw o in u . Multiplying by ∆ α ; B ( x, y , z ) and integrating, w e obtain d du     u =0 Ξ α ; B ( g u ) = Z [0 , 1] 3 ∆ α ; B ( x, y , z )  φ ( x, y ) g ( y , z ) g ( z , x ) + g ( x, y ) φ ( y , z ) g ( z , x ) + g ( x, y ) g ( y , z ) φ ( z , x )  dx dy dz. (3.43) The right-hand side of ( 3.43 ) is the sum of three triple integrals, whic h can b e written in the same form, using the symmetry of ∆ α ; B and g : d du     u =0 Ξ α ; B ( g u ) = (3.44) 3 Z [0 , 1] 2 φ ( x, y )  Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz  dx dy . (3.45) Com bining ( 3.39 ), ( 3.40 ), ( 3.41 ), and ( 3.45 ), we find that for every b ounded symmetric φ , d du F ( B ) α , h ( g u )     u =0 = Z [0 , 1] 2 φ ( x, y ) " 1 2 Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz + 1 2 h B ( x, y ) − 1 2 ln g ( x, y ) 1 − g ( x, y ) # dx dy = 0 . Cho osing φ ( x, y ) equal to the brack eted term (which is b ounded, as g is b ounded aw a y from 0 and 1), yields that the brack et is 0 a.e., i.e. ln g ( x, y ) 1 − g ( x, y ) = h B ( x, y ) + Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz . (3.46) This pro v es ( 2.17 ), b y recalling ( 2.18 ). 2. W e now mov e to the pro of of Item 1 . Fix p ∈ (0 , 1) and define the truncation p erturbation g p,u ( x, y ) := (1 − u ) g ( x, y ) + u max { g ( x, y ) , p } , u ∈ [0 , 1] . Then g p,u ∈ W and d du g p,u ( x, y )     u =0 = ( p − g ( x, y )) + . (3.47) Set φ p ( x, y ) := max { g ( x, y ) , p } − g ( x, y ) = ( p − g ( x, y )) + , 24 where ( · ) + denotes the p ositive part. Then, for every u ∈ [0 , 1] and a.e. ( x, y ) ∈ [0 , 1] 2 , g p,u ( x, y ) = g ( x, y ) + u φ p ( x, y ) . (3.48) Note that φ p is measurable, symmetric, and b ounded with 0 ≤ φ p ( x, y ) ≤ p . The same steps p erformed in Item 2 for computing the directional deriv ativ e at u = 0, com bined with ( 3.47 ), yield d du F ( B ) α , h ( g p,u )     u =0 = Z [0 , 1] 2 " 1 2 Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz (3.49) + 1 2 h B ( x, y ) − 1 2 ln g ( x, y ) 1 − g ( x, y ) # ( p − g ( x, y )) + dx dy . (3.50) W e adopt the conv en tion that the integrand equals + ∞ when g ( x, y ) = 0, and equals 0 when g ( x, y ) = 1. No w, we observe that ∥ ∆ α ; B ∥ ∞ ≤ max i,j,ℓ ∈ [ k ] | α ij ℓ | =: α ∞ < ∞ , (3.51) and 0 ≤ g ( x, z ) g ( y , z ) ≤ 1 a.e. Hence, for a.e. ( x, y ), Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz ≥ − Z 1 0 | ∆ α ; B ( x, y , z ) | dz ≥ − α ∞ . (3.52) Also, h B is blo ck-constan t with v alues h ij , so it is b ounded: ∥ h B ∥ ∞ = max i,j ∈ [ k ] | h ij | =: h ∞ < ∞ , (3.53) Com bining ( 3.52 ) and ( 3.53 ), w e get the following point wise lo wer b ound: for a.e. ( x, y ), 1 2 Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz + 1 2 h B ( x, y ) ≥ − 1 2 ( α ∞ + h ∞ ) =: − κ, (3.54) where κ ≡ κ ( α , h ) > 0. No w, assume by contradiction that λ ( { g < p } ) > 0, where λ denotes the Leb esgue measure. On the set A p := { ( x, y ) ∈ [0 , 1] 2 : g ( x, y ) < p } we ha ve ( p − g ( x, y )) + > 0. Moreo v er, for ( x, y ) ∈ A p , ln g ( x, y ) 1 − g ( x, y ) ≤ ln p 1 − p , b ecause the map u 7→ ln u 1 − u is strictly increasing on (0 , 1). Therefore, using ( 3.54 ), for a.e. ( x, y ) ∈ A p the square brack et in ( 3.49 ) satisfies 1 2 Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz + 1 2 h B ( x, y ) − 1 2 ln g ( x, y ) 1 − g ( x, y ) ≥ − κ − 1 2 ln p 1 − p . (3.55) 25 No w choose p ∈ (0 , 1) so small that − κ − 1 2 ln p 1 − p > 0. Then, ( 3.55 ) im- plies that the brac ket is strictly p ositive on A p , and hence the in tegrand in ( 3.49 ) is strictly p ositive on A p . Since λ ( A p ) > 0, it follows that d du     u =0 F ( B ) α , h ( g p,u ) > 0 . (3.56) Recall that ˜ g ⋆ w as assumed to b e a maximizer of ( 3.38 ). By construction, g p,u ∈ W for all u ∈ [0 , 1], and satisfies g p, 0 = g for some represen tativ e g ∈ ˜ g ⋆ . It follows that the map u 7− → F ( B ) α , h ( g p,u ) attains its maximum at u = 0. In particular its right-deriv ative at 0 m ust satisfy d du     u =0 + F ( B ) α , h ( g p,u ) ≤ 0 , whic h contradicts ( 3.56 ). Hence λ ( { g < p } ) = 0, i.e. g ≥ p a.e. A similar argumen t sho ws the upp er b ound in ( 2.16 ), th us completing the pro of. ■ 3.3 Scalar problem for the free energy Pr o of of Thm. 2.14 , Items 1 & 2 . W e use the non-linear part (i.e. the trian- gle term) of the functional in ( 2.15 ) to identify the structural form of the maximizer. Recall the cell-restricted subgraph density ( 2.10 ). The following H¨ older inequalit y applies: t ( B ) ℓ 1 ℓ 2 ··· ℓ m ( H , g ) = Z [ B ℓ 1 ×B ℓ 2 ×···×B ℓ m ] Y { i,j }∈E ( H ) g ( x i , x j ) dx 1 . . . dx m . ≤ Y { i,j }∈E ( H ) " Z [ B i ×B j ×···×B ℓ ] g E ( H ) ( x i , x j ) dx 1 . . . dx m # 1 |E ( H ) | . W e now specialize to the case where H is a triangle (i.e., in our notation, H ≡ H 2 ); to ease notation we relab eled the indices ℓ 1 , ℓ 2 , ℓ 3 as i, j, ℓ . W e get: t ( B ) ij ℓ ( H 2 , g ) ≤ b i Z B j ×B ℓ g 3 ( x, y ) dxdy ! 1 / 3  b j Z B ℓ × B i g 3 ( x, y ) dxdy  1 / 3 × b ℓ Z B i ×B j g 3 ( x, y ) dxdy ! 1 / 3 = Y π ∈ Cyc( ij ℓ ) b π ( i ) Z B π ( j ) ×B π ( ℓ ) g ( x, y ) 3 dx dy ! 1 / 3 , (3.57) 26 where Cyc( ij ℓ ) denotes the set of cyclic permutations of ( i, j, ℓ ). The ab o v e displa y , together with the non-negativit y of α , yields the following inequalit y: k X i,j,ℓ =1 α ij ℓ t ( B ) ij ℓ ( H 2 , g ) ≤ k X i,j,ℓ =1 α ij ℓ Y π ∈ Cyc( ij ℓ ) b π ( i ) Z B π ( j ) ×B π ( ℓ ) g ( x, y ) 3 dx dy ! 1 / 3 . (3.58) Instead, for the edge term t ( B ) ij ( H 1 , g ) equality is reached in ( 3.57 ). Then, the argumen t runs as follows: the ob jectiv e functional in ( 2.15 ), and consequen tly its suprem um, is alw ays less than or equal to the one obtained b y replacing the triangle term with the righ t–hand side of ( 3.58 ) (notice that the edge term re- mains unc hanged under H¨ older inequality). If w e can exhibit a function e g ⋆ ( B ) for which the tw o suprema coincide, then the conclusion follows. Indeed, by construction, the supremum of the first functional is b ounded ab ov e by that of the second; on the other hand, equality at e g ⋆ ( B ) implies that the first supre- m um is also b ounded b elo w by the second. Combining the t wo inequalities, the t w o v ariational problems must therefore coincide. Now, equality in ( 3.58 ) holds if and only if ( 3.57 ) is an equalit y for eac h triple ( i, j, ℓ ), as the difference b et w een the left and right-hand side of ( 3.57 ) is nonp ositive. By the charac- terization of the equality case in H¨ older inequality (see, e.g., [ 16 , Thm. 6.2]), this is equiv alent to the existence of a measurable function W ( x, y , z ) ≥ 0 and constan ts a ij ℓ , b ij ℓ , c ij ℓ > 0 suc h that, for every i, j, ℓ ∈ [ k ], the identities g ( x, y ) 3 = a ij ℓ W ( x, y , z ) , (3.59) g ( y , z ) 3 = b ij ℓ W ( x, y , z ) , (3.60) g ( z , x ) 3 = c ij ℓ W ( x, y , z ) , (3.61) hold for a.e. ( x, y , z ) ∈ B i × B j × B ℓ . No w fix x ∈ B i , y ∈ B j and v ary z ∈ B ℓ . F rom ( 3.59 ) w e get g ( x,y ) 3 g ( y ,z ) 3 = a ij ℓ b ij ℓ =: µ and similarly for the other ratios. Therefore conditions ( 3.59 )–( 3.61 ) translate in to the fact that equality in ( 3.58 ) is reached if and only if, for some constan ts λ, µ ∈ R + g ( x, y ) 3 = µg ( y , z ) 3 = λg ( z , x ) 3 a.e. on B i × B j × B l . (3.62) Fix y ∈ B j . F rom ( 3.62 ) we obtain g ( y ) ( x ) 3 = µ g ( y ) ( z ) 3 for a.e. ( x, z ) ∈ B i × B ℓ . By F ubini’s theorem, for a.e. z ∈ B ℓ the ab o v e relation holds for a.e. x ∈ B i . Fix suc h a z . Then for a.e. x 1 , x 2 ∈ B i , g ( y ) ( x 1 ) 3 = µg ( y ) ( z ) 3 = g ( y ) ( x 2 ) 3 , so g ( y ) ( x ) is constan t in x on B i (a.e.). Similarly , w e get that g ( y ) ( z ) is constan t in z on B ℓ (a.e.). Hence, for a.e. y ∈ B j , there exist constants a ( y ) , b ( y ) such that g ( x, y ) = a ( y ) for a.e. x ∈ B i , g ( y , z ) = b ( y ) for a.e. z ∈ B ℓ . (3.63) 27 W e no w use the second identit y in ( 3.62 ), g ( x, y ) 3 = λg ( z , x ) 3 a.e. on B i × B j × B ℓ . Fix ( x, z ) ∈ B i × B ℓ outside a null set suc h that the ab o v e holds for a.e. y ∈ B j . Since the right-hand side do es not dep end on y , it follows that g ( x, y ) is constant in y on B j (a.e.). Th us, for a.e. ( x, z ) ∈ B i × B ℓ , there exists a constan t c ( x, z ) such that g ( x, y ) = c ( x, z ) for a.e. y ∈ B j . (3.64) No w fix x ∈ B i and pick z 1 , z 2 ∈ B ℓ outside a null set such that ( 3.64 ) holds for a.e. y ∈ B j with z = z 1 and with z = z 2 . Now, we may choose y ∈ B j suc h that b oth identities hold, and therefore c ( x, z 1 ) = g ( x, y ) = c ( x, z 2 ) . Therefore c ( x, z ) is constan t in z on B ℓ (a.e.), and we may write it as g ( x, y ) = c ( x ) for a.e. y ∈ B j . (3.65) Let X 0 ⊂ B i and Y 0 ⊂ B j b e full-measure sets where ( 3.63 ) and ( 3.65 ) hold. F or all ( x, y ) ∈ X 0 × Y 0 , a ( y ) = g ( x, y ) = c ( x ) . Hence for any y 1 , y 2 ∈ Y 0 and x ∈ X 0 , a ( y 1 ) = c ( x ) = a ( y 2 ) , so a ( y ) is constant on Y 0 . Since the argumen t is carried out for fixed blo cks B i and B j , we denote this constant b y c ij . It then follows that c ( x ) = c ij ∈ [0 , 1] for all x ∈ X 0 . Therefore, g ( x, y ) = c ij for a.e. ( x, y ) ∈ B i × B j . By symmetry of the graphon we hav e c ij = c j i . This shows that an y optimizer has the form g C ( x, y ) := P 1 ≤ i,j ≤ k c ij 1 B i ( x ) 1 B j ( y ) . Substituting this ansatz in to ( 2.15 ), w e obtain the finite-dimensional problem ( 2.23 ), whic h admits at least one solution C ⋆ since the functional is con tin uous, and C sym is compact. ■ Remark 3.6. Supp ose that α ij ℓ = 1 for al l i, j, ℓ ∈ [ k ] . By applying again 28 H¨ older ine quality to the r.h.s. of ( 3.58 ) , we get k X i,j,ℓ =1 Y π ∈ Cyc( ij ℓ ) b π ( i ) Z B π ( j ) ×B π ( ℓ ) g ( x, y ) 3 dx dy ! 1 / 3 ≤   k X i,j,ℓ =1 b i Z B j ×B ℓ g ( x, y ) 3 dx dy   1 / 3 ×   k X i,j,ℓ =1 b j Z B i ×B ℓ g ( x, y ) 3 dx dy   1 / 3   k X i,j,ℓ =1 b ℓ Z B i ×B j g ( x, y ) 3 dx dy   1 / 3 = Z [0 , 1] 2 g 3 ( x, y ) dxdy , wher e the last e quality fol lows sinc e, for e ach factor, the sum P k r =1 |B r | c an b e factor e d out and e quals 1 , as ( B r ) k r =1 is a p artition of [0 , 1] . In p articular, when k = 1 (that is, in the single-c ommunity c ase) the first and last terms in the ab ove chain c oincide. Pr o of of Thm. 2.14 , Item 3 . This is just a corollary of Thm. 2.13 , Item 2 . Consider g C as in ( 2.22 ), i.e. g C ( x, y ) = P k i,j =1 c ij 1 B i ( x ) 1 B j ( y ) . It is easy to chec k that for all i, j ∈ [ k ], Eq. ( 2.17 ) reduces to c ij = exp  h ij + P k ℓ =1 b ℓ c iℓ c j ℓ α ij ℓ  1 + exp  h ij + P k ℓ =1 b ℓ c iℓ c j ℓ α ij ℓ  . (3.66) W e do this via ( 3.46 ). Fix x ∈ B i and y ∈ B j . F or z ∈ B ℓ , we hav e g ( x, z ) = c iℓ and g ( y , z ) = c j ℓ . Recalling from ( 2.19 ) that ∆ α ; B ( x, y , z ) = P k i,j,ℓ =1 α ij ℓ 1 B i ( x ) 1 B j ( y ) 1 B ℓ ( z ), we get Z 1 0 ∆ α ; B ( x, y , z ) g ( x, z ) g ( y , z ) dz = k X ℓ =1 Z B ℓ ∆ α ; B ( x, y , z ) c iℓ c j ℓ dz = k X ℓ =1 b ℓ c iℓ c j ℓ α ij ℓ . Since h B ( x, y ) = h ij and g ( x, y ) = c ij for a.e. ( x, y ) ∈ B i × B j , plugging ev erything in to ( 3.46 ) yields ln c ij 1 − c ij = h ij + k X ℓ =1 b ℓ c iℓ c j ℓ α ij ℓ , (3.67) and the fixed-p oint equation ( 3.66 ) follows. ■ 29 Remark 3.7 (Reduction to standard edge-triangle) . Equation ( 3.66 ) is a c ouple d fixe d-p oint system for the matrix C . It gener alizes the standar d fixe d- p oint e quation u = exp( h + αu 2 ) 1+exp( h + αu 2 ) 5 by r eplacing the sc alar u 2 with the biline ar form P k ℓ =1 b ℓ c iℓ c j ℓ and by al lowing blo ck-dep endent triangle weights thr ough α ij ℓ . In p articular, the e quations for differ ent p airs ( i, j ) ar e not indep endent. The pro of Cor. 2.18 is omitted from this section, as it is a straigthforw ard consequence of Thms. 2.14 – 2.15 . 3.4 Uniqueness Pr o of of Thm. 2.15 . Since [0 , 1] k × k is a closed subset of the Banac h space ( R k × k , ∥ · ∥ ∞ ), it is complete. The map S ( B ) α ; h is well defined on [0 , 1] k × k and maps it into itself; hence Banach fixed-p oint theorem applies provided it is a con traction. W e no w estimate its Lipsc hitz constan t. Let σ ( x ) = e x 1+ e x . Then σ is C 1 and σ ′ ( x ) = e x (1 + e x ) 2 = σ ( x )  1 − σ ( x )  . Since u (1 − u ) ≤ 1 4 for all u ∈ [0 , 1], w e obtain 0 ≤ σ ′ ( x ) ≤ 1 4 for all x ∈ R . Hence, b y the mean v alue theorem, | σ ( x ) − σ ( x ′ ) | ≤ 1 4 | x − x ′ | for all x, x ′ ∈ R . (3.68) Fix C, b C ∈ [0 , 1] k × k and i, j ∈ [ k ]. With a sligh t abuse of notation, we use the same symbol T ( B ) α to denote both the operator acting on graphons (recall ( 2.18 )) and its restriction to B –blo ck graphons, iden tified with matrices C ∈ [0 , 1] k × k . Define ( T ( B ) α ( C )) ij := k X ℓ =1 b ℓ c iℓ c j ℓ α ij ℓ , (3.69) so that ( S ( B ) α ; h ( C )) ij = σ ( h ij + ( T ( B ) α ( C )) ij ) (recall ( 2.25 )). Using ( 3.68 ) with x = h ij + ( S ( B ) α ; h ( C )) ij and x ′ = h ij + ( S ( B ) α ; h ( b C )) ij , w e get | ( S ( B ) α ; h ( C )) ij − ( S ( B ) α ; h ( b C )) ij | ≤ 1 4 | ( T ( B ) α ( C )) ij − ( T ( B ) α ( b C )) ij | . (3.70) 5 Obtained from ( 3.66 ) for k = 1; first analyzed in [ 7 , Lem. 12]. 30 Next, b y linearit y and the triangle inequality , | ( T ( B ) α ( C )) ij − ( T ( B ) α ( b C )) ij | =      k X ℓ =1 b ℓ α ij ℓ  c iℓ c j ℓ − ˆ c iℓ ˆ c j ℓ       ≤ k X ℓ =1 b ℓ | α ij ℓ | | c iℓ c j ℓ − ˆ c iℓ ˆ c j ℓ | ≤ ∥ α ∥ ∞ k X ℓ =1 b ℓ | c aℓ c bℓ − c ′ aℓ c ′ bℓ | , (3.71) where we recall ∥ α ∥ ∞ = max i,j,ℓ ∈ [ k ] | α ij ℓ | . W e now b ound the pro duct differ- ence. F or each ℓ , c iℓ c j ℓ − ˆ c iℓ ˆ c j ℓ = ( c iℓ − ˆ c iℓ ) c j ℓ + ˆ c iℓ ( c j ℓ − ˆ c j ℓ ) , hence | c iℓ c j ℓ − ˆ c iℓ ˆ c j ℓ | ≤ | c iℓ − ˆ c iℓ | c j ℓ + ˆ c iℓ | c j ℓ − ˆ c j ℓ | . (3.72) Since C, b C ∈ [0 , 1] k × k , w e ha v e c j ℓ ≤ 1 and ˆ c iℓ ≤ 1, while | c iℓ − ˆ c iℓ | ≤ ∥ C − b C ∥ ∞ and | c j ℓ − ˆ c j ℓ | ≤ ∥ C − b C ∥ ∞ Plugging these b ounds into ( 3.72 ) yields | c iℓ c j ℓ − ˆ c iℓ ˆ c j ℓ | ≤ 2 ∥ C − C ′ ∥ ∞ . (3.73) Com bining ( 3.71 ) and ( 3.73 ) (and recalling ( 3.32 )) giv es | ( T ( B ) α ( C )) ij − ( T ( B ) α ( b C )) ij | ≤ 2 ∥ α ∥ ∞ ∥ C − b C ∥ ∞ . (3.74) Finally , injecting ( 3.74 ) into ( 3.70 ) yields, for all i, j , | ( S ( B ) α ; h ( C )) ij − ( S ( B ) α ; h ( b C )) ij | ≤ 1 2 ∥ α ∥ ∞ ∥ C − b C ∥ ∞ . T aking the maximum ov er ( i, j ) pro ves the Lipschitz b ound with constan t L = 1 2 ∥ α ∥ ∞ , yielding ( 2.26 ). If L < 1, then S ( B ) α ; h is a con traction on the complete metric space [0 , 1] k × k . By Banac h’s fixed-p oint theorem, S ( B ) α ; h admits a unique fixed p oint C ⋆ ≡ C ⋆ ( α , h ) ∈ [0 , 1] k × k . Finally , b y definition ( 2.25 ), the fixed-p oint equation S ( B ) α ; h ( C ) = C is exactly the comp onent wise system ( 2.24 ). Therefore the fixed p oin t C ⋆ is the unique solution of ( 2.24 ). ■ 3.5 SLLN Pr o of of The or em 2.19 . Define the finite-size free energy f ( B ) n ; α , h := 1 n 2 ln Z ( B ) n ; α , h , where Z ( B ) n ; α , h is the partition function asso ciated with the Hamiltonian ( 2.3 ). F or s ∈ R , w e in tro duce the scaled cum ulan t generating function asso ciated with the edge count E n , defined as c ( B ) n ( s ) := 2 n 2 ln E ( B ) n ; α , h  exp  s E n  , (3.75) 31 where E ( B ) n ; α , h denotes exp ectation with resp ect to the Gibbs measure induced b y the Hamiltonian H ( B ) n ; α , h . W e prov e exponential conv ergence of the edge densit y via the sequence of functions ( 3.75 ). By [ 14 , I I.6.3], this b oils do wn to pro ving that the limiting cumulan t generating function c ( B ) ( s ) (see ( 3.79 ) b elo w) is differentiable at s = 0 and that its deriv ative at the origin coincides with the claimed limit. A direct calculation yields c ( B ) n ( s ) = 2 n 2 ln X X ∈A ( B ) n exp  H ( B ) n ; α , h ( X ) + s E n ( X )  Z ( B ) n ; α , h . (3.76) Since E n ( X ) = P i,j ∈ [ k ] P u ∈ B ( n ) i , v ∈ B ( n ) j us g 2 ( x a r , x b r ) , s = 0 , 1 , . . . , E , thus obtaining the de c omp osition (c.f. ( A.11 ) ) | t ( B ) ℓ ( H , g 1 ) − t ( B ) ℓ ( H , g 2 ) | = E X s =1 I s . (A.18) Each I s is the absolute value of an inte gr al whose inte gr and c ontains exactly one factor ( g 1 − g 2 )( x a s , x b s ) and al l other factors ar e in [0 , 1] . T r e ating ( x a s , x b s ) as fixe d p ar ameters and inte gr ating over the r emaining m − 2 vari- ables, one obtains, for e ach s ∈ E , a r epr esentation analo gous to ( A.17 ) : I s ≤ ∥ g 1 − g 2 ∥ □ . Final ly, fr om ( A.18 ) we get   t ( B ) ℓ ( H , g 1 ) − t ( B ) ℓ ( H , g 2 )   ≤ |E ( H ) | ∥ g 1 − g 2 ∥ □ , and ( A.3 ) al lows to c onclude as b efor e. 37 A.3 Alternativ e deriv ation of the free energy Pr o of of Thm. 2.12 (fr e e ener gy). W e aim to show that lim n →∞ 1 n 2 ln Z ( B ) n ; α , h = f ( B ) α , h , where f ( B ) α , h coincides with the r.h.s. of ( 2.15 ). This conclusion migh t b e already suggested by ( 3.15 ), as an application of Laplace’s metho d (see, e.g. [ 28 , Subsec. 3.3.2]). A rigorous and more direct argument, ho w ever, can b e obtained b y adapting the pro of in [ 8 , Thm. 3.1]. Indeed, the tw o k ey ingredien ts of the strategy , namely the contin uity of subgraph densities and the large deviation principle for the sequence of Erd˝ os-R´ enyi measures, remain v alid in our setting. First, w e observe that combining represen tation ( 3.13 ) together with ( 3.11 ), we get exp [ − 2 γ η n ( B ) n 2 ] X X ∈A ( B ) n exp [ n 2 U ( B ) α , h ( X )] ≤ Z ( B ) n ; α , h = X X ∈A ( B ) n exp [ n 2 U ( B n ) α , h ( X )] (A.19) = X X ∈A ( B ) n exp [ n 2 U ( B ) α , h ( X )] exp [ n 2 R n ; α , h ( X )] (A.20) ≤ exp [2 γ η n ( B ) n 2 ] X X ∈A ( B ) n exp [ n 2 U ( B ) α , h ( X )] . (A.21) Since η n ( B ) → 0 as n → ∞ , it suffices to analyze the term Ψ n := 1 n 2 ln X X ∈A ( B ) n exp [ n 2 U ( B ) α , h ( X )] , as the contribution of the remaining terms v anishes in the limit n → ∞ . Fix ε > 0. Recall that U ( B ) α , h can b e viewed as a function f W ( k ) B → R , and that it is b ounded. Therefore, there is a finite set A suc h that the in terv als { ( a, a + ε ) : a ∈ A } co v er the range of U ( B ) α , h . F or each a ∈ A , let e F ( a, B ) :=  U ( B ) α , h  − 1 ([ a, a + ε ]) ⊆ f W ( k ) B (A.22) e F ( a, B ) n := { X ( B ) ∈ A ( B ) n : τ ( g X B ) ∈ e F ( a, B ) } . (A.23) By the contin uit y of U ( B ) α , h , eac h e F ( a, B ) is closed. Now, e n 2 Ψ n = X X ∈A ( B ) n e n 2 U ( B ) α , h ( g X B ) = X a ∈ A X X ( B ) ∈ e F ( a, B ) n e n 2 U ( B ) α , h ( g X B ) ≤ X a ∈ A e n 2 ( a + ε ) | e F ( a, B ) n | ≤ | A | sup a ∈ A e n 2 ( a + ε ) | e F ( a, B ) n | . (A.24) By Lem. A.1 (i.e. the Erd˝ os-R´ enyi LDP on the space f W ( k ) B ), lim sup n →∞ ln | e F ( a, B ) n | n 2 ≤ ln 2 2 − inf ˜ h ∈ e F ( a, B ) I ( B ) 1 2 ( ˜ h ) = − 1 2 inf ˜ h ∈ e F ( a, B ) I ( B ) ( ˜ h ) . (A.25) 38 Hence, w e get lim sup n →∞ Ψ n = lim sup n →∞ 1 n 2 ln e n 2 Ψ n ( A.24 ) ≤ lim sup n →∞ ln( A ) n 2 + sup a ∈ A " lim sup n →∞ 1 n 2 ln e n 2 ( a + ε ) + lim sup n →∞ ln | e F ( a, B ) n | n 2 # ( A.25 ) = sup a ∈ A  a + ε − 1 2 inf ˜ h ∈ e F ( a, B ) I ( B ) ( ˜ h )  . (A.26) No w, from ( A.22 ), each ˜ h ∈ e F ( a, B ) satisfies U ( B ) α , h ( ˜ h ) ≥ a . Consequently , sup ˜ h ∈ e F ( a, B )  U ( B ) α , h ( ˜ h ) − 1 2 I ( B ) ( ˜ h )  ≥ sup ˜ h ∈ e F ( a, B )  a − 1 2 I ( B ) ( ˜ h )  = a + 1 2 sup ˜ h ∈ e F ( a, B ) [ −I ( B ) ( ˜ h )] = a − 1 2 inf ˜ h ∈ e F ( a, B ) I ( B ) ( ˜ h ) , and rearranging terms − 1 2 inf ˜ h ∈ e F ( a, B ) I ( B ) ( ˜ h ) ≤ − a + sup ˜ h ∈ e F ( a, B )  U ( B ) α , h ( ˜ h ) − 1 2 I ( B ) ( ˜ h )  . (A.27) Substituting ( A.27 ) into ( A.26 ) w e get lim sup n →∞ Ψ n ≤ ε + sup a ∈ A sup ˜ h ∈ e F ( a, B )  U ( B ) α , h ( ˜ h ) − 1 2 I ( B ) ( ˜ h )  = ε + sup ˜ h ∈ f W ( k ) B  U ( B ) α , h ( ˜ h ) − 1 2 I ( B ) ( ˜ h )  . (A.28) Similarly , for each a ∈ A , one can define e O ( a, B ) :=  U ( B ) α , h  − 1 (( a, a + ε )). By retracing the same steps (and using in ( A.25 ) the opp osite inequalit y for the lim inf of op en sets provided by the LDP) one finally concludes that lim sup n →∞ Ψ n ≥ − ε + sup ˜ h ∈ f W ( k ) B  U ( B ) α , h ( ˜ h ) − 1 2 I ( B ) ( ˜ h )  . (A.29) Since ε is arbitrary in ( A.28 ) and ( A.29 ), this completes the pro of. ■ References [1] R. Abraham, J.-F. c. Delmas, and J. W eib el. Probability-graphons: limits of large dense weigh ted graphs. Innov. Gr aph The ory , 2:25–117, 2025. [2] D. Aristoff and C. Radin. Emergent structures in large net works. J. Appl. Pr ob ab. , 50(3):883–888, 2013. MR3102521 . 39 [3] A. Bianc hi, F. Collet, and E. 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