Consistency of community detection in networks under degree-corrected stochastic block models

The Annals of Statistics 2012, V ol. 40, No. 4, 2266–22 92 DOI: 10.1214 /12-AOS1036 c  Institute of Mathematical Statistics , 2 012 CONSISTENCY OF COMMUNITY DETE CTION IN NETW ORKS UNDER DEGREE-CORREC TED STOCHASTIC BLOCK MODELS By Yunpeng Zhao, Eliza vet a Levina 1 and Ji Zhu 2 Ge o r ge M ason University, U niversity of Michigan and University of Michigan Comm unity detection is a fundamental problem in netw ork anal- ysis, with applications in man y diverse areas. The sto chastic block mod el is a common to ol for model-based co mmunity detection, and asymptotic tools for c hecking consistency of comm unity d etection under the block model hav e been recen tly developed. How ever , the block mod el is limited by its assumption that all nod es within a comm unity are sto chas tically equiva lent, and pro vides a p o or fit to netw orks with hubs or highly v arying no de degrees within comm uni- ties, whic h are common in practice. The degree-corrected sto chas tic block mo del was p rop osed to add ress this shortcoming and allow s v ariation in n od e degrees w ithin a communit y while preserving the o verall blo ck community structure. I n t his pap er w e establish general theory for c hec king consistency of comm unity detection und er the degree-corrected sto chastic blo ck mod el and compare several com- munit y detection criteria under b oth the stand ard and the d egree- corrected mo dels. W e sho w which criteria are consistent under which mod els and constraints , as wel l as compare their relative performance in practice. W e find that m et h od s based on th e degree-corrected b lock mod el, which includ es the standard blo ck m o del as a sp ecial case, are consistent under a wider class of mo dels and that mod ularity- type meth od s require parameter constrain ts for consistency , whereas lik eliho o d-based metho ds do not. On th e other hand, in p ractice, the degree correction inv olv es estimating many more parameters, and empirically we fin d it is only worth doing if the no de d egrees within comm unities are ind eed h ighly v ariable. W e illustrate th e metho ds on simulated netw orks and on a n etw ork of p olitical b logs. Received Novem b er 2011; rev ised July 2012. 1 Supp orted in part by NSF Grants DMS-08-05798, DMS- 01106772 and D MS-1159005 2 Supp orted in part by NSF Grant DMS-07-48389 and NIH Grant R01-GM-096194. AMS 2000 subje ct classific ations. 62G20. Key wor ds and phr ases. Comm unity d etection, degree-corrected sto chastic blo ck mo d- els, consistency . This is an electronic repr int of the origina l ar ticle published by the Institute of Mathematical Statistics in The Annals of Statistics , 2012, V ol. 40, No. 4, 2266–229 2 . This repr int differs from the or iginal in pagination and t yp ogr aphic detail. 1 2 Y. ZH AO, E. LEV INA AND J. ZHU 1. In tro duction. Net w orks ha ve b ecome one of the more common forms of data, and net w ork analysis has rece ive d a lot of atten tion in computer science, physic s, so cial sciences, biology and statist ics (see [ 13 , 15 , 25 ] for reviews). The applications are man y and v aried, includ in g so cial n et w orks [ 31 , 37 ], gene regulatory net works [ 33 ], recommender systems and secur ity monitoring. One of the fundamenta l problems in netw ork analysis is com- m unity detection, where comm unities are groups o f nod es that are, in some sense, more similar to eac h other than to other no des. The precise defin ition of comm un it y , lik e that o f a cluster in multiv ariate analysis, is difficult to formalize, but many metho ds ha v e b een dev elop ed to address this problem (see [ 11 , 15 , 23 ] for compr eh ensiv e r ecen t reviews), often relying on the in- tuitiv e notion of communit y as a group of no des with man y links b et w een themselv es and few er links to the rest of the netw ork. Three groups of m etho ds for comm u nit y d etection can b e loosely iden- tified in the literature. A n umber of greedy algorithms such as hierarc hical clustering ha v e b een pr op osed (see [ 22 ] for a review), wh ic h we will not fo cus on in this pap er. The second class of metho ds in vo lv es optimiz ation of so me “reasonable” global criteria o v er all p ossible net work partitions and includes graph cuts [[ 34 ], [ 38 ]], sp ectral clustering [ 28 ] an d mo dularit y [ 23 , 26 ], the latter discussed in detail b elo w. Finally , mo d el-based method s rely on fit- ting a p robabilistic mo del for a n et w ork with comm unities. Perhaps the b est kno wn suc h mo del is the sto c hastic blo c k mod el, which w e will also refer to as simply the blo c k mo del [ 18 , 29 , 35 ]. O ther mo d els include a recen tly in- tro duced degree-co rr ected sto chastic blo c k mod el [ 20 ], mixture mo d els for directed n et w orks [ 27 ], m ultiv ariate laten t v ariable mod els [ 16 ], laten t fea- ture m o dels [ 17 ] and mixed mem b ership sto c hastic blo c k mo d els for model- ing o verlapping comm unities [ 2 ]. F rom the algo rithmic p oin t of view, many mo del-based metho ds also lead to criteria to b e optimized o ve r all partitions, suc h as the profile like liho o d under the assum ed mod el. The la rge num b er o f a v ailable metho ds leads to the question of ho w to compare them in a p rincipled manner, other th an on ind ividual examples. There has b een little theoretical analysis of communit y detect ion method s unt il v ery recen tly , when a consistency framework for comm unity detec- tion w as in tro duced b y Bic k el and C hen [ 5 ]. They dev elop ed general theory for c hecking the consistency of detection criteria under the stoc hastic blo ck mo del (discussed in detail b elo w) as the num b er of no des grows and the num- b er of comm unities remains fixed, and their result has b een generalized to allo w the n umber of comm unities to grow in [ 7 ]; see also [ 32 ]. The sto chasti c blo c k mo del, how ever, h as serious limitati ons in practice : it trea ts all no des within a communit y as stochastic ally equiv alent, and thus do es not allo w for the exist ence of “hubs,” h igh-d egree no des at the cent er of m an y comm u ni- ties o bserved in real data. T o add ress this issue, Karrer and Newman [ 20 ] CONSISTENCY OF COMMUNITY DETECTION 3 prop osed the degree-correcte d sto c hastic blo c k mo del, wh ic h can a ccommo- date hubs (a similar mo del for a directed net work w as previously pr op osed in [ 36 ], bu t they did not fo cus on comm unity detectio n and assumed kn o wn comm unit y memb ership). In [ 20 ], the authors ga ve several examples sho wing this mo del fits d ata with hubs muc h b etter than the blo c k mo del; ho w ev er, there are no consistency results av ailable under this new mo del, and thus no w a y to compare method s in general. In this pap er we generalize the consistency fr amew ork of [ 5 ] to the degree- corrected sto c hastic blo ck mod el and obta in a general theorem for comm u- nit y detection consistency . S ince the degree-co rr ected mo del includes the regular blo c k mo d el as a sp ecial case, co nsistency results under the block mo del follo w automatica lly . W e then ev aluate t w o t yp es of mo dularity and the tw o criteria derived from the b lo c k mo del and the degree-corrected blo ck mo del using this general framework. One of our goa ls is to emphasize the difference b etw een assu m ed mo dels (needed for theoretical analysis) an d cr i- teria for fin ding the optimal partitio n, whic h ma y o r ma y not be motiv ated b y a particular mo d el. What we ultimately sh o w agrees with statistical com- mon sen s e: cr iteria der ived from a p articular mo d el are consisten t when this mo del is assumed, but not necessarily consistent if the mo del do es not hold. F urther , if a criterion relies imp licitly on an assump tion ab out the mo del parameters (e.g., mo d ularit y implicitly assumes that links within communi- ties are s tr onger than b et w een), then it will b e co nsistent only if the model parameters are co nstrained to satisfy this assumption. W e m ak e all of the ab o ve s tatemen ts precise later in the pap er. The rest of the article is organized as follo ws. W e set up all notation and define the relev an t mo d els and criteria in S ection 2 . Consistency results under the regular and the degree-corrected stoc hastic blo ck mo dels for all of the criteria in Section 2 are stated in Section 3 . The general consistency theorem which implies all of these results is presen ted in Section 4 . In Section 5 we compare the p erforman ce of these criteria on simulated netw orks, and in Section 6 w e illustrate the m etho ds on a net work of p olitical blogs. Sectio n 7 concludes with a summ ary and discussion. All pro ofs are given in the App end ix . 2. Net work mo dels and communit y d etectio n crit eria. Before w e p ro- ceed to discuss sp ecific criteria and m o dels, w e introd uce some basic nota- tion. A net w ork N = ( V , E ), where V is the set of no des (vertice s), | V | = n , and E is the set of edges, can b e repr esen ted b y its n × n adjacency matrix A = [ A ij ], where A ij = 1 if ther e is an edge from i to j , and A ij = 0 other- wise. W e only co nsid er u n wei ghte d and undirected net w orks here, and thus A is a binary symmetric matrix. The c ommunit y detectio n problem can b e form ulated as findin g a disjoin t partition V = V 1 ∪ · · · ∪ V K or, equiv alen tly , 4 Y. ZH AO, E. LEV INA AND J. ZHU a set of no de lab els e = { e 1 , . . . , e n } , w h ere e i is the lab el of no de i and tak es v alues in { 1 , 2 , . . . , K } . F or a ny set of lab el assignmen ts e , let O ( e ) b e the K × K matrix defined b y O k l ( e ) = X ij A ij I { e i = k , e j = l } , where I is the in dicator function. F urther, let O k ( e ) = X l O k l ( e ) , L = X ij A ij . F or k 6 = l , O k l is the total num b er of edges b etw een comm unities k and l ; O k is th e sum of no de degrees in comm unit y k , and L is the sum of all degrees in the net w ork. If self-loops are not allo w ed (i.e., A ii = 0 is enforced), then w e can also interpret O k k as t w ice the total num b er of edges w ithin communit y k and L as t wice the n umber of edges in the whole netw ork. Finally , let n k ( e ) = P i I { e i = k } b e the n umber of no des in the k th communit y , and f ( e ) = ( n 1 n , n 2 n , . . . , n K n ) T . The stoc hastic blo c k mo del, whic h is p erhaps the most commonly u s ed mo del for n et w orks with comm unities, p ostulates that, giv en no de lab els c = { c 1 , . . . , c n } , the edge v ariables A ij ’s are indep endent Bernoulli rand om v ariables with E [ A ij ] = P c i c j , (2.1) where P = [ P ab ] is a K × K symmetric matrix. W e will use this form ulation throughout the pap er, whic h allo ws for self-loops. While it is also common to exclude self-lo ops, sometimes they are pr esen t in the data (as in our example in Section 6 ) and allo wing them leads to simpler notation. In pr in ciple, all of our results go through for the v ersion of the mo dels with s elf-lo ops excluded, with appropriate mod ifi cations mad e to the proofs. Under the mo del ( 2.1 ), all no des with the same lab el are sto chastic ally equiv alen t to eac h other, whic h in practice limits the app licabilit y of the sto c hastic blo ck mo del, as p oint ed out in [ 20 ]. Th e alternativ e p r op osed in [ 20 ], the degree- corrected sto c hastic blo c k mod el, is to r eplace ( 2.1 ) with E [ A ij ] = θ i θ j P c i c j , (2.2) where θ i is a “degree parameter” asso ciated with nod e i , reflecting its in- dividual prop ensit y to form ties. The degree parameters ha ve to sati sfy a constrain t to be iden tifiable, whic h in [ 20 ] w as set to P i θ i I ( c i = k ) = 1, for eac h k (other constraint s are p ossible). F u r ther, they replaced the Bernoulli CONSISTENCY OF COMMUNITY DETECTION 5 lik eliho o d by the Poisson, to simplify tec hnical d eriv ations. With these as- sumptions, a profile lik eliho o d can b e d eriv ed b y maximizing o v er θ and P , giving the fol lo wing criterion to b e optimized o v er all p ossible partitions: Q DCBM ( e ) = X k l O k l log O k l O k O l . (2.3) W e ha v e compared the p erformance of this criterion in practice to its sligh tly more complicated v ersion b ased on th e (correct) Bernoulli likeli ho o d instead of the Po isson and found no difference in the solutio ns these t w o methods pro du ce. The Bern oulli distribution with a small mean is w ell appro ximated b y the P oisson distrib ution, and most real netw orks are sparse, so one can exp ect the approxi mation to work w ell; see also a more detailed d iscu ssion of this in [ 30 ]. W e w ill use ( 2.3 ) in all further analysis, to b e consisten t w ith [ 20 ] and ta ke a dv an tage of the simpler form. The degree-corrected model includes the regular stochastic blo ck mo del as a sp ecial case, w ith all θ i ’s equal. Enforcing this a dd itional constrain t on the profile like liho o d leads to the follo wing criterion to b e optimized ov er all partitions: Q BM ( e ) = X k l O k l log O k l n k n l . (2.4) Lik e crite rion ( 2.3 ), th is is based on the P oisson assumption but gi ve s iden- tical r esults to the Bernoulli version in practice. Here w e use the form ( 2.4 ) for consistency with ( 2.3 ) and with [ 20 ]. A differen t t yp e of criterion u sed for communit y detectio n is mo du larit y , in tro du ced in [ 26 ]; see also [ 23 ] and [ 24 ]. The basic idea of modu larit y is to compare the num b er of observed edges within a comm un it y to the num b er of exp ected ed ges under a null mo del and maximize this difference o v er all p ossible communit y partitions. Thus, the ge neral form of a mod ularit y criterion is Q ( e ) = X ij [ A ij − P ij ] I ( e i = e j ) , (2.5) where P ij is the (estimated) pr ob ab ility of an edge falling b et wee n i and j under the null m o del. The con v en tion in the physics literature is to d ivide Q b y L , which we omit here, since it do es not change the solution. The c hoice of the n ull mo del, that is, of a mo del with no comm unities ( K = 1), determines the exact f orm of mo d ularit y . The sto c hastic blo ck mo del with K = 1 is simply the Erd os–Ren yi random graph, where P ij is a constan t whic h can b e estimated b y L/n 2 . Plu gging P ij = L/n 2 in to ( 2.5 ) giv es what w e will call the Erdos–Ren yi mo d ularit y (ERM), Q ERM ( e ) = X k  O k k − n 2 k n 2 L  . (2.6) 6 Y. ZH AO, E. LEV INA AND J. ZHU If instead we take th e degree-corrected mo del with K = 1 as the n ull mo del, it p ostulates that P ij ∝ θ i θ j , where θ i is the degree p arameter. This is essentially the w ell-kno wn exp ected degree rand om graph, also kn o wn as the configuration mo del. In this case, P ij can b e estimated by d i d j /L , where d i = P j A ij is the degree of no de i . Sub stituting this into ( 2.5 ) giv es the p opular Newman–Girv an mod ularit y (NGM) , introd uced in [ 26 ]: Q NGM ( e ) = X k  O k k − O 2 k L 2 L  . (2.7) The four d ifferen t criteria for comm unit y detection are summ arized in T a- ble 1 . Note that the t w o lik eliho o d-based criteria, BM and DCBM, tak e in to accoun t all links within and b etw een comm unities, and wh ic h comm unities they connect; whereas the mo d u larities w ould n ot c hange if all the links connecting differen t communities were randomly p ermute d (a s long as they did n ot b ecome links within comm unities). F ur ther, note that the degree correction amoun ts to s u bstituting O k for n k and L for n , b oth for mo du- larit y and likeli ho o d-based criteria. Th us, if all no d es within a comm unity are treated as equ iv alen t, their n umber s u ffices to w eigh communit y stren gth appropriately; and if the no des are allo wed to ha ve differen t exp ected de- grees, then the num b er of edges b ecomes the correct w eigh t. Both of these features mak e sense intuitiv ely and, as w e will see later, will fit in naturally with consistency conditions. Our analysis indicates that Newman–Girv an mo d ularit y and degree-correcte d blo c k mo del criteria are consistent under the more general degree-c orrected mo dels but Erdos–Ren yi modularity and blo c k mo del criteria are not, ev en though they are consistent un der the regular blo c k mo del. F urther, we sho w that lik eliho o d-based m etho ds are consisten t und er their assu med m o del with no restrictions on parameters, wher eas mo dularities are only consis- ten t if the mo del parameters are constrained to satisfy a “stronger lin ks within than b et w een” condition, which is the basis of m o dularity deriv a- tions. In short, w e sho w that a criterion is consisten t when the underlyin g mo del and a ssu m ptions are c orrect, and not nece ssarily otherwise. T able 1 Summary of c ommunity dete ction crite ria Block mo del Degree-corrected blo ck mo del Modu larit y P k ( O kk − n 2 k n 2 L ) (ERM) P k ( O kk − O 2 k L 2 L ) (NGM) Likel iho o d P kl O kl log O kl n k n l (BM) P kl O kl log O kl O k O l (DCBM) CONSISTENCY OF COMMUNITY DETECTION 7 3. Consistency of comm unit y det ection criteria. Here w e present all the consistency results for the four differen t criteria d efined in S ection 2 . All these results follo w from the general consistency theorem in Section 4 ; th e pro ofs are giv en in the App end ix . The notion of consistency of comm unit y detection as the n umber of no des grows was in tro du ced in [ 5 ]. They d efi ned a comm unit y detectio n criterion Q to b e consistent if the n o de lab els obtained b y maximizing the criterio n, ˆ c = arg max e Q ( e ), satisfy P [ ˆ c = c ] → 1 as n → ∞ . (3.1) Strictly sp eaking, this defin ition suffers from an identifiabilit y problem, sin ce most reasonable criteria, including all the ones d iscussed ab ov e, are inv arian t under a p ermutation of c ommunit y la b els { 1 , . . . , K } . Thus, a b etter w a y to define consistency is to replace the equalit y ˆ c = c with the requiremen t that ˆ c and c b elong to the same equiv alence class of lab el p ermutat ions. F or simplicit y of notation, w e still write ˆ c = c in all consistency results in the rest of the pap er, but tak e them to mean that ˆ c and c are equal up to a p ermutatio n of lab els. The notion o f consistency in ( 3.1 ) is very strong, sin ce it requires asymp- totical ly no errors. One can also define wh at w e w ill call w eak consistency , ∀ ε > 0 P " 1 n n X i =1 1(ˆ c i 6 = c i ) ! < ε # → 1 as n → ∞ , (3.2) where equalit y is also interpreted to mean members hip in the same equiv- alence class with resp ect to lab el p ermutat ions. In [ 6 ], cond itions were es- tablished for a criterion to b e w eakly consistent under the sto chastic blo c k mo del. All other assumptions b eing equal, weak consistency only r equires that the exp ected degree of the graph λ n → ∞ , whereas strong consistency requires λ n / log n → ∞ . Here, we will analyze both strong and we ak consis- tency under the d egree-correct ed sto c hastic blo c k mo del. F or the asymp totic analysis, we us e a sligh tly differen t formulatio n o f th e degree-correcte d mo del than that giv en by [ 20 ]. The main difference is that w e treat true communit y lab els c and deg ree parameters θ = ( θ 1 , . . . , θ n ) as laten t rand om v ariables r ather than fi xed p arameters. Note, ho w ev er, that the crite ria w e a nalyze w er e obtained as profile likeli ho o ds with parameters treated as constan ts. This is one of th e standard approac hes to random effects mo dels, k n o wn as conditional like liho o d (see page 234 of [ 21 ]). Th e net w ork model we use f or consistency analysis can b e describ ed as follo w s: (1) Eac h no de is in dep en d en tly assigned a pair of laten t v ariables ( c i , θ i ), where c i is the comm unity lab el taking v alues in 1 , . . . , K , and θ i is a discrete “degree v ariable” taking v alues in x 1 ≤ · · · ≤ x M . W e do not assume that c i is indep end en t of θ i . 8 Y. ZH AO, E. LEV INA AND J. ZHU (2) The marginal distrib ution of c is m ultinomial with p arameter π = ( π 1 , . . . , π K ) T , a nd θ satisfies E [ θ i ] = 1 for ident ifiabilit y . (3) Giv en c and θ , the edges A ij are indep enden t Bernoulli ran d om v ari- ables with E [ A ij | c , θ ] = θ i θ j P c i c j , where P = [ P ab ] is a K × K symmetric matrix. F or simplicit y , w e allo w self-loops in the net w ork, that is, E [ A ii | c , θ ] = θ 2 i P c i c i . Otherw ise diagonal terms of A ha v e to b e treated sep arately , whic h ultimately make s no difference for the analysis but makes notatio n more a wkw ard. T o en s ure that all probab ilities are alw a ys less than 1, we requ ire the mo del to s atisfy the co nstraint x 2 M max a,b P ab ≤ 1. W e al so need to consider ho w the mo del c hanges with n . If P ab remains fixed as n gro ws , the exp ected degree λ n will b e p rop ortional to n , which makes th e net w ork unrealistically dense. In stead, w e a llo w the matrix P to scal e with n and, in a slight abuse of notatio n, reparameterize it as P n = ρ n P , where ρ n = P ( A ij = 1) → 0 and P is fixed. W e then sp ecify the rate of c t he exp ected degree λ n = nρ n , whic h has to satisfy λ n log n → ∞ for s tr ong c onsistency and λ n → ∞ for wea k consistency . Let Π b e the K × M matrix r epresent ing the join t distribution of ( c i , θ i ) with P ( c i = a, θ i = x u ) = Π au . F urth er, define ˜ π a = P u x u Π au . Note that P a ˜ π a = 1 since E ( θ i ) = 1. Moreo v er, we ha v e ˜ π a = π a if c and θ are inde- p end en t, or if θ i ≡ 1 (blo c k mo dels). Thus, we can view ˜ π as an adjusted v ersion of π . Next, we state our consistency r esu lts for the t wo t yp es of mo d ularities under b oth the degree-corrected and the standard block mo del. Theorem 3.1. Under the d e gr e e- c orr e cte d sto c hastic blo ck mo del, if the p ar ameters satisfy ˜ E aa > 0 , ˜ E ab < 0 for al l a 6 = b, wher e ˜ P 0 = P ab ˜ π a ˜ π b P ab , ˜ W ab = ˜ π a ˜ π b P ab ˜ P 0 , ˜ E = ˜ W − ( ˜ W 1 )( ˜ W 1 ) T , the Newman– Girvan mo dularity is str ongly c onsistent when λ n / log n → ∞ and we akly c onsistent when λ n → ∞ . The paramete r constraints in Th eorem 3.1 require, essen tially , that the links within comm unities are more like ly th an the links b et we en. This is particularly easy to see when K = 2, in whic h case the constraint s im p lifies to P 11 P 22 > P 2 12 . T aking θ i ≡ 1, we immed iately obtain the follo wing. CONSISTENCY OF COMMUNITY DETECTION 9 Corollar y 3.1 (Established in [ 5 ]). Under the standar d sto chastic blo ck mo del with p ar ameters satisfying The or em 3.1 c onstr aints with ˜ π r eplac e d by π , Newman–Girva n mo dularity is str ongly c onsistent when λ n / log n → ∞ and we akly c onsistent when λ n → ∞ . F or Erdos–Ren yi mo du larit y , whic h has not b een stud ied theoretically b efore, w e can also show consistency u nder the standard blo ck mo del, alb eit with a sligh tly s tronger condition on links within comm unities b eing more lik ely than the links b et w een: Theorem 3.2. Under the standar d sto chastic blo ck mo del, if the p ar am- eters satisfy P aa > P 0 , P ab < P 0 for al l a 6 = b, wher e P 0 = P ab π a π b P ab , the Er dos–R enyi mo dularity criterion ( 2.6 ) is str ongly c onsistent when λ n / log n → ∞ and we akly c onsistent when λ n → ∞ . Ho w ev er, th e Erd os–Ren yi mo dularit y is n ot consisten t u nder the degree- corrected mo del, at least not u nder the s ame parameter constraint . The Erdos–Ren yi mo dularity prefers to group n o des with similar degrees to- gether, whic h ma y not agree w ith true comm un ities when the v ariance in no de degrees is large. Here is a counter-e xample demonstrating this. Let K = 2 , π = (1 / 2 , 1 / 2) T , ρ n = 1 (so that the graph b ecomes dense as n → ∞ ), and P =  0 . 1 0 . 05 0 . 05 0 . 1  . F urther , θ is indep end en t o f c and tak es only tw o v alues, 1 . 6 and 0 . 4, with probabilit y 1 / 2 eac h. If we assign all n o des their true lab els, th e p opula- tion v ersion of th e criterion (where all random quantit ies are replaced b y their exp ectations under the true m o del) giv es Q ERM = 0 . 012 5. How ever, b y grouping no des with the same v alue of θ i ’s toget her, w e get the p opulation v ersion of Q ERM = 0 . 0135, higher than the v alue f or the tr u e partitio n, and this solution will therefore b e p referred in th e limit. Once again, the r esult mak es sense intuitiv ely , since the Erd os–Ren yi mo d- ularit y uses the regular blo c k mo del as its n ull hypothesis, and the parameter constrain t matc h es the “few er links b et w een than within” notion. F rom the algorithmic p oin t of view, the main difference b et we en Erdos–Ren yi mo d- ularit y and Newman–Girv an modu larit y is that the la tter dep ends on the edge matrix O only and “w eighs” communities by the num b er of edges, whereas th e former weig hs comm unities b y the n um b er of n o des n k (whic h, under the blo c k m o del, is p rop ortional to the n umb er of edges, but und er the degree-co rrected mo del is not) . 10 Y. ZH AO, E. LEV INA AND J. ZHU Next we state the consistency results for the t w o criteria deriv ed from profile lik eliho o ds, DCBM ( 2.3 ) and BM ( 2.4 ). These require no parameter constrain ts. Theorem 3.3. Under th e de gr e e-c orr e cte d sto chastic blo ck mo del (and ther efor e under the r e gular mo del as wel l), the de gr e e-c orr e cte d criterion ( 2.3 ) is str ongly c onsistent when λ n / log n → ∞ and we akly c onsistent when λ n → ∞ . Theorem 3.4. Under the sto chastic blo c k mo del, the blo c k mo del crite- rion ( 2.4 ) is str ongly c onsistent when λ n / log n → ∞ and we akly c onsistent when λ n → ∞ . Theorem 3.4 was pr ov ed in [ 5 ] for a sligh tly different form of the pr ofile lik eliho o d (Bernoulli rather than the P oisson). Under the degree-corrected blo c k mo d el, criterion ( 2.4 ) is not n ecessarily consisten t—the same counter- example can b e used to demonstrate this. As w as the case with mo dularities, the crite rion consisten t under the d egree-correcte d b lo c k mod el dep ends on O only , whereas the criterion consisten t only under the regular blo c k mo del also dep ends on n k . The theoret ical results suggest that th e likel iho o d-based criteria are al - w a ys preferable o v er the mo dularit y-based criteria, and th at criteria based on the d egree-correcte d mo del are alwa ys p referred to the criteria based on the regular b lo c k mo del, since they are consisten t under w eake r conditions. In practice, ho wev er, th is ma y not alw a ys hold. Computationally , mo du lar- it y t yp e criteria can b e appro x im ately optimized b y solving an eigen v alue problem [ 24 ], whereas lik eliho o d typ e criteria ha v e n o suc h appro ximations and thus h av e to b e optimized by s lo w er heuristic searc h algorithms, as w as done in [ 5 ] and [ 20 ]. Moreo v er, fitting the degree-corrected blo c k mo d el re- quires estimating man y m ore parameters than fitting a blo c k mo del and creates the usual trade-off b etw een mo del complexit y and go o d ness of fit. If the no de degrees w ithin communities do not v ary w id ely , fitting a blo c k mo del ma y pro vide a better solution; see more on this in Section 5 . 4. A general theorem on consistency under d egree-co rrected sto chastic blo c k m o dels. Here w e prov e a general theorem for chec king consistency under degree-correcte d sto chasti c b lo c k mo d els for any criterion defi ned b y a reasonably nice fu nction. All consistency results for sp ecific metho ds discussed in Sectio n 3 are c orollaries of this theorem. A la rge class of comm u nit y detection criteria c an b e writte n as Q ( e ) = F  O ( e ) µ n , f ( e )  , (4.1) CONSISTENCY OF COMMUNITY DETECTION 11 where µ n = n 2 ρ n . F or instance, man y graph cut methods (mincut, ratio cut [ 38 ], n ormalized cut [ 34 ]) ha v e this form and use functions that are designed to minimize th e n umb er of edges b etw een comm unities. All criteria discussed in Section 3 can also b e wr itten in this form. Ou r goal here is to establish conditions for consistency of a criterion of this form und er degree-corrected blo c k mo dels. A natural condition for consistency is that the “p opulation ve rsion” of Q ( e ) should b e maximize d b y the correct communit y assignment, as in M - estimation. T o define th e p opulation ve rsion of Q , we first defin e fun ctions H ( S ) and h ( S ) corresp ondin g to p opulation v ersions of O ( e ) and f ( e ), re- sp ectiv ely (the p recise meaning of “p opu lation version” is clarified in Pr op o- sition 4.1 b elo w). F or any generic arr a y S = [ S k au ] ∈ R K × K × M , d efine a K × K matrix H ( S ) = [ H k l ( S )] by H k l ( S ) = X abuv x u x v P ab S k au S lbv , and a K -dimen sional v ector h ( S ) = [ h k ( S )] by h k ( S ) = X au S k au . Also define R ( e ) ∈ R K × K × M b y R k au ( e ) = 1 n n X i =1 I ( e i = k , c i = a, θ i = x u ) . Then w e ha v e the follo wing: Pr opo s ition 4.1. 1 µ n E [ O k l | c , θ ] = H k l ( R ( e )) , (4.2) f k ( e ) = h k ( R ( e )) . (4.3) Prop osition 4.1 explains the precise meaning of “p opulation v ersion”: w e tak e the conditional exp ectations give n c and θ and write them as fun ctions of a generic v ariable S instead of R ( e ). The p opulation v ersion of Q is defined as F ( H ( S ) , h ( S )). No w w e can sp ecify t he k ey sufficien t condition as fol lo ws: ( ∗ ) F ( H ( S ) , h ( S )) is uniquely maximized o v er S = { S : S ≥ 0 , P k S k au = Π au } b y S = D , with D k au = Π au E k a , for an y a and u , w here E is an y ro w p ermuta tion of a K × K ident it y matrix. 12 Y. ZH AO, E. LEV INA AND J. ZHU The matrix E deals w ith the p ermutatio n equiv alence class. Since R ( c ) → D as n → ∞ , S = D imp lies eac h class k exactly matc hes a comm unit y in the p opu lation. F or simp licit y , in what follo w s we assume that E is in fact the iden tit y m atrix itself. W e will elab orate on this condition b elo w. In addition, w e need some r egularit y conditions, analog ous to those in [ 5 ]: (a) F is Lipsc hitz in its arguments; (b) Let W = H ( D ). The directional d eriv ativ es ∂ 2 F ∂ ε 2 ( M 0 + ε ( M 1 − M 0 ) , t 0 + ε ( t 1 − t 0 )) | ε =0+ are con tinuous in ( M 1 , t 1 ) for all ( M 0 , t 0 ) in a neigh b orh o o d of ( W , π ) ; (c) Let G ( S ) = F ( H ( S ) , h ( S )). Th en on S , ∂ G ((1 − ε ) D + εS ) ∂ ε | ε =0+ < − C < 0 for all π , P . No w w e are ready to state the main theorem. Theorem 4.1. F or any Q ( e ) of the form ( 4.1 ), if π , P , F satisfy ( ∗ ), (a)–(c ) , then Q is str ongly c onsistent under de gr e e-c orr e cte d sto chastic blo c k mo dels if λ n log n → ∞ and we akly c onsistent if λ n → ∞ . The pro of is giv en in the App end ix . This theorem is a generalization of Theorem 1 in [ 5 ] from the standard sto chastic block mo d els to degree - corrected mo dels, and it implies all of the consistency results in S ection 3 . Finally , w e return to the key condition ( ∗ ). If Q ( e ) is maximized by the true comm u nit y lab els c , then as n → ∞ , F ( H ( S ) , h ( S )), the p opulation v ersion of Q ( e ), s hould also b e maximized by the true partition S = D , since R ( c ) → D and Q ( c ) → F ( H ( D ) , h ( D )), making ( ∗ ) a natural condi- tion. F urther, since for an y e , P k R k au ( e ) → Π au , the limit S of R ( e ) m ust satisfy P k S k au = Π au . Therefore, we only need to consider maximizers of F ( H ( S ) , h ( S )) satisfying this constraint. 5. Numerical ev aluation. In this section w e compare the p erformance of the four communit y detection criteria from S ection 2 on simulated data, gen- erated from the regular or th e d egree-correcte d blo c k mo del. The criteria are maximized o ver partitions using a greedy lab el-switc hing algo rithm called tabu search [ 4 , 14 ]. The ke y idea of tabu searc h is that once a n o de lab el has b een switc hed, it w ill b e “tabu” and not av ailable for switching for a certain n umb er of iterations, to pr ev en t being trapp ed in a lo cal maxim um. Ev en though tabu searc h cann ot guarante e con v ergence to the global maximum, it per f orms w ell in practic e. Moreo ve r, w e run the searc h for a n umber of initial v alues and d ifferen t orderings of nod es, to help av oid lo cal maxima. T o compare th e solution to the true lab els, w e u se the adjusted Rand index [ 19 ], a measur e of similarit y b et we en partitions commonly used in cluster- ing. W e ha ve a lso computed th e normalize d m utual information, a measure CONSISTENCY OF COMMUNITY DETECTION 13 more commonly used b y p h ysicists in the net works literature, which giv es v ery similar results (not rep orted to sa ve space). Th e adjusted Rand index is scaled so that 1 corresp ond s to the p erfect matc h and 0 to the exp ected difference b etw een tw o random partitions, with higher v alues ind icating b et- ter ag reement. The figur es in this sectio n all presen t the median adjusted Rand index o v er 100 replications. In all examples b elo w, w e generate net wo rks with n = 1000 n o des and K = 2 comm unities. Th e no de lab els are generated indep endent ly with P ( c i = 1) = π , P ( c i = 2) = 1 − π . By v arying π , w e can in v estigate robu stness of the metho d s to unbal anced comm unit y sizes. The probability matrix for th e blo c k mo del and the d egree-correcte d blo c k mo del is set to P = ρ  4 1 1 4  , where w e v ary ρ to obtain different exp ected degrees λ . 5.1. The d e gr e e- c orr e cte d sto chastic blo ck mo del. F or this sim ulation, we generate data f rom the degree-corrected mo del with t wo possib le v alues for the degree parameter θ . The degree parameters are generated indep endently from the la b els, with P ( θ i = mx ) = P ( θ i = x ) = 1 / 2 , whic h implies x = 2 m +1 , since we need to ha ve E ( θ i ) = 1. W e v ary the ratio m from 1 (the regular blo c k mo del) to 10, wh ic h allo ws us to s tu dy the effect of mo d el missp ecification on the regular blo ck mo d el. In this sim ulation, the comm unit y sizes are balanced ( π = 0 . 5). Figure 1 s ho ws the r esults f or three d ifferen t exp ected degrees λ . F or the densest net work with λ = 125 in Figure 1 (a), the degree-corrected b lo c k mo del and Newman–Girv an mo dularit y p erform th e b est o verall, as they assume the correct mo d el and the metho ds are consistent . A t m = 1, the Fig. 1. R esults for the de gr e e-c orr e cte d sto chastic blo ck mo del with two values for t he de gr e e p ar ameters, π = 0 . 5 , m varies. 14 Y. ZH AO, E. LEV INA AND J. ZHU Fig. 2. R esults for the standar d sto chastic blo ck mo del, m = 1 , π varies. regular blo c k mo del is just as go o d , b ut its p erformance d eteriorates r apidly as m increases. The Erdos–Ren yi mo du larit y also p erforms p erfectly for m = 1, and it tak es larger v alues of m for its p erformance to deteriorate than for blo ck mo del lik eliho o d, so we can co nclude that the Erdos–Ren y i mo dularity is more robust to v ariation in degrees. F or b oth of them, p o or results are due to grou p ing no des with similar degrees toge ther. The o v erall trend for sparser n et w orks [Figure 1 (b) and (c)] is s imilar, bu t all metho ds p erform worse, as with f ew er links there is effectiv ely less d ata to use for fitting the mo del, and the effect is more pronoun ced f or large m , when degrees ha v e higher v ariance. 5.2. The st o chastic blo ck mo del. Here we fo cus on the s tandard sto c has- tic b lo c k mo del ( m = 1 ) and v ary π to assess robustn ess to unbala nced comm unit y sizes. All the four criteria are consisten t in this case, but, in practice, the closer π is to 0.5, the b etter they p erform (Figure 2 ), with the exception of the blo ck mo d el lik eliho o d in the dense case ( λ = 125), where it p erforms per f ectly for all π . Ov erall, the blo ck mo d el lik eliho o d p erforms b est, whic h is natur al b ecause it is the maxim um lik eliho o d estimator of the correct mo del. The Erdos–Ren yi mo d ularit y also p erforms b etter than the other t w o criteria, whic h o v erfit the data b y assuming the degree-corrected mo del and account ing for v ariation in observe d degrees, whic h in this case only adds n oise. 5.3. Unb alanc e d c ommunity si ze s. In this simulation we consider the degree-correcte d sto c hastic blo c k mo d el with un balanced comm unity sizes. W e fix π = 0 . 3 and v ary the r atio m in Figure 3 . F or a d en se net wo rk [ λ = 125, Figure 3 (a)], the p erformance with π = 0 . 3 is similar to the balanced case with π = 0 . 5 [Figure 1 (a)]. Ho wev er, in sparser net works mo dularit y p er- forms m uch worse with un balanced comm u nit y sizes. Th is can also b e seen in Figure 2 f or the case m = 1. The failure of mo dularity to d eal with un bal- anced comm unity sizes w as also recen tly pointed out b y [ 39 ]. Note also that CONSISTENCY OF COMMUNITY DETECTION 15 Fig. 3. R esults for the de gr e e-c orr e cte d sto chastic blo ck mo del with two values for t he de gr e e p ar ameters, π = 0 . 3 , m varies. in the sparsest case ( λ = 12, Figure 3 ), th e degree-correct ed mo del su ffers from o ver-fitting when m = 1, as w as al so seen in Fig ur e 2 . 5.4. A differ ent de gr e e distribution. In the last sim ulation w e test the sensitivit y of all m etho ds, but in particular the degree-corrected mo del, to the assumption of a d iscrete degree distribu tion. Here w e sample the deg ree parameters θ i indep en den tly from the follo wing distribu tion: θ i =    η i , w.p. α , 2 / ( m + 1) , w.p. (1 − α ) / 2, 2 m/ ( m + 1) , w.p. (1 − α ) / 2, where η i is uniformly distributed on th e in terv al [0 , 2]. The v ariance o f θ i is equal to α/ 3 + (1 − α )( m − 1) 2 / ( m + 1) 2 . In this sim ulation, w e fix m = 10, whic h mak es the v ariance a decreasing function of α , and v ary α fr om 0 to 1. W e also fix π = 0 . 5. The r esults in Figure 4 s h o w that the d egree-corrected blo c k mo del lik eli- ho o d and Newman–Girv an mo du larit y still p erform well, which su ggests that Fig. 4. R esults for the de gr e e-c orr e cte d sto chastic blo ck mo del with a mixtur e de gr e e dis- tribution, m = 10 , π = 0 . 5 , mixtur e p ar ameter α va ries. 16 Y. ZH AO, E. LEV INA AND J. ZHU T able 2 Statistics of no de de gr e es in the p olitic al blo gs network Mean Median Mi n 1st Qt. 3rd Qt. Max 27.36 13.00 1.00 3.00 36.00 351. 00 the discreteness of θ is not a crucial assum p tion. The regular blo ck mo del fails in this case, as w e would exp ect from earlier results since m = 10, but the p erforman ce of the E rdos–Ren yi mo d ularit y improv es as α incr eases, whic h agrees with our earlier observ ation on its relativ e robus tn ess to v ari- ation in degrees. 6. Example: The p olitical blogs net work. In this sectio n we analyz e a real net w ork of p olitical blogs co mpiled by [ 1 ]. The no des of this net w ork are blogs ab out US p olitics and th e edges are hyp erlinks b et wee n th ese blogs. T he data were collected righ t after the 2004 presiden tial election and demonstrate strong divisions; eac h b log w as man ually lab eled as lib eral or conserv ativ e by [ 1 ], whic h we tak e as ground truth. F ollo win g the analysis in [ 20 ], w e ignore d irections of the h yp erlinks and fo cus on the largest connected comp onent of this net w ork, whic h contai ns 1222 no des, 16,7 14 edges and has the a verag e degree of app r o ximately 27. S ome summary statistic s of the no de d egrees are giv en in T ab le 2 , which sho ws th at the degree d istribution is hea vily sk ewed to the right. W e compare the partitions into t w o co mmunities found by the four dif- feren t comm un it y detection criteria with the true lab els u sing the adjusted Rand ind ex. The Newman–Girv an m o dularity and the degree-corrected mo del find v ery similar partitions (they d iffer ov er only four no des an d ha ve the same adjus ted Rand index v alue of 0.819, the highest of all m etho ds). The partition found by the E r dos–Ren yi mo dularit y has a sligh tly w orse agree- men t with th e truth (adjusted Rand index of 0.793). The blo c k mo del likeli - ho o d divides the no d es int o t wo groups of lo w degree and high degree, with the adj u sted Rand index of n early 0, w hic h is equiv alent to r andom guessing. The results are sho wn in Figure 5 (d ra wn using the igraph pac k age in R [ 9 ] with the F ruch terman and Reingold la y out [ 12 ]). These are consisten t with what we observ ed in sim ulation studies: the Newman–Girv an mo d u larit y and the degree-correct ed blo c k mo del like liho o d p er f orm b etter in a n et- w ork with high d egree v ariation, and th e Erdos–Ren yi mo du larit y is more robust to degree v ariation than th e bloc k mo del lik eliho o d. All criteria w ere m aximized b y tabu searc h , but for modu larities w e also computed the sol utions based on the eigendecomp osition of the modu larit y matrix. Both solutions we re worse that those found by tabu searc h, bu t w h ile for Newman–Girv an mo dularit y the difference wa s sligh t (the adjusted Rand CONSISTENCY OF COMMUNITY DETECTION 17 Fig. 5. Politic al blo gs data. No de ar e a is pr op ortional to the lo garithm of its de gr e e and the c olors r epr esent c ommunity lab els. 18 Y. ZH AO, E. LEV INA AND J. ZHU index of 0.781 ins tead of 0.819), eigendecomp osition of the Erdos–Renyi mo dularity yielded a p o or r esult similar to that of block mo d el likel iho o d (with adjusted Rand ind ex v alue of 0.092 instead of 0.819 b y tabu searc h). This sugge sts that Erdos–Renyi mo dularity is numerically less stable und er high degree v ariation, in addition to b eing theoretically not consisten t. More analysis of the eigendecomp osition-based solutions is needed f or b oth t yp es of mo du larities to un derstand conditions under whic h these appr o ximations w ork w ell. 7. Summary and discussion. In this pap er w e deve lop ed a general to ol for chec king consistency of comm unit y detection criteria u n der the degree- corrected sto c hastic b lo c k m o del, a m ore general an d p ractical mo d el than the stand ard sto chastic b lo c k mo del for whic h such theory was previously a v ailable [ 5 ]. This general tool allo wed us to obtain co nsistency results for four differen t communit y detectio n criteria, and, to the b est of our kno wledge for th e fi r st time in the net works literature, to clearly separate the effects of the mo d el assumed for criteria deriv ation from the mo del assumed tru e for analysis of the criteria. What w e ha v e sho wn is, essent ially , statistical common sense: metho ds are consisten t when the mo del they assume holds for the data. The parameter constrain ts are n eeded w hen m etho ds implicitly rely on th em, although we found that th e t w o d ifferen t m o dularity metho d s, while using the same constrain t in sp irit, require somewhat differen t con- ditions on parameters to be consisten t. The theoretical analysis agrees w ell with b oth sim ulation studies and the data analysis, whic h also indicate that the m etho ds w ith b etter theoretical consistency prop erties d o n ot alwa ys p erform b est in pr actice: there is a cost asso ciated with fi tting the extra complexit y of the degree-corrected mod el, and if there is not enough d ata for that, or the data do es not h a v e m uc h v ariation in node degrees, simpler metho ds based on the standard sto c hastic blo c k mo del will in fact do b etter. There are man y questions that require further in v estigation here, ev en in the con text of mo del-based comm unity detecti on w h en a mod el is assumed true. F or example, w e assum ed that K is known, whic h is not unr easonable in some cases (e.g., divid ing p olitical blogs into lib eral and conserv ativ e), but is in general a difficult op en problem in comm unity d etection. Standard metho ds suc h as AIC and BIC do n ot seem to lend themselv es easily to this case, b ecause of p arameters d isap p earing in nonstandard wa ys w hen going from K + 1 to K b lo c ks. A p ermutat ion test w as prop osed in [ 40 ], but clearly more work is needed. Th ere is also the qu estion of what happ ens if K is allo w ed to gro w w ith n , wh ic h is p robably more realistic than fixed K ; for the stoc hastic blo ck mod el, this case has b een co nsid er ed by [ 7 ] and [ 32 ], but their analysis is sp ecific to the particular metho ds they co nsid ered and d o es n ot extend easily to the degree-co rrected b lo c k mo del. Another op en question is the prop erties of appro ximate but more easily computable CONSISTENCY OF COMMUNITY DETECTION 19 solutions based on th e eigendecomp osition, as opp osed to the p r op erties of global maximizers we s tudied h ere. F or the s to c hastic blo ck mo del, part of th is analysis wa s p erf ormed in [ 32 ]. Our p r actical exp erience su ggests that the b eh avior of eige nv ectors can b e quite complicate d, and it is n ot understo o d at this p oin t wh en this appro ximation w orks w ell. Finally , the sparse case λ n = O (1) is an op en problem in general, although results for some sp ecial cases of the sto c hastic b lo c k mo d el h a v e b een recen tly obtained [ 8 , 10 ]. APPENDIX W e start from summ arizing notation. Let R ( e ) , V ( e ) ∈ R K × K × M , ˆ Π ∈ R K × M , f ( e ) , f 0 ( e ) ∈ R K , where R k au ( e ) = 1 n n X i =1 I ( e i = k , c i = a, θ i = x u ) , V k au ( e ) = P n i =1 I ( e i = k , c i = a, θ i = x u ) P n i =1 I ( c i = a, θ i = x u ) , ˆ Π au = 1 n n X i =1 I ( c i = a, θ i = x u ) , f k ( e ) = 1 n n X i =1 I ( e i = k ) = X au V k au ( e ) ˆ Π au , f 0 k ( e ) = X au V k au ( e )Π au . Ev en though the arbitrary lab eling e is not random, in tuitiv ely one can think of R as the empirical joint d istribution of e , c , and θ , V as the c onditional distribution of e giv en c and θ . F urther, ˆ Π is the emp irical joint d istribution of c and θ , and th u s an estima te of their true join t distrib ution Π, f is the empirical marginal “distribution” of e , and f 0 is the s ame marginal bu t with the empirical joint distr ibution ˆ Π replaced by its p opulation v ersion Π. Then P k V k au ( e ) = 1, and V k au ( c ) = I ( k = a ) for all u . F urther, define ˆ T ( e ) ∈ R K × K to b e a r escaled exp ectation of th e matrix O cond itional o n c and θ , ˆ T k l ( e ) = 1 µ n E [ O k l | c , θ ] . F rom Prop osition 4.1 , ˆ T k l ( e ) = X abuv x u x v P ab R k au ( e ) R lbv ( e ) 20 Y. ZH AO, E. LEV INA AND J. ZHU = X abuv x u x v P st V k au ( e ) ˆ Π au V lbv ( e ) ˆ Π bv . Replacing ˆ Π b y its exp ectation ˆ Π, w e define T ( e ) ∈ R K × K b y T k l ( e ) = X abuv x u x v P st V k au ( e )Π au V lbv ( e )Π bv . Also defin e X ( e ) ∈ R K × K to b e the rescaled difference b et wee n O and its conditional exp ectation, X k l ( e ) = O k l ( e ) µ n − ˆ T k l ( e ) . These qu an tities will b e u sed in the pr o of of the general Theorem 4.1 , where w e fir st app ro ximate 1 µ n O k l b y ˆ T k l ( e ) and then appr o ximate ˆ T k l ( e ) by T k l ( e ). Pr oof of Proposition 4.1 . W e only proof ( 4 .2 ) since ( 4.3 ) is trivial. 1 µ n E [ O k l | c , θ ] = 1 µ n X ij X abuv E [ A ij I ( e i = k , c i = a, θ i = x u ) I ( e j = l , c j = b, θ j = x v ) | c , θ ] = X abuv x u x v P ab R k au ( e ) R lbv ( e ) = H k l ( R ( e )) .  Before we pro ceed to the general theorem, we state a lemma b ased on Bernstein’s inequalit y . Lemma A.1. L et k X k ∞ = max k l | X k l | and | e − c | = P n i =1 I ( e i 6 = c i ) . Then P  max e k X ( e ) k ∞ ≥ ε  ≤ 2 K n +2 exp  − 1 8 C ε 2 µ n  (A.1) for ε < 3 C , wher e C = max { x u x v P ab } . P  max | e − c |≤ m k X ( e ) − X ( c ) k ∞ ≥ ε  ≤ 2  n m  K m +2 exp  − 3 8 εµ n  (A.2) for ε ≥ 6 C m/n . P  max | e − c |≤ m k X ( e ) − X ( c ) k ∞ ≥ ε  ≤ 2  n m  K m +2 exp  − n 16 mC ε 2 µ n  (A.3) for ε < 6 C m/n . CONSISTENCY OF COMMUNITY DETECTION 21 This lemma is similar to Lemma 1.1 of [ 5 ], with a few minor errors cor- rected. T he pro of can b e foun d in the electronic supplement to this article [ 41 ]. Pr oof o f Theore m 4.1 . The pro of is divid ed int o three steps. Step 1: sho w that F ( O ( e ) µ n , f ( e )) is uniformly close to its p opulation v er- sion. More precisely , w e need to prov e that there exists ε n → 0, suc h that P  max e     F  O ( e ) µ n , f ( e )  − F ( T ( e ) , f 0 ( e ))     < ε n  → 1 if λ n → ∞ . (A.4) Since     F  O ( e ) µ n , f ( e )  − F ( T ( e ) , f 0 ( e ))     ≤     F  O ( e ) µ n , f ( e )  − F ( ˆ T ( e ) , f ( e ))     + | F ( ˆ T ( e ) , f ( e )) − F ( T ( e ) , f 0 ( e )) | , it is su fficien t to b ound these t w o terms uniformly . By Lipsc hitz co ntin u it y ,     F  O ( e ) µ n , f ( e )  − F ( ˆ T ( e ) , f ( e ))     ≤ M 1 k X ( e ) k ∞ . (A.5) By ( A.1 ), ( A.5 ) con v erges to 0 uniformly if λ n → ∞ , and | F ( ˆ T ( e ) , f ( e )) − F ( T ( e ) , f 0 ( e )) | (A.6) ≤ M 1 k ˆ T ( e ) − T ( e ) k ∞ + M 2 k f ( e ) − f 0 ( e ) k where k · k is the Euclidean norm for v ectors. F urth er , | ˆ T k l ( e ) − T k l ( e ) | =     X abuv x u x v P ab V k au ( e ) V lbv ( e )( ˆ Π au ˆ Π bv − Π au Π bv )     (A.7) ≤ X abuv x u x v P ab | ˆ Π au ˆ Π bv − Π au Π bv | , and | f k ( e ) − f 0 k ( e ) | =     X au V k au ( e )( ˆ Π au − Π au )     ≤ X au | ˆ Π au − Π au | . (A.8) Since ˆ Π P → Π, ( A.6 ) con verges to 0 uniform ly . Th us, ( A.4 ) holds. Step 2: Prov e that there exists δ n → 0, suc h that P  max { e : k V ( e ) − I k 1 ≥ δ n } F  O ( e ) µ n , f ( e )  < F  O ( c ) µ n , f ( c )  → 1 , (A.9) where k W k 1 = P k au | W k au | for W ∈ R K × K × M . 22 Y. ZH AO, E. LEV INA AND J. ZHU By con tinuit y and ( ∗ ), there exists δ n → 0, suc h that F ( T ( c ) , f 0 ( c )) − F ( T ( e ) , f 0 ( e )) > 2 ε n if k V ( e ) − I k 1 ≥ δ n , where I = V ( c ). Thus, from ( A.4 ), P  max { e : k V ( e ) − I k 1 ≥ δ n } F  O ( e ) µ n , f ( e )  < F  O ( c ) µ n , f ( c )  ≥ P      max { e : k V ( e ) − I k 1 ≥ δ n } F  O ( e ) µ n , f ( e )  − max { e : k V ( e ) − I k 1 ≥ δ n } F ( T ( e ) , f 0 ( e ))     < ε n ,     F  O ( c ) µ n , f ( c )  − F ( T ( c ) , f 0 ( c ))     < ε n  → 1 . ( A.9 ) implies P ( k V ( ˆ c ) − I k < δ n ) → 1 . Since 1 n | e − c | = 1 n n X i =1 I ( c i 6 = e i ) = X au Π au (1 − V aau ( e )) ≤ X au (1 − V aau ( e )) = 1 2  X au (1 − V aau ( e )) + X au X k 6 = a V k au ( e )  = 1 2 k V ( e ) − I k 1 , w eak consistency follo ws. Step 3: In order to pr o v e strong consistency , w e need to sh ow that P  max { e : 0 < k V ( e ) − I k 1 <δ n } F  O ( e ) µ n , f ( e )  < F  O ( c ) µ n , f ( c )  → 1 . (A.10) Note that c ombining ( A.9 ) and ( A. 10 ), w e ha ve P  max { e : e 6 = c } F  O ( e ) µ n , f ( e )  < F  O ( c ) µ n , f ( c )  → 1 , whic h implies the strong consistency . Here w e closely follo w the deriv ation giv en in [ 3 ]. T o p r o v e ( A.10 ), note that by Lipschitz con tin uity and the cont inuit y of deriv ativ es of F with resp ect to V ( e ) in the n eigh b orh o o d of I , we ha ve F  O ( e ) µ n , f ( e )  − F  O ( c ) µ n , f ( c )  (A.11) = F ( ˆ T ( e ) , f ( e )) − F ( ˆ T ( c ) , f ( c ) ) + ∆ ( e , c ) , CONSISTENCY OF COMMUNITY DETECTION 23 where | ∆( e , c ) | ≤ M ′ ( k X ( e ) − X ( c ) k ∞ ), and F ( T ( e ) , f 0 ( e )) − F ( T ( c ) , f 0 ( c )) (A.12) ≤ − C ′ k V ( e ) − I k 1 + o ( k V ( e ) − I k 1 ) . Since th e deriv ative of F is con tin uous with resp ect to V ( e ) in the neigh- b orho o d of I , there exist s a δ ′ suc h that F ( ˆ T ( e ) , f ( e )) − F ( ˆ T ( c ) , f ( c ) ) (A.13) ≤ − ( C ′ / 2) k V ( e ) − I k 1 + o ( k V ( e ) − I k 1 ) holds when k ˆ Π − Π k ∞ ≤ δ ′ . Since ˆ Π → Π, ( A.13 ) holds with p robabilit y approac hing 1. Combining ( A.11 ) and ( A.13 ), it is easy to see that ( A.10 ) follo w s if w e can sho w P  max { e 6 = c } | ∆( e , c ) | ≤ C ′ k V ( e ) − I k 1 / 4  → 1 . (A.14) Again note that 1 n | e − c | ≤ 1 2 k V ( e ) − I k 1 . So f or eac h m ≥ 1, P  max | e − c | = m | ∆( e , c ) | > C ′ k V ( e ) − I k 1 / 4  (A.15) ≤ P  max | e − c |≤ m k X ( e ) − X ( c ) k ∞ > C ′ m 2 M ′ n  = I 1 . Let α = C ′ / 2 M ′ , if α ≥ 6 C , b y ( A. 2 ), I 1 ≤ 2 K m +2 n m exp  − α 3 m 8 n µ n  = 2 K 2 [ K exp( log n − αµ n / (8 / 3 n )) ] m . If α < 6 C , b y ( A.3 ), I 1 ≤ 2 K m +2 n m exp  − α 2 m 16 C n µ n  = 2 K 2 [ K exp(log n − α 2 µ n / (16 C n )) ] m . In b oth cases, since λ n / log n → ∞ , P  max { e 6 = c } | ∆( e , c ) | > C ′ k V ( e ) − I k 1 / 4  = ∞ X m =1 P  max | e − c | = m | ∆( e , c ) | > C ′ k V ( e ) − I k 1 / 4  → 0 as n → ∞ , whic h completes the pro of.  24 Y. ZH AO, E. LEV INA AND J. ZHU Pr oof of Theore m 3.2 . The regularit y conditions are ea sy to v erify . T o c hec k the k ey condition ( ∗ ), note that und er the b lo c k mo del assu mption, ( ∗ ) b ecomes ( ∗∗ ) F ( H ( S ) , h ( S )) is u niquely maximized o ver S = { S : S ≥ 0 , P k S k a = π a } b y S = D , with D = diag( π ) , where S is a generic K b y K matrix. Up to a constan t, the p opulation v ersion of Q ERM is F ( H ( S ) , h ( S )) = X k ( H k k − h 2 k P 0 ) . Using the id entit y , X k ( H k k − h 2 k P 0 ) + X k 6 = l ( H k l − h k h l P 0 ) = X k l H k l −  X k h k  2 P 0 = 0 , and define ∆ k l =  1 , if k = l , − 1 , if k 6 = l . Then w e ha v e F ( H ( S ) , h ( S )) = 1 2 X k l ∆ k l ( H k l − h k h l P 0 ) = 1 2 X k l ∆ k l  X ab S k a S lb P ab − X ab S k a S lb P 0  = 1 2 X k l X ab S k a S lb ∆ k l ( P ab − P 0 ) ≤ 1 2 X k l X ab S k a S lb ∆ ab ( P ab − P 0 ) = 1 2 X ab ∆ ab π a π b ( P ab − P 0 ) = F ( H ( D ) , h ( D )) . No w it remains to sho w the d iagonal m atrix D (up to a p ermuta tion) is the unique maximizer of F . Th is follo ws from Lemma 3.2 in [ 5 ], sin ce equalit y holds only if ∆ k l = ∆ ab when S k a S lb > 0 and ∆ does not ha v e t w o iden tical columns.  Pr oof of Theorem 3.1 . The consistency of Newman–Girv an mo du - larit y un der the blo c k mo d el has already b een sho wn in [ 5 ]. T o extend this CONSISTENCY OF COMMUNITY DETECTION 25 result to the degree-corrected blo ck m o del, define ˜ S k a = P u x u S k au . Then ˜ π a = X k ˜ S k a , H k l = X abuv x u x v P ab S k au S lbv = X ab ˜ S k a ˜ S lb P ab , H k = X l H k l = X as ˜ S k a ˜ π s P as . The p opulation v ersion of Q NGM is F ( H ( S )) = X k  H k k ˜ P 0 −  H k ˜ P 0  2  . Using the id entit y X k  H k k ˜ P 0 −  H k ˜ P 0  2  + X k 6 = l  H k l ˜ P 0 − H k H l ˜ P 2 0  = X k l H k l ˜ P 0 −  X k H k ˜ P 0  2 = 0 , w e obtain F ( H ( S )) = 1 2 X k l ∆ k l  P ab ˜ S k a ˜ S lb P ab ˜ P 0 − ( P as ˜ S k a ˜ π s P as )( P bt ˜ S lb ˜ π t P bt ) ˜ P 2 0  = 1 2 X k l X ab ˜ S k a ˜ S lb ∆ k l  P ab ˜ P 0 − ( P s ˜ π s P as )( P t ˜ π t P bt ) ˜ P 2 0  ≤ 1 2 X k l X ab ˜ S k a ˜ S lb ∆ ab  P ab ˜ P 0 − ( P s ˜ π s P as )( P t ˜ π t P bt ) ˜ P 2 0  = 1 2 X ab ∆ ab ˜ π a ˜ π b  P ab ˜ P 0 − ( P s ˜ π s P as )( P t ˜ π t P bt ) ˜ P 2 0  = F ( H ( D )) . Similar to Theorem 3.2 , D is the uniqu e maximizer of F ( H ( ˜ S )) , so it is enough to sh o w S = D whenever ˜ S = D to pro v e uniqueness. ˜ S = D implies ˜ S k a = 0, if k 6 = a . Since x u > 0, we o btain S k au = 0 if k 6 = a , wh ic h giv es the result. W e n ote that this argumen t cannot b e applied to pr o v e the consistency of Erdos–Ren yi mo d u larit y under degree-corrected blo c k mo dels, b ecause in that case h k = P au S k au 6 = P a ( P u x u S k au ) = P a ˜ S k a , when w e use the transformation ˜ S k a = P u x u S k au .  Pr oof of Theorem 3.4 . Up to a constan t, th e p opulation ve rsion of Q BL is F ( H ( S ) , h ( S )) = X k l  H k l log H k l h k h l − H k l  . 26 Y. ZH AO, E. LEV INA AND J. ZHU Let g k l = H k l / ( h k h l ), F ( H ( S ) , h ( S )) = X k l ( H k l log g k l − h k h l g k l ) = X abkl S k a S lb ( P ab log g k l − g k l ) ≤ X ab X k l S k a S lb ( P ab log P ab − P ab ) = X ab ( π a π b P ab log P ab − π a π b P ab ) = F ( H ( D ) , h ( D )) . Since the inequalit y h olds if and only if g k l = P ab when S k a S lb > 0, uniqu eness follo w s from Lemma A.2 , stated next.  Lemma A.2. L et g , P , S b e K × K matric es with nonne gative entries. Assume that: (a) P and g ar e symmetric; (b) P do e s not have two identic al c olumns; (c) ther e exists at le ast one nonzer o entry in e ach c olumn of S ; (d) for 1 ≤ k , l , a, b ≤ K, g k l = P ab whenever S k a S lb > 0 . Then S is a diagonal matrix or a r ow/c olumn p ermutation of a diagonal matrix. This lemma is a generalization of Lemma 3.2 in [ 5 ]. Th e pro of is giv en in the elect ronic su pplement [ 41 ]. Pr oof of Theorem 3.3 . Up to a constan t, th e p opulation ve rsion of Q DCBM is F ( H ( S )) = X k l  H k l log H k l H k H l − H k l  , (A.16) where w e only c hec k ( ∗∗ ) [the form ( ∗ ) tak es under th e blo c k mo del]. Th e generalizat ion to the degree-corrected block mo del is similar to the pro of of Theorem 3.1 and is omitted. Let g k l = H k l / ( H k H l ), and F ( H ( S )) = X k l ( H k l log g k l − H k H l g k l ) = X k l  X ab S k a S lb P ab log g k l −  X as S k a π s P as  X bt π t S lb P tb  g k l  = X k l X ab S k a S lb  P ab log g k l −  X s π s P as  X t π t P tb  g k l  = I 2 . CONSISTENCY OF COMMUNITY DETECTION 27 Since arg max x ( c 1 log x − c 2 x ) = c 1 /c 2 , replacing g k l b y P ab ( P s π s P as )( P t π t P tb ) , w e obtain I 2 ≤ X k l X ab S k a S lb  P ab log P ab ( P s π s P as )( P t π t P tb ) − P ab  = X ab  π a π b P ab log P ab ( P s π s P as )( P t π t P tb ) − π a π b P ab  = F ( H ( D )) .  Ac kno wledgment s. 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[41] Zhao, Y. , Levina , E. and Zh u, J. (20 12). Supplement to “Consis tency of com- munit y d etection in net works under degree-corrected stochastic block mo dels.” DOI: 10.1214 /12-AOS1036SUPP . Y. Zhao Dep a rt ment of S t at istics George Ma son University 4400 University Drive, MS 4A7 F airf ax, Virginia 2203 0-4444 USA E-mail: yzhao15@gm u.edu E. Levina J. Zhu Dep a rt ment of S t at istics University of Michiga n 439 West Hall 1085 S. University A ve. Ann Arbor, Michig an 48109-1 107 USA E-mail: elevina@umic h.edu jizhu@umic h.edu The Annals of Statistics 2015, V ol. 43, No. 1, 462–466 DOI: 10.1214 /14-AOS1271 c  Institute of Mathematical S tatistics , 2015 CORRECTION TO THE PR OOF OF C ONSISTENCY OF COMMUNITY DETECTION By Peter J. Bickel, Aiyou Chen , Yunpeng Zhao, Eliza vet a Levina and J i Zhu University of California, Berkeley, Go o gle Inc, Ge or ge Mason Univ ersity, University of Michigan and U niversity of Michigan This n ote corrects an error in tw o related pro ofs of consistency of comm unity detection: under stochastic block mo dels by Bic kel and Chen [ Pr o c. Natl. A c ad. Sci. USA 106 (2009) 21068–2107 3] and under degree-corrected stochastic block mod el b y Zhao, Levina and Zh u [ A nn. Statist. 40 (2012) 2266–2 292]. This note pro vides a correction to the p r o of of consistency of comm un it y detection under degree-co rrected sto c hastic blo ck models [ 2 ], publish ed in this journal. The same error app eared earlier in the pro of of consistency under the sto c hastic b lo c k m o dels [ 1 ]. In this note, w e provide the correction for the pro of of [ 2 ], using the notation of that pap er, since the case of the degree-correcte d sto c hastic b lo c k mod els is more general and includes the regular s to c hastic blo c k mo d els as a sp ecial case. V ery similar arguments can b e used to correct the pro of of [ 1 ] directly . W e start b y ve ry br iefly r estating notation. L et e b e an arbitrary set of lab el assignment s, c b e the true lab el assignmen ts and ˆ c b e the maximizer of a comm unity detecti on criterion. Let O ( e ) ∈ R K × K , V ( e ) ∈ R K × K × M , ˆ Π ∈ R K × M , f ( e ) ∈ R K , where O k l ( e ) = X ij A ij I { e i = k , e j = l } , V k au ( e ) = P n i =1 I ( e i = k , c i = a, θ i = x u ) P n i =1 I ( c i = a, θ i = x u ) , Received Au gust 2014; revised Septemb er 2014. AMS 2000 subje ct classific ations. 62G20. Key wor ds and phr ases. Netw ork comm unities, stochastic blo ck model, degree- corrected sto chastic blo ck mod el, consistency of communit y d etection. This is an electronic repr int of the origina l ar ticle published by the Institute of Mathematical Statistics in The Annals of Statistics , 2015, V ol. 43, No. 1, 462–466 . This repr int differs from the o r iginal in pagination and t yp ogr aphic detail. 1 2 P . J. BICKEL ET AL. ˆ Π au = 1 n n X i =1 I ( c i = a, θ i = x u ) , f k ( e ) = 1 n n X i =1 I ( e i = k ) = X au V k au ( e ) ˆ Π au . W e considered comm unit y detection criteria that can b e wr itten in the form Q ( e ) = F  O ( e ) µ n , f ( e )  , where µ n = n 2 ρ n and ρ n → 0 is the a verage probability of an edge in the net w ork. F or any mat rix B , k B k ∞ = max k l | B k l | . The statemen t | ∆( e , c ) | ≤ M 1 ( k X ( e ) − X ( c ) k ∞ ) b elo w (A.11) in [ 2 ] is incorrect. (W e ha v e r eplaced M ′ and C ′ in the original with M 1 and C 1 in this correction since w e will need more constan ts.) F or the pro of to go through, w e need a different wa y of provi ng P  max 1 ≤| e − c |≤ δ n n | ∆( e , c ) | − C 1 k V ( e ) − I k 1 / 4 ≤ 0  → 1 , (1.1) where δ n → 0. Note that ( 1.1 ) is similar to the (A.14) in [ 2 ], w ith an extra constrain t | e − c | ≤ δ n n . Since w e ha v e already pro v ed P ( 1 n | ˆ c − c | ≤ δ n ) → 1 in [ 2 ], ( 1.1 ) will complete the p ro of, and the conclusion of Theorem 4.1 in [ 2 ] remains v alid. W e first need a lemma based on Bernstein’s inequalit y . Lemma 1.1. F or m ∈ { 1 , . . . , n } , P  max | e − c |≤ m k X ( e ) k ∞ ≥ ε  ≤ 2  n m  K m +2 exp  − 3 µ n ε 2 4( ε + 3)  . (1.2) The pr o of of Lemma 1.1 closely follo ws the pro of of (A.2) and (A.3) in [ 2 ] and h ence is omitted h er e. Pro of of ( 1.1 ): By T a ylor’s expansion, F  O ( e ) µ n , f ( e )  − F ( ˆ T ( e ) , f ( e )) = ∂ F ∂ M     M = ˆ T ( e ) , t = f ( e ) v ec( X ( e )) + O ( k X ( e ) k 2 ∞ ) , where ∂ F ∂ M is the p artial d eriv ativ e o v er th e first comp onen t (v ectorize d) of F ( M , t ). Similarly , F  O ( c ) µ n , f ( c )  − F ( ˆ T ( c ) , f ( c )) CORRECTION TO CONS ISTENCY OF COMMUNITY DETECTION 3 = ∂ F ∂ M     M = ˆ T ( c ) , t = f ( c ) v ec( X ( c )) + O ( k X ( c ) k 2 ∞ ) . Since ∂ F ∂ M is con tin uous with resp ect to M and t , and ˆ T ( e ) and f ( e ) are con tin uous with resp ect to e , ∂ F ∂ M     M = ˆ T ( e ) , t = f ( e ) = ∂ F ∂ M     M = ˆ T ( c ) , t = f ( c ) + O ( k V ( e ) − I k 1 ) . (1.3) Therefore, since ∆( e , c ) = F  O ( e ) µ n , f ( e )  − F ( ˆ T ( e ) , f ( e )) − F  O ( c ) µ n , f ( c )  + F ( ˆ T ( c ) , f ( c )) = ∂ F ∂ M     M = ˆ T ( c ) , t = f ( c ) v ec( X ( e ) − X ( c )) + O ( k V ( e ) − I k 1 ) vec( X ( e )) + O ( k X ( e ) k 2 ∞ ) + O ( k X ( c ) k 2 ∞ ) , w e ha v e | ∆( e , c ) | ≤ M 1 k X ( e ) − X ( c ) k ∞ + M 2 k V ( e ) − I k 1 k X ( e ) k ∞ + M 3 k X ( e ) k 2 ∞ + M 4 k X ( c ) k 2 ∞ . No w w e pro v e ( 1.1 ), whic h holds if the follo wing four stateme nts hold: P  max 1 ≤| e − c |≤ δ n n M 1 k X ( e ) − X ( c ) k ∞ − C 1 k V ( e ) − I k 1 / 16 ≤ 0  → 1 , (1.4) P  max 1 ≤| e − c |≤ δ n n M 2 k X ( e ) k ∞ − C 1 / 16 ≤ 0  → 1 , (1.5) P  max 1 ≤| e − c |≤ δ n n M 3 k X ( e ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 ≤ 0  → 1 , (1.6) P  max 1 ≤| e − c |≤ δ n n M 4 k X ( c ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 ≤ 0  → 1 . (1.7) The pro of of ( 1.4 ) is similar to the p ro of of (A.15) in [ 2 ]. No te th at 1 n | e − c | ≤ 1 2 k V ( e ) − I k 1 . So f or eac h m ≥ 1, P  max | e − c | = m M 1 k X ( e ) − X ( c ) k ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  ≤ P  max | e − c |≤ m k X ( e ) − X ( c ) k ∞ > C 1 m 8 M 1 n  = I 1 . Let α = C 1 / 8 M 1 if α ≥ 6 C , by (A.2) in [ 2 ], I 1 ≤ 2 K m +2 n m exp  − α 3 m 8 n µ n  = 2 K 2 [ K exp( log n − αµ n / (8 / 3 n )) ] m . 4 P . J. BICKEL ET AL. If α < 6 C , b y (A.3) in [ 2 ], I 1 ≤ 2 K m +2 n m exp  − α 2 m 16 C n µ n  = 2 K 2 [ K exp(log n − α 2 µ n / (16 C n )) ] m . In b oth cases, since λ n / log n → ∞ ( λ n = nρ n ), P  max 1 ≤| e − c |≤ δ n n M 1 k X ( e ) − X ( c ) k ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  ≤ ∞ X m =1 P  max | e − c | = m M 1 k X ( e ) − X ( c ) k ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  → 0 , as n → ∞ , whic h completes the pro of of ( 1.4 ). Equation ( 1.5 ) simply foll o ws (A.1) in [ 2 ]. W e next pro v e ( 1.6 ). F or eac h 1 ≤ m ≤ δ n n , P  max | e − c | = m M 3 k X ( e ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  ≤ P  max | e − c |≤ m k X ( e ) k 2 ∞ > C 1 m 8 M 3 n  = I 2 . Let ε = q C 1 m 8 M 3 n , α = C 1 / 64 M 3 . Then f r om Lemma 1.1 , I 2 ≤ 2 K m +2 n m exp  − 3 µ n ε 2 4( ε + 3)  ≤ 2 K m +2 n m exp  − µ n ε 2 8  = 2 K m +2 n m exp  − α µ n n m  = 2 K 2  K exp  log n − α µ n n  m . Since λ n / log n → ∞ , P  max 1 ≤| e − c |≤ δ n n M 3 k X ( e ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  ≤ ∞ X m =1 P  max | e − c | = m M 3 k X ( e ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  → 0 , as n → ∞ , whic h completes the pro of of ( 1.6 ). CORRECTION TO CONS ISTENCY OF COMMUNITY DETECTION 5 W e no w complete the p ro of b y sho w ing ( 1.7 ). F or ea c h 1 ≤ m ≤ δ n n , P  max | e − c | = m M 4 k X ( c ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  = P  k X ( c ) k 2 ∞ > C 1 m 8 M 4 n  = I 3 . Let ε = q C 1 m 8 M 4 n , α = C 1 / 64 M 4 . Then f r om Bernstein’s inequalit y , I 3 ≤ 2 K 2 exp  − 3 µ n ε 2 4( ε + 3)  ≤ 2 K 2 exp  − α µ n n m  . (1.8) Therefore, P  max 1 ≤| e − c |≤ δ n n M 4 k X ( c ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 > 0  ≤ ∞ X m =1 P ( M 4 k X ( e ) k 2 ∞ − C 1 k V ( e ) − I k 1 / 16 > 0) → 0 as n → ∞ . Ac kno wledgemen ts. W e are v ery grateful to Emma Jingfei Zhang, a for- mer Ph.D. stud en t at Univ ersit y of Illinois at Urbana-Champaign no w at Univ ersit y of Miami, who discov ered the err or and p ersisted in trac king do wn its root cause. REFERENCES [1] Bickel, P. J. and Chen, A. (2009 ). A nonparametric view of netw ork models and Newman-Girv an and other modu larities. Pr o c. Nat l. A c ad. Sci. USA 106 21068– 21073. [2] Zhao, Y . , Levina, E. and Zhu, J. (2012). Consistency of communit y detection in netw orks under degree-corrected sto chastic blo ck mo dels. Ann. Statist . 40 2266– 2292. MR3059083 P. J. Bickel Dep a r tm ent of St at istics University of California, Berkeley 367 Ev ans Hall Berkeley, California 94720-3 860 USA E-mail: bic kel@stat.berkeley .edu A. Chen Google Inc 1600 Amphithea tre Pkwy Mount ain View, California 94043 USA E-mail: aiyo uchen @go ogle.com Y. Zhao Dep a r tm ent of St at istics George Ma son University 1714 Engineering Building 4400 University Drive F airf ax, Virginia 2203 0-4444 USA E-mail: yzhao15@gm u.edu E. Levina J. Zhu Dep a r tm ent of St at istics University of Michiga n 311 West Hall 1085 S. University A ve. Ann Arbor, Michig an 48109-1 107 USA E-mail: elevina@umic h.edu jizhu@umic h.edu