Higher-order clustering in networks

Higher-order clustering in net w orks Hao Yin ∗ Institute for Computational and Mathemat i c al Engine ering, Stanfor d Uni versity, Stanfor d, CA, 94305, USA Austin R. Benson † Dep artment of Computer Sc ienc e, Cornel l University, Ithac a, NY, 14850, USA Jure Lesko vec ‡ Computer Scienc e Dep art m ent, Stanfor d University, Stanfor d, CA, 94305, USA (Dated: Jan uary 8, 2 018) A fund amen tal prop ert y of complex n et works is the ten dency for edges to cluster. The extent of the clustering is t y pically quan tified b y the clustering co efficien t, which is the probabilit y th at a length-2 path is closed, i.e., induces a triangle in th e netw ork. How ever, higher-order cliques b eyo nd triangles are crucial to understanding complex n et w ork s, and th e clustering b ehavior with respect to suc h higher-order netw ork structu res is not well understoo d. Here we in tro duce h igher- order clustering co efficients that measure th e closure probability of higher-order netw ork cliqu es and provide a more comprehensive view of how the edges of complex netw orks cluster. Ou r higher- order clustering coefficients are a natural generali zation of th e traditional clustering coefficient. W e derive sev eral properties about higher-order clustering coefficients and analyze th em under common random graph mo dels. Finally , we use h igher-order clustering co efficien ts to gain n ew insights into the structure of real-wo rld netw orks from sev eral domains. I. INTRO D UCTION Net works are a fundamental to ol for understa nding and mo deling complex physical, so cial, informationa l, and biolo gical systems [ 1 ]. Althoug h such netw or ks are t ypically sparse, a recur r ing trait of netw o rks through- out all of these domains is the tendency of edges to ap- pea r in small clus ters o r cliques [ 2 , 3 ]. In many cases, such clustering can b e explained by lo cal evolutionary pro cesses. F or example, in so cial netw orks, cluster s ap- pea r due to the formation of triangles where t wo indi- viduals who sha re a common friend ar e more likely to bec ome friends themselves, a pro cess known as triadic closur e [ 2 , 4 ]. Similar triadic closures o ccur in other net- works: in citation netw orks , tw o reference s app earing in the same publica tion are mor e likely to be on the sa me topic and he nce more likely to cite each other [ 5 ] and in co-author s hip netw orks , scien tists with a mutual collab- orator a re mo re likely to collab orate in the future [ 6 ]. In other cas es, lo c al clustering arises from hig hly connected functional units o per ating within a large r system, e.g., metab olic net works a re or ganized by densely co nnected mo dules [ 7 ]. The clustering c o efficient quantifies the extent to which edges o f a netw ork cluster in terms of tria ngles. The clustering co efficient is defined as the fraction of length-2 paths, or we dges , that ar e close d with a trian- gle [ 3 , 8 ] (Fig. 1 , row C 2 ). In other w o rds, the clustering co efficient mea s ures the proba bilit y of triadic closur e in ∗ yinh@stanford.edu † arb@cs.cornell.edu ‡ jure@cs.stanford.edu 1. Start with 2. Find an adjacent edge 3. Check for an an ℓ - clique to form an ℓ -wedge ( ℓ + 1)-clique C 2 C 3 C 4 FIG. 1. Overview of higher-order clustering co efficients as clique expansion probabilities. The ℓ th-order clustering co- efficien t C ℓ measures the probabilit y that an ℓ -clique and an adjacen t edge, i.e., an ℓ -wedge, is closed, meaning t hat the ℓ − 1 possible edges b etw een the ℓ -clique and the outside no de in the adjacen t edge ex ist to form an ( ℓ + 1)-clique. the netw or k . How ever, the clustering co efficient is in- herently restr ictiv e as it mea sures the closure probabil- it y of just o ne simple structure—the triangle. Mo reov er , higher-or der s tructures such as larger cliques are crucia l to the struc tur e and function of complex netw or ks [ 9 – 11 ]. F or example, 4-cliques reveal comm unit y structur e in word asso ciation and protein-pr otein in ter action net- works [ 12 ] and cliques of sizes 5– 7 are more frequent than triangles in ma n y rea l-world net works with r espe c t to cer- tain null models [ 13 ]. How ever, the extent of clustering of such higher-or der str uctures has not been w ell under- sto o d nor quantified. Here, we provide a framework to q uant ify hig her-order clustering in ne tw orks b y measuring the no rmalized fre- 2 quency at which higher- o rder cliques are closed, which we ca ll higher-or der clustering c o efficients . W e derive our higher-or der c lustering co efficient s b y extending a nov el int erpretation of the class ical clustering co efficient as a form of clique expa nsion (Fig. 1 ). W e then derive sev- eral prop erties ab out higher-o rder clustering co efficient s and a nalyze them under the G n,p and small-world null mo dels. Using our theoretical analysis as a guide, we analy z e the higher-o rder c lustering behavior of real-world net- works from a v ariety of domains. W e find tha t each domain of netw or k s has its own hig her-order clustering pattern, whic h the traditional c lustering co efficient do es not show o n its own. Con ven tional wisdom in netw o r k science p osits that pr actically all real-world netw orks e x- hibit clustering; how ever, we find that no t a ll netw orks exhibit higher -order clustering . Mo r e sp ecifically , o nce we control for the clustering as measur ed by the cla s sical clustering co efficient, some net works do not show signifi- cant clustering in terms of higher-o rder cliques . In addi- tion to the theoretica l proper ties and empirical findings exhibited in this pap er, our related w or k also demon- strates a co nnection betw e e n higher- order cluster ing and communit y detection [ 14 ]. II. DERIV A T ION OF HIGHER-OR DER CLUSTERING COEFFICIENTS In this s e c tion, we derive our higher-o rder clustering co efficients a nd some o f their basic prop erties. W e first present an alter na tiv e interpretation of the clas sical clus- tering co efficient and then show how this nov el interpre- tation s eamlessly ge neralizes to ar rive at our definition of higher-orde r clustering co efficients. W e then pr ovide some probabilistic int erpretations of higher-or der clus- tering co efficients that will b e useful for our subsequent analysis. A. Alternative interpretation of the cl as sical clustering coeffici ent Here we give an alterna tive in terpretation of the clus- tering co efficient that will later allow us to ge ner alize it and qua ntify clustering of higher - order netw ork struc- tures (this in terpr e tation is summar iz ed in Fig. 1 ). Our int erpretation is based o n a no tion of clique expansio n. First, we consider a 2-c lique K in a gra ph G (that is , a single edge K ; s ee Fig. 1 , row C 2 , column 1). Next, we exp and the clique K by considering a n y edge e adja- cent to K , i.e., e and K shar e exactly one no de (Fig. 1 , row C 2 , column 2). This expanded subgraph forms a wedge, i.e., a length-2 path. The clas sical globa l cluster - ing co efficien t C of G (sometimes called the tra nsitivity of G [ 15 ]) is then defined as the fraction of wedges that are close d , meaning that the 2-cliq ue and a djacent edge induce a (2 + 1)-clique, or a triangle (Fig. 1 , row C 2 , co l- umn 3) [ 8 , 16 ]. The nov elty of our interpretation o f the clustering co efficient is considering it as a form of cliq ue expansion, rather tha n a s the clos ure o f a length-2 path, which is key to our genera liz ations in the next section. F o rmally , the clas sical global clustering co efficient is C = 6 | K 3 | | W | , (1) where K 3 is the set o f 3-cliques (triangles), W is the set of wedges, and the co efficient 6 c o mes from the fact that each 3-clique closes 6 wedges—the 6 ordered pairs of edges in the tria ngle. W e can a lso reinterpret the loc al clustering co effi- cient [ 3 ] in this wa y . In this case , each wedge again consists o f a 2- c lique and adjacen t edge (Fig . 1 , row C 2 , column 2), and we ca ll the unique no de in the in ter section of the 2-clique and adjacent edg e the c enter of the w edge . The lo c al clustering clu stering c o efficient of a no de u is the fraction of wedges cent ered at u that ar e closed: C ( u ) = 2 | K 3 ( u ) | | W ( u ) | , (2) where K 3 ( u ) is the set o f 3 -cliques containing u and W ( u ) is the set of wedges with center u (if | W ( u ) | = 0 , w e say that C ( u ) is undefined). The aver age clustering c o effi- cient ¯ C is the mea n of the lo cal clustering co efficients, ¯ C = 1 | e V | X u ∈ e V C ( u ) , (3) where e V is the set of no de s in the netw or k wher e the lo cal clustering co efficient is defined. B. Generalizing to hi gher-order clustering coe ffi cients Our alter native interpretation of the clus tering co ef- ficient , describ ed ab ov e as a for m o f clique expansio n, leads to a natural gener alization to higher-o rder cliques . Instead o f expanding 2-cliques to 3-cliq ues, we expand ℓ -cliques to ( ℓ + 1)-c liq ues (Fig. 1 , rows C 3 and C 4 ). F or- mally , we define an ℓ -wedge to consist of an ℓ -clique and an adjacent edg e for ℓ ≥ 2. Then w e define the globa l ℓ th- order clustering co efficie nt C ℓ as the fra ction of ℓ -wedges that are closed, meaning that they induce a n ( ℓ + 1)-clique in the netw or k. W e can wr ite this as C ℓ = ( ℓ 2 + ℓ ) | K ℓ +1 | | W ℓ | , (4) where K ℓ +1 is the se t of ( ℓ + 1)-cliques, and W ℓ is the se t of ℓ -wedges. The co efficient ℓ 2 + ℓ comes fro m the fact that each ( ℓ + 1)-clique closes that many wedges: each ( ℓ + 1)-clique contains ℓ + 1 ℓ -cliques, and each ℓ -clique contains ℓ no des which may serve as the cen ter of a n ℓ - wedge. Note that the cla ssical definition of the g lobal 3 clustering co efficien t given in Eq . 1 is equiv alent to the definition in Eq . 4 when ℓ = 2. W e also define higher-or der lo cal cluster ing co efficients: C ℓ ( u ) = ℓ | K ℓ +1 ( u ) | | W ℓ ( u ) | , (5) where K ℓ +1 ( u ) is the s et of ( ℓ + 1)-cliques co n taining no de u , W ℓ ( u ) is the set of ℓ -wedges with center u (where the center is the unique no de in the intersection of the ℓ -clique and adjacent edge co mprising the wedge; see Fig. 1 ), and the co efficient ℓ co mes from the fact that each ( ℓ + 1)-clique containing u clo ses that man y ℓ -wedges in W ℓ ( u ). The ℓ th- o rder clustering co efficien t of a node is defined for any node that is the center of at least one ℓ -wedge, and the average ℓ th-order clustering co efficient is the mean of the lo ca l clustering co efficients: ¯ C ℓ = 1 | e V ℓ | X u ∈ e V ℓ C ℓ ( u ) , ( 6) where e V ℓ is the set of no des that are the centers of at least one ℓ -wedge. T o understand how to compute higher -order clustering co efficients, we substitute the following useful identit y | W ℓ ( u ) | = | K ℓ ( u ) | · ( d u − ℓ + 1) , (7) where d u is the degr ee o f no de u , into Eq . 5 to get C ℓ ( u ) = ℓ · | K ℓ +1 ( u ) | ( d u − ℓ + 1) · | K ℓ ( u ) | . (8) F r om E q . 8 , it is easy to see that we ca n c o mpute a ll lo cal ℓ th-o rder cluster ing co efficients by enumerating all ( ℓ + 1)-cliques a nd ℓ -cliques in the gra ph. The compu- tational complexity of the alg orithm is th us b ounded b y the time to enumerate ( ℓ + 1)-cliques and ℓ -cliques. Using the Chiba and Nishizeki algo rithm [ 17 ], the co mplexit y is O ( ℓa ℓ − 2 m ), wher e a is the ar bo ricity of the graph, and m is the num b er of edges. The ar bor icit y a may b e as large a s √ m , so this algorithm is only g uaranteed to take po lynomial time if ℓ is a constant. In general, determin- ing if there exists a single cliq ue with at leas t ℓ no des is NP-complete [ 18 ]. F o r the global clustering co efficient, note that | W ℓ | = X u ∈ V | W ℓ ( u ) | . (9) Thu s, it suffices to enum erate ℓ -cliques (to compute | W ℓ | using Eq. 7 ) a nd to count the total num b e r of ℓ -cliques. In practice , we use the Chiba a nd Nishizeki to enumerate cliques and simultaneously co mpute C ℓ and C ℓ ( u ) for all no des u . This suffices for our clustering analy s is with ℓ = 2 , 3 , 4 on net works with ov er a h undr ed million edges in Section IV . C. Probabilistic interpr e tations of higher-order clustering coeffici ent s T o facilitate understanding of higher-o r der clus ter ing co efficients and to aid our a nalysis in Section II I , we present a few pr obabilistic interpretations of the quanti- ties. Fir s t, we can interpret C ℓ ( u ) as the probability that a w edge w chosen uniformly at random from all w edg es centered at u is closed: C ℓ ( u ) = P [ w ∈ K ℓ +1 ( u )] . (10) The v ariant of this interpretation for the clas sical clus- tering case of ℓ = 2 has b een useful for gra ph algor ithm developmen t [ 19 ]. F o r the nex t probabilistic interpretation, it is us eful to analyze the structure o f the 1- hop neighborho o d gra ph N 1 ( u ) of a given node u (not co n taining no de u ). The vertex set o f N 1 ( u ) is the set of all nodes adjacent to u , and the edge set consists of all edges betw een neighbor s of u , i.e., { ( v , w ) | ( u, v ) , ( u, w ) , ( v , w ) ∈ E } , where E is the edge set of the graph. An y ℓ -clique in G cont aining no de u co rresp onds to a unique ( ℓ − 1)-clique in N 1 ( u ), a nd sp ecifically for ℓ = 2 , any edge ( u, v ) corresp onds to a no de v in N 1 ( u ). There- fore, each ℓ -wedge centered at u corresp onds to an ( ℓ − 1)- clique K and o ne of the d u − ℓ + 1 no des outside K (i.e., in N 1 ( u ) \ K ). Thus, Eq. 8 can b e re-written as ℓ · | K ℓ ( N 1 ( u )) | ( d u − ℓ + 1) · | K ℓ − 1 ( N 1 ( u )) | , (11) where K k ( N 1 ( u )) denotes the num b er of k - cliques in N 1 ( u ). If we uniformly at r andom select an ( ℓ − 1)-cliq ue K from N 1 ( u ) and then a lso uniformly at rando m select a no de v from N 1 ( u ) outside of this clique, then C ℓ ( u ) is the probability that thes e ℓ no des form a n ℓ - clique: C ℓ ( u ) = P [ K ∪ { v } ∈ K ℓ ( N 1 ( u ))] . (12) Moreov er, if we condition on obs e rving an ℓ - clique from this sampling pro cedure, then the ℓ -clique itself is se- lected uniformly at random fro m all ℓ -cliques in N 1 ( u ). Therefore, C ℓ − 1 ( u ) · C ℓ ( u ) is the proba bilit y that an ( ℓ − 1)-clique and tw o nodes sele cted unifor mly a t r an- dom from N 1 ( u ) form an ( ℓ + 1 )- clique. Applying this recursively g iv es ℓ Y j =2 C j ( u ) = | K ℓ ( N 1 ( u )) |  d u ℓ  . (13) In other words, the pro duct of the higher-or der lo ca l clus- tering co efficients of node u up to order ℓ is the ℓ -clique density amongs t u ’s neighbors. 4 u u u C 2 ( u ) 1 d 2( d − 1) ≈ 1 2 d − 2 4 d − 4 ≈ 1 4 C 3 ( u ) 1 0 d − 4 2 d − 4 ≈ 1 2 C 4 ( u ) 1 0 d − 6 2 d − 6 ≈ 1 2 FIG. 2. Ex ample 1-hop n eigh b orho o ds of a node u with degree d with different higher-order clustering. Left: F or cliques, C ℓ ( u ) = 1 for any ℓ . Middle: If u ’s neighbors form a complete bipartite graph, C 2 ( u ) is constan t while C ℓ ( u ) = 0, ℓ ≥ 3. Right: If half of u ’s neighbors form a star and half form a clique with u , then C ℓ ( u ) ≈ p C 2 ( u ), which is th e u pp er b ound in Prop osition 1 . II I. THEORETICAL ANAL YSIS AND HIGHER-ORDER CLUSTERING IN RANDOM GRAPH MODELS W e now pr ovide some theoretical a na lysis of our higher-or der clustering co efficients. W e first give s ome extremal bounds on the v alues that higher-orde r clus- tering co efficients can take giv en the v alue o f the tradi- tional (second-or der) clustering co efficient . After, we a n- alyze the v a lues o f higher-or der clustering coefficients in t wo co mmon random graph mo dels—the G n,p and small- world mo dels. The theory from this sec tio n w ill b e a useful guide for interpreting the clustering behavior of real-world netw or ks in Section IV . A. Extremal bounds W e first analyze the relationships b etw een lo cal higher- order cluster ing co efficients of differe nt order s. Our tech- nical r esult is Pr op o sition 1 , which provides essentially tight low e r and upp er b ounds for hig her-order lo cal clus- tering c oe fficien ts in terms of the traditional lo ca l cluster- ing co efficient. The main idea s of the pro of are illustrated in Fig. 2 . Prop osition 1. F or any fixe d ℓ ≥ 3 , 0 ≤ C ℓ ( u ) ≤ p C 2 ( u ) . (14) Mor e over, 1. Ther e exists a fin ite gr aph G with a no de u such that the lower b ound is tight and C 2 ( u ) is within ǫ of any pr escrib e d value in [0 , ℓ − 2 ℓ − 1 ] . 2. Ther e exists a fin ite gr aph G with a no de u such that C ℓ ( u ) is within ǫ of t he u pp er b ound for any pr escrib e d value of C 2 ( u ) ∈ [0 , 1] . Pr o of. Clearly , 0 ≤ C ℓ ( u ) if the lo cal c lustering co efficient is well defined. This bo und is tigh t when N 1 ( u ) is ( ℓ − 1 )- partite, as in the middle column of Fig. 2 . In the ( ℓ − 1)- partite cas e, C 2 ( u ) = ℓ − 2 ℓ − 1 . By r emoving edg e s from this extremal ca se in a sufficien tly larg e graph, w e can ma ke C 2 ( u ) arbitra rily close to any v alue in [0 , ℓ − 2 ℓ − 1 ]. T o deriv e the upp er bo und, consider the 1-hop neigh- bo rho o d N 1 ( u ), and let δ ℓ ( N 1 ( u )) = | K ℓ ( N 1 ( u )) |  d u ℓ  (1 5 ) denote the ℓ -clique density of N 1 ( u ). The Kr usk al- Katona theorem [ 20 , 21 ] implies that δ ℓ ( N 1 ( u )) ≤ [ δ ℓ − 1 ( N 1 ( u ))] ℓ/ ( ℓ − 1) δ ℓ − 1 ( N 1 ( u )) ≤ [ δ 2 ( N 1 ( u ))] ( ℓ − 1) / 2 . Combining this with Eq. 8 gives C ℓ ( u ) ≤ [ δ ℓ − 1 ( N 1 ( u ))] 1 ℓ − 1 ≤ p δ 2 ( N 1 ( u )) = p C 2 ( u ) , where the las t equality uses the fact that C 2 ( u ) is the edge density of N 1 ( u ). The upp e r bo und b ecomes tight when N 1 ( u ) consists of a clique and isolated no des (Fig. 2 , right) and the neighborho o d is sufficiently large. Sp ecifically , let N 1 ( u ) consist of a clique of size c and b isolated no des. When ℓ = 2 , C ℓ ( u ) =  c 2   c + b 2  = ( c − 1) c ( c + b − 1)( c + b ) →  c c + b  2 and by E q. 11 , when 3 ≤ ℓ ≤ c , C ℓ ( u ) = ℓ ·  c ℓ  ( c + b − ℓ + 1) ·  c ℓ − 1  = c − ℓ + 1 c + b − ℓ + 1 → c c + b . By a djusting the ratio c/ ( b + c ) in N 1 ( u ), we can con- struct a family o f graphs s uc h that C 2 ( u ) takes a n y v a lue in the interv al [0 , 1] a s d u → ∞ and C ℓ ( u ) → p C 2 ( u ) as d u → ∞ . The seco nd part of the result requires the neighbor - ho o ds to be sufficiently lar ge in order to reach the upper bo und. How ever, we will see later that in some real-world data, there are no des u for which C 3 ( u ) is clos e to the upper bound p C 2 ( u ) for several v a lues of C 2 ( u ). Next, we ana lyze higher-order clustering co efficients in t wo common random graph mo dels: the Erd˝ os-R´ enyi mo del with edg e pro babilit y p (i.e., the G n,p mo del [ 22 ]) and the small-world mo del [ 3 ]. B. Analysi s for the G n,p mo del Now, we analyze higher-or der clustering co efficients in classical Erd˝ os-R´ enyi ra ndo m gra ph mo del, where 5 each edge exists indep enden tly with probability p (i.e., the G n,p mo del [ 2 2 ]). W e implicitly assume that ℓ is small in the following a nalysis so that there should b e at least o ne ℓ -wedge in the g r aph (with high proba bil- it y a nd n lar ge, there is no clique o f size gr eater than (2 + ǫ ) lo g n/ log(1 /p ) for any ǫ > 0 [ 23 ]). Therefore, the global and lo cal cluster ing coefficients a re well-defined. In the G n,p mo del, we first observe that any ℓ -w e dg e is closed if and only if the ℓ − 1 pos sible edges b et ween the ℓ -clique and the o utside no de in the a djacent e dge exist to form a n ( ℓ + 1)-clique. Ea c h of the ℓ − 1 edges exist independently with probability p in the G n,p mo del, which means that the higher -order cluster ing co efficients should scale as p ℓ − 1 . W e formalize this in the following prop osition. Prop osition 2. L et G b e a r andom gr aph dr awn fr om the G n,p mo del. F or c onstant ℓ , 1. E G [ C ℓ ] = p ℓ − 1 2. E G [ C ℓ ( u ) | W ℓ ( u ) > 0 ] = p ℓ − 1 for any n o de u 3. E G  ¯ C ℓ  = p ℓ − 1 Pr o of. W e prove the firs t part by conditio ning on the set of ℓ -wedges, W ℓ : E [ C ℓ ] = E G [ E W ℓ [ C ℓ | W ℓ ]] = E G h E W ℓ h 1 | W ℓ | P w ∈ W ℓ P [ w is closed] ii = E G h E W ℓ h 1 | W ℓ | P w ∈ W ℓ p ℓ − 1 ii = E G  p ℓ − 1  = p ℓ − 1 . As noted ab ov e, the seco nd equality is w ell defined (with high probability) for s mall ℓ . The third eq ua lit y comes from the fa c t that any ℓ -wedge is clos ed if and only if the ℓ − 1 pos sible edge s be tw een the ℓ -clique and the outside no de in the adjacent edge exist to form an ( ℓ + 1)-clique. The pro of of the second pa rt is essentially the same, except we condition over the set of poss ible cases where W ℓ ( u ) > 0. Recall that e V is the set of no des at the center of at least one ℓ -wedge. T o prov e the third part, we tak e the conditional expe c ta tion o ver e V a nd use our result from the second part. The ab ov e res ults say that the global, lo cal, and av- erage ℓ th o rder cluster ing co efficie nts decrease ex p onen- tially in ℓ . It turns out that if we also condition on the second-o rder cluster ing coefficient having some fixed v alue, then the higher -order clustering co efficients still decay exp onentially in ℓ for the G n,p mo del. This will b e useful for in terpreting the distribution of lo ca l clustering co efficients on real- w orld net works. Prop osition 3. L et G b e a r andom gr aph dr awn fr om the G n,p mo del. Then for c onst ant ℓ , E G [ C ℓ ( u ) | C 2 ( u ) , W ℓ ( u ) > 0] =  C 2 ( u ) − (1 − C 2 ( u )) · O (1 /d 2 u )  ℓ − 1 ≈ ( C 2 ( u )) ℓ − 1 . Pr o of. Similar to the pr o of of Prop ositio n 3 , we look at the conditional exp ectation over W ℓ ( u ) > 0: E G [ C ℓ ( u ) | C 2 ( u ) , W ℓ ( u ) > 0] = E G  E W ℓ ( u ) > 0 [ C ℓ ( u ) | C 2 ( u ) , W ℓ ( u )]  = E G h E W ℓ ( u ) > 0 h 1 | W ℓ ( u ) | P w ∈ W ℓ ( u ) P [ w closed | C 2 ( u )] ii . Now, note that N 1 ( u ) has m = C 2 ( u ) ·  d u 2  edges. Know- ing that w ∈ W ℓ ( u ) acco un ts for  ℓ − 1 2  of these edges. By symmetry , the other q = m −  ℓ − 1 2  edges appear in an y of the re maining r =  d u 2  −  ℓ − 1 2  pairs of nodes uniformly at random. There are  r q  wa ys to place these edges, of which  r − ℓ +1 q − ℓ +1  would clo se the w edge w . Thus, P [ w is clo sed | C 2 ( u )] = ( r − ℓ +1 q − ℓ +1 ) ( r q ) = ( r − ℓ +1)! q ! ( q − ℓ + 1)! r ! = ( q − ℓ + 2)( q − ℓ +3) ·· · q ( r − ℓ +2)( r − ℓ +3) ··· r . Now, for any small nonnegative integer k , q − k r − k = C 2 ( u ) · ( d u 2 ) − ( ℓ − 1 2 ) − k ( d u 2 ) − ( ℓ − 1 2 ) − k = C 2 ( u ) − (1 − C 2 ( u ))  ( ℓ − 1 2 ) + k ( d u 2 ) − ( ℓ − 1 2 ) − k  = C 2 ( u ) − (1 − C 2 ( u )) · O (1 /d 2 u ) . (Recall that ℓ is co nstant by assumption, so the big- O no- tation is a ppropriate). The ab ov e expressio n approaches ( C 2 ( u )) ℓ − 1 when C 2 ( u ) → 1 as well a s when d u → ∞ . Prop osition 3 says that e ven if the second-order lo cal clustering co efficient is lar ge, the ℓ th-orde r clustering co - efficient will still decay exp onentially in ℓ , at leas t in the limit a s d u grows large. By examining hig her-order clique closures, this allows us to distinguish b etw een no des u whose neigh b or ho o ds are “dense but random” ( C 2 ( u ) is large but C ℓ ( u ) ≈ ( C 2 ( u )) ℓ − 1 ) o r “dense and s tructured” ( C 2 ( u ) is larg e and C ℓ ( u ) > ( C 2 ( u )) ℓ − 1 ). Only the latter case exhibits higher-order clustering . W e use this in our analysis of rea l-world net works in Section IV . C. Analysi s for the s m all-wor ld mo del W e also study higher -order cluster ing in the small- world random gra ph mo del [ 3 ]. The mo del b egins with a ring netw ork where each no de connects to its 2 k nearest neighbors. Then, for each node u a nd each of the k edges ( u, v ) with v fo llowing u clo ckwise in the ring, the edge is rewired to ( u , w ) with pro bability p , where w is chosen uniformly at ra ndom. With no rewir ing ( p = 0 ) and k ≪ n , it is kno wn that ¯ C 2 ≈ 3 / 4 [ 3 ]. As p increases, the a verage clustering co ef- ficient ¯ C 2 slightly decr eases until a pha se tra nsition near p = 0 . 1, wher e ¯ C 2 decays to 0 [ 3 ] (a lso s ee Fig. 3 ). Here, we generalize these results for higher -order clustering co- efficients. 6 10 -3 10 -2 10 -1 10 0 Rewiring probability (p) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Avg. clust. coeff. C 2 - C 3 C 4 - - FIG. 3. Average higher-order clustering co efficien t ¯ C ℓ as a function of rewiring probabilit y p in small-w orld n et works for ℓ = 2 , 3 , 4 ( n = 20 , 000, k = 5). Prop osition 4 sho ws that the ℓ th-order clustering co efficien t when p = 0 p redicts that the clustering should decrease mo destly as ℓ increases. Prop osition 4. In the smal l-world mo del without r ewiring ( p = 0 ) , ¯ C ℓ → ( ℓ + 1) / (2 ℓ ) for any c onst ant ℓ ≥ 2 as k → ∞ and n → ∞ while 2 k < n . Pr o of. Applying Eq . 8 , it suffices to show that | K ℓ ( u ) | = ℓ ( ℓ − 1)! · k ℓ − 1 + O ( k ℓ − 2 ) (16) as C ℓ ( u ) = ℓ · ( ℓ +1) k ℓ ℓ ! (2 k − ℓ + 1) · ℓk ℓ − 1 ( ℓ − 1)! , which appr oaches ℓ +1 2 ℓ as k → ∞ . Now we give a deriv ation of Eq. 16 . W e fir st label the 2 k neig h bo rs of u as 1 , 2 , . . . , 2 k by their clo ckwise order- ing in the ring. Since 2 k < n , these no des are unique. Next, define the sp an of a n y ℓ -cliq ue containing u as the difference b etw een the larg est and s mallest labe l of the ℓ − 1 no des in the clique o ther than u . The spa n s of any ℓ -clique satisfies s ≤ k − 1 since any no de is directly connected with a no de of lab el differ e nce no grea ter than k − 1 . Also, s ≥ ℓ − 2 sinc e there a re ℓ − 1 no des in a n ℓ - clique o ther than u . F or each span s , we ca n find 2 k − 1 − s pairs o f ( i, j ) suc h that 1 ≤ i , j ≤ 2 k and j − i = s . Fi- nally , for every s uch pair ( i , j ), there a re  s − 1 ℓ − 3  choices of ℓ − 3 no des betw e e n i and j whic h will form an ℓ -clique together with no des u , i , and j . Therefor e, | K ℓ ( u ) | = P k − 1 s = ℓ − 2 (2 k − 1 − s ) ·  s − 1 ℓ − 3  = P k − 1 s = ℓ − 2 (2 k − 1 − s ) · ( s − 1)( s − 2) ··· ( s − ℓ +3) ( ℓ − 3)! = P k − ℓ +2 t =1 (2 k + 2 − t − ℓ ) · t ( t +1) ··· ( t + ℓ − 4) ( ℓ − 3)! . If we ignore lower-order terms k and note that t = O ( k ), we get | K ℓ ( u ) | = P k t =1 h (2 k − t ) t ℓ − 3 ( ℓ − 3)! + O ( k ℓ − 3 ) i = 1 ( ℓ − 3)! P k t =1 (2 k t ℓ − 3 − t ℓ − 2 ) + O ( k ℓ − 2 ) . = 1 ( ℓ − 3)! h 2 k · k ℓ − 2 ℓ − 2 − k ℓ − 1 ℓ − 1 i + O ( k ℓ − 2 ) , = ℓ ( ℓ − 1)! · k ℓ − 1 + O ( k ℓ − 2 ) . Prop osition 4 shows that, when p = 0, ¯ C ℓ decreases as ℓ increases. F urthermo r e, via sim ulation, we obser ve the sa me behavior as for ¯ C 2 when adjusting the rewiring probability p (Fig. 3 ). Rega rdless of ℓ , the phase tra n- sition ha ppens near p = 0 . 1. Ess e n tia lly , once there is enough r ewiring, all loc al clique structure is lost, a nd clustering a t all orders is lost. This is partly a conse- quence of Prop osition 1 , which says that C ℓ ( u ) → 0 as C 2 ( u ) → 0 for any ℓ . IV. EXPERIMENT AL RESUL TS ON REAL-W ORLD NETWORKS W e now analyze the hig he r -order clustering of rea l- world netw orks. W e first study how the higher-o rder global and a verage clus tering co efficients v ary as w e in- crease the or de r ℓ of the clustering co efficient on a collec- tion of 20 net works from several domains. After, w e co n- centrate on a few r epresentativ e net works and compa re the higher-o rder clustering of rea l-world net works to null mo dels. W e find that only so me netw ork s exhibit higher- order clustering once the traditional clustering co efficient is controlled. Finally , we ex a mine the lo ca l clustering of real-world netw or ks. A. Higher-order global and av erage clustering W e compute and a nalyze the higher-or de r clustering for net works fro m a v ariety of domains (T a ble I ). W e briefly descr ibe the co llection o f netw or ks and their cat- egoriza tion b elow: 1. Two s yn thetic netw o rks—a rando m instance of an Erd˝ os-R´ enyi graph with n = 1 , 0 00 no des and edge probability p = 0 . 2 and a sma ll- w orld net work with n = 20 , 00 0 no des, k = 1 0 , and rewiring probability p = 0 . 1; 2. F our neural netw orks—the co mplete neural sy s- tems of the nemato de worms P. p acificus and C. ele gans as well as the neural co nnections of the Drosophila medulla and mouse retina; 3. F our online so cial netw orks —t wo F aceb o ok friend- ship netw o r ks b et ween s tudents at universities fro m 2005 (fb-Stanford, fb-Cornell) and t wo complete online friendship net works (P o kec and Orkut); 7 Netw ork No des Edges C 2 C 3 C 4 ¯ C 2 ¯ C 3 ¯ C 4 | e V 2 | / | V | | e V 3 | / | V | | e V 4 | / | V | Erd˝ os-R´ enyi [ 22 ] 1,000 99,831 0.200 0.040 0.008 0.200 0.040 0.008 1.000 1.000 1.000 Small-w orld [ 3 ] 20,000 100,000 0.480 0.359 0.229 0.489 0.350 0.205 1.000 1.000 0.999 P. p acificus [ 24 ] 50 576 0.015 0.051 0.035 0.073 0.052 0.034 0.880 0.580 0.440 C. ele gans [ 3 ] 297 2,148 0.181 0.080 0.056 0.30 8 0.137 0.062 0.949 0.926 0.808 Drosophila-medulla [ 25 ] 1,781 32,311 0.000 0.002 0.001 0.116 0.061 0.024 0.803 0.616 0.425 mouse-retina [ 26 ] 1,076 577,350 0.008 0.038 0.029 0.033 0.100 0.085 0.998 0.996 0.994 fb-Stanford [ 27 ] 11,621 568,330 0.157 0.107 0.116 0.253 0.181 0.157 0.955 0.922 0.877 fb-Cornell [ 27 ] 1 8,660 790,777 0.136 0.106 0.121 0.225 0.169 0.148 0.973 0.951 0.923 P okec [ 28 ] 1,632,8 03 22,30 1,964 0.047 0.044 0.046 0.122 0.084 0.061 0.900 0.675 0.508 Orkut [ 29 ] 3,072,4 41 117,185,083 0.04 1 0.022 0.019 0.170 0.131 0.110 0.978 0.949 0.878 arxiv-HepPh [ 30 ] 12,008 118,505 0.659 0.749 0.788 0.698 0.586 0.519 0.876 0.723 0.567 arxiv-AstroPh [ 30 ] 18,77 2 198,050 0.318 0.326 0.359 0.677 0.609 0.561 0.932 0.839 0.740 congress-committees [ 31 ] 871 248,848 0.037 0.080 0.063 0.082 0.142 0.126 1.000 1.000 1.000 DBLP [ 32 ] 317,080 1,049,8 66 0.306 0.634 0.821 0.732 0.613 0.517 0.864 0.675 0.489 email-Enron-core [ 33 ] 148 1356 0.383 0.245 0.192 0.496 0.363 0.277 0.966 0.946 0.946 email-Eu-core [ 14 , 30 ] 1005 16064 0.267 0.170 0.135 0.450 0.329 0.264 0.887 0.847 0.784 CollegeM sg [ 34 ] 1,899 41,579 0.004 0.005 0.003 0.053 0.017 0.006 0.829 0.591 0.332 wiki-T alk [ 35 ] 2,394,3 85 4,659,565 0.002 0.011 0.010 0.201 0.081 0.051 0.262 0.077 0.027 oregon2-01052 6 [ 36 ] 11,461 32,730 0.037 0.085 0.097 0.494 0.294 0.300 0.711 0.269 0.121 as-caida-200711 05 [ 36 ] 26,475 53,381 0.007 0.012 0.015 0.333 0.159 0.134 0.625 0.171 0.060 p2p-Gnutella31 [ 30 , 37 ] 62,586 147,892 0.004 0.003 0.000 0.010 0.001 0.000 0.542 0.067 0.001 as-skitter [ 36 ] 1,696,4 15 11,09 5,298 0.005 0.007 0.011 0.296 0.126 0.109 0.871 0.633 0.335 T ABLE I. Higher-order clustering coefficien ts on random g raph models, neural connections, online social net works, co llaboration netw orks, human comm un ication, and technological systems. Broadly , netw orks from the same domain hav e similar higher- order clustering characteristics . S ince e V ℓ is the set of no des at the center of at leas t one ℓ -w edge (see Eq. 6 ), | e V ℓ | / | V | is the fraction of no des at th e center of at least one ℓ -wedge ( the higher-order av erage clustering coefficient ¯ C ℓ is only measured ov er those no des participating in at least one ℓ -wedge). 4. F o ur collab oration netw o rks—tw o co- authorship net works constructed fro m arxiv submission ca t- egories (arx iv-AstroPh a nd arxiv -HepPh), a co- authorship netw o rk c o nstructed fro m DBLP , and the co-co mmittee member s hip netw ork of United States congres s per sons (congr ess-committees); 5. F o ur human communication netw orks—tw o ema il net works (ema il-Enron-co re, email-Eu-core), a F a cebo o k-like messa g ing netw ork from a college (CollegeMsg), and the edits of user talk pa ges b y other users on Wikip edia (wik i- T a lk); and 6. F o ur technological systems net works—three au- tonomous systems (oreg o n2-010 5 26, as-caida - 20071 105, as -skitter) a nd a peer-to -pe e r connection net work (p2p-Genutella31). In a ll cases, we ta k e the edge s as undirected, ev en if the original netw ork data is directed. T able I lists the ℓ th-order global and av er age clustering co efficients for ℓ = 2 , 3 , 4 as well as the fraction of no des that are the center of at least one ℓ -wedge (recall that the average clustering co efficient is the mean only ov er higher-or der lo cal cluster ing co efficients of no des partici- pating in a t least one ℓ -wedge; see Kaiser [ 38 ] for a discus- sion on ho w this can a ffect netw or k analyses). W e high- light so me impor tant trends in the raw clustering co effi- cients, a nd in the next section, we fo cus on higher-or der clustering compar ed to what one g ets in a null mo del. Prop ositions 2 a nd 4 say that we should exp ect the higher-or der global and av e r age clustering co efficients to decreas e as we increase the order ℓ for b oth the Erd˝ os-R´ enyi and small-world mo dels, and indeed ¯ C 2 > ¯ C 3 > ¯ C 4 for these netw or ks. This trend also holds for most of the rea l- w orld networks (mouse-retina , co ngress- committees, and o regon2- 01052 6 a r e the exceptions). Thu s, when averaging ov er no des, higher -order cliques are ov er all less likely to close in bo th the s yn thetic and real-world netw or ks. The rela tionship b etw een the higher-ordre r globa l clus- tering co efficient C ℓ and the order ℓ is le s s uniform ov er the datas ets. F o r the three co-authorship net- works (arxiv - HepPh, arxiv-Astr oPh, a nd DBLP) a nd the three a utonomous systems netw orks (oreg on2-010 526, as- caida-20 07110 5, and as - skitter), C ℓ increases with ℓ , al- though the base clus tering levels ar e muc h higher for co- authorship netw or k s. This is not s imply due to the pres- ence of cliques—a clique has the same clustering for any order (Fig. 2 , left). Instead, these datasets hav e no des that serve a s the center o f a star and a ls o participate in a cliq ue (Fig . 2 , rig h t; see also Prop ositio n 1 ). On the other hand, C ℓ decreases with ℓ for the t wo email net works and the tw o nemato de worm neural net works. Finally , the change in C ℓ need not b e mono tonic in ℓ . 8 C. ele gans fb-Stanford arxiv-AstroPh email-Enron-core oregon2-01052 6 original CM MRCN original CM MRCN original CM MRC N original CM MRCN original CM MRC N ¯ C 2 0 . 31 0 . 15 ∗ 0 . 31 0 . 25 0 . 03 ∗ 0 . 25 0 . 68 0 . 01 ∗ 0 . 68 0 . 50 0 . 23 ∗ 0 . 50 0 . 49 0 . 25 ∗ 0 . 49 ¯ C 3 0 . 14 0 . 04 ∗ 0 . 17 † 0 . 18 0 . 00 ∗ 0 . 14 ∗ 0 . 61 0 . 00 ∗ 0 . 60 0 . 36 0 . 08 ∗ 0 . 35 0 . 29 0 . 10 ∗ 0 . 14 ∗ T ABLE I I. Average higher-order clustering coefficients for five netw orks as well as th e clustering with resp ect to tw o null mod els: a Configuration Mo del ( CM) th at samples random graphs with the same degree d istribution [ 39 , 40 ], and Maximally Random Clustered Net w ork s (MRC N) that preserv e degree distribution as w ell as ¯ C 2 [ 41 , 42 ]. F or t he random netw orks, we rep ort the mean ov er 100 samples. An asterisk ( ∗ ) denotes when th e v alue in the original netw ork is at least five stand ard deviations ab o ve the mean and a d agger ( † ) denotes when the v alue in the original netw ork is at least five standard dev iations b elo w the mean. A lthough all netw orks exhibit clustering with resp ect t o CM, only some of the netw orks exhibit h igher-order clustering when contro lling for ¯ C 2 with MRCN. In three of the four online soc ia l netw ork s, C 3 < C 2 but C 4 > C 3 . Overall, the trends in the higher -order clustering co - efficients ca n be different within one of our dataset cat- egories, but tend to be uniform within s ub-categorie s : the change o f ¯ C ℓ and C ℓ with ℓ is the same for the tw o nemato de worms within the neural netw or ks, the tw o email netw or ks within the communication netw or ks, and the three co-a utho rship netw ork s within the collab or a- tion netw o rks. The s e trends hold even if the (clas sical) second-or der clustering co efficients differ substantially in absolute v alue. While the raw clustering v alues are informa tiv e , it is also useful to co mpare the clustering to wha t one exp ects from null models. W e find in the next sec tio n that this reveals additiona l insight s int o our data. B. Comparison against null mo dels F o r one rea l-world netw o rk fro m each data set categ o ry , we also meas ure the hig her-order clus tering co efficients with resp ect to tw o null mo dels (T able II ). Fir st, w e com- pare against the Configuration Model (CM) that sa mples uniformly from simple graphs with the same degr ee dis- tribution [ 39 , 40 ]. In r eal-world netw o rks, ¯ C 2 is muc h larger than expec ted with resp ect to the CM n ull model. W e find that the same holds for ¯ C 3 . Second, w e us e a null mo del that samples gr aphs pre- serving b oth degre e distribution and ¯ C 2 . Specifica lly , these ar e sa mples from an ensem ble of exp onential gr aphs where the Hamiltonian measur es the abso lute v a lue of the difference b et ween the o riginal ne tw ork and the sa m- pled netw or k [ 41 ]. Suc h samples are r eferred to as a s Maximally Random Clustered Net works (MRCN) a nd are sampled with a sim ulated annea ling pro cedur e [ 42 ]. Comparing ¯ C 3 betw een the real-world a nd the n ull net- work, we obs erve different behavior in higher-order clus- tering acr oss our da tasets. C o mpared to the MRCN n ull mo del, C. ele gans has significantly less than exp ected higher-or der clus tering (in terms of ¯ C 3 ), the F a cebo o k friendship and a utonomous sys tem netw orks hav e sig nif- icantly mor e than expe c ted higher-orde r clustering, and the co - authorship and ema il netw orks have slight ly (but not significantly) more than expec ted higher-orde r clus- tering (T a ble II ). Put a nother wa y , all real-world net- works exhibit clustering in the classic a l sense of tr iadic closure. How ever, the higher-or der clustering co e fficien ts reveal that the fr iendship and a utonomous sy s tems net- works exhibit significant cluster ing be y ond what is given by tria dic clo sure. These results suggest the need for mo dels that directly acco un t fo r closur e in no de neigh- bo rho o ds [ 43 , 44 ]. Our finding ab out the la c k of hig her-order cluster - ing in C. ele gans ag r ees with previous results that 4- cliques are under-expressed, while open 3-w edg es re- lated to co op erative information propaga tion are ov er - expressed [ 9 , 4 5 , 4 6 ]. This also provides credence for the “3-layer” mo del o f C. ele gans [ 46 ]. The observed clus- tering in the friendship netw ork is consistent with prio r work s howing the relative infrequency of o pen ℓ -wedges in many F aceb o ok netw ork subgr aphs with r esp e ct to a n ull mo del a ccounting for triadic closure [ 47 ]. C o - authorship netw ork s and email netw orks are b oth co n- structed from “even ts” tha t create m ultiple edges—a pa - per with k authors induces a k -clique in the co-authorship graph and an ema il sent from one address to k o thers in- duces k edges. This even t-driven gr aph constr uction c r e- ates enough closure structure so that the average third- order clustering co efficient is not muc h la rger than ran- dom gra phs where the classica l second-or der clustering co efficient and degr ee se quence is kept the s ame. W e emphasize that simple clique counts are not suffi- cient to o bta in these results. F or example, the discrep- ancy in the third-or der av er age clustering of C. ele gans and the MRCN n ull mo del is not s imply due to the presence o f 4 -cliques. The original neura l netw ork ha s nearly twice as man y 4-cliques (2,01 0) than the samples from the MRCN mo del (mean 1006 .2, standa rd deviatio n 73.6), but the third-or der clustering co efficient is la rger in MR CN. The reason is that clustering co efficients normal- ize clique counts w ith resp ect to opp ortunities for closure. Thu s far, we have analyzed global and a verage higher- order clustering , which b oth summarize the clustering of the entire netw o rk. In the next section, we lo ok at more lo calized prop erties , na mely the distribution of higher- order lo cal clustering co efficients and the higher-or der av era ge clustering co efficient a s a function of no de degree. 9 10 0 10 1 10 2 10 3 Degree 10 -3 10 -2 10 -1 10 0 Avg. clust. coeff. oregon2_010526 10 0 10 1 10 2 10 3 Degree 10 -2 10 -1 10 0 Avg. clust. coeff. arxiv-AstroPh 10 0 10 1 10 2 10 3 Degree 10 -2 10 -1 10 0 Avg. clust. coeff. fb-Stanford 10 0 10 1 10 2 Degree 10 -3 10 -2 10 -1 10 0 Avg. clust. coeff. C. elegans C 2 - C 3 - C 4 - 10 1 10 2 Degree 10 -1 10 0 Avg. clust. coeff. email-Enron-core A B C D E FIG. 4. T op ro w: Joint distributions of ( C 2 ( u ), C 3 ( u )) for (A) C. ele gans (B) F aceb ook friendship, (C) arxiv co-authorship, (D) email, and (E) autonomous systems n et w ork s. Eac h blue dot rep resen ts a n ode, and the red cu rve tracks the a verag e ov er logarithmic bins. The upp er trend line is th e b ound in Eq. 14 , and the low er trend line is exp ected Erd˝ os-R´ enyi b ehavior from Proposition 3 . Bottom ro w: Average higher-order clustering coefficients as a function of degree. 10 2 10 3 Degree 10 -3 10 -2 10 -1 10 0 Avg. clust. coeff. Erd s Rényi 10 1 10 2 Degree 10 -2 10 -1 10 0 Avg. clust. coeff. Small-world C 2 - C 3 - C 4 - A B FIG. 5. Analogous plots of Fig. 4 for synthetic (A) Erd˝ os-R´ enyi and (B) small-w orld netw orks. T op row: Joint distributions of ( C 2 ( u ), C 3 ( u )). Bottom row : Average h igher- order clustering coefficients as a function of degree. C. Higher-order lo cal clustering co efficients and degree depende nci es W e now examine more lo calized clus ter ing prop erties of our netw orks. Figure 4 (top) plots the joint distribu- tion of C 2 ( u ) and C 3 ( u ) for the five net works analyze d in T a ble II , and Fig. 5 (top) provides the ana logous plots for the Erd˝ os-R´ enyi and small-world netw or ks. In these plots, the lo wer dashed tr e nd line repres en ts the exp ected Erd˝ os-R´ enyi b ehavior, i.e., the exp ected clustering if the edges in the neig h bo rho o d of a no de were configured ran- domly , as formalize d in Prop ositio n 3 . The upp er das he d trend line is the maximum p ossible v alue of C 3 ( u ) given C 2 ( u ), as given by Prop osition 1 . F o r man y no des in C. ele gans , lo cal clustering is nearly random (Fig. 4A , top), i.e., r esembles the Erd˝ os-R´ enyi joint distribution (Fig. 5 A , top). In other words, there are many no des tha t lie on the low er tr end line. This provides further evidence that C. ele gans lac ks higher- order cluster ing. In the arxiv co-author s hip net work, there are many no des u with a la rge v a lue of C 2 ( u ) that hav e an even lar g er v alue of C 3 ( u ) near the upp er b ound of Eq. 14 (see the inset o f Fig. 4C , top). This implies that some no des app ear in b oth cliques a nd a lso as the center of star-like pa tterns, as in Fig. 2 . On the other hand, only a handful of no des in the F a cebo o k friendships, En- ron email, and O regon autonomous systems netw orks are close to the upp er b ound (insets of Fig s. 4B , 4D , and 4E , top). Figures 4 and 5 (b ottom) plot higher -order av erag e clustering a s a function of node degree in the r eal-world and synthetic net works. In the Erd˝ os-R´ enyi , small-world, C. ele gans , and Enron email ne tw orks, there is a distinct gap betw een the average higher -order clustering co effi- cients for no des of all degrees. Thus, our previous find- ing that the average clustering co efficient ¯ C ℓ decreases with ℓ in these net works is independent of deg ree. In the F a cebo o k friendship netw ork , C 2 ( u ) is lar ger than C 3 ( u ) and C 4 ( u ) on average for no des of all degrees, but C 3 ( u ) and C 4 ( u ) are roughly the same for no des of all degrees, which means that 4 -cliques and 5-cliques close at roughly the sa me rate, indep enden t of degr e e, alb eit at a smaller rate than tr aditional triadic clo s ure (Fig. 4B , bo ttom). In the co -authorship ne tw ork, no des u hav e r oughly the same C ℓ ( u ) for ℓ = 2, 3 , 4, which means that ℓ -c liq ues close at a bout the same r ate, independent of ℓ (Fig. 4C , bo ttom). In the Orego n autonomous systems netw ork, we see that, on av era ge, C 4 ( u ) > C 3 ( u ) > C 2 ( u ) for no des with la rge degr e e (Fig. 4E , b ottom). This explains how the global cluster ing co efficient increa ses with the order, but the av er age clustering do es not, as o bserved in T able I . 10 V. DISCUSSION W e hav e prop osed higher-order clustering co efficients to study highe r -order c losure patterns in netw orks, whic h generalizes the widely used clustering co efficient that measures triadic clos ure. Our work compliments other recent developmen ts on the imp ortance of higher -order information in netw o rk navigation [ 11 , 48 ] a nd on tem- po ral communit y structure [ 49 ]; in contrast, w e exa mine higher-or der clique clo sure and only implicitly consider time as a motiv ation for c lo sure. Prior efforts in gener a lizing clustering co efficients hav e fo cused o n shortest paths [ 50 ], cycle formation [ 51 ], a nd triangle frequency in k -hop neighbo r ho o ds [ 52 , 53 ]. Such approaches fail to capture closur e patterns of cliques, suf- fer fro m challenging computationa l issues, a nd are dif- ficult to theoretically analyze in random graph mo dels more so phis tica ted than the E r d˝ os-R´ enyi model. On the other ha nd, o ur higher-o rder clustering co efficients are simple but effective meas urements that ar e ana ly z- able and easily computable (we o nly rely clique enumer- ation, a well-studied algorithmic task). F urthermore , our metho dology provides new insights into the clustering b e- havior of several real- w orld net works and random graph mo dels, and our theore tica l analysis provides intuition for the way in which higher- o rder clustering co efficients describ e lo cal clustering in g raphs. Finally , w e fo cused on higher-or der clustering co effi- cients a s a g lobal netw or k measurement a nd as a no de- level meas urement , and in related work we also show that lar ge higher- o rder clustering implies the e x istence of mesoscale clique-dense communit y structure [ 14 ]. ACKNO WLEDGMENTS This resear ch has bee n suppor ted in par t b y NSF I IS- 11498 37, AR O MURI, D ARP A, ONR, Huaw ei, and Stan- ford Data Science Initiative. 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