$beta$ models for random hypergraphs with a given degree sequence

β mo dels for random h yp ergraphs with a giv en degree sequence Despina Stasi 1 , Ka yv an Sadeghi 2 , Alessandro Rinaldo 3 , Sonja P etro vi ´ c 4 , Stephen E. Fien b erg 5 Abstract: W e introduce the beta model for random hypergraphs in order to represen t the o ccurrence of m ulti-wa y interactions among agen ts in a so cial netw ork. This mo del builds up on and generalizes the well-studied beta mo del for random graphs, which in- stead only considers pairwise interactions. W e pro vide t wo algorithms for fitting the mo del parameters, IPS (iterativ e prop ortional scaling) and fixed point algorithm, prov e that b oth algorithms conv erge if maximum likelihoo d estimator (MLE) exists, and pro- vide algorithmic and geometric w ays of dealing the issue of MLE existence. 1. In tro duction So cial net w ork mo dels [ 8 ] are statistical mo dels for the join t o ccurrence of random edges in a graph, as a means to model so cial in teractions among agen ts in a population of in terest. These mo dels t ypically fo cus on represen ting only binary relations b et ween individuals. As a result, when one is in terested in higher-order ( k -ary) interactions, statistical mo dels based on graphs ma y b e ineffective or inadequate. Examples of k -ary relations are plentiful, and include forum or committee mem b ership, co-authorship on scien tific pap ers, or pro ximit y of groups of people in photographs. These datasets hav e b een studied by replacing eac h k -dimensional group with a num b er of binary relations (in particular,  k 2  of them, which form a clique), thus extracting binary information from the data, and then mo deling and studying the resulting graph. Suc h a pro cess inevitably causes information loss. F or instance, let us consider statisticians Adam ( A ), Barbara ( B ), Cassandra ( C ), and Da vid ( D ), see Figure 1 . Supp ose the authors wrote three pap ers in follo wing groups: ( A, B , C ), ( A, D ), ( C, D ). Represen ting this information as a graph with edges b et w een any t wo individuals who ha ve co-authored a paper provides a graph G with edges { ( A, B ) , ( B , C ) , ( C, D ) , ( A, C ) } . A hypergraph H representing this information w ould instead use the exact groups as h yp eredges and, unlike G , w ould b e able to represen t additional prop erties of suc h in teractions, including ho w many pap ers w ere coauthored by 1 despina.stasi@gmail.c om , Illinois Institute of T ec hnology 2 kayvans@andr ew.cmu.e du , Carnegie Mellon Universit y 3 arinaldo@stat.cmu.e du , Carnegie Mellon Universit y 4 sonja.p etr ovic@iit.e du , Illinois Institute of T ec hnology 5 fienb er g@stat.cmu.e du , Carnegie Mellon Universit y 1 these four individuals; see Figure 1 . If, in addition, A is more lik ely to write a 3-author pap er than a 2-author pap er, this requires mo deling separately the probabilities of these collab orations. Despite the growing needs of practical v alues, mo dels for random hypergraphs are relatively few and simple. Random h yp ergraphs hav e b een studied ([ 7 ]) as generalizations of the simple Erd¨ os-R ´ en yi model [ 4 ] for netw orks; [ 5 ] considers an application of random tripartite hypergraphs to Flickr photo-tag data. A B C D B A D C B A D C 1 Fig 1: Distinct hypergraphs H and H 0 reduced to same graph G (left, right, middle). In this paper w e in tro duce a simple and natural class of statistical models for random h yp er- graphs, whic h w e term h yp ergraph b eta mo dels, that allo ws one to mo del directly sim ultaneous higher-order (and not only binary) in teractions among individuals in a netw ork. As its name suggests, our mo del arises as a natural extension of the w ell-studied b eta mo del for random graphs, the exp onen tial family for undirected net w orks which assumes indep enden t edges and whose minimal sufficient statistics v ector is the degree sequence of the graph. It is a special class of the more general of p 1 mo dels [ 6 ] which assume independent edges and parametrize the probabilit y of eac h edge b y the prop ensit y of the t wo endp oint no des. This mo del has b een studied extensiv ely; see [ 2 , 3 , 9 , 10 , 11 ], which giv e, among other results, methods for model fitting. Belo w w e formalize the class of the beta models for h yp ergraphs. Just like the graph b eta mo del, these are natural exp onen tial random graph mo dels o v er hypergraphs whic h p ostulate indep enden t edges and whose sufficien t statistics are the (h yp ergraph) degree sequences. Our con tributions are tw o-fold: first we formalize three classes of linear exp onen tial families for random h yp ergraphs of increasing degree of complexit y and deriv e the corresp onding sufficien t statistics and moment equations for obtaining the maxim um likelihoo d estimator (MLE) of the mo del parameters. Secondly , we design tw o iterative algorithms for fitting these mo dels that do not require ev aluating the gradient or Hessian of the lik eliho o d function and can therefore b e applied to large data: a v arian t of the IPS algorithm and a fixed p oin t iterative algorithm to compute the MLE of the edge probabilities and of the natural parameters, resp ectively . W e sho w that b oth algorithms will con v erge if the MLE exists. Finally , w e illustrate our results 2 and metho ds with some simulations. As our analysis reveals, the study of the theoretical and asymptotic prop erties of h yp er- graph b eta mo dels is esp ecially c hallenging, more so than with the ordinary beta model. The complexit y of the new mo dels, in turn, leads to the problem of optimizing a complex lik eliho o d function. Indeed, when the MLE do es not exist, optimizing the lik eliho o d function b ecomes highly non-trivial and, to a large exten t, unsolved for our mo del as well as for man y other discrete linear exp onen tial families. T o this end, w e describ e a geometric w a y for dealing with the issue of existence of the MLE for these mo dels and gain further insights into this difficult problem with sim ulation exp eriments. 2. The h yp ergraph b eta mo del: three v arian ts A hyp er gr aph H is a pair ( V , F ), where V = { v 1 , . . . , v n } is a set of no des (vertices) and F is a family of non-empty subsets of V of cardinality different than 1; the elements of F are called the hyp er e dges (or simply e dges ) of H . In a k -uniform h yp ergraph, all edges are of size k . W e restrict ourselv es to the set H n of h yp ergraphs on n no des, where no des ha ve a distinctive lab eling. Let E = E n b e the set of all realizable h yp eredges for a hypergraph on n no des. While E can in principle b e the set of all p ossible hyperedges, b elow we will consider more parsimonious mo dels in which E is restricted to b e a structured subset of edges. Thus w e may write a h yp ergraph x = ( V , F ) ∈ H n as the zero/one vector x = { x e , e ∈ E } , where x e = 1 for e ∈ F and x e = 0 for e ∈ E \ F . The degree of a no de in x is the num b er of edges it b elongs to; the degree information for x is summarized in the degree sequence vector whose i th en try is the degree d i ( x ) of no de i in x . Hyp ergraph b eta mo dels are families of probabilit y distributions ov er H n whic h p ostulate that the h yp eredges o ccur indep endently . In details, let p = { p e : e ∈ E n } b e a vector of probabilities whose e th co ordinate indicates the probability of observing the h yp eredge e . W e will assume p e ∈ (0 , 1). Every such vector p parametrizes a b eta-h yp ergraph mo del as follows: the probability of observing the hypergraph x = { x e , e ∈ E } is P ( x ) = Y e ∈ E p x e e (1 − p e ) 1 − x e . (1) The graph b eta mo del is a simple instance of this model, with E = { ( i, j ) , 1 ≤ i < j ≤ n } . The  n 2  edge probabilities are parametrized as p i,j = e β i + β j / (1 + e β i + β j ), for i < j and some real vector β = ( β 1 , . . . , β n ). V arious so cial netw ork mo deling considerations for no de interactions require a flexible class of mo dels adaptable to those settings. Thus, w e introduce three v ariants of the b eta mo del for h yp ergraphs with indep endent edges in the form of linear exp onen tial families: b eta mo dels for uniform hyp er gr aphs , for gener al hyp er gr aphs , and for layer e d uniform hyp er gr aphs . F or 3 eac h, w e provide an exp onential family parametrization in minimal form and describ e the corresp onding minimal sufficien t statistics. Uniform h yp ergraphs. The probabilit y of a size- k h yp eredge e = i 1 . . . i k app earing in the h yp ergraph is parametrized by a vector β ∈ R n as follows: p i 1 ,...,i k = e β i 1 + ... + β i k 1 + e β i 1 + ... + β i k (2) with q i 1 ,...,i k = 1 − p i 1 ,...,i k = 1 1+ e β i 1 + ... + β i k , for all i 1 < · · · < i n . In terms of odds ratios, log p i 1 ,...,i k q i 1 ,...,i k = β i 1 + . . . + β i k . (3) In order to write the mo del in exp onen tial family form, w e abuse notation and define for each h yp eredge e ∈ F , ˜ β e = P i ∈ e β i . In addition, let  [ n ] k  b e the set of all subsets of size k of the set { 1 , . . . , n } . By using ( 1 ), w e obtain P β ( x ) = exp n P e ∈ ( [ n ] k ) ˜ β e x e o Q e ∈ ( [ n ] k ) 1 + e ˜ β e = exp ( X i ∈ V d i ( x ) β i − ψ ( β ) ) , where d i is the degree of the no de i in x . Then it is clear that the sufficient statistics for the k − uniform b eta mo del are the en tries of the degree sequence v ector of the h yp ergraph, ( d 1 ( x ) , . . . , d n ( x )), and the normalizing constan t is ψ ( β ) = P e ∈ ( [ n ] k ) log(1 + e ˜ β e ). (4) La y ered uniform h yp ergraphs. Allowing for v arious size edges has the adv an tage of con trolling the prop ensity of eac h individual to b elong to a size- k group indep endently for distinct k ’s. Let r b e the (natural b ound for the) maximum size of a hyperedge that app ears in x . This mo del is then parametrized b y r − 1 vectors in R n as follows: p i 1 ,i 2 ,...,i k = e β ( k ) i 1 + β ( k ) i 2 + ... + β ( k ) i k 1 + e β ( k ) i 1 + β ( k ) i 2 + ... + β ( k ) i k where, for each k = 2 , . . . , r , β ( k ) = ( β ( k ) 1 , . . . , β ( k ) n ). There are ( r − 1) n parameters in this parametrization. By using ( 1 ) again, w e obtain P β ( x ) = r Y k =2 Y e ∈ ( [ n ] k ) e ˜ β ( k ) e x e 1 + e ˜ β ( k ) e = exp ( r X k =2 X i ∈ V d ( k ) i ( x ) β ( k ) i − ψ ( β ) ) , 4 where d ( k ) i is the n um b er of h yp eredges of size k to which no de i b elongs in x . Notice that the v ector of sufficien t statistics in this case is d = ( d (2) 1 ( x ) , . . . , d (2) n ( x ) , d (3) 1 ( x ) , . . . , d (3) n ( x ) , . . . , d ( r ) 1 ( x ) , . . . , d ( r )( x ) n ), and the normalizing constan t is ψ ( β ) = P r k =2 P e ∈ ( [ n ] k ) log(1 + e ˜ β ( k ) e ). (5) General h yp ergraphs. In the third v ariant of the mo del we define one parameter for each no de, con trolling the prop ensit y of that no de to b e in a relation of an y size. The probabilit y of observing a hypergraph x is thus P β ( x ) = exp n P r k =2 P e ∈ ( [ n ] k ) ˜ β e x e o Q r k =2 Q e ∈ ( [ n ] k ) 1 + e ˜ β e = exp ( X i ∈ V d i ( x ) β i − ψ ( β ) ) . The vector of sufficient statistics is then d = ( d 1 ( x ) , . . . d n ( x )), where d i ( x ) = P r k =2 d ( k ) i ( x ), and the normalizing constant is ψ ( β ) = P r k =2 P e ∈ ( [ n ] k ) log(1 + e ˜ β e ). 3. P arameter estimation Iterativ e prop ortional scaling algorithms. F rom the theory of exp onen tial families, it is known that the MLE ˆ β satisfies the follo wing system of equations: ∂ ψ ( ˆ β ) ∂ ˆ β i = ¯ d i , for i ∈ { 1 , . . . , n } , (6) where ¯ d is the av erage observ ed degree sequence. By using ( 4 ), we then obtain X s ∈ ( [ n ] \{ i } k − 1 ) e ˆ ˜ β s + ˆ β i 1 + e ˆ ˜ β s + ˆ β i = ¯ d i , for i ∈ { 1 , . . . , n } , (7) whic h is itself equiv alent to P s ∈ ( [ n ] \{ i } k − 1 ) ˆ p s,i = ¯ d i , for i ∈ { 1 , . . . , n } . Iterativ e prop ortional scaling (IPS) algorithms fit the necessary margins of a pro vided ta- ble, whose elements corresp ond to the mean-v alue parameters (in this case probabilities of observing an edge). W e design the following IPS algorithm for computing ˆ p . Algorithm 3.1. Define A = ( a i 1 ,...,i k ) to b e an n × · · · × n k -w ay table with margins ¯ d 1 , . . . , ¯ d n for all its la yers. Set the following structural zeros on the table: a i 1 ,...,i k = 0 if i a = i b for at least one pair a 6 = b , 1 ≤ a, b ≤ k . (Note that there are n ( n − 1) . . . ( n − ( k − 1)) non-zero 5 elemen ts in the table.) Place 2 ¯ e/ ( n ( n − 1) . . . ( n − ( k − 1))) on all other elemen ts of the matrix, where 2 ¯ e = P n i =1 ¯ d i . Then apply the following iterativ e ( t + 1)st step for every elemen t a i 1 ,...,i k : a ( t +1) i 1 ,...,i k = a ( t ) i 1 ,...,i k ( F ( t ) i 1 . . . F ( t ) i k ) 1 /k , where F i b ( t ) = d i b / P s ∈ ( [ n ] \{ i b } k − 1 ) a ( s ) i b ,i s . IPS algorithms are known to conv erge to elemen ts of the limiting matrix ( ˆ p i 1 ,...,i k ) whic h are unique and preserve all the marginals (see e.g. [ 1 ]). Solving the system ( 3 ) for every 1 ≤ i 1 < · · · < i k ≤ n pro vides ˆ β . Algorithm 3.1 can b e adjusted for lay ered uniform and general hypergraph b eta mo dels. F or lay ered k -uniform h yp ergraphs, by using ( 12 ) and ( 5 ) we obtain for i ∈ { 1 , . . . , n } and k ∈ { 2 , . . . , r } , X s ∈ ( [ n ] \{ i } k − 1 ) ˆ p s,i = ¯ d ( k ) i . (8) Therefore, w e can apply Algorithm 3.1 to ( r − 1) k -w a y tables similar to those of the k -uniform case, where k ranges from 2 to r . F or general h yp ergraphs, we similarly obtain r X k =2 X s ∈ ( [ n ] \{ i } k − 1 ) ˆ p s,i = ¯ d i , for i ∈ { 1 , . . . , n } . (9) In this case w e apply the IPS algorithm to the follo wing table: Define A = ( a i 1 ,...,i k ) to b e a k -w ay table of size ( n + 1) × ( n + 1) × · · · × ( n + 1) consisting of lab els ( ∅ , 1 , 2 , . . . , n ) with margins ¯ d ∅ , ¯ d 1 , . . . , ¯ d n for all its la y ers, where ¯ d ∅ do es not need to b e kno wn or calculated. W e also set the following structural zeros in the table: a i 1 ,...,i k = 0 if (1) i a = i b 6 = ∅ for at least one pair a 6 = b , 1 ≤ a, b ≤ k ; (2) i 1 = · · · = i k = ∅ except p ossibly for one i b . W e apply Algorithm 3.1 as in the k -uniform case except the fact that we do not fit the ¯ d ∅ margins. W e read the elemen ts of the limiting matrix of from, ˆ p ∅ ,s as ˆ p s , which corresp onds to a lo w er dimensional probability . Fixed P oin t Algorithms. An alternativ e metho d for computing MLE is based on [ 3 ]. In the k -uniform case, for i ∈ { 1 , . . . , n } , Equation ( 7 ) can b e rewritten as ˆ β i = log d i − log X s ∈ ( [ n ] \{ i } k − 1 ) e ˆ ˜ β s 1 + e ˆ ˜ β s + ˆ β i := ϕ i  ˆ β  . (10) Therefore, in order to find ˆ β , it is sufficient to find the fixed p oint of the function ϕ . Algorithm 3.2. Start from any ˆ β (0) and define ˆ β ( l +1) = ϕ ( ˆ β ( l ) ) for l = 0 , 1 , 2 , . . . . 6 Theorem 3.3. If the MLE exists, Algorithm 3.2 c onver ges ge ometric al ly fast; if the MLE do es not exist ther e is a diver ging subse quenc e in { ˆ β ( i ) } . The proof is omitted due to space limitations. F or the other mo dels, the ab o v e theory can be easily generalized. F or the la y ered mo dels and general h yp ergraph mo dels, w e apply the same algorithm to obtain the fixed points of the following functions resp ectively for i ∈ { 1 , . . . , n } and k ∈ { 2 , . . . , r } and i ∈ { 1 , . . . , n } . ϕ i ( ˆ β ( k ) ) := log d ( k ) i − log X s ∈ ( [ n ] \{ i } k − 1 ) e ˆ β ( k ) s 1 + e ˆ β ( k ) s + ˆ β ( k ) i ; (11) ϕ i ( ˆ β ) := log d i − log r X k =2 X s ∈ ( [ n ] \{ i } k − 1 ) e ˆ ˜ β s 1 + e ˆ ˜ β s + ˆ β i . (12) 4. Sim ulations and Analysis MLE. W e use the fixed p oin t algorithm to estimate the natural parameters for h yp ergraph b eta mo dels, examine non-existence of MLE and compare the la yered and general v arian ts of the model on simulated data. Note that most dense h yp ergraphs, when reduced to binary relations give the complete graph, for which the MLE do es not exist. In contrast, MLE is exp ected to exist for the h yp ergraph b eta mo del in this case. Example 4.1. W e sim ulate a hypergraph H = ( V , F ) drawn from the b eta mo del for 3- uniform h yp ergraphs on 10 vertices with β = ( − 5 . 05 , − 0 . 57 , 2 . 87 , 4 . 85 , 1 . 98 , − 6 . 69 , − 3 . 95 , 5 . 97 , − 6 . 61 , − 4 . 24). The av erage simulated degree sequence of h yp ergraphs drawn from this mo del is ¯ d = (6 . 28 , 10 . 70 , 17 . 59 , 20 . 81 , 16 . 55 , 4 . 41 , 7 . 47 , 23 . 02 , 4 . 50 , 7 . 17) , and the av erage sim ulated densit y of the corresp onding h yp ergraph is 0.33. Algorithm 3.2 pro vides the following MLE es- timate using ¯ d as the sufficient statistic: ˆ β = ( − 4 . 94 , − 0 . 58 , 2 . 81 , 4 . 76 , 1 . 94 , − 6 . 55 , − 3 . 86 , 5 . 86 , − 6 . 48 , − 4 . 15). Note that || β − ˆ β || ∞ = 0 . 14. F or a larger example, we select a β v alue giving rise to 3-uniform hypergraphs on 100 vertices with density 0 . 44, and obtain a closer estimate: || β − ˆ β || ∞ = 0 . 12. Example 4.2. Theorem 3.3 guaran tees that if ˆ β is the solution to the ML equations ( 10 ), ( 12 ), or ( 12 ), then the sequence of β -estimates that the fixed p oin t algorithm pro duces will con v erge to ˆ β ; else there will b e a div ergen t subsequence. T o detect a divergen t sequence in practice, w e either lo ok for a p erio dic subsequence, or for a n um b er with large absolute v alue in the sequence that seems to b e gro wing, sometimes quite slowly . F rom ( 2 ), since e β i k / (1 + e β i k ) con verges to 1 quic kly ( e 10 / (1 + e 10 ) > 0 . 9999), for graphs with small num b er of no des (i.e. far from the asymptotic b ehavior), it is plausible to conclude that the corresponding 7 0.2 0.3 0.4 0.5 0.6 B e ta m o d e l fo r 3 -u n i fo r m h y p e r g r a p h s o n 2 5 v e r ti c e s Ed g e D e n si t y o f H yp e rg ra p h ML E Exi st e n ce No Yes 0.2 0.3 0.4 0.5 0.6 B e ta m o d e l fo r 3 -u n i fo r m h y p e r g r a p h s o n 5 0 v e r ti c e s Ed g e D e n si t y o f H yp e rg ra p h ML E Exi st e n ce No Yes 0.2 0.3 0.4 0.5 0.6 Ed g e D e n si t y o f H yp e rg ra p h ML E Exi st e n ce No Yes 0.2 0.3 0.4 0.5 0.6 B e ta m o d e l fo r g e n e r a l h y p e r g r a p h s w i th { 2 , 3 } -e d g e s o n 5 0 v e r ti c e s Ed g e D e n si t y o f H yp e rg ra p h ML E Exi st e n ce No Yes 0.2 0.3 0.4 0.5 0.6 B e ta m o d e l fo r 4 -u n i fo r m h y p e r g r a p h s o n 2 5 v e r ti c e s Ed g e D e n si t y o f H yp e rg ra p h ML E Exi st e n ce No Yes 0.1 0.2 0.3 0.4 0.5 0.6 B e ta m o d e l fo r 2 -u n i fo r m h y p e r g r a p h s o n 2 5 v e r ti c e s Ed g e D e n si t y o f H yp e rg ra p h ML E Exi st e n ce No Yes Fig 2: MLE existence (v ertical axes) for the k -uniform b eta mo del on n vertices against edge density (horizon tal axes). T op row: k = 3; n = 25 (left), n = 50 (middle), n = 100 (right). Bottom row: k = 2 and n = 25 (left); k = 4 and n = 25 (middle); k = { 2 , 3 } and n = 50 (right). mean v alue parameter is approximately 0 or 1, and hence the MLE do es not exist. Figure 2 demonstrates MLE existence against edge densities for random hypergraphs with a fixed edge- densit y . Interestingly , in this restricted class, our sim ulations give evidence of a transition from non-existence of the MLE to existence as the densit y of the hypergraphs increases. The transition p oint seems to dep end on both the n umber of v ertices and the edge sizes allo wed in the mo del. Mo del fitting: Lay ered v ersus general hypergraph b eta mo dels. Consider the t wo v arian ts of the b eta mo del for non-uniform h yp ergraphs: the general mo del, with one parameter β i p er no de i , and the lay ered mo del, with one parameter β ( k ) i p er no de i and edge size k . Since the former can b e considered a submo del of the latter b y setting certain constrain ts on β ( k ) i , k ∈ { 1 , . . . , r } , w e compare the fit of these t w o mo dels using the lik eliho o d ratio test with test statistics λ = 2 log L ( ˆ β lay ered ) − 2 log L ( ˆ β general ). Our exp erimen ts indicate that the la y ered mo del fits significan tly b etter than the general case. Using 100 random sequences on 10 vertices, with allo wed edge-sizes 2 and 3, w e obtain the av erage observed test statistics 53 . 649, in the critical region for 0 . 005 significance lev el, (25 . 188 , ∞ ), for c hi-square with 10 degrees of freedom. The la yered mo del fits significantly b etter for significance lev el 0.05 in all 8 100 cases, and 97 and 94 times b etter for significance levels 0.01 and 0.005, resp ectiv ely . References [1] Y. M. M. Bishop, S. E. Fien b erg, and P . W. Holland. Discr ete Multivariate A nalysis: The ory and Pr actic e . MIT Press, Cam bridge, Mass.-London, 1975. 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