Sharp Impossibility Results for Hyper-graph Testing

1 Introduction . The hypergraph is a useful representation of social relationships beyond pairwise interactions [ 5 , 11 ] . For example , the co-authorship hypergraph is often used to analyze the co-authorship topology of authors , and it provides more information than a co-authorship graph ( where an m-author paper is treated as an m-clique ) . The community detection on a hypergraph [ 10 ] is a problem of great interest ( communities in a hypergraph are clusters of nodes that have more hyperedges within than across ) . It has many applications in social network analysis [ 15 ] and machine learning [ 1 , 17 , 18 , 23 ] . We are interested in the problem of global testing , where we test whether the hypergraph has one community or multiple communities . It has applications in measuring co-authorship and citation diversity [ 12 ] and discovering non-obvious social groups and patterns [ 4 ] . It is also useful for understanding other problems such as community detection and change-point detection in dynamic hypergraphs . For instructional purpose only , we start with the 3-uniform hypergraphs ( i.e. , each hyperedge consists of 3 nodes ) , but our results cover both higher-order hypergraphs and non-uniform hypergraphs . Let A be the adjacency tensor of a uniform and symmetric order-3 hypergraph with n nodes , where Ai1i2i3 equals 1 if i1 , i2 , i3 share a hyperedge and equals 0 otherwise , ( 1.1 ) for 1 ≤ i1 , i2 , i3 ( distinct ) ≤ n. Since the tensor is symmetric , Ai1i2i3 = Aj1j2j3 for two sets of indices { i1 , i2 , i3 } and { j1 , j2 , j3 } if one is a permutation of the other . We do not consider self hyperedges , hence , Ai1i2i3 = 0 whenever i1 , i2 , i3 are non-distinct . Real world hypergraphs have several noteworthy features . First , there may be severe degree heterogeneity ( i.e. , the degree of one node is many times higher than that of another ) . Second , the overall sparsity levels may vary significantly from one hypergraph to another . Last , a node may have mixed-memberships across multiple communities ( i.e. , nonzero weights on more than one community ) . To accommodate these features , we adopt the Degree-Corrected Mixed-Membership ( tensor-DCMM ) model . The notations below are frequently used in tensor analysis . Definition 1.1 . ( matricization , slicing , and slice aggregation ) . Let A be a 3-symmetric tensor of ndimension . First , we call the n×n2 matrix A the matricization of A , defined by Ai , j+n ( k−1 ) = Aijk , 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 1 ≤ i , j , k ≤ n. Second , for 1 ≤ k ≤ n , we use A : :k to denote the n× n matrix whose row-i-andcolumn-j is Aijk , 1 ≤ i , j ≤ n , and call it the k-th slice of A . Last , for any n× 1 vector x , we use ( Ax ) to denote the matrix ∑n k=1 xkA : :k , which is an aggregation of the slices . Now , suppose the tensor has K perceivable communities . Let P be a symmetric ( 3-uniform ) tensor of K-dimension that models the community structure , let θ = ( θ1 , θ2 , . . . , θn ) ′ be positive parameters that model the degree heterogeneity of nodes , and π1 , π2 , . . . , πn be K-dimensional membership vectors where πi ( k ) = the weight node i puts on community k , 1 ≤ k ≤ K. We assume for all 1 ≤ i1 , i2 , i3 ≤ n that are three distinct indicies , Ai1i2i3 are independent Bernoulli random variables with P ( Ai1i2i3 = 1 ) = θi1θi2θi3 ∑K k1 , k2 , k3=1 Pk1 , k2 , k3πi1 ( k1 ) πi2 ( k2 ) πi3 ( k3 ) , where by Definition 1.1 , the right hand side equals to θi1θi2θi3π ′ i1 ( Pπi3 ) πi2 . Introduce a non-stochastic 3-uniform tensor Q of n-dimension where Qi1i2i3 = θi1θi2θi3π′i1 ( Pπi3 ) πi2 for 1 ≤ i1 , i2 , i3 ≤ n. Let diag ( Q ) be the tensor with the same size of Q where ( diag ( Q ) ) i1i2i3 = Qi1i2i3 if i1 , i2 , i3 are non-distinct , and ( diag ( Q ) ) i1i2i3 = 0 otherwise . It follows that E ( A ) = Q− diag ( Q ) , where Qi1i2i3 = θi1θi2θi3π′i1 ( Pπi3 ) πi2 . ( 1.2 ) For identifiability , let P ∈ RK , K2 be the matricization of P ( see Definition 1.1 ) . We assume rank ( P ) = K , and Piii = 1 for all 1 ≤ i ≤ K. ( 1.3 ) Definition 1.2 . We call ( 1.1 ) - ( 1.3 ) the tensor-DCMM model for 3-uniform hypergraphs . We call Q and P the Bernoulli probability tensor and community structure tensor for DCMM , respectively . Later in Section 3 , we introduce the non-uniform tensor-DCMM as a more sophisticated model . In tensor-DCMM , if we require all πi to be degenerate ( i.e. , one entry is 1 , all other entries are 0 ) , then tensor-DCMM reduces to the Degree-Corrected Block Model ( tensor-DCBM ) [ 15 ] . If we further require θ1 = . . . = θn ( but the second condition in ( 1.3 ) can be removed ) , then tensor-DCBM further reduces to the Stochastic Block Model ( tensor-SBM ) [ 9 ] . For simplicity , we may drop “ tensor '' in these terms if there is no confusion . The global testing problem above is then to test H0 : K = 1 vs. H1 : K > 1 . ( 1.4 ) Our primary goals are ( a ) to find a sharp information lower bound for 3-uniform DCMM , and especially , to fully characterize the lower bound by a simple quantity to be discovered , and ( b ) extend the results to more sophisticated non-uniform hypergraphse ( see Section 3 ) . A good understanding of the problem greatly helps us understand the fundamental limits of many other problems ( e.g. , community detection , determining the number of communities K , dynamic hypergraphs ) . For example , for community detection ( e.g. , [ 10 ] ) , we either assume K as known or estimate it first . Note that in parameter regions where we can not tell whether K = 1 or K > 1 , we can not estimate K consistently , so we can not have consistent community detection either . Therefore , a lower bound for global testing is always a valid lower bound for estimating K and for community detection . To facilitate the lower bound study , we frequently adopt a Random Mixed-Membership ( RMM ) model . Introduce a subset V0 = { x ∈ RK : xk ≥ 0 , ∑K k=1 xk = 1 } , and let F be a K-variate distribution with support contained in V0 . We assume πi iid∼ F ; ( let h = EF [ πi ] ) . ( 1.5 ) Moreover , let V ∗0 = { e1 , e2 , . . . , eK } ⊂ V0 , where ek is the k-th basis vector of RK . Similarly , RMM-DCMM reduces to RMM-DCBM if we require supp ( F ) = V ∗0 , and reduces to RMM-SBM if we further require θ1 = θ2 = . . . = θn ( but the second condition in ( 1.3 ) can be removed ) . Let θ = ( θ1 , θ2 , . . . , θn ) ′ . We allow ( θ , P , h , F ) to vary with n to cover a variety of settings where we allow for severe degree heterogeneity , mixed-memberships , flexible sparsity levels , and weak signals . Example 1 ( 2-parameter SBM [ 2 , 22 , 19 ] ) . This model is a special case of DCMM , where θ1 = . . . = θn = αn ( no degree heterogeneity ) , all πi are degenerate ( no mixed membership ) , and Pijk = 1 if i = j = k and Pijk = ρn otherwise , for two parameters ( αn , ρn ) . Also , Lin et al . [ 20 ] studied a 3-parameter SBM , which is the same as above except that they assume a different form of P , where Pijk equals to 1 , ρn , or τn if i , j , k take 1 , 2 , or 3 distinct values , respectively , for three parameters ( αn , ρn , τn ) . Compared to DCMM , these models are much narrower : they do not accommodate severe degree heterogeneity or mixed-memberships , and P is parametrized by 2 or 3 different values . How to derive a sharp lower bound for global testing ( and especially , to identify a simple quantity that fully characterizes the lower bound ) in our setting is a rather challenging problem . Our model is a non-uniform hypergraph model ( see Section 3 ) , which consists of hypergraphs of order 2 , 3 , . . . , M , and each layer consists of many unknown parameters ( θ , P , h , F ) . Existing works on lower bounds have been focused on uniform hypergraphs , and non-uniform hypergraphs are much less studied . Even for uniform hypergraphs , existing works have been focused on the the special SBM as in Example 1 , not the more general DCMM model . For example , Yuan et al . [ 22 ] derived the lower bounds for global testing with the 2-parameter SBM , focusing on the extremely sparse case . Ahn et al . [ 2 ] provided lower bound results for exactly recovering the communities ( see Liang et al . [ 19 ] and Kim et al . [ 16 ] for related settings ) with a similar model . Lin et al . [ 20 ] and Chien et al . [ 7 ] used the 3-parameter SBM in Example 1 for study of the lower bounds for community detection . While these papers are very interesting , their lower bounds are characterized by only 2 or 3 parameters ( i.e. , αn , ρn , τn ) assumed in their models . For the much broader tensor-DCMM model considered here , we have many parameters θ , P , h , F ( θ is an n-vector and P is a tensor ) , and how to extend existing results to the tensor-DCMM setting here is a challenging problem . Our contributions . The main challenge in lower bound study is that , since each DCMM model ( no matter what K is ) has a large number of unknown parameters , so given an alternative hypothesis ( K > 1 ) , it is hard to identify the null hypothesis ( K = 1 ) that is most difficult to distinguish from the alternative hypothesis . As our main contribution , we approach this by proposing a degree matching strategy , where for any given DCMM model with K > 1 , we pair it with a DCMM model with K = 1 in a way so that for each node , the expected degree under the null matches with that under the alternative . This way , it is hard to separate the two hypotheses by a naive degree-based statistic . We show ( a ) the degree matching is always possible by using a tensor scaling technique [ 3 , 8 ] , and ( b ) the pair of hypotheses we construct this way lead to sharp results on lower bounds . See Section 2 . We have the following results . Consider the 3-uniform hypergraph first . We first present the standard minimax theory . Define a class of RMM-DCMM models with K > 1 and µ22∥θ∥2∥θ∥1 → 0 ( µ2 is the second singular value of P ) . In this class , we can find an RMM-DCMM model ( the alternative ) , and pair it with a DCMM model with K = 1 ( the null ) , such that the χ2-divergence between the pair converges to 0 as n → ∞ . Therefore , in this class , there exists an alternative model that is asymptotically inseparable from the null . The standard minimax theory only claims that there exists an alternative model within a specified class that is inseparable from the null . It is desirable to show a much stronger result where for any alternative in the class , we can pair it with a null so that the χ2-divergence of the pair goes to 0 as n → ∞ . In detail , we show that in the parameter space ( θ , P , h , F ) of RMM-DCBM , there is a Region of Impossibility defined by µ22∥θ∥2∥θ∥1 → 0 ; for any alternative in Region of Impossibility , we can pair it with a null such that the χ2-divergence between the pair goes 0 as n→ ∞ . Compared with existing results on minimax lower bounds , these results are more informative and theoretically more satisfactory . The proof is also different from the proof of minimax lower bounds : we have used the tensor scaling theory [ 3 , 8 ] and the “ degree matching strategy '' aforementioned . We also extend such results to the broader RMM-DCMM case ( Section 2.3 ) and discuss some major differences on the Region of Impossibility between DCBM and the more restrictive SBM ( Section 2.4 ) . Next , we generalize the results to higher-order and non-uniform hypergraphs . Fix M ≥ 2 and consider a non-uniform hypergraph ( e.g. , see [ 10 ] ) that consists of m-uniform hypergraphs for all m = 2 , . . . , M , each following a DCMM model with individual ( θ ( m ) , P ( m ) ) but the common π1 , . . . , πn . Let ℓm = ∥θ ( m ) ∥m−21 ∥θ ( m ) ∥2 ( µ ( m ) 2 ) 2 ( to be defined in Section 3 ) . We show that ( a ) for the m-uniform hypergraph case , the Region of Impossibility for the hypothesis testing ( 1.4 ) is fully characterized by the condition of ℓm → 0 , and ( b ) for the non-uniform hypergraph case , the Region of Impossibility for the hypothesis testing ( 1.4 ) is fully characterized by the condition of max2≤m≤M { ℓm } → 0 . Last , we show that our lower bounds are tight . Consider the non-uniform hypergraph above . We propose a new test statistic and show that the sum of Type I error and Type II error → 0 if max2≤m≤M { ℓm } ≥ log ( n ) 1+δ for a constant δ > 0 ( taking δ = 0.1 will work ) . Therefore , except for a logarithmic factor here , our lower bounds are tight . In summary , existing results on lower bounds are largely focused on more restrictive settings ( e.g. , uniform hypergraphs without degree heterogeneity or mixed membership ) . We provide sharp lower bounds for a much broader setting . Our study is highly non-trivial because we need ( i ) a novel degree matching strategy to construct least favorable hypothesis pairs , ( ii ) to identify a simple quantity that is able to fully characterize the lower bounds , ( iii ) delicate analysis of the χ2-divergence between the null and alternative , ( iv ) a carefully designed test that leads to tight upper bounds . 2 Sharp Lower Bounds for 3-Uniform Hypergraphs For notational simplicity , we focus on 3-uniform hypergraphs in this section . The study of higherorder and non-uniform hypergraphs is deferred to Section 3 . Consider a ( 3-uniform ) DCMM model . Recall that h = EF [ πi ] and that P ∈ RK , K 2 is the matricization of the tensor P . In this paper , we use C > 0 as a generic constant which may vary from occurrence to occurrence . We assume ∥P∥ ≤ C , θmax ≡ max { θ1 , . . . , θn } ≤ C , max 1≤k≤K { hk } ≤ C min 1≤k≤K { hk } , ∥θ∥2∥θ∥1 → ∞ . ( 2.6 ) The first condition is mild , because the model identifiability already requires that all diagonal entries of P are 1 . The second one is also mild ( note that while the largest possible value of θmax is O ( 1 ) , θmax is allowed to tend to 0 relatively fast ) . The third one assumes that the community memberships are balanced , which is also mild . For the last condition , we will see soon that if ∥θ∥2∥θ∥1 → 0 , then the signal is so weak that successful global testing is impossible , so this condition is also mild .

Summary: This paper introduces and studies a hypergraph community detection model called RMM-DCMM. In this model, each vertex may have a different degree (degree-corrected) and may also belong to many different communities (mixed-membership).  A main takeaway from the paper is that possibility/impossibility of testing whether this complicated model has K = 1 or K > 1 communities boils down to the quantity ||\theta||^2 ||\theta||_1 \mu_2^2.   The main results are: 1) Region of impossibility result for RMM-DCBM (a subset of RMM-DCMM). This shows that given any: 	(a) degree-correction factors \theta for a 1-community model 	(b) tensor P representing connection strengths between subsets of the K communities, 	(c) distribution F over mixed membership vectors, supported on {e_1,...,e_K}, 	if ||\theta||^2 ||\theta||_1 \mu_2^2 --> 0 (where \mu_2 is the second eigenvalue of the matricization of P), then there is a choice of degree-correction factors \theta^* such that the K-community model given  by (P, \theta^*, F) is information-theoretically indistinguishable from the 1-community model given by \theta  2) Region of possibility result for RMM-DCBM. For any K-community model (P,\theta,F) with K > 2, such that ||\theta||^2 ||\theta||_1 \mu_2^2 / log(n)^{1.1} --> \infty, there is a test (based on estimating the degree-correction vector \theta assuming that the model is null) to distinguish it from any 1-community model.  These results extend to RMM-DCMM under some additional conditions.