Title: The Graph Pencil Method: Mapping Subgraph Densities to Stochastic Block Models
Venue: NeurIPS 2023
Accepted: yes
Keywords: Stochastic block model

SBM

graphons

matrix pencil method

method of moments
Abstract: In this work, we describe a method that determines an exact map from a finite set of subgraph densities to the parameters of a stochastic block model (SBM) matching these densities. Given a number K of blocks, the subgraph densities of a finite number of stars and bistars uniquely determines a single element of the class of all degree-separated stochastic block models with K blocks. Our method makes it possible to translate estimates of these subgraph densities into model parameters, and hence to use subgraph densities directly for inference. The computational overhead is negligible; computing the translation map is polynomial in K, but independent of the graph size once the subgraph densities are given.

The Graph Pencil Method: Mapping Subgraph Densities to Stochastic Block Models

Abstract

In this work, we describe a method that determines an exact map from a finite set of subgraph densities to the parameters of a stochastic block model (SBM) matching these densities. Given a number K of blocks, the subgraph densities of a finite number of stars and bistars uniquely determines a single element of the class of all degree-separated stochastic block models with K blocks. Our method makes it possible to translate estimates of these subgraph densities into model parameters, and hence to use subgraph densities directly for inference. The computational overhead is negligible; computing the translation map is polynomial in K , but independent of the graph size once the subgraph densities are given.

1 Introduction

The class of infinitely exchangeable random graphs is a large class of network models that contains all graphon models [6, 21] and stochastic block models (SBMs) [1, 2, 12]. The key statistics of such random graphs are subgraph densities: each fixed, finite graph defines a subgraph density, and it is well known that every exchangeable graph distribution is completely determined by the collection of all its densities of finite subgraphs [12, 21].

In general, an infinite number of densities is required to uniquely determine an infinitely exchangeable graph distribution. For a subclass of models, a finite number of densities suffice. In combinatorics, these random graph distributions are called finitely forcible [21]. In particular, every stochastic block model is completely determined by a finite number of subgraph densities [21]. Such a finite number of subgraph densities can be consistently estimated at a guaranteed asymptotic rate from a single graph-data as this graph grows large [3, 17, 30].

It has long been recognized that the role subgraph densities play for exchangeable network models is analogous to that played by moments for estimation problems on Euclidean sample spaces [5, 7, 12, 22]. On Euclidean space, the method of moments [24, 26] is a powerful inference tool, in particular for mixture models [4, 14]. This suggests that subgraph densities should be well-suited for inference in stochastic block models in particular, since these models are a network analogue of mixture models and only a finite number of densities must be estimated.

Nonetheless, most inference algorithms used in practice (e.g., [8, 9, 13, 18, 25]) do no rely on subgraph densities, but rather on some combination of maximum likelihood estimation, MCMC sampling, variational inference, clustering of the nodes, or other heuristics. The reason is that estimating subgraph densities is only a first step in the inference process - once the densities are known, they must be translated into an estimate of a graphon or stochastic block model.

This paper introduces a new method that does so by generalizing a classical strategy for inferring latent sources (sometimes known as Prony's method [26]) to graph data. The classical Prony's

method is an example of an algorithm that recovers a sparse signal from noisy data, and is related to compressed sensing [28] and the notion of a matrix pencil [23].

Our method proceeds in two steps. The first step (section 3.2) is essentially an application of Prony's method to estimate properties of individual latent blocks, such as their normalized degrees and their relative sizes. The second step (section 4.2) is entirely new; by leveraging the properties inferred in the first step, it uses a generalization of Prony's method to infer properties of pairs of the latent blocks, such as the connection properties of a stochastic block model.

The proposed method solves several problems simultaneously: it estimates parameters without laborious and delicate numerical fitting, and makes it possible to efficiently sample from the model without the frequently-encountered problem of degeneracy that plagues many intuitively attractive network models [15, 16, 19]. In contrast to other algorithms for fitting stochastic block models (e.g., [8, 10, 11, 25]) and graphons (e.g, [2, 20, 29]), our method requires essentially zero computational overhead once the subgraph densities have been estimated.

2 Notation and Definitions

We use lower-case bold symbols to denote vectors, and upper-case bold symbols to denote matrices and higher-order tensors. Non-bold versions of these symbols refer to particular entries of the bold versions, indicated by their subscripts. We use ◦ to denote Hadamard product (i.e., element-wise multiplication), · for inner product, and angled brackets 〈 〉 to denote expectation.

A graph G is defined by a set of vertices/nodes V ( G ) and a set of edges E ( G ) ⊆ V ( G ) × V ( G ) denoting pairwise connections between nodes. For simplicity, here we focus on undirected, unweighted graphs, with no self-loops or parallel edges. Indexing the N = | V ( G ) | nodes by the positive integers [ N ] , we can represent G by its N -byN adjacency matrix A , where A ij = A ji = 1 if there is an edge between nodes i and j , and 0 otherwise.

Astochastic block model SBM ( π , B ) is defined by: π , a probability vector of length K , which assigns each node to one of K (unobserved) latent blocks/communities; and B , a K -byK connectivity matrix, whose entries B kk ′ give the probability of an edge connecting a node in block k to a node in block k ′ . As we are considering undirected graphs, B is symmetric.

Sampling a graph from a SBM ( π , B ) proceeds in two steps. First, for each node n ∈ [ N ] , sample its latent block k ( n ) ∈ [ K ] independently from π . Then, for each pair of nodes ( n, n ′ ) , include an edge between them independently with probability B k ( n ) k ( n ′ ) . Thus, a stochastic block model defines a distribution over graphs with N nodes for all choices of N , and can be identified with the limit of this sequence of distributions as N →∞ .

As mentioned in the introduction, such infinitely exchangeable graph distributions can be characterized by their homomorphism densities 1 µ ( g ) . Indexed by (some family of) subgraphs g , homomorphism densities are the graphical analogue of the classical moments of a distribution [5, 12, 22]. For a stochastic block model, one can compute µ ( g ) by summing over all possible assignments of the vertices of g to the blocks of SBM ( π , B ) [21]:

In the following two sections (sections 3 and 4), we describe how to recover the parameters of an SBM from its subgraph densities. Later, in section 5, we describe how to infer these subgraph densities using a graph sampled from such an SBM.

3 Mapping Star Densities to Block Degrees

As the first step of our method, we obtain the normalized degrees of the blocks d (equation (4)), as well as their relative sizes π . This can be seen as an application of the classical matrix pencil method to the degree distribution, for which the density of the star subgraphs serve as sufficient statistics.

3.1 Classical Coin Collecting

Before we explain our graph pencil method, let us consider the simpler case of a mixture model for infinitely exchangeable sequences of binary variables. Also known as the Bernoulli mixture model [14, 27], such a distribution can be thought of as the outcomes of (some number of) flips of (some number of) biased coins, where each coin is sampled i.i.d. from a (possibly unequal) mixture of K different biases.

To recover the parameters of this model, one must infer the K (unobserved) latent biases b k , as well as the fraction π k of coins with each bias. Note that one does not know which of the coins have the same latent bias (otherwise the inference problem would be trivial).

The moments of this distribution are the expectation of powers of these latent biases, which are indexed by the exponents r ∈ N :

Note that these moments can be consistently estimated from the observed data, as we are averaging over the latent variables. From these moments, one can systematically infer the mixture proportion π and the biases b using the matrix pencil method.

In short, construct two matrices C and C ′ , with entries C i j = 〈 b i + j 〉 and C ′ i j = 〈 b i + j +1 〉 :

While not immediately obvious, it is easy to show that the eigenvalues of C ′ C -1 are the entries of b . From these, it is straightforward to obtain the associated entries of π . In the next section, we show why this is the case, while superficially replacing the biases b of the latent coin types with the normalized average degrees d of the latent node blocks.

3.2 Distilling the Degree Distribution

The first step of our graph pencil method is to apply the standard matrix pencil method to the obtain the average normalized degree of each of the K latent blocks (that is, the likelihood that a random node shares an edge with a node in block k ). Denoted by d , the entries of this vector are:

These d k are latent parameters of the blocks, so we can solve for them in exactly the same way as we did for b k in the previous coin example. To this end, we consider the analogous moments:

As before, while π k and d k are unobserved, the resulting moments can be consistently estimated from observations. In particular, they are precisely the homomorphism densities of the star subgraphs:

We then define the two matrices C and C d , with entries C i j = 〈 d i + j 〉 and C d i j = 〈 d i + j +1 〉 , and the eigenvalues of C d ( C ) -1 are the normalized degrees of the K blocks. To see why, notice that C can be written as a sum of K rank-1 matrices (weighted by π ):

where we have defined V as the matrix with entries V jk = d j -1 k , i.e.:

By decomposing C d in the same manner, we get:

where diag ( π d ) is the diagonal matrix with entries π k d k .

We now see that C d ( C ) -1 can be diagonalized as follows:

Hence, the eigenvalues of C d ( C ) -1 are indeed the normalized degrees d of the K latent blocks:

Remark 1 ( Degree separated assumption ) . This is where the assumption of degree-separated blocks is used; if the normalized degrees of the blocks are not unique, then V will not be invertible.

From the entries of d , we know the entries of V , and can solve a linear system of equations for the corresponding entries of π :

4 From Bistar Densities to a Stochastic Block Model

This second step of our graph pencil method is the main insight of our paper. It uses the results from the previous step to reconstruct the connection probabilities B k,k ′ .

4.1 Gluing Graphs: An Algebra

We now describe a natural algebra over rooted subgraphs (see Chapter 6 of [21]). We find that using this notation makes our method easier to understand, and facilitates reasoning about its generalizations. For a longer introduction, see appendix A.

Recall the definition of homomorphism subgraph densities of an SBM in equation (1). For a given distribution SBM ( π , B ) , each subgraph g corresponds to a scalar density µ ( g ) . Rooted homomorphism subgraph densities can be seen as a generalization of this notion. In addition to the subgraph g , (singly-)rooted densities also require a designated 'rooted' vertex v ∈ V ( g ) . And instead of a single scalar, they correspond to (a vector of) K scalars µ ( g, v ) , indexed by the block k ∈ [ K ] to which the rooted vertex v is mapped:

The sum is now over maps ϕ that send the rooted vertex v to a particular block k , and the probability of assigning the vertices V ( g ) to blocks includes only the remaining 'unrooted' vertices. All the edges E ( g ) contribute to the product exactly as in equation (1).

A rooted vertex with a single edge corresponds to the normalized degrees of the K blocks:

Rooted subgraphs may be combined via the gluing product, denoted by ◦ . The gluing product corresponds to taking their disjoint union, then merging (i.e., 'gluing') the rooted vertices:

Algebraically, the gluing product corresponds to entrywise multiplication of the vectors, also known as the Hadamard product.

Finally, to convert a rooted subgraph into an observable subgraph density, the rooted vertex may be 'unrooted' by taking the dot product with π :

Before describing how to recover the entries of B , let us summarize the method for recovering the normalized degrees from the previous section using the language of rooted graphs.

For an SBM with K blocks, define a vector v of (at least) K rooted subgraphs, and take the outer product of this vector with itself where entries are combined via the gluing product. For example:

While the rooted subgraph densities are not directly observable, their unrooted counterparts are, and the entries of C (equation (3), left) are given by the densities of these unrooted versions:

(The glyph containing an unrooted vertex and no edges evaluates to 1 .)

To obtain the second matrix C d (equation (3), right), glue the rooted subgraph to all entries before unrooting:

Indeed, the reason that the spectrum of C ( C d ) -1 gives the normalized degrees d of the latent blocks is precisely because we glued an extra copy of the corresponding rooted subgraph in the construction of C d . The next section follows a similar recipe.

4.2 Extracting the Edge Expectations

The previous matrices C and C d were obtained by taking a dot product that sums over the K blocks. This allowed us to obtain properties of a single block (e.g., its degree d k ). The first main insight of this paper is that the matrix pencil method can also be used to obtain properties of pairs of blocks (i.e., their connection probability B kk ′ ) by summing over such pairs.

The extension of the gluing algebra to birooted subgraphs is relatively straightforward; a birooted subgraph ( g ; u, v ) has two (distinct) rooted vertices, and their gluing product is given by taking the disjoint union then separately merging the vertices rooted u and the vertices rooted v .

(

)

(

)

For an SBM with K blocks, there are K + ( K 2 ) degrees of freedom for the entries of B . So we define a vector v of (at least) K + ( K 2 ) birooted subgraphs. To this end, we use the symmetric polynomials, with exponent at most K , in the two variables and , corresponding to the normalized degrees of the blocks corresponding to the two roots.

In this example for K = 2 , the second row of v has been symmetrized with respect to the two rooted vertices, resulting in a formal linear combination of elements in the gluing algebra.

As before, C is obtained by unrooting the entries, while for the construction of C B , we glue the rooted subgraph corresponding to entries of B .

Note that, once unrooted, isomorphic graphs are equivalent, e.g.: = =

As promised, the eigenvalues of C B ( C ) -1 are precisely the entries of B

Moreover, the structure of their corresponding eigenvectors allows us to select specific entries B kk ′

Thus, for each pair of blocks k and k ′ with normalized degrees d k and d k ′ , we can estimate their connection probability B kk ′ , estimate the eigenvalue of its corresponding eigenvector:

Remark 2 ( Sufficiency of subgraph densities ) . Suppose we fix a finite number K of blocks, and consider all SBMs with K blocks. The 2 K -1 subgraph densities defined by all stars with up to 2 K -1 edges then completely determines the degrees and sizes of blocks. Adding the subgraph densities of all bistars with maximum degree at most K then completely determines the connection probabilities between blocks (as a function of their degrees), and therefore the entire distribution of the random graph. In other words, the vector of these densities of stars and bistars is a sufficient statistic for the class of degree-separated stochastic block models with K blocks.

4.3 Beyond Degree Correlations: Adding 'Two-Hop' Subgraphs

The method described above can be thought of as a 'basic' minimum working example of a much more general family of graph pencil methods. The subgraph densities in C and C d are sensitive only to the degree distribution, and the subgraphs in C and C B contain information about the degree-degree correlations. While this is sufficient to recover the parameters of an SBM when the normalized degrees of the blocks are well-separated, the method can be made more robust by adding additional subgraphs.

In particular, define the 'two-hop' matrix of connection probabilities Λ = B · π B , where · π denotes the matrix product weighted by π , i.e.:

These birooted subgraphs interact the others via gluing and unrooting (see table 3).

(

)

(

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We can use this to add more columns to C and C B :

Remarkably, we can use these larger matrices in exactly the same way as before (using the MoorePenrose pseudoinverse). As shown in in figure 1, including these additional subgraphs can make the recovery of the SBM more robust.

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5 From an Observed Graph to a Stochastic Block Model

In order to apply the graph pencil method to an observed graph, we need to obtain unbiased estimators of the homomorphism densities used to construct the C matrices. In particular, for a subgraph g and a (large) graph G , we want to compute the fraction of injective maps ϕ from V ( g ) to V ( G ) , such that if ( u, v ) ∈ E ( g ) , then ( ϕ ( u ) , ϕ ( v )) ∈ E ( G ) .

5.1 Quickly Counting (Injective) Homomorphisms

We start with the adjacency matrix A , the identity matrix I , and the all-ones matrix 1 . Then, we perform our only difficult matrix multiplication: ∑ k A ik A kj = D i j + Λ i j , where D is the diagonal

matrix of node degrees, and Λ is the (traceless) 'two-hop adjacency' matrix. From these, we can recursively count all the relevant subgraphs using only entrywise multiplication.

We index our subgraphs with a tuple of non-negative integers ( /lscript , c, r ) , corresponding to: the number of edges incident to only the 'left' node, the number of two-hop paths between the 'left' and 'right' nodes, and the number of edges incident to only the 'right' node j . First, we add the two-hop paths:

Define the matrix of 'left' degrees to be L = D · ( 1 -I ) -A (i.e., the degree of node i if node j = i were deleted, and zero when i = j ), and the matrix of 'right' degrees to be its transpose R = L /latticetop . Next, we add single edges to node i , then to node j :

/negationslash

Finally, if an edge connecting the left and right nodes is to be included, we put a mark on the middle integer:

To obtain the counts of a subgraph g in the entire graph, simply sum the entries of the corresponding matrix M ( /lscript ,c,r ) . This is analogous to the 'unrooting' operation from before.

For example, /llbracket A /rrbracket is twice the number of edges, as the injective homomorphisms count both orientations. The injective homomorphism densities are obtained by dividing this count by the number of injective maps

6 Discussion

The map from a stochastic block model to the density of a given subgraph can be computed analytically (e.g., [12]), or approximated by simulation (sample a graph from the model, count the number of occurrences of the given subgraph, and normalize appropriately). In contrast, the map from the subgraph densities to the stochastic block model is rather nontrivial. This paper provides a way to compute this map.

Since asymptotic rates are known for subgraph densities (see, e.g., [5] and [3]), and the mapping we construct here is differentiable in the interior of its domain, we should expect similar asymptotic rates for recovery of the model parameters. We do not currently know how the rate behaves at the boundaries. At the boundary, the method may in principle output invalid probabilities, as there is at present no explicit constraint enforcing the connection probabilities to be between 0 and 1 . Hence, as many other methods for inferring SBMs, our method is well-behaved in the interior.

Finally, we note that the method presented here can be extended in a variety of ways, for example to directed edges, weighted edges, directed weighted edges, etc.

Code

Code implementing our method is available at https://github.com/TheGravLab/TheGraphPencilMethod.

Acknowledgments and Disclosure of Funding

LMG and PO were funded by the Gatsby Charitable Foundation.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper adapts the matrix pencil method to estimate parameters in models with prescribed subgraph counts, and to simulate from them. A prime example are edge counts and the stochastic blockmodel.\n\nSTRENGTHS:\nThe idea to use the matrix pencil method for parameter estimation is interesting and the paper addresses an important problem.\n\nWEAKNESSES:\nMy main concern is that I do not think that (6) is correct. Focusing on d^3, we have \n\n\\langle d^3 \\rangle = \\sum_k  \\pi_k (\\sum_j \\pi_j B_{jk})^3 \n\nExpanding the right hand side gives \n\n \\sum_k  \\pi_k \\sum_j \\pi_j B_{jk} \\sum_r B_{rk} \\sum_s B_{sk}\n\nHowever\n\nmu (3-star) = \\sum_k \\pi_k \\sum_{j, r, s distinct} B_{jk} B_{rk} B_{sk} \n\nso the two expressions do not coincide. As equality (6) is the foundation of the proposed approach, I am not convinced of the method. \n\nMore generally the paper is not well presented. The abstract mentions exponential random graph models but they do not appear in the main paper at all. A particular version of a stochastic blockmodel is introduced which gives exchangeable edge indicators, and then it is claimed that this can be identified with the limit as the number of vertices tends to infinity. This is not clear at all; in which sense is the limit taken? Two SBMs, one on N vertices and the other one on N+1 vertices, need not be related at all. Is there a coupling construction which maintains exchangeability?\n\nQUESTIONS:\nThe homomorphism density does not seem to take care of automorphisms of the counts; is that not an issue? \n\nIs the underlying graph supposed to be finite or infinite? All matrices appear to be finite; why is the notion of a limit important? Are there any theoretical guarantees regarding consistency of the estimation? \n\nThere is a lot of literature on graph with prescribed degree distributions, see for example \n\nBritton, T., Deijfen, M., & Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. Journal of statistical physics, 124, 1377-1397\n\nand \n\nVan Koevering, K., Benson, A. and Kleinberg, J., 2021, April. Random graphs with prescribed k-core sequences: A new null model for network analysis. In Proceedings of the Web Conference 2021 (pp. 367-378).\n\nIn Figure 2 in the supplementary material, how many numbers of nodes were used? Was it a step size of 1? \n\nThe graphs for Figure 2 are very dense; how does the proposed method work for sparser graphs?\n\nLIMITATIONS:\nThere is no mention of any limitation of the method; there is no mention of model mis-specification, and there is no discussion of the variability in the estimates.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 1, "DATE": "2023-06-26"}

comments: SUMMARY:
The paper studies how to make inference sampling from a stochastic blocking model (SBM) given its corresponding subgraph densities. They first estimate the normalized degrees and the relative sizes of the latent blocks of a SBM, and then infer the connection properties of the SBM using generalized Prony's method.

STRENGTHS:
- It is nontrivial study to consider the inverse map from subgraph densities to a stochastic blocking model.
- Overall, the paper is not hard to follow. The proposed method and its properties may have some impacts on its community. As the authors claim, the proposed method can be well generalized to directed, weighted graphs.

WEAKNESSES:
Typos:
L63: entries, give -> entry, gives
L79: explain(ing)
L117: Recall(ing)
L141: allowed -> allows

Suggestions:
Eq.10 looks a bit lengthy, and can be moved to Supplement.

QUESTIONS:
How to generalize your method to capture hypergraphs with multi-edges between two vertices, signed graphs? 
Can you discuss more on how does the method capture large sparse graphs, and graph with structured sparsity?

LIMITATIONS:
I do not find distinguished limitations of the method. Nonetheless, I would think it could be incremented in the supplement if the method can be easily generalized to directed, weighted graphs.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nGiven the subgraph densities of the stochastic block model (SBM), the authors consider the problem of obtaining SBM's parameters (node and edge probabilities). The authors observe a connection between this estimation problem and estimating parameters of an exponential signal. So, they cleverly teleported the classical Prony's method for obtaining signal parameters to obtaining the SBM's parameters. This observation by the authors is the main noteworthy contribution of the paper. Finally, the authors provide analytical formulae for computing the SBM parameters via eigendecompositions of matrices constructed from the subgraph densities of the SBM.\n\nSTRENGTHS:\nThe main strength of the paper is to develop a simple and elegant solution for obtaining the parameters of SBM using classical Prony's method. The glue algebra and its connection to extracting edge probabilities seem interesting and intriguing.\n\nWEAKNESSES:\nThe are three main weaknesses in the paper: \n1. it expects a lot of background knowledge from the reader ranging from SBM and its subgraph densities to glue algebra--for e.g., Equations in (1) and (13) appear out of the blue. How are the densities related to SBM?\n2. Some of the technical details appear to be heuristic rather than rigorous proof--for e.g., The use of glyphs in matrices and the glue algebra explanation in Sections 3.3 and 3.4 appear more like a heuristic rather than a rigorous proof. Why are the identities in equations (13)-(18) true?\n3. The title, the narrative in the abstract and introduction, and the actual problem solved are disconnected. Moreover, the current story of exponential random graphical models (which is not even discussed); titbits on statistical significance (lines 14 -21); and concluding remarks on summary statistics are very broad and unrelated. As an example, the last sentence in the conclusion, “...stepping-stones over two centuries old.” is meaningless for the reader who does not have a background in Prony’s analysis or its historical context. \nThe paper develops a procedure for obtaining parameters of the SBM model using subgraph densities. I suggest the authors rewrite the introduction/abstract/conclusion directly stating this message.\n\nQUESTIONS:\n1. The authors claim to have generalized Prony's method for their network problem. This generalization needs to be clarified and concrete. The \"quoted\" generalization in Sections 3.3 and 3.4 seems like using Prony's method for extracting edges. Please justify. \n2. It is a well-known fact in probability theory that the moments generally do not specify the distribution completely. In that sense, how can the subgraph densities (analogous to moments) determine the SBM parameters?  \n3. At the high level, is unlabelling vertices (in Eq (16)) similar to the law of total probability? That is, first consider conditional probabilities (labeled homomorphism densities) and then add them up to get the final probability (unlabelled densities). \n4. Eq (34) is a random quantity obtained from a realization drawn from SBM. The authors didn't provide any statistical guarantees on this quantity. For example, how accurate is this quantity compared to the true sub-graph density? \n5. I did not completely understand the details in Section 34. In particular, the glue algebra seems hard to understand. Could the authors provide a layman's explanation for extracting the edge expectations?\n\nLIMITATIONS:\nThe authors have not discussed the limitations of their current approach. However, I think the proposed method is limited to only SBMs. I suggest the authors to comment on generalizing their method to exponential random graph models (if there is any).", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-25"}