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2018 IEEE International Symposium on Information Theory (ISIT) Comparing Massive Networks via Moment Matrices Hayoung Choi, Yifei Shen, and Yuanming Shi School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China Email: {hchoi,shenyf,shiym}@shanghaitech.edu.cn Abstract—In this paper, a novel similarity measure for com- similarity between massive networks based on the spectral paring massive complex networks based on moment matrices is distribution of the corresponding adjacency matrix in the given proposed. We consider the corresponding adjacency matrix of state. Specifically, we first compute the corresponding positive a graph as a real random variable of the algebraic probability space with a state. It is shown that the spectral distribution of definite moment matrix whose entries consist of the first few the matrix can be expressed as a unique discrete probability number of moments of the spectrum distribution. Our pro- measure. Then we use the geodesic distance between positive posed distance between networks is obtained by the geodesic definite moment matrices for comparing massive networks. It is distance of the moment matrices (called GDMM). We show proved that this distance is graph invariant and sub-structure that this distance is graph invariant and sub-structure invariant. invariant. Numerical simulations demonstrate that the proposed method outperforms state-of-art method in collaboration network GDMM is scalable to extremely massive networks and highly classification and its computational cost is extremely cheap. parallelable. Numerical Simulations demonstrate that GDMM not only has better performance over the competing methods, I. INTRODUCTION but also outperforms the state-of-art method in collaboration Network is one of the most common representations of network classification. complex data and plays an indispensable role in diversified research areas. Over the past several decades, enormous II. BACKGROUND AND PRELIMINARY breakthroughs have been made while many fundamental prob- Let M(n; C) (resp. M(n; R)) be the set of n × n complex lems about networks are remaining to be solved. Comparing (resp. real) matrices. Denote N the set of nonnegative integer networks is one of the most important problems with a numbers. In general, we will follow the notations and defini- very long history [1]. In practice, the similarity measure of tions in [9]. networks is widely applied in social science, biology, and chemistry. For instance, the similarity measure of networks A. Graph can be used to classify ego networks [2], distinguish between Let V be the set of vertices, and let fx; yg denote the edge neurological disorders [3], and discover molecules with similar connecting two points x; y 2 V .A undirected graph is a pair properties [4]. In order to measure similarity between networks G = (V; E) where the set V of vertices is finite, and the set effectively, several definitions of distance and similarity have E of edges is a subset of the set ffx; yg : x; y 2 V g. We been proposed. Graph edit distances are the minimum cost say that two vertices x; y 2 V are adjacent if fx; yg 2 E, for transforming one network to another by the distortion of denoted by x ∼ y.A degree of a vertex x 2 V is defined by nodes and edges [5]. These definitions only pay attention to the deg(x) = jfy 2 V : y ∼ xgj. In this paper we consider a finite similarities of the nodes and edges but lacks the information of undirected graph. Two graphs G = (V; E) and G0 = (V 0;E0) topological structures of the networks. For the purpose of ad- are isomorphic if there is a bijection f : V −! V 0, such dressing this limitation, frequency subgraph mining algorithms that any two vertices u; v 2 V are adjancent in G if and [6], graph kernels [7] and methods based on moments [8] only if f(u) and f(v) are adjacent in V 0. For m 2 N, a have been proposed. However, these methods are not scalable finite sequence of vertices x0; x1; : : : ; xm 2 V is called a to massive networks containing millions of edges, which are walk of length m if x0 ∼ x1 ∼ · · · ∼ xm, where some common in today’s applications. As a result, effective and of x0; x1; : : : ; xm may coincide. A graph G = (V; E) is scalable methods for massive network comparison is urgently connected if every pair of distinct vertices x; y 2 V (x 6= y) needed. are connected by a walk. If there is a walk connecting two In this paper, we propose a novel similarity measure for distinct vertices x; y 2 V , the graph distance between x comparing massive complex networks. We consider the adja- and y is the minimum length of a walk connecting x and cency matrix of the network as a probability random variable y, denoted by @(x; y). If there is no such a walk, we define of the algebraic probability space with the proposed state. @(x; y) = 1. For x = y we define @(x; x) = 0. For graphs We show that the spectral distribution of the matrix can be Gi = (Vi;Ei), i = 1; 2 with V1 \ V2 = ;, the direct sum of expressed as a unique discrete probability measure. Then G1 and G2 is defined as G = (V1 [ V2;E1 [ E2), denoted we propose an efficient and scalable method to measure the by G = G1 t G2. From now on, without loss of generality we assume that V = f1; 2; : : : ; ng. Any graph G = (V; E) is This work was partly supported by the National Nature Science Foundation n×n of China under Grant No. 61601290, and the Shanghai Sailing Program under represented by the adjacency matrix A 2 f0; 1g where Grant No. 16YF1407700. Ai;j = 1 if and only if fi; jg 2 E. Every permutation 978-1-5386-4780-6/18/$31.00©2018 IEEE 656 2018 IEEE International Symposium on Information Theory (ISIT) π : f1; 2; : : : ; ng −! f1; 2; : : : ; ng is associated with a Theorem 2. Let (A ;') be an algebraic probability space. For corresponding permutation matrix P . Given an adjacency a real random variable a = a∗ 2 A there exists a probability matrix A, graphs corresponding to adjacency matrix A and measure µ 2 B(R) such that P AP > are isomorphic, i.e., they represent the same graph Z k k structure. A property of graph is called graph invariant if '(a ) = x dµ(x) for all k 2 N0: the property does not change under the transformation of R reordering of vertices. Note that the adjacency matrix of a Such µ is called the spectral distribution of a in ' [9]. graph includes the full information about a graph. For x; y 2 V It is noted that M(n; C) with the usual operators is a unital ∗-algebra. The following is a typical example of a state. For and m 2 N let Wm(x; y) denote the number of m-step walks A = [A ] 2 M(n; ) the normalized trace connecting x and y. Remark that W0(x; y) = 0 if x 6= y and ij C , the is defined m by W0(x; y) = 1 if x = y. It is noted that (A )ij = Wm(i; j) n 1 1 X for all i; j 2 V and m 2 N. ' (A) = tr(A) = A : (II.1) tr n n ii Let A (G) be the unital algebra generated by A (the algebra i=1 generated by A and the identity matrix I = A0, i.e., A (G) = Note that the normalized trace is a state on M(n; ), implying ff(A): f 2 [x]g, where [x] is the set of all polynomials C C C (M(n; );' ) becomes an algebraic probability space. with complex coefficients. Moreover, the involution is defined C tr m ∗ m by (cA ) =c ¯A for c 2 C. Then A (G) becomes a unital III. MAIN RESULTS ∗-algebra. We call A (G) adjacency algebra of G. n Denote the vector of all ones by e 2 C . Define a function Proposition 1. Let s(G) denote the number of distinct eigen- 'e : M(n; C) −! C by value of G For a connected finite graph G we have 1 ' (A) = he; Aei (III.2) e n s(G) = dimA (G) ≥ diam(G) + 1: for all A 2 M(n; C). Then it is clear that 'e is a vector state on M(n; ), implying that (M(n; );' ) is an algebraic B. Quantum Probability C C e probability space. Let A be a unital ∗-algebra over the complex number field Let G = (V; E) be a graph and let ' be a state given C with the multiplication unit 1A . A function ' : A −! C on the adjacency algebra A (G). Since the adjacency matrix is called a state on A if A 2 M(n; C) of G can be regarded as a real random variable of the algebraic probability space (M(n; );'e), by Theorem ∗ C (i) ' is linear; (ii) '(a a) ≥ 0; (iii) '(1A ) = 1. 2 it follows that there exists the spectral distribution of A in The pair (A ;') is called an algebraic probability space. the state 'e such that Note that a state ' on a unital ∗-algebra is a ∗-map, i.e., Z A ' (Ak) = xkdµ(x) for all k 2 : (III.3) '(a∗) = '(a). Let ( ;') be an algebraic probability space. e N A R An element a 2 A is called an algebraic random variable Note that or a random variable for short. A random variable a 2 A ∗ k 1 k k is called real if a = a . For a random variable a 2 A the 'e(A ) = he; A ei = [A e]; n E quantity of the form: 1 Pn where E(v) = i=1 vi is the average of entries of vector "1 "2 "m n '(a a ··· a );"1;"2;:::;"m 2 f1; ∗}; k k v.
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