MOSS-5: A Fast Method of Approximating Counts of 5-Node Graphlets in Large Graphs

Abstract

Counting 3-, 4-, and 5-node graphlets in graphs is important for graph mining applications such as discovering abnormal/evolution patterns in social and biology networks. In addition, it is recently widely used for computing similarities between graphs and graph classification applications such as protein function prediction and malware detection. However, it is challenging to compute these metrics for a large graph or a large set of graphs due to the combinatorial nature of the problem. Despite recent efforts in counting triangles (a 3-node graphlet) and 4-node graphlets, little attention has been paid to characterizing 5-node graphlets. In this paper, we develop a computationally efficient sampling method to estimate 5-node graphlet counts. We not only provide fast sampling methods and unbiased estimators of graphlet counts, but also derive simple yet exact formulas for the variances of the estimators which is of great value in practice-the variances can be used to bound the estimates' errors and determine the smallest necessary sampling budget for a desired accuracy. We conduct experiments on a variety of real-world datasets, and the results show that our method is several orders of magnitude faster than the state-of-the-art methods with the same accuracy. Index Terms-graphlet kernel, subgraph sampling, graph mining. ! 1 INTRODUCTION F Or complex networks such as online social networks (OSNs), computer networks, and biological networks, designing tools for estimating the counts (or frequencies) of 3-, 4-, and 5-node connected subgraph patterns (i.e., graphlets) shown in Fig. 1 is fundamental for detecting evolution and anomaly patterns in a large graph and computing graph similarities for graph classification, which have been widely used for a variety of graph mining and learning tasks. To explore patterns in a large graph, Milo et al. [1] defined network motifs as graphlets occurring in networks at numbers that are significantly larger than those found in random networks. Network motifs have been used for pattern recognition in gene expression profiling [2], evolution patterns in OSNs [3]-[6], and Internet traffic classification and anomaly detection [7,8]. In addition to mining a single large graph, graphlet counts also have been used to classify a large number of graphs. The graphlet kernel [9] (the dot product of two vectors of normalized graphlet counts) and RGF-distance [10]

Motif Counting Beyond Five Nodes

“…Counting graphlets is a well-studied problem in graph mining and social-networks analysis [1,3,7,8,11,14,18,20,[27][28][29]32]. Given an input graph, the problem asks to count the frequencies of all induced connected subgraphs (called graphlets), up to isomorphism, of a certain size.…”

Sampling Methods for Counting Temporal Motifs

“…Furthermore, the counts of motifs have also been used to automatically uncover fundamental design principles in complex systems [37,40,41]. The scale of graph datasets has led to a number of algorithms for estimating the frequency of motif counts [2,7,12,24,63]. For example, just the task of estimating the number of triangles in a graph has garnered a substantial amount of attention [4,11,35,52,56,58].…”

The Power of Pivoting for Exact Clique Counting

“…The problem of counting cliques (and variants such as counting maximal cliques) has received much attention both from the applied and theoretical computer science communities [3,8,9,33]. Classic techniques like color-coding [6,37] and path sampling [17,25,34] have been employed for counting cliques up to size 5.…”