Waddling Random Walk: Fast and Accurate Mining of Motif Statistics in Large Graphs

Abstract

Algorithms for mining very large graphs, such as those representing online social networks, to discover the relative frequency of small subgraphs within them are of high interest to sociologists, computer scientists and marketeers alike. However, the computation of these network motif statistics via naive enumeration is infeasible for either its prohibitive computational costs or access restrictions on the full graph data. Methods to estimate the motif statistics based on random walks by sampling only a small fraction of the subgraphs in the large graph address both of these challenges. In this paper, we present a new algorithm, called the Waddling Random Walk (WRW), which estimates the concentration of motifs of any size. It derives its name from the fact that it sways a little to the left and to the right, thus also sampling nodes not directly on the path of the random walk. The WRW algorithm achieves its computational efficiency by not trying to enumerate subgraphs around the random walk but instead using a randomized protocol to sample subgraphs in the neighborhood of the nodes visited by the walk. In addition, WRW achieves significantly higher accuracy (measured by the closeness of its estimate to the correct value) and higher precision (measured by the low variance in its estimations) than the current state-of-the-art algorithms for mining subgraph statistics. We illustrate these advantages in speed, accuracy and precision using simulations on well-known and widely used graph datasets representing real networks. [13]. For example, the clustering coefficient (the number of triangles in relation to the number of wedges) has long served as an important metric in sociometry and social network analysis [14], [15]. In fact, the relative frequencies of network motifs are indicative of important properties of graphs such as modularity, the tendency of nodes in a network to form tightly interconnected communities, and even play a role in the organization and evolution of networks [6]. Knowledge of these motif statistics combined with homophily, the tendency of similar nodes to connect to one another, add to the ability of businesses such as Facebook to better mine their graphs and monetize their social platforms through targeted advertisements [16].Computing motif statistics, however, is rendered difficult by two challenges: one computational and the other having to do with restricted access to the full graph data. The computational challenge arises because accurate computation of the relative frequencies of different motifs requires enumeration of all the induced subgraphs and checking each for isomorphism to known motif types. The time complexity of enumerating all induced subgraphs of size k in a graph with V vertices and E edges is exponential in k with an upper bound of O(E k ) and a lower bound of O(V c k−1 ) [17]. Even when k is as small as 4, in a graph with only millions of edges, the number of motifs can reach hundreds of billions. The other problem is one of restricted access because the data on...

Estimating Graphlet Statistics via Lifting

“…The variance of sampling algorithms can be decomposed into two parts, an independent sample variance component and a between sample covariance component. As we have seen the independent variance component is based on the properties of π resulting from the procedure (see (11)). The covariance component will be low if the samples are not highly dependent on the past, namely that the Markov chain is well mixed.…”

Mining Frequent Patterns in Evolving Graphs

“…Approximate counting of 3-, 4-, 5-vertex graphlets in static graphs has received much more attention than exact counting, which has an exponential cost. Most of the literature on approximate counting of graphlets uses randomwalks to collect a uniform sample of graphlets on static graphs [3,8,16,37]. Alternatively, Bressan et al [6] proposed a color coding based scheme for estimating k-vertex graphlet statistics.…”

Multi-MotifGAN (MMGAN): Motif-Targeted Graph Generation And Prediction

“…Relation to Existing Work: MMGAN uses multiple techniques for learning on graphs and combines them into a motif-aware model. Random walks on graphs are widely used to learn the local and global topology of a graph [17][18][19], while biased random walks are used to characterize higher-order network structures like hyperedges and network motifs [8,[20][21][22][23][24]. Generative Adversarial Networks (GAN) are highly effective at learning implicit features of a data set and using these to generate realistic data samples.…”