Identifying Relevant Subgraphs in Large Networks

Abstract

Structural relationships between objects are used to model as graphs in many applications. In this paper, we study the problem of identifying relevant subgraphs in large networks. Relevant subgraphs in large networks contain network elements which are maintained by network administrators. We formalize the problem and propose a framework consisting of two major phases. The relevance scores of all vertex pairs are computed in the offline phase, while relevant subgraphs are identified in the online phase. We analyze the relevance score measure carefully and design an efficient algorithm for relevant subgraph identification by repeatedly expanding candidate subgraphs and merging overlapping ones. Our experiments based on real data sets show that our relevant subgraphs are of high quality and can be found efficiently, which are useful for network administrators during network operation and maintenance.

Appendices

Appendix

A Relationship Between Expected f-Distance and Random Walk with Restart

The vertex relevance matrix using the expected f-distance is not much different from one using random walk with restart. The proof is presented below. Based on the iterative form of the definition of random walk with restart, the vertex relevance score matrix ${\varPi '^l}$ of graph $G_i$ can be expressed as following.

$$\begin{aligned} \begin{aligned} {\varPi '}^l&= (1-c) {\varPi '}^{l-1} P + cI ,\\ \end{aligned} \end{aligned}$$

(6)

where c is the restart probability, P is the transition matrix of G and I is identity matrix. Then we have

$$\begin{aligned} \begin{aligned} {\varPi '}^l&= (1-c) {\varPi '}^{l-1} P + cI \\&= (1-c) ( (1-c) {\varPi '}^{l-2} P + cI)P + cI \\&= (1-c)^l P^l + c \sum _{\gamma =1}^{l-1}(1-c)^\gamma P^\gamma + cI \\&= c(1-c)^l P^l + c \sum _{\gamma =1}^{l-1}(1-c)^\gamma P^\gamma + (1-c)^{l+1}P^l + cI \\&= \sum _{\gamma =1}^{l}c(1-c)^\gamma P^\gamma + (1-c)^{l+1}P^l + cI \\&= \varPi ^l + (1-c)^{l+1}P^l + cI .\\ \end{aligned} \end{aligned}$$

(7)

The last line of Eq. (7) contains three items. The first item is the vertex relevance matrix ${\varPi ^l_i}$ using the expected f-distance. The third item cI affects only the diagonal entries of the vertex relevance matrix, which is ignored since we do not consider the vertex self-relevance. Then, the difference using random walk with restart and the expected f-distance results in the second item $(1-c)^{l+1}P^l_i$. When l goes to infinity, the vertex relevance matrices using expected f-Distance and random walk with restart are the same except the diagonal entries. Even when l is small, the corresponding entries of two matrices do not differ so much since $(1-c)^{l+1}P^l$ is very small comparing with $\varPi ^l = \sum _{\gamma =1}^{l}c(1-c)^\gamma P^\gamma$.