Abstract
Graph clustering, a fundamental technique in network science for understanding structures in complex systems, presents inherent problems. Though studied extensively in the literature, graph clustering in large systems remains particularly challenging because massive graphs incur a prohibitively large computational load. The heat kernel PageRank provides a quantitative ranking of nodes, and a local cluster can be efficiently found by performing a sweep over the heat kernel PageRank vector. But computing an exact heat kernel PageRank vector may be expensive, and approximate algorithms are often used instead. Most approximate algorithms compute the heat kernel PageRank vector on the whole graph, and thus are dependent on global structures. In this paper, we present an algorithm for approximating the heat kernel PageRank on a local subgraph. Moreover, we show that the number of computations required by the proposed algorithm is sublinear in terms of the expected size of the local cluster of interest, and that it provides a good approximation of the heat kernel PageRank, with approximation errors bounded by a probabilistic guarantee. Numerical experiments verify that the local clustering algorithm using our approximate heat kernel PageRank achieves state-of-the-art performance.
Highlights
Graph clustering, a fundamental technique in network science for understanding structures in complex systems, presents inherent problems
We introduced a local algorithm for computing approximate heat kernel PageRank vectors, and based on this algorithm we devised a novel local clustering algorithm
We restricted our computation of heat kernel PageRank vectors to within a local subgraph near the seed node, discarding simulated random walks that depart from this subgraph
Summary
Most local clustering algorithms calculate an approximate heat kernel PageRank vector on a whole graph, and the number of computations is dependent on the size of the whole graph. We calculate an approximate heat kernel PageRank vector by simulating random walks in the sampled subgraph Through this process, we can ensure that the number of computations of the heat kernel PageRank approximation is dependent only on the size of the local cluster, and not on the size of the whole graph. Input : Seed node s, graph G, expected volume ς of local cluster, constant α Output: Sampled subgraph S 1 Initiate S as s and all neighbors of s; 2 while vol(S) < ας do 3 Find all neighbors of S, namely all nodes with connections to any member of S; 4 Denote as κv the fraction of connections to S in total connections dv of a neighboring node v; 5 Denote as κ as the maximum of fractions κv among all neighbors v of S; 6 Add all neighbors with κ fractions into S
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