Based on the definition of local spectral subspace, we propose a novel approach called LOSP for local overlapping community detection. Using the power method for a few steps, LOSP finds an approximate invariant subspace, which depicts the embedding of the local neighborhood structure around the seeds of interest. LOSP then identifies the local community expanded from the given seeds by seeking a sparse indicator vector in the subspace where the seeds are in its support. We provide a systematic investigation on LOSP, and thoroughly evaluate it on large real world networks across multiple domains. With the prior information of very few seed members, LOSP can detect the remaining members of a target community with high accuracy. Experiments demonstrate that LOSP outperforms the Heat Kernel and PageRank diffusions. Using LOSP as a subroutine, we further address the problem of multiple membership identification, which aims to find all the communities a single vertex belongs to. High F1 scores are achieved in detecting multiple local communities with respect to arbitrary single seed for various large real world networks.
A local graph partitioning algorithm finds a cut near a specified starting vertex, with a running time that depends largely on the size of the small side of the cut, rather than the size of the input graph. In this paper, we present a local partitioning algorithm using a variation of PageRank with a specified starting distribution. We derive a mixing result for PageRank vectors similar to that for random walks, and show that the ordering of the vertices produced by a PageRank vector reveals a cut with small conductance. In particular, we show that for any set C with conductance and volume k, a PageRank vector with a certain starting distribution can be used to produce a set with conductance O ( k ). We present an improved algorithm for computing approximate PageRank vectors, which allows us to find such a set in time proportional to its size. In particular, we can find a cut with conductance at most o , whose small side has volume at least 2b, in time O ( 2^b ^2 m o^2 ) where m is the number of edges in the graph. By combining small sets found by this local partitioning algorithm, we obtain a cut with conductance o and approximately optimal balance in time O ( m ^4 m o^2 ). The heat kernel is a type of graph diffusion that, like the much-used personalized PageRank diffusion, is useful in identifying a community nearby a starting seed node. We present the first deterministic, local algorithm to compute this diffusion and use that algorithm to study the communities that it produces. Our algorithm is formally a relaxation method for solving a linear system to estimate the matrix exponential in a degree-weighted norm. We prove that this algorithm stays localized in a large graph and has a worst-case constant runtime that depends only on the parameters of the diffusion, not the size of the graph. On large graphs, our experiments indicate that the communities produced by this method have better conductance than those produced by PageRank, although they take slightly longer to compute. On a real-world community identification task, the heat kernel communities perform better than those from the PageRank diffusion. Expanding a seed set into a larger community is a common procedure in link-based analysis. We show how to adapt recent results from theoretical computer science to expand a seed set into a community with small conductance and a strong relationship to the seed, while examining only a small neighborhood of the entire graph. We extend existing results to give theoretical guarantees that apply to a variety of seed sets from specified communities. We also describe simple and flexible heuristics for applying these methods in practice, and present early experiments showing that these methods compare favorably with existing approaches. We present an efficient algorithm for solving linear systems with a boundary condition by computing the Green’s function of a connected induced subgraph S of a graph. Different from previous linear solvers, we introduce the method of using the Dirichlet heat kernel pagerank of the induced graph to approximate the solution to diagonally dominant linear systems satisfying given boundary conditions. Our algorithm runs in time O(1), with the assumption that a unit time allows a step in a random walk or a sampling of a specified distribution, where the big-O term depends on the error term and the boundary condition.   The concept of pagerank was first started as a way for determining the ranking of Web pages by Web search engines. Based on relations in interconnected networks, pagerank has become a major tool for addressing fundamental problems arising in general graphs, especially for large information networks with hundreds of thousands of nodes. A notable notion of pagerank, introduced by Brin and Page and denoted by PageRank, is based on random walks as a geometric sum. In this paper, we consider a notion of pagerank that is based on the (discrete) heat kernel and can be expressed as an exponential sum of random walks. The heat kernel satisfies the heat equation and can be used to analyze many useful properties of random walks in a graph. A local Cheeger inequality is established, which implies that, by focusing on cuts determined by linear orderings of vertices using the heat kernel pageranks, the resulting partition is within a quadratic factor of the optimum. This is true, even if we restrict the volume of the small part separated by the cut to be close to some specified target value. This leads to a graph partitioning algorithm for which the running time is proportional to the size of the targeted volume (instead of the size of the whole graph).   