8,042 research outputs found

    Where Graph Topology Matters: The Robust Subgraph Problem

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    Robustness is a critical measure of the resilience of large networked systems, such as transportation and communication networks. Most prior works focus on the global robustness of a given graph at large, e.g., by measuring its overall vulnerability to external attacks or random failures. In this paper, we turn attention to local robustness and pose a novel problem in the lines of subgraph mining: given a large graph, how can we find its most robust local subgraph (RLS)? We define a robust subgraph as a subset of nodes with high communicability among them, and formulate the RLS-PROBLEM of finding a subgraph of given size with maximum robustness in the host graph. Our formulation is related to the recently proposed general framework for the densest subgraph problem, however differs from it substantially in that besides the number of edges in the subgraph, robustness also concerns with the placement of edges, i.e., the subgraph topology. We show that the RLS-PROBLEM is NP-hard and propose two heuristic algorithms based on top-down and bottom-up search strategies. Further, we present modifications of our algorithms to handle three practical variants of the RLS-PROBLEM. Experiments on synthetic and real-world graphs demonstrate that we find subgraphs with larger robustness than the densest subgraphs even at lower densities, suggesting that the existing approaches are not suitable for the new problem setting.Comment: 13 pages, 10 Figures, 3 Tables, to appear at SDM 2015 (9 pages only

    The Densest k-Subhypergraph Problem

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    The Densest kk-Subgraph (DkkS) problem, and its corresponding minimization problem Smallest pp-Edge Subgraph (SppES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both DkkS and SppES from graphs to hypergraphs. We consider the Densest kk-Subhypergraph problem (given a hypergraph (V,E)(V, E), find a subset WVW\subseteq V of kk vertices so as to maximize the number of hyperedges contained in WW) and define the Minimum pp-Union problem (given a hypergraph, choose pp of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n4(43)/13+ϵ)O(n0.697831+ϵ)O(n^{4(4-\sqrt{3})/13 + \epsilon}) \leq O(n^{0.697831+\epsilon})-approximation (for arbitrary constant ϵ>0\epsilon > 0) for Densest kk-Subhypergraph and an O~(n2/5)\tilde O(n^{2/5})-approximation for Minimum pp-Union. We also give an O(m)O(\sqrt{m})-approximation for Minimum pp-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.Comment: 21 page

    Fully Dynamic Algorithm for Top-kk Densest Subgraphs

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    Given a large graph, the densest-subgraph problem asks to find a subgraph with maximum average degree. When considering the top-kk version of this problem, a na\"ive solution is to iteratively find the densest subgraph and remove it in each iteration. However, such a solution is impractical due to high processing cost. The problem is further complicated when dealing with dynamic graphs, since adding or removing an edge requires re-running the algorithm. In this paper, we study the top-kk densest-subgraph problem in the sliding-window model and propose an efficient fully-dynamic algorithm. The input of our algorithm consists of an edge stream, and the goal is to find the node-disjoint subgraphs that maximize the sum of their densities. In contrast to existing state-of-the-art solutions that require iterating over the entire graph upon any update, our algorithm profits from the observation that updates only affect a limited region of the graph. Therefore, the top-kk densest subgraphs are maintained by only applying local updates. We provide a theoretical analysis of the proposed algorithm and show empirically that the algorithm often generates denser subgraphs than state-of-the-art competitors. Experiments show an improvement in efficiency of up to five orders of magnitude compared to state-of-the-art solutions.Comment: 10 pages, 8 figures, accepted at CIKM 201

    Robust Densest Subgraph Discovery

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    Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph G=(V,E)G=(V,E) with an edge-weight vector w=(we)eEw=(w_e)_{e\in E}, we aim to find SVS\subseteq V that maximizes the density, i.e., w(S)/Sw(S)/|S|, where w(S)w(S) is the sum of the weights of the edges in the subgraph induced by SS. Although the densest subgraph problem is one of the most well-studied optimization problems for dense subgraph discovery, there is an implicit strong assumption; it is assumed that the weights of all the edges are known exactly as input. In real-world applications, there are often cases where we have only uncertain information of the edge weights. In this study, we provide a framework for dense subgraph discovery under the uncertainty of edge weights. Specifically, we address such an uncertainty issue using the theory of robust optimization. First, we formulate our fundamental problem, the robust densest subgraph problem, and present a simple algorithm. We then formulate the robust densest subgraph problem with sampling oracle that models dense subgraph discovery using an edge-weight sampling oracle, and present an algorithm with a strong theoretical performance guarantee. Computational experiments using both synthetic graphs and popular real-world graphs demonstrate the effectiveness of our proposed algorithms.Comment: 10 pages; Accepted to ICDM 201
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