Large Very Dense Subgraphs in a Stream of Edges
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We study the detection and the reconstruction of a large very dense subgraph in a social graph with n nodes and m edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when m=O(n. log n). A subgraph S is very dense if it has Ω(|S|^2) edges. We uniformly sample the edges with a Reservoir of size k=O(√(n).log n). Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size Ω(√(n)), then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.
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