Large Very Dense Subgraphs in a Stream of Edges

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 $m=O(n. łog n)$. A subgraph is very dense if its edge density is comparable to a clique. We uniformly sample the edges with a Reservoir of size $k=O(\sqrtn.łog n)$. The detection algorithm of a large very dense subgraph checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $Ømega(\sqrtn )$, 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.