Large very dense subgraphs in a stream of edges

Abstract

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 $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$ . Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $\Omega(\sqrt{n})$ , 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.

Computer Science and Information Technology Trends

“…component in the Reservoir with high probability. It is the basis of the approach of [12] to detect clusters on a stream of edges without storing the entire graph but only K > c. √ n. log n γ.δ edges. The detection algorithm detects γ-clusters of size larger than δ.…”

“…We consider a specific regime of graphs defined by a stream of edges e 1 , ..., e m , ... which follow a power-law degree distribution µ and use a sublinear space algorithm. A Reservoir sampling [23] with K = O( √ n. log n/γ • δ) edges can detect (γ, δ)-large dense subgraphs (clusters) with high probability [12], on the specific class of graphs taken from µ. We use a one-sided stochastic randomized algorithm A to detect the existence of a cluster:…”

Giant Components in Texts Generated by a Stream

“…component in the Reservoir with high probability. It is the basis of the approach of [12] to detect clusters on a stream of edges without storing the entire graph but only K > c. √ n. log n γ.δ edges. The detection algorithm detects γ-clusters of size larger than δ.…”

“…We consider a specific regime of graphs defined by a stream of edges e 1 , ..., e m , ... which follow a power-law degree distribution µ and use a sublinear space algorithm. A Reservoir sampling [23] with K = O( √ n. log n/γ • δ) edges can detect (γ, δ)-large dense subgraphs (clusters) with high probability [12], on the specific class of graphs taken from µ. We use a one-sided stochastic randomized algorithm A to detect the existence of a cluster:…”