Large Very Dense Subgraphs in a Stream of Edges

TL;DR

This paper presents methods for detecting and reconstructing large very dense subgraphs in streaming social graphs with power law degree distributions, using reservoir sampling and analysis of giant components.

Contribution

It introduces a new detection algorithm based on reservoir sampling and giant component analysis, and a model for power law graphs with dense subgraphs, including dynamic graph extensions.

Findings

Detection algorithm is almost surely correct for large dense subgraphs.

Power law graphs almost surely lack large very dense subgraphs.

Reconstruction of dense subgraphs is possible with high probability on the new model.

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.