Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure

TL;DR

This paper introduces algorithms for detecting small dense bipartite-like subgraphs with low bipartiteness ratio, providing approximation guarantees and a spectral characterization, useful for analyzing complex network structures.

Contribution

It presents a bicriteria approximation algorithm and a local algorithm for identifying small bipartite-like subgraphs, along with a spectral method for their characterization.

Findings

The algorithms find subgraphs with controlled volume and bipartiteness ratio.

The local algorithm operates efficiently, independent of graph size.

Spectral analysis characterizes the structure of these subgraphs.

Abstract

We study the problem of finding and characterizing subgraphs with small \textit{bipartiteness ratio}. We give a bicriteria approximation algorithm \verb|SwpDB| such that if there exists a subset $S$ of volume at most $k$ and bipartiteness ratio $\theta$, then for any $0<\epsilon<1/2$, it finds a set $S'$ of volume at most $2k^{1+\epsilon}$ and bipartiteness ratio at most $4\sqrt{\theta/\epsilon}$. By combining a truncation operation, we give a local algorithm \verb|LocDB|, which has asymptotically the same approximation guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness ratio of the output set, and runs in time $O(\epsilon^2\theta^{-2}k^{1+\epsilon}\ln^3k)$, independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the $k$th \textit{largest} eigenvalue of the Laplacian of the graph.