Fast computation of distance-generalized cores using sampling

TL;DR

This paper introduces a sampling-based randomized algorithm for efficiently approximating distance-generalized core decompositions in large graphs, significantly reducing computation time while maintaining theoretical guarantees.

Contribution

It presents the first efficient randomized approximation algorithm for $(k, h)$-core decomposition with provable guarantees, addressing computational challenges in large graphs.

Findings

The algorithm achieves $O(rac{1}{\e^2}hm ( ext{log}^2 n - ext{log} \delta))$ runtime.

Approximate decompositions are significantly faster than exact solutions on large networks.

The method provides theoretical guarantees and complements exact algorithms in practical scenarios.

Abstract

Core decomposition is a classic technique for discovering densely connected regions in a graph with large range of applications. Formally, a $k$-core is a maximal subgraph where each vertex has at least $k$ neighbors. A natural extension of a $k$-core is a $(k, h)$-core, where each node must have at least $k$ nodes that can be reached with a path of length $h$. The downside in using $(k, h)$-core decomposition is the significant increase in the computational complexity: whereas the standard core decomposition can be done in $O(m)$ time, the generalization can require $O(n^2m)$ time, where $n$ and $m$ are the number of nodes and edges in the given graph. In this paper we propose a randomized algorithm that produces an $\epsilon$-approximation of $(k, h)$ core decomposition with a probability of $1 - \delta$ in $O(\epsilon^{-2} hm (\log^2 n - \log \delta))$ time. The approximation is based on sampling the neighborhoods of nodes, and we use Chernoff bound to prove the approximation guarantee. We also study distance-generalized dense subgraphs, show that the problem is NP-hard, provide an algorithm for discovering such graphs with approximate core decompositions, and provide theoretical guarantees for the quality of the discovered subgraphs. We demonstrate empirically that approximating the decomposition complements the exact computation: computing the approximation is significantly faster than computing the exact solution for the networks where computing the exact solution is slow