Finding Subgraphs in Highly Dynamic Networks

TL;DR

This paper presents deterministic algorithms for finding specific subgraphs like cliques and cycles in highly dynamic networks with limited message sizes, achieving constant amortized rounds, and establishes complexity bounds for other subgraph problems.

Contribution

It introduces the robust 2-hop and 3-hop neighborhoods for efficient subgraph detection in dynamic networks, and provides tight bounds for various subgraph listing problems.

Findings

Deterministic $O(1)$-amortized algorithms for $k$-clique, 4-cycle, and 5-cycle listing.

Introduction of robust neighborhoods that can be maintained efficiently in dynamic networks.

Impossibility results showing near-linear or higher complexity for other subgraph listing problems.

Abstract

In this paper we consider the fundamental problem of finding subgraphs in highly dynamic distributed networks - networks which allow an arbitrary number of links to be inserted / deleted per round. We show that the problems of $k$-clique membership listing (for any $k\geq 3$), 4-cycle listing and 5-cycle listing can be deterministically solved in $O(1)$-amortized round complexity, even with limited logarithmic-sized messages. To achieve $k$-clique membership listing we introduce a very useful combinatorial structure which we name the robust $2$-hop neighborhood. This is a subset of the 2-hop neighborhood of a node, and we prove that it can be maintained in highly dynamic networks in $O(1)$-amortized rounds. We also show that maintaining the actual 2-hop neighborhood of a node requires near linear amortized time, showing the necessity of our definition. For $4$-cycle and $5$-cycle listing, we need edges within hop distance 3, for which we similarly define the robust $3$-hop neighborhood and prove it can be maintained in highly dynamic networks in $O(1)$-amortized rounds. We complement the above with several impossibility results. We show that membership listing of any other graph on $k\geq 3$ nodes except $k$-clique requires an almost linear number of amortized communication rounds. We also show that $k$-cycle listing for $k\geq 6$ requires $\Omega(\sqrt{n} / \log n)$ amortized rounds. This, combined with our upper bounds, paints a detailed picture of the complexity landscape for ultra fast graph finding algorithms in this highly dynamic environment.