Finding Subgraphs in Highly Dynamic Networks

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.