Finding Subgraphs in Highly Dynamic Networks
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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≥ 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≥ 3 nodes except k-clique requires an almost linear number of amortized communication rounds. We also show that k-cycle listing for k≥ 6 requires Ω(√(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.
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