About complexity of complex networks

Abstract

Abstract We derive complexity estimates for two classes of deterministic networks: the Boolean networks S(Bn, m), which compute the Boolean vector-functions Bn, m, and the classes of graphs $G(V_{P_{m,\,l}}, E)$ G ( V P m , l , E ) , with overlapping communities and high density. The latter objects are well suited for the synthesis of resilience networks. For the Boolean vector-functions, we propose a synthesis of networks on a NOT, AND, and OR logical basis and unreliable channels such that the computation of any Boolean vector-function is carried out with polynomial information cost.All vertexes of the graphs $G(V_{P_{m,\,l}}, E)$ G ( V P m , l , E ) are labeled by the trinomial (m2±l,m)-partitions from the set of partitions Pm, l. It turns out that such labeling makes it possible to create networks of optimal algorithmic complexity with highly predictable parameters. Numerical simulations of simple graphs for trinomial (m2±l,m)-partition families (m=3,4,…,9) allow for the exact estimation of all commonly known topological parameters for the graphs. In addition, a new topological parameter—overlapping index—is proposed. The estimation of this index offers an explanation for the maximal density value for the clique graphs $G(V_{P_{m,\,l}}, E)$ G ( V P m , l , E ) .