| Summary: | Abstract We derive complexity estimates for two classes of deterministic networks: the Boolean networks S(B n, m ), which compute the Boolean vector-functions B n, m , and the classes of graphs G(VPm,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(VPm,l,E) $G(V_{P_{m,\,l}}, E)$ are labeled by the trinomial (m 2±l,m)-partitions from the set of partitions P m, 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 (m 2±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(VPm,l,E) $G(V_{P_{m,\,l}}, E)$.
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