Network measures that reflect the most salient properties of complex large-scale networks are in high demand in the network research community. In this paper we adapt a combinatorial measure of negative curvature (also called hyperbolicity) to parametrized finite networks, and show that a variety of biological and social networks are hyperbolic. This hyperbolicity property has strong implications on the higher-order connectivity and other topological properties of these networks. Specifically, we derive and prove bounds on the distance among shortest or approximately shortest paths in hyperbolic networks. We describe two implications of these bounds to crosstalk in biological networks, and to the existence of central, influential neighborhoods in both biological and social networks.

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Widespread usage of complex interconnected social networks such as Facebook, Twitter and LinkedIn in modern internet era has also unfortunately opened the door for privacy violation of users of such networks by malicious entities. In this article we investigate, both theoretically and empirically, privacy violation measures of large networks under active attacks that was recently introduced in (Information Sciences, 328, 403-417, 2016). Our theoretical result indicates that the network manager responsible for prevention of privacy violation must be very careful in designing the network if its topology does not contain a cycle. Our empirical results shed light on privacy violation properties of eight real social networks as well as a large number of synthetic networks generated by both the classical Erdös-Rényi model and the scale-free random networks generated by the Barábasi-Albert preferential-attachment model.

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The strong metric dimension of a graph was first introduced by Sebö and Tannier (Mathematics of Operations Research, 29(2), 383-393, 2004) as an alternative to the (weak) metric dimension of graphs previously introduced independently by Slater (Proc. 6 th Southeastern Conference on Combinatorics, Graph Theory, and Computing, 549-559, 1975) and by Harary and Melter (Ars Combinatoria, 2, 191-195, 1976), and has since been investigated in several research papers. However, the exact worst-case computational complexity of computing the strong metric dimension has remained open beyond being NP-complete. In this communication, we show that the problem of computing the strong metric dimension of a graph of n nodes admits a polynomial-time 2-approximation, admits a O * 2 0.287 n -time exact computation algorithm, admits a O 1.2738 k + n k -time exact computation algorithm if the strong metric dimension is at most k, does not admit a polynomial time (2 − ε)-approximation algorithm assuming the unique games conjecture is true, does not admit a polynomial time (10 √ 5 − 21 − ε)-approximation algorithm assuming P NP, does not admit a O * 2 o(n) -time exact computation algorithm assuming the exponential time hypothesis is true, and does not admit a O * n o(k) -time exact computation algorithm if the strong metric dimension is at most k assuming the exponential time hypothesis is true.