Structure Entropy and Resistor Graphs
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We propose the notion of resistance of a graph as an accompanying notion of the structure entropy to measure the force of the graph to resist cascading failure of strategic virus attacks. We show that for any connected network G, the resistance of G is R(G)=H^1(G)-H^2(G), where H^1(G) and H^2(G) are the one- and two-dimensional structure entropy of G, respectively. According to this, we define the notion of security index of a graph to be the normalized resistance, that is, θ (G)=R(G)/H^1(H). We say that a connected graph is an (n,θ)- resistor graph, if G has n vertices and has security index θ (G)≥θ. We show that trees and grid graphs are (n,θ)-resistor graphs for large constant θ, that the graphs with bounded degree d and n vertices, are (n,2/d-o(1))-resistor graphs, and that for a graph G generated by the security model LLPZ2015, LP2016, with high probability, G is an (n,θ)-resistor graph, for a constant θ arbitrarily close to 1, provided that n is sufficiently large. To the opposite side, we show that expander graphs are not good resistor graphs, in the sense that, there is a global constant θ_0<1 such that expander graphs cannot be (n,θ)-resistor graph for any θ≥θ_0. In particular, for the complete graph G, the resistance of G is a constant O(1), and hence the security index of G is θ (G)=o(1). Finally, we show that for any simple and connected graph G, if G is an (n, 1-o(1))-resistor graph, then there is a large k such that the k-th largest eigenvalue of the Laplacian of G is o(1), giving rise to an algebraic characterization for the graphs that are secure against intentional virus attack.
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