Network reliability: The effect of local network structure on diffusive processes

Abstract

This paper re-introduces the network reliability polynomial – introduced by Moore and Shannon in 1956 – for studying the effect of network structure on the spread of diseases. We exhibit a representation of the polynomial that is well-suited for estimation by distributed simulation. We describe a collection of graphs derived from Erdős-Rényi and scale-free-like random graphs in which we have manipulated assortativity-by-degree and the number of triangles. We evaluate the network reliability for all these graphs under a reliability rule that is related to the expected size of a connected component. Through these extensive simulations, we show that for positively or neutrally assortative graphs, swapping edges to increase the number of triangles does not increase the network reliability. Also, positively assortative graphs are more reliable than neutral or disassortative graphs with the same number of edges. Moreover, we show the combined effect of both assortativity-by-degree and the presence of triangles on the critical point and the size of the smallest subgraph that is reliable.

Using the network reliability polynomial to characterize and design networks

“…We can also take advantage of constraints including monotonicity and the finite set of possible values for P k . We find that the entire set of coefficients can be estimated to a useful uniform, absolute precision on a laptop for graphs with millions of edges [17]. Furthermore, the estimation procedure is an embarrassingly parallel computation, since the sampling and rule evaluation can be performed independently and asynchronously, with very little inter-processor communication.…”

“…For experiments in a previous paper [17], we generated a large set of graphs with 992 edges, 341 vertices, and carefully controlled degree distributions, assortativity-by-degree, and number of triangles. Each graph was created by swapping edges in one of two original graphs, a connected Erdős-Rényi graph (“GNM”) drawn by choosing edges independently at random and rejecting any graphs that were not connected, and a hand-designed connected, scale-free-like (“SFL”) graph.…”

“…Figure 2 shows constrained edge-swapping operations that adjust the assortativity-by-degree and number of triangles while maintaining the degree distribution unchanged. See [17] for more details on the ranges of assortativity and triangles in these graphs.…”

“…In large part, this is due to well-established complexity results for both its evaluation and approximation. However, as the authors have recently observed, estimation of the polynomial to within practically useful confidence intervals is easy, even without taking into consideration any properties of the graph [4, 17]. Estimation is also embarrassingly parallel, and we have extended this previous work by developing a distributed implementation, available to researchers as a web service, that scales to graphs with tens of millions of edges, limited only by the memory available to each computing element.…”

Speeding up of genetic algorithm for network topology optimization with use of cumulative updating of network reliability

“…Apart this index several other indicies are studied by various authors: average pairwise connectivity [1], expectation of connected nodes [2], and others. The problem of computing this characteristic is known to be NP-hard [3], just like the problems of other network reliability measures computing.…”