We demonstrate a general method to analyze the sensitivity of attack rate in a network model of infectious disease epidemiology to the structure of the network. We use Moore and Shannon’s “network reliability” statistic to measure the epidemic potential of a network. A number of networks are generated using exponential random graph models based on the properties of the contact network structure of one of the Add Health surveys. The expected number of infections on the original Add Health network is significantly different from that on any of the models derived from it. Because individual-level transmissibility and network structure are not separately identifiable parameters given population-level attack rate data it is possible to re-calibrate the transmissibility to fix this difference. However, the temporal behavior of the outbreak remains significantly different. Hence any estimates of the effectiveness of time dependent interventions on one network are unlikely to generalize to the other. Moreover, we show that in one case even a small perturbation to the network spoils the re-calibration. Unfortunately, the set of sufficient statistics for specifying a contact network model is not yet known. Until it is, estimates of the outcome of a dynamical process on a particular network obtained from simulations on a different network are not reliable.

This paper uses the reliability polynomial, introduced by Moore and Shannon in 1956, to analyze the effect of network structure on diffusive dynamics such as the spread of infectious disease. We exhibit a representation for the reliability polynomial in terms of what we call structural motifs that is well suited for reasoning about the effect of a network's structural properties on diffusion across the network. We illustrate by deriving several general results relating graph structure to dynamical phenomena.

We introduce a new method to characterize the network reliability polynomial of graphs – and hence the graph itself – using only a few parameters. Exact evaluation of the reliability polynomial is almost impossible for large graphs; estimating the polynomial’s coefficients is feasible but requires significant computation. Furthermore, the information required to specify the polynomial scales with the size of the graph. Thus, we aim to develop a way to characterize the polynomial well with as few parameters as possible. We show that the error function provides a two-parameter family of functions that can closely reproduce reliability polynomials of both random graphs and synthetic social networks. These parameter values can be used as statistics for characterizing the structure of entire networks in ways that are sensitive to dynamical properties of interest.

We consider methods for solving certain network characterization and design problems that arise in network epidemiology. We argue that the network reliability polynomial introduced by Moore and Shannon is a useful framework in which to reason about these problems. Specifically, we show how efficient estimation of the polynomial permits characterizing and distinguishing very large networks in ways that are are dynamically relevant. Furthermore, a generalization of flows and cuts to structures that determine the reliability suggests a new measure of edge or vertex centrality that we call criticality. We describe how criticality is related to the more common notion of betweenness and illustrate its application to targeting interventions to control outbreaks of infectious disease. Although our applications are to infectious disease outbreaks, the methods we develop are applicable to many other diffusive dynamical systems over complex networks.

We demonstrate how to compare networks using a novel family of measures of an edge's contribution to particular aspects of diffusion dynamics on a network. These centrality measures are based on membership in structural motifs that determine the network's reliability, as introduced by Moore and Shannon in 1956. Moore:56 The measures are analogous to between ness, but are more directly related to specific dynamical phenomena. They can be estimated efficiently in a distributed, scalable way with readily controlled precision. We illustrate how these measures can be used to understand which differences between two networks over the same set of vertices are most important, and exactly how those differences affect the dynamics. We focus here on phenomena of interest in network epidemiology, but the methods can easily be generalized to other diffusive processes on a network.

This paper reintroduces the network reliability polynomial, introduced by Moore and Shannon [Moore and Shannon, J. Franklin Inst. 262, 191 (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 of 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.

Strongly interacting atoms trapped in optical lattices can be used to explore phase diagrams of Hubbard models. Spatial inhomogeneity due to trapping typically obscures distinguishing observables. We propose that measures using boson double occupancy avoid trapping effects to reveal two key correlation functions. We define a boson core compressibility and core superfluid stiffness in terms of double occupancy. We use quantum Monte Carlo on the Bose-Hubbard model to empirically show that these quantities intrinsically eliminate edge effects to reveal correlations near the trap center. The boson core compressibility offers a generally applicable tool that can be used to experimentally map out phase transitions between compressible and incompressible states.