Exact Network Reconstruction from Complete SIS Nodal State Infection Information Seems Infeasible

Abstract

The SIS dynamics of the spread of a virus crucially depend on both the network topology and the spreading parameters. Since neither the topology nor the spreading parameters are known for the majority of applications, they have to be inferred from observations of the viral spread. We propose an inference method for both topology and spreading parameters based on a maximum-a-posteriori estimation approach for the sampled-time Markov chain of an SIS process. The resulting estimation problem, given by a mixed-integer optimisation problem, results in exponential computational time if a brute-force approach is employed. By introducing an efficient and accurate, polynomial-time heuristic, the topology of the network can almost always be exactly reconstructed. Notwithstanding, reconstructing the network with a reasonably high accuracy requires a subexponentially increasing number of observations and an exponentially increasing computation time with respect to the number of nodes N. Such long observation periods are hardly realistic, which justifies the claim in the title.

Backbone reconstruction in temporal networks from epidemic data

“…In the technical literature, the problem of link reconstruction and prediction has been studied from a variety of angles, mostly relying on the direct observations of contacts [32][33][34]. Dealing with observations of nodal dynamics, several methods have been proposed to reconstruct patterns of interactions [35], including the use of similarity [36], information theory [37], belief propagation [38], likelihood maximization [39,40], compressed sensing [41,42], optimization [43], nonparametric Bayesian methods [44], and data-driven approaches [45,46]. However, these strategies are of limited use when strong and weak ties coexist, thereby presently challenging the inference of backbone networks from observations of node dynamics.…”

Inferring network properties based on the epidemic prevalence

“…Much work on the inverse problem exists (Mateos et al 2019;Dong et al 2019). Most of the papers focus on reconstructing the underlying graphs by measuring the timedependent dynamical state of each node (Shandilya and Timme 2011;Berry et al 2012;Timme and Casadiego 2014;Nitzan et al 2017;Prasse and Van Mieghem 2018;Netrapalli and Sanghavi 2012;Myers and Leskovec 2010;Sefer and Kingsford 2015;Gomez Rodriguez et al 2010). With the complete dynamics of each node, the network may be approximately reconstructed by different heuristic algorithms, e.g., the Bayesian methods (Friston 2002;Pajevic and Plenz 2009), the conflict-based method (Ma et al 2015), statistical inference based method (Ma et al 2018) and the compressed sensing or lasso methods (Shen et al 2014).…”

Maximum-Likelihood Network Reconstruction for SIS Processes is NP-Hard

“…We consider the network reconstruction of the sampled-time susceptible-infected-susceptible (SIS) process in a maximum-likelihood (ML) sense as introduced in [1]. We assume that the infection rate β and the curing rate δ are known and that no self-infections occur; hence, the self-infection rate is ǫ = 0.…”

“…The network reconstruction problem for sampled-time SIS process is stated in the ML sense [1]. In contrast to the true adjacency matrix A, which generated the viral states x[k], the optimisation variable in the ML estimation problem is denoted as Â.…”

“…To stress the dependency of the ML estimate A ML on a given viral state sequence x [1], ..., x[n], we may also denote the ML estimate by A ML (x [1], ..., x [n]). The SIS network reconstruction (1) gives rise to two fundamental problems:…”