Reconstructing networks from simple and complex contagions

TL;DR

The paper tackles network reconstruction from binary time-series under both simple and complex contagions by a nonparametric Bayesian framework that jointly infers the adjacency matrix $A$ and contagion rules from observed states $X$. The model expresses contagion as a nonparametric infection function $c(\nu)$ with $\nu$ infected neighbors, and yields a closed-form marginal posterior $P(A|X)$ via conjugate Beta priors and an edge-flip Markov Chain Monte Carlo to sample $A$, producing edge probabilities $Q$. Key finding: reconstruction accuracy depends on contagion complexity and saturation; complex contagions outperform simple ones in dense or saturated regimes, while simple contagions perform better when saturation is low, with the basic reproduction number $R_0 = \beta \sigma(A) / \gamma$ guiding the regime. The work demonstrates that varying data streams and contagion rules can optimize reconstructability, offering practical guidance for epidemiological and information-spread inference and advancing nonparametric network reconstruction.

Abstract

Network scientists often use complex dynamic processes to describe network contagions, but tools for fitting contagion models typically assume simple dynamics. Here, we address this gap by developing a nonparametric method to reconstruct a network and dynamics from a series of node states, using a model that breaks the dichotomy between simple pairwise and complex neighborhood-based contagions. We then show that a network is more easily reconstructed when observed through the lens of complex contagions if it is dense or the dynamic saturates, and that simple contagions are better otherwise.

Figures (5)

• Figure 1: Overview of the problem. (a) A contagion spreads on a network for $T$ time steps, and we observe the resulting sequence of states $\bm{X}$. The probability that a susceptible node (white) becomes infected (red) at the next time step is a function $c(\nu)$ of the number of infected neighbors it has, e.g., $\nu=4$ for the square node highlighted in blue. (b) We compute a nonparametric Bayesian estimate of the contagion function $c(\nu)$. Here, we show an estimate of $c(\nu)$ obtained from a single short realization of the dynamics when the network is known. Error bars show the 50% highest-density posterior interval (HDPI) of $c(\nu)$. (c) We estimate the network and the contagion function $c(\nu)$ simultaneously using the marginals of the posterior distribution, Eq. \ref{['eq:joint_posterior']}. The reconstruction error goes to 0 as the amount of data $T$ goes to infinity. The shaded regions indicate the 50% HDPI, and lines show the median AUROC across $10^3$ repetitions. (d) The reconstruction quality is determined by the shape of the contagion function, here demonstrated by varying its overall infectivity $\beta$ and the level of complexity $\omega\in[0,1]$. We use the parametrization $c(\nu,\beta,\omega) = (1 - \omega) g + \omega h$ where $g(\nu, \beta) = 1 - (1-\beta)^{\nu}$ describes a simple contagion model, and $h(\nu, \beta) =\beta \bm{1}_{\nu \geq 2}$ describes a complex threshold model.
• Figure 2: The average difference in reconstruction performance between simple and complex contagions for the SIS model and the threshold model with $\tau=2$. Red denotes regions where complex contagions outperform simple contagions. Blue denotes regions where simple contagions outperform complex contagions. The first row visualizes a sample from each network model; the second row compares the performance of the network reconstruction; and the third row compares the estimation of the network density. The gray lines show the basic reproduction number $R_0=\beta\sigma(\bm{A})/\gamma\in\{1,11,21,...\}$diekmann_definition_1990, where $\sigma(\bm{A})$ is the spectral radius of $\bm{A}$, and grow when moving from the lower left corner to the upper right.
• Figure 3: Difference in reconstruction performance of the network SIS contagion process and the threshold contagion process with threshold $\tau=2$, for nodes of various coreness and dynamical parameters identical to those chosen in Fig. \ref{['fig:framework']}(c). This illustrates that for intermediate amounts of infection data, complex contagions outperform simple contagions due to their recovery of $k \geq 2$-core nodes. The shaded regions represent the 50% HDPI, and the lines are the median of $10^3$ realizations.
• Figure S1: The average difference in reconstruction performance between simple and complex contagion for the SIS model and the threshold model with $\tau=3$. Red denotes regions where complex contagions outperform simple contagions. Blue denotes regions where simple contagions outperform complex contagions. The first row visualizes a sample from each network model; the second row compares the performance of the network reconstruction; and the third row compares the estimation of the network density. The gray lines show the basic reproduction number $R_0=\beta\sigma(\bm{A})/\gamma\in\{1,11,21,...\}$, and grow when moving from the lower left corner to the upper right.
• Figure S2: The performance of network reconstruction for several common generative network models. For each network model, we plot the AUROC with respect to the infectivity and a characteristic structural parameter for three common contagion models: the network SIS model and a variant of the threshold contagion model with two different thresholds, $\tau=2$ and $\tau=3$.