Propagation on networks: an exact alternative perspective

An "opinion reproduction number" for infodemics in a bounded-confidence content-spreading process on networks

We study the spreading dynamics of content on networks. To do this, we use a model in which content spreads through a bounded-confidence mechanism. In a bounded-confidence model (BCM) of opinion dynamics, the agents of a network have continuous-valued opinions, which they adjust when they interact with agents whose opinions are sufficiently close to theirs. Our content-spreading model, which one can also interpret as an independent-cascade model, introduces a twist into BCMs by using bounded confidence for the content spread itself. We define an analog of the basic reproduction number from disease dynamics that we call an opinion reproduction number. A critical value of the opinion reproduction number indicates whether or not there is an "infodemic" (i.e., a large content-spreading cascade) of content that reflects a particular opinion. By determining this critical value, one can determine whether or not an opinion dies off or propagates widely as a cascade in a population of agents. Using configuration-model networks, we quantify the size and shape of content dissemination by calculating a variety of summary statistics, and we illustrate how network structure and spreading-model parameters affect these statistics. We find that content spreads most widely when agents have a large expected mean degree or a large receptiveness to content. When the spreading process slightly exceeds the infodemic threshold, there can be longer dissemination trees than for larger expected mean degrees or receptiveness (which both promote content sharing and hence help push content spread past the infodemic threshold), even though the total number of content shares is smaller.

Propagation of interacting diseases on multilayer networks

The study on the dynamics of interacting diseases has attracted considerable attention in recent years. This paper proposes a model for two interacting epidemics spreading concurrently on a two-layer network, where both the epidemic thresholds and the dynamics of disease outbreaks are investigated. The analytical expression of the epidemic threshold shows strong coupling between the two interacting epidemics. Moreover, two metrics, the maximum prevalence and the highest spreading speed, are proposed to describe the outbreak process. Theoretical analysis together with extensive simulations illustrate the functions of various factors, including the network topological parameters, percentage of overlapped network links, vulnerable individuals, and the reciprocity of the two diseases. It is found that the seemingly important factor, i.e., the percentage of overlapped links, possesses no effect on the propagation, while the frequently overlooked factor, i.e., the percentage of vulnerable individuals, has significant effects. For the interaction of the two diseases, the recovery state of one disease is more influential than the other in both the mutually enhanced and the mutually impaired situations.

Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.

Networks are all around. They describe the flow of information, the movement of people and goods via multiple modes of transportation, and the spread of disease across interconnected populations. Traditionally, networks have been studied as if they were a single layer, which flattens out hierarchies such as social circles. Multilayer networks, which consider each of those circles as a layer, are more accurate descriptions of real-world networks and their use can have deep implications for understanding the dynamics of the system. Using the spread of disease as a model, we have developed a mathematical framework that accounts for the multilayer structure, and we have identified several behaviors that emerge from this analysis.

The framework relies on tensors, mathematical objects that allow us to represent multidimensional data in a compact way. Through mathematical analysis and numerical simulations, we find a number of interesting features such as the existence of multiple epidemic thresholds and transmission rates beyond which the number of individuals that catch a disease is non-negligible. We also show the existence of disease localization, a scenario in which the disease cannot escape a layer and jump to another.

Our work provides a unifying mathematical approach to studying disease transmission among multilayered populations. There are still many aspects to investigate such as how to use these results to help contain an epidemic as well as how the picture changes in more complex scenarios. Disease-like models can also be used to explore other networks such as the propagation of information.

Epidemics on networks with large initial conditions or changing structure

In this paper we extend previous work deriving dynamic equations governing infectious disease spread on networks. The previous work has implicitly assumed that the disease is initialized by an infinitesimally small proportion of the population. Our modifications allow us to account for an arbitrarily large initial proportion infected. This helps resolve an apparent paradox in earlier work whereby the number of susceptible individuals could increase if too many individuals were initially infected. It also helps explain an apparent small deviation that has been observed between simulation and theory. An advantage of this modification is that it allows us to account for changes in the structure or behavior of the population during the epidemic.

Coexistence of phases and the observability of random graphs

In a recent Letter, Yang et al. [Phys. Rev. Lett. 109, 258701 (2012)] introduced the concept of observability transitions: the percolationlike emergence of a macroscopic observable component in graphs in which the state of a fraction of the nodes, and of their first neighbors, is monitored. We show how their concept of depth-L percolation--where the state of nodes up to a distance L of monitored nodes is known--can be mapped onto multitype random graphs, and use this mapping to exactly solve the observability problem for arbitrary L. We then demonstrate a nontrivial coexistence of an observable and of a nonobservable extensive component. This coexistence suggests that monitoring a macroscopic portion of a graph does not prevent a macroscopic event to occur unbeknown to the observer. We also show that real complex systems behave quite differently with regard to observability depending on whether they are geographically constrained or not.

Competition-induced criticality in a model of meme popularity

Heavy-tailed distributions of meme popularity occur naturally in a model of meme diffusion on social networks. Competition between multiple memes for the limited resource of user attention is identified as the mechanism that poises the system at criticality. The popularity growth of each meme is described by a critical branching process, and asymptotic analysis predicts power-law distributions of popularity with very heavy tails (exponent α<2, unlike preferential-attachment models), similar to those seen in empirical data.

Epidemic fronts in complex networks with metapopulation structure

Infection dynamics have been studied extensively on complex networks, yielding insight into the effects of heterogeneity in contact patterns on disease spread. Somewhat separately, metapopulations have provided a paradigm for modeling systems with spatially extended and "patchy" organization. In this paper we expand on the use of multitype networks for combining these paradigms, such that simple contagion models can include complexity in the agent interactions and multiscale structure. Using a generalization of the Miller-Volz mean-field approximation for susceptible-infected-recovered (SIR) dynamics on multitype networks, we study the special case of epidemic fronts propagating on a one-dimensional lattice of interconnected networks-representing a simple chain of coupled population centers-as a necessary first step in understanding how macroscale disease spread depends on microscale topology. Applying the formalism of front propagation into unstable states, we derive the effective transport coefficients of the linear spreading: asymptotic speed, characteristic wavelength, and diffusion coefficient for the leading edge of the pulled fronts, and analyze their dependence on the underlying graph structure. We also derive the epidemic threshold for the system and study the front profile for various network configurations.

Large deviations of cascade processes on graphs

Simple models of irreversible dynamical processes such as bootstrap percolation have been successfully applied to describe cascade processes in a large variety of different contexts. However, the problem of analyzing nontypical trajectories, which can be crucial for the understanding of out-of-equilibrium phenomena, is still considered to be intractable in most cases. Here we introduce an efficient method to find and analyze optimized trajectories of cascade processes. We show that for a wide class of irreversible dynamical rules, this problem can be solved efficiently on large-scale systems.