Diffusion-annihilation processes in complex networks

We present a detailed analytical study of the $A+A\to\emptyset$ diffusion-annihilation process in complex networks. By means of microscopic arguments, we derive a set of rate equations for the density of $A$ particles in vertices of a given degree, valid for any generic degree distribution, and which we solve for uncorrelated networks. For homogeneous networks (with bounded fluctuations), we recover the standard mean-field solution, i.e. a particle density decreasing as the inverse of time. For heterogeneous (scale-free networks) in the infinite network size limit, we obtain instead a density decreasing as a power-law, with an exponent depending on the degree distribution. We also analyze the role of finite size effects, showing that any finite scale-free network leads to the mean-field behavior, with a prefactor depending on the network size. We check our analytical predictions with extensive numerical simulations on homogeneous networks with Poisson degree distribution and scale-free networks with different degree exponents.Comment: 9 pages, 5 EPS figure

Diffusion-annihilation processes in complex networks

We present a detailed analytical study of the $A+A\to\emptyset$ diffusion-annihilation process in complex networks. By means of microscopic arguments, we derive a set of rate equations for the density of $A$ particles in vertices of a given degree, valid for any generic degree distribution, and which we solve for uncorrelated networks. For homogeneous networks (with bounded fluctuations), we recover the standard mean-field solution, i.e. a particle density decreasing as the inverse of time. For heterogeneous (scale-free networks) in the infinite network size limit, we obtain instead a density decreasing as a power-law, with an exponent depending on the degree distribution. We also analyze the role of finite size effects, showing that any finite scale-free network leads to the mean-field behavior, with a prefactor depending on the network size. We check our analytical predictions with extensive numerical simulations on homogeneous networks with Poisson degree distribution and scale-free networks with different degree exponents.Comment: 9 pages, 5 EPS figure

Detecting the Influence of Spreading in Social Networks with Excitable Sensor Networks

Detecting spreading outbreaks in social networks with sensors is of great significance in applications. Inspired by the formation mechanism of human's physical sensations to external stimuli, we propose a new method to detect the influence of spreading by constructing excitable sensor networks. Exploiting the amplifying effect of excitable sensor networks, our method can better detect small-scale spreading processes. At the same time, it can also distinguish large-scale diffusion instances due to the self-inhibition effect of excitable elements. Through simulations of diverse spreading dynamics on typical real-world social networks (facebook, coauthor and email social networks), we find that the excitable senor networks are capable of detecting and ranking spreading processes in a much wider range of influence than other commonly used sensor placement methods, such as random, targeted, acquaintance and distance strategies. In addition, we validate the efficacy of our method with diffusion data from a real-world online social system, Twitter. We find that our method can detect more spreading topics in practice. Our approach provides a new direction in spreading detection and should be useful for designing effective detection methods

Activation thresholds in epidemic spreading with motile infectious agents on scale-free networks

We investigate a fermionic susceptible-infected-susceptible model with mobility of infected individuals on uncorrelated scale-free networks with power-law degree distributions $P (k) \sim k^{-\gamma}$ of exponents $2<\gamma<3$. Two diffusive processes with diffusion rate $D$ of an infected vertex are considered. In the \textit{standard diffusion}, one of the nearest-neighbors is chosen with equal chance while in the \textit{biased diffusion} this choice happens with probability proportional to the neighbor's degree. A non-monotonic dependence of the epidemic threshold on $D$ with an optimum diffusion rate $D_\ast$, for which the epidemic spreading is more efficient, is found for standard diffusion while monotonic decays are observed in the biased case. The epidemic thresholds go to zero as the network size is increased and the form that this happens depends on the diffusion rule and degree exponent. We analytically investigated the dynamics using quenched and heterogeneous mean-field theories. The former presents, in general, a better performance for standard and the latter for biased diffusion models, indicating different activation mechanisms of the epidemic phases that are rationalized in terms of hubs or max $k$-core subgraphs.Comment: 9 pages, 4 figure

Bosonic reaction-diffusion processes on scale-free networks

Reaction-diffusion processes can be adopted to model a large number of dynamics on complex networks, such as transport processes or epidemic outbreaks. In most cases, however, they have been studied from a fermionic perspective, in which each vertex can be occupied by at most one particle. While still useful, this approach suffers from some drawbacks, the most important probably being the difficulty to implement reactions involving more than two particles simultaneously. Here we introduce a general framework for the study of bosonic reaction-diffusion processes on complex networks, in which there is no restriction on the number of interacting particles that a vertex can host. We describe these processes theoretically by means of continuous time heterogeneous mean-field theory and divide them into two main classes: steady state and monotonously decaying processes. We analyze specific examples of both behaviors within the class of one-species process, comparing the results (whenever possible) with the corresponding fermionic counterparts. We find that the time evolution and critical properties of the particle density are independent of the fermionic or bosonic nature of the process, while differences exist in the functional form of the density of occupied vertices in a given degree class k. We implement a continuous time Monte Carlo algorithm, well suited for general bosonic simulations, which allow us to confirm the analytical predictions formulated within mean-field theory. Our results, both at the theoretical and numerical level, can be easily generalized to tackle more complex, multi-species, reaction-diffusion processes, and open a promising path for a general study and classification of this kind of dynamical systems on complex networks.Comment: 15 pages, 7 figure

Phase transitions in contagion processes mediated by recurrent mobility patterns

Human mobility and activity patterns mediate contagion on many levels, including the spatial spread of infectious diseases, diffusion of rumors, and emergence of consensus. These patterns however are often dominated by specific locations and recurrent flows and poorly modeled by the random diffusive dynamics generally used to study them. Here we develop a theoretical framework to analyze contagion within a network of locations where individuals recall their geographic origins. We find a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is affected. This transition cannot be uncovered by continuous deterministic models due to the stochastic features of the contagion process and defines an invasion threshold that depends on mobility parameters, providing guidance for controlling contagion spread by constraining mobility processes. We recover the threshold behavior by analyzing diffusion processes mediated by real human commuting data.Comment: 20 pages of Main Text including 4 figures, 7 pages of Supplementary Information; Nature Physics (2011

Communities, Knowledge Creation, and Information Diffusion

In this paper, we examine how patterns of scientific collaboration contribute to knowledge creation. Recent studies have shown that scientists can benefit from their position within collaborative networks by being able to receive more information of better quality in a timely fashion, and by presiding over communication between collaborators. Here we focus on the tendency of scientists to cluster into tightly-knit communities, and discuss the implications of this tendency for scientific performance. We begin by reviewing a new method for finding communities, and we then assess its benefits in terms of computation time and accuracy. While communities often serve as a taxonomic scheme to map knowledge domains, they also affect how successfully scientists engage in the creation of new knowledge. By drawing on the longstanding debate on the relative benefits of social cohesion and brokerage, we discuss the conditions that facilitate collaborations among scientists within or across communities. We show that successful scientific production occurs within communities when scientists have cohesive collaborations with others from the same knowledge domain, and across communities when scientists intermediate among otherwise disconnected collaborators from different knowledge domains. We also discuss the implications of communities for information diffusion, and show how traditional epidemiological approaches need to be refined to take knowledge heterogeneity into account and preserve the system's ability to promote creative processes of novel recombinations of idea