Controlling Contagion Processes in Activity Driven Networks

Abstract

The vast majority of strategies aimed at controlling contagion processes on networks considers the connectivity pattern of the system as either quenched or annealed. However, in the real world many networks are highly dynamical and evolve in time concurrently to the contagion process. Here, we derive an analytical framework for the study of control strategies specifically devised for time-varying networks. We consider the removal/immunization of individual nodes according the their activity in the network and develop a block variable mean-field approach that allows the derivation of the equations describing the evolution of the contagion process concurrently to the network dynamic. We derive the critical immunization threshold and assess the effectiveness of the control strategies. Finally, we validate the theoretical picture by simulating numerically the information spreading process and control strategies in both synthetic networks and a large-scale, real-world mobile telephone call dataset.The spreading of infectious diseases, malwares, scientific ideas, memes are just few example of real world phenomena that can be modeled as contagion processes on networks [1][2][3][4]. It has long been acknowledged that network structures and connectivity patterns are a relevant factor in determining the properties of spreading processes. A number of strategies aimed at controlling them have been proposed with the aim of improving on the random removal strategies that originally defined the concept of herd immunity. Those strategies target nodes according the number of connections (degree), the k-core or the betweenness of nodes, just to mention a few examples [5,6]. The efficiency of each strategy is then measured by its effect on the contagion process when a fraction p of nodes is removed from the system. More precisely, the smaller is the fraction p of nodes removed in order to halt the contagion process and the more effective is the strategy. Although most real world networks show a high level of dynamic activity, the large majority of theoretical results concerning the control of contagion processes have been obtained by using a complete timescale separation between the contagion process, τ P , and the change in network's structure, τ G . In these approaches the dynamical process takes place in either static (τ P τ G ) or annealed (τ P τ G ) networks. However, when the two time scales are comparable these convenient approximations might introduce uncontrolled biases in the correct characterization of the dynamical properties of the contagion process . Here we investigate the effect of time-varying connectivity pattern of networks on contagion control strategies by considering the specific class of activity driven network models [23]. In particular, we consider the susceptibleinfected-susceptible (SIS) contagion model [30] and derive analytically its critical immunization threshold in three different control strategies. We also validate the findings obtained in synthetic networks by studying the effect of each strategy...

Correlation function for generalized Pólya urns: Finite-size scaling analysis

“…The contagion process is one of the most studied topics in statistical physics and has attracted the attention of many researchers from various disciplines [1][2][3][4][5][6][7][8]. Pólya urn is one of the simplest models for this process [9][10][11].…”

Control Strategies of Contagion Processes in Time-Varying Networks

“…Thus, the Eq. 6 becomes: First, let us consider the random strategy (RS) in which a fraction p of nodes is immunized with an uniform probability [83]. In this case, the system of equations describing the dynamic process in activity-driven networks can be obtained by setting R a = pN a .…”

“…This method is equivalent to fix a value a c so that any node with activity a ≥ a c is immune to the contagion process 1 . Also, for this scheme, it is possible to derive the analytic expression for the epidemic threshold [83]:…”

“…Following Ref. [83] and what presented above for annealed and static networks we will consider three main strategies: random, global and local. In all the cases, we introduce a fraction p of nodes as immunized.…”

“…The formulation is then a good approximation for small f and T , when the probability that a probe is selected more than once is very small. Replacing the expression for the removed/immunized individuals in the basic SIS equations yields the following epidemic threshold for the egocentric sampling strategy (ESS) [83]:…”