Diffusion and Cascading Behavior in Random Networks

The spread of new ideas, behaviors or technologies has been extensively studied using epidemic models. Here we consider a model of diffusion where the individuals' behavior is the result of a strategic choice. We study a simple coordination game with binary choice and give a condition for a new action to become widespread in a random network. We also analyze the possible equilibria of this game and identify conditions for the coexistence of both strategies in large connected sets. Finally we look at how can firms use social networks to promote their goals with limited information. Our results differ strongly from the one derived with epidemic models and show that connectivity plays an ambiguous role: while it allows the diffusion to spread, when the network is highly connected, the diffusion is also limited by high-degree nodes which are very stable

Contagions in Random Networks with Overlapping Communities

We consider a threshold epidemic model on a clustered random graph with overlapping communities. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the matrix whose largest eigenvalue gives the phase transition.Comment: Minor modifications for the second version: added comments (end of Section 3.2, beginning of Section 5.3); moved remark (end of Section 3.1, beginning of Section 4.1); corrected typos; changed titl

Coordination in Network Security Games: a Monotone Comparative Statics Approach

Malicious softwares or malwares for short have become a major security threat. While originating in criminal behavior, their impact are also influenced by the decisions of legitimate end users. Getting agents in the Internet, and in networks in general, to invest in and deploy security features and protocols is a challenge, in particular because of economic reasons arising from the presence of network externalities. In this paper, we focus on the question of incentive alignment for agents of a large network towards a better security. We start with an economic model for a single agent, that determines the optimal amount to invest in protection. The model takes into account the vulnerability of the agent to a security breach and the potential loss if a security breach occurs. We derive conditions on the quality of the protection to ensure that the optimal amount spent on security is an increasing function of the agent's vulnerability and potential loss. We also show that for a large class of risks, only a small fraction of the expected loss should be invested. Building on these results, we study a network of interconnected agents subject to epidemic risks. We derive conditions to ensure that the incentives of all agents are aligned towards a better security. When agents are strategic, we show that security investments are always socially inefficient due to the network externalities. Moreover alignment of incentives typically implies a coordination problem, leading to an equilibrium with a very high price of anarchy.Comment: 10 pages, to appear in IEEE JSA

Viral Marketing On Configuration Model

We consider propagation of influence on a Configuration Model, where each vertex can be influenced by any of its neighbours but in its turn, it can only influence a random subset of its neighbours. Our (enhanced) model is described by the total degree of the typical vertex, representing the total number of its neighbours and the transmitter degree, representing the number of neighbours it is able to influence. We give a condition involving the joint distribution of these two degrees, which if satisfied would allow with high probability the influence to reach a non-negligible fraction of the vertices, called a big (influenced) component, provided that the source vertex is chosen from a set of good pioneers. We show that asymptotically the big component is essentially the same, regardless of the good pioneer we choose, and we explicitly evaluate the asymptotic relative size of this component. Finally, under some additional technical assumption we calculate the relative size of the set of good pioneers. The main technical tool employed is the "fluid limit" analysis of the joint exploration of the configuration model and the propagation of the influence up to the time when a big influenced component is completed. This method was introduced in Janson & Luczak (2008) to study the giant component of the configuration model. Using this approach we study also a reverse dynamic, which traces all the possible sources of influence of a given vertex, and which by a new "duality" relation allows to characterise the set of good pioneers

Topics in random graphs, combinatorial optimization, and statistical inference

The manuscript is made of three chapters presenting three differenttopics on which I worked with Ph.D. students. Each chapter can be read independently of the others andshould be relatively self-contained. Chapter 1 is a gentle introduction to the theory of random graphswith an emphasis on contagions on such networks. In Chapter 2, I explain the main ideas of the objectivemethod developed by Aldous and Steele applied to the spectral measure of random graphs and themonomer-dimer problem. This topic is dear to me and I hope that this chapter will convince the readerthat it is an exciting field of research. Chapter 3 deals with problems in high-dimensional statistics whichnow occupy a large proportion of my time. Unlike Chapters 1 and 2 which could be easily extended inlecture notes, I felt that the material in Chapter 3 was not ready for such a treatment. This field ofresearch is currently very active and I decided to present two of my recent contributions

On the behavior of threshold models over finite networks

We study a model for cascade effects over finite networks based on a deterministic binary linear threshold model. Our starting point is a networked coordination game where each agent's payoff is the sum of the payoffs coming from pairwise interaction with each of the neighbors. We first establish that the best response dynamics in this networked game is equivalent to the linear threshold dynamics with heterogeneous thresholds over the agents. While the previous literature has studied such linear threshold models under the assumption that each agent may change actions at most once, a study of best response dynamics in such networked games necessitates an analysis that allows for multiple switches in actions. In this paper, we develop such an analysis. We establish that agent behavior cycles among different actions in the limit, we characterize the length of such limit cycles, and reveal bounds on the time steps required to reach them. We finally propose a measure of network resilience that captures the nature of the involved dynamics. We prove bounds and investigate the resilience of different network structures under this measure.Irwin Mark Jacobs and Joan Klein Jacobs Presidential FellowshipSiebel ScholarshipUnited States. Air Force Office of Scientific Research (Grant FA9550-09-1-0420)United States. Army Research Office (Grant W911NF-09-1-0556