On threshold models over finite networks

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 109-110).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 thesis, we develop such an analysis and construct a combinatorial framework to understand the behavior of the model. To this end, we establish that the agents behavior cycles among different actions in the limit and provide three sets of results. We first characterize the limiting behavioral properties of the dynamics. We determine the length of the limit cycles and reveal bounds on the time steps required to reach such cycles for different network structures. We then study the complexity of decision/counting problems that arise within the context. Specifically, we consider the tractability of counting the number of limit cycles and fixed-points, and deciding the reachability of action profiles. 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.by Elie M. Adam.S.M

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

Networks of conforming or nonconforming individuals tend to reach satisfactory decisions

Binary decisions of agents coupled in networks can often be classified into two types: “coordination,” where an agent takes an action if enough neighbors are using that action, as in the spread of social norms, innovations, and viral epidemics, and “anticoordination,” where too many neighbors taking a particular action causes an agent to take the opposite action, as in traffic congestion, crowd dispersion, and division of labor. Both of these cases can be modeled using linear-threshold–based dynamics, and a fundamental question is whether the individuals in such networks are likely to reach decisions with which they are satisfied. We show that, in the coordination case, and perhaps more surprisingly, also in the anticoordination case, the agents will indeed always tend to reach satisfactory decisions, that is, the network will almost surely reach an equilibrium state. This holds for every network topology and every distribution of thresholds, for both asynchronous and partially synchronous decision-making updates. These results reveal that irregular network topology, population heterogeneity, and partial synchrony are not sufficient to cause cycles or nonconvergence in linear-threshold dynamics; rather, other factors such as imitation or the coexistence of coordinating and anticoordinating agents must play a role