We consider the convergence time for solving the binary consensus problem using the interval consensus algorithm proposed by B ' en ' ezit, Thiran and Vetterli (2009). In the binary consensus problem, each node initially holds one of two states and the goal for each node is to correctly decide which one of these two states was initially held by a majority of nodes. We derive an upper bound on the expected convergence time that holds for arbitrary connected graphs, which is based on the location of eigenvalues of some contact rate matrices. We instantiate our bound for particular networks of interest, including complete graphs, paths, cycles, star-shaped networks, and Erd " os-R ' enyi random graphs; for these graphs, we compare our bound with alternative computations. We find that for all these examples our bound is tight, yielding the exact order with respect to the number of nodes. We pinpoint the fact that the expected convergence time critically depends on the voting margin defined as the difference between the fraction of nodes that initially held the majority and the minority states, respectively. The characterization of the expected convergence time yields exact relation between the expected convergence time and the voting margin, for some of these graphs, which reveals how the expected convergence time goes to infinity as the voting margin approaches zero. Our results provide insights into how the expected convergence time depends on the network topology which can be used for performance evaluation and network design. The results are of interest in the context of networked systems, in particular, peer-to-peer networks, sensor networks and distributed databases.
Viral marketing takes advantage of preexisting social networks among customers to achieve large changes in behaviour. Models of influence spread have been studied in a number of domains, including the effect of "word of mouth" in the promotion of new products or the diffusion of technologies. A social network can be represented by a graph where the nodes are individuals and the edges indicate a form of social relationship. The flow of influence through this network can be thought of as an increasing process of active nodes: as individuals become aware of new technologies, they have the potential to pass them on to their neighbours. The goal of marketing is to trigger a large cascade of adoptions. In this paper, we develop a mathematical model that allows to analyze the dynamics of the cascading sequence of nodes switching to the new technology. To this end we describe a continuous-time and a discrete-time models and analyse the proportion of nodes that adopt the new technology over time. The flow of information or influence through a large social network can be thought of as unfolding with the dynamics of an epidemic: as individuals become aware of new ideas, technologies, fads, rumors, or gossip, they have the potential to pass them on to their friends and colleagues, causing the resulting behavior to cascade through the network. We consider a collection of probabilistic and game-theoretic models for such phenomena proposed in the mathematical social sciences, as well as recent algorithmic work on the problem by computer scientists. Building on this, we discuss the implications of cascading behavior in a number of on-line settings, including word-of-mouth effects (also known as “viral marketing”) in the success of new products, and the influence of social networks in the growth of on-line