We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 − α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value α c which does not depend on u, with ρ ≈ u for α > α c and ρ ≈ 0 for α < α c . In case (ii), the transition point α c ðuÞ depends on the initial density u. For α > α c ðuÞ, ρ ≈ u, but for α < α c ðuÞ, we have ρðα,uÞ ¼ ρðα,1∕2Þ. Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.I n recent years, a variety of research efforts from different disciplines have combined with established studies in social network analysis and random graph models to fundamentally change the way we think about networks. Significant attention has focused on the implications of dynamics in establishing network structure, including preferential attachment, rewiring, and other mechanisms (1-5). At the same time, the impact of structural properties on dynamics on those networks has been studied, (6), including the spread of epidemics (7-10), opinions (11-13), information cascades (14-16), and evolutionary games (17,18). Of course, in many real-world networks the evolution of the edges in the network is tied to the states of the vertices and vice versa. Networks that exhibit such a feedback are called adaptive or coevolutionary networks (19,20). As in the case of static networks, significant attention has been paid to evolutionary games (21)(22)(23)(24) and to the spread of epidemics (25-29) and opinions (30-35), including the polarization of a network of opinions into two groups (36,37). In this paper, we examine two closely related variants of a simple, abstract model for coevolution of a network and the opinions of its members. Holme-Newman ModelOur starting point is the model of Holme and Newman (38-41). They begin with a network of N vertices and M edges, where each vertex x has an opinion ξðxÞ from a set of G possible opinions and the number of people per opinion γ N ¼ N∕G stays bounded as N gets large. On each step of the process, a vertex x is picked at random. If its degree dðxÞ ¼ 0, nothing happens. For dðxÞ > 0, (i) with probability α an edge attached to vertex x is selected and the other end of that edge is moved to a vertex chosen at random from those with o...

We introduce a new kind of percolation on finite graphs called jigsaw percolation. This model attempts to capture networks of people who innovate by merging ideas and who solve problems by piecing together solutions. Each person in a social network has a unique piece of a jigsaw puzzle. Acquainted people with compatible puzzle pieces merge their puzzle pieces. More generally, groups of people with merged puzzle pieces merge if the groups know one another and have a pair of compatible puzzle pieces. The social network solves the puzzle if it eventually merges all the puzzle pieces. For an Erdős-Rényi social network with n vertices and edge probability pn, we define the critical value pc(n) for a connected puzzle graph to be the pn for which the chance of solving the puzzle equals 1/2. We prove that for the n-cycle (ring) puzzle, pc(n) = Θ(1/ log n), and for an arbitrary connected puzzle graph with bounded maximum degree, pc(n) = O(1/ log n) and ω(1/n b ) for any b > 0. Surprisingly, with probability tending to 1 as the network size increases to infinity, social networks with a power-law degree distribution cannot solve any bounded-degree puzzle. This model suggests a mechanism for recent empirical claims that innovation increases with social density, and it might begin to show what social networks stifle creativity and what networks collectively innovate.

A number of real world problems in many domains (e.g. sociology, biology, political science and communication networks) can be modeled as dynamic networks with nodes representing entities of interest and edges representing interactions among the entities at different points in time. A common representation for such models is the snapshot modelwhere a network is defined at logical time-stamps. An important problem under this model is change point detection. In this work we devise an effective and efficient three-step-approach for detecting change points in dynamic networks under the snapshot model. Our algorithm achieves up to 9X speedup over the state-of-the-art while improving quality on both synthetic and real world networks.

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Place a car independently with probability p at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when p ≥ 1/2, and only finitely many times otherwise.2010 Mathematics Subject Classification. 60K35, 82C22,82B26.