Abstract. We investigate how social reinforcement drives the spread of permanent innovations and transient fads. We account for social reinforcement by endowing each individual with M + 1 possible awareness states 0, 1, 2, . . . , M , with state M corresponding to adopting an innovation. An individual with awareness k < M increases to k + 1 by interacting with an adopter. Starting with a single adopter, the time for an initially unaware population that consists of N individuals to adopt an innovation grows as ln N for M = 1, and as N 1−1/M for M > 1. When individuals can abandon the innovation at rate λ, the population fraction that remains clueless about the fad undergoes a phase transition at λ c ; this transition is second order for M = 1 and first order for M > 1, with macroscopic fluctuations accompanying the latter. The time for the fad to disappear has an intriguing non-monotonic dependence on λ.

We analyze record-breaking events in time series of continuous random variables that are subsequently discretized by rounding to integer multiples of a discretization scale ∆ > 0. Rounding leads to ties of an existing record, thereby reducing the number of new records. For an infinite number of random variables that are drawn from distributions with a finite upper limit, the number of discrete records is finite, while for distributions with a thinner than exponential upper tail, fewer discrete records arise compared to continuous variables. In the latter case the record sequence becomes highly regular at long times.

We introduce the confident voter model, in which each voter can be in one of two opinions and can additionally have two levels of commitment to an opinion -confident and unsure. Upon interacting with an agent of a different opinion, a confident voter becomes less committed, or unsure, but does not change opinion. However, an unsure agent changes opinion by interacting with an agent of a different opinion. In the meanfield limit, a population of size N is quickly driven to a mixed state and remains close to this state before consensus is eventually achieved in a time of the order of ln N . In two dimensions, the distribution of consensus times is characterized by two distinct times -one that scales linearly with N and another that appears to scale as N 3/2 . The longer time arises from configurations that fall into long-lived states that consist of two (or more) single-opinion stripes before consensus is reached. These stripe states arise from an effective surface tension between domains of different opinions.

In certain parliamentary democracies, there are two major parties that move in and out of power every few elections, and a third minority party that essentially never governs. We present a simple model to account for this phenomenon, in which minority party supporters sometimes vote ideologically (for their party) and sometimes strategically (against the party they like the least). The competition between these disparate tendencies reproduces the empirical observation of two parties that frequently exchange majority status and a third party that is almost always in the minority. PACS numbers: 89.75.Fb (structures and organization in complex systems), 02.50.-r (probability theory)

We investigate the first-passage properties of bursty random walks on a finite one-dimensional interval of length L, in which unit-length steps to the left occur with probability close to one, while steps of length b to the right-'bursts'-occur with small probability. This stochastic process provides a crude description of the early stages of virus spread in an organism after exposure. The interesting regime arises when b/L 1, where the conditional exit time to reach L, corresponding to an infected state, has a non-monotonic dependence on initial position. Both the exit probability and the infection time exhibit complex dependencies on the initial condition due to the interplay between the burst length and interval length.