Mathematical analysis of Markov models for social processes

We present Markov models for two social processes: the spread of rumors and the change in the spatial distribution of a population over time. For the spread of rumors, we present two models. The first is for the situation in which all particles are identical but one initially knows the rumor. The second is for a situation in which there are two kinds of particles: spreaders, who can spread the rumor, and ordinary particles, who only can learn the rumor. We find that the limiting distribution for the first model is the convolution of two double exponential distributions and for the second model is a double exponential distribution. The stochastic dynamics for our model of the change in the spatial distribution of a population over time include the four basic demographic processes: birth, death, migration, and immigration. We allow interaction between particles only inasmuch as the immigration rate can depend on the existing configuration of particles. We focus on the critical case of constant mean density, under the conditions of long jumps migration, immigration in which distant particles have a positive effect, or both. We prove, under these conditions, the existence of ergodic limiting behavior: the point process is stationary in space and time. Without the strong mixing due to these conditions, the population vanishes due to infinite clusterization

Inequalities in network structures

ICS University of Groningen. We use a model of continuous attachments in networks to generate propositions concerning inequalities in network structures, and test the propositions on data from organizational settings. Our network model, inspired by that of [Gould, Roger 2002. The origins of status hierarchies: A formal theory and empirical test. American Journal of Sociology 107, 1143–1178], is based on a theoretically informed actor model, in which each network member sets attachment strengths based on perceived partner quality, reciprocity, influence from others, attribute homophily, and attachment resistance. A computer algorithm finds the single robust equilibrium configuration of attachment strengths. This allows us to generate six propositions concerning inequalities at the individual, dyadic, triadic, and network levels. We test the propositions on network data for four kinds of attachments over four waves for five organizations, and find that the results generally support the propositions. The results suggest that partner quality, reciprocity, and attachment resistance are the most important elements in the network members’ choices.