Bounds for the expected value of one-step processes

Mean-field models are often used to approximate Markov processes with large state-spaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space $\{0,1,\ldots,N\}$ and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model.Comment: 14 pages, 4 figures, revise

The Impact of Network Flows on Community Formation in Models of Opinion Dynamics

We study dynamics of opinion formation in a network of coupled agents. As the network evolves to a steady state, opinions of agents within the same community converge faster than those of other agents. This framework allows us to study how network topology and network flow, which mediates the transfer of opinions between agents, both affect the formation of communities. In traditional models of opinion dynamics, agents are coupled via conservative flows, which result in one-to-one opinion transfer. However, social interactions are often non-conservative, resulting in one-to-many transfer of opinions. We study opinion formation in networks using one-to-one and one-to-many interactions and show that they lead to different community structure within the same network.Comment: accepted for publication in The Journal of Mathematical Sociology. arXiv admin note: text overlap with arXiv:1201.238