Diffusion in countably infinite networks

We investigate the phenomenon of diffusion in a countably infinite society of individuals interacting with their neighbors. At a given time, each individual is either active (i.e., has the status or opinion 1) or inactive (i.e., has the status or opinion 0). The configuration of the society describes active and inactive individuals. The diffusion mechanism is based on an aggregation function, which leads to a Markov process with an uncountable set of states, requiring the involvement of σ-fields. We focus on two types of aggregation functions - strict, and Boolean. We determine absorbing, transient and irreducible sets under strict aggregation functions. We show that segregation of the society cannot happen and its state evolves towards a mixture of infinitely many active and infinitely many inactive agents. In our analysis, we mainly focus on the network structure. We distinguish networks with a blinker (periodic class of period 2) and those without. ø-irreducibility is obtained at the price of a richness assumption of the network, meaning that it should contain infinitely many complex stars and have enough space for storing local configurations. When considering Boolean aggregation functions, the diffusion process becomes deterministic and the contagion model of Morris (2000) can be seen as a particular case of our framework with aggregation functions. In this case, consensus and non trivial absorbing states as well as cycles can exist.