Ising model on clustered networks: A model for opinion dynamics

We study opinion dynamics on networks with a nontrivial community structure,assuming individuals can update their binary opinion as the result of theinteractions with an external influence with strength $h\in [0,1]$ and withother individuals in the network. To model such dynamics, we consider the Isingmodel with an external magnetic field on a family of finite networks with aclustered structure. Assuming a unit strength for the interactions inside eachcommunity, we assume that the strength of interaction across differentcommunities is described by a scalar $\epsilon \in [-1,1]$, which allows aweaker but possibly antagonistic effect between communities. We are interestedin the stochastic evolution of this system described by a Glauber-type dynamicsparameterized by the inverse temperature $\beta$. We focus on thelow-temperature regime $\beta\rightarrow\infty$, in which homogeneous opinionpatterns prevail and, as such, it takes the network a long time to fully changeopinion. We investigate the different metastable and stable states of thisopinion dynamics model and how they depend on the values of the parameters$\epsilon$ and $h$. More precisely, using tools from statistical physics, wederive rigorous estimates in probability, expectation, and law for the firsthitting time between metastable (or stable) states and (other) stable states,together with tight bounds on the mixing time and spectral gap of the Markovchain describing the network dynamics. Lastly, we provide a fullcharacterization of the critical configurations for the dynamics, i.e., thosewhich are visited with high probability along the transitions of interest.