Ising model on clustered networks: A model for opinion dynamics

Abstract

We study opinion dynamics on networks with a nontrivial community structure, assuming individuals can update their binary opinion as the result of the interactions with an external influence with strength $h\in [0,1]$ and with other individuals in the network. To model such dynamics, we consider the Ising model with an external magnetic field on a family of finite networks with a clustered structure. Assuming a unit strength for the interactions inside each community, we assume that the strength of interaction across different communities is described by a scalar $\epsilon \in [-1,1]$, which allows a weaker but possibly antagonistic effect between communities. We are interested in the stochastic evolution of this system described by a Glauber-type dynamics parameterized by the inverse temperature $\beta$. We focus on the low-temperature regime $\beta \to \infty$, in which homogeneous opinion patterns prevail and, as such, it takes the network a long time to fully change opinion. We investigate the different metastable and stable states of this opinion dynamics model and how they depend on the values of the parameters $\epsilon$ and $h$. More precisely, using tools from statistical physics, we derive rigorous estimates in probability, expectation, and law for the first hitting time between metastable (or stable) states and (other) stable states, together with tight bounds on the mixing time and spectral gap of the Markov chain describing the network dynamics. Lastly, we provide a full characterization of the critical configurations for the dynamics, i.e., those which are visited with high probability along the transitions of interest.

arXiv comment31 pages, 8 figures

References in corpus (22)

• Rational Group Decision Making. A random field Ising model at T=0
• Opinion dynamics: models, extensions and external effects
• Freezing and Slow Evolution in a Constrained Opinion Dynamics Model
• Tunneling and Metastability of continuous time Markov chains
• Fragmentation versus Stability in Bimodal Coalitions
• Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States
• Metastability for reversible probabilistic cellular automata with self--interaction
• Metastable states, quasi-stationary distributions and soft measures
• Metastability for general dynamics with rare transitions: escape time and critical configurations
• Hitting times asymptotics for hard-core interactions on grids
• Competitive nucleation in reversible Probabilistic Cellular Automata
• Sum of exit times in a series of two metastable states
• Ising-PageRank model of opinion formation on social networks
• Kawasaki dynamics with two types of particles: stable/metastable configurations and communication heights
• Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics
• Low-temperature behavior of the multicomponent Widom-Rowlison model on finite square lattices
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version
• Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamics
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions
• Metastability for Kawasaki dynamics on the hexagonal lattice
• Metastability of synchronous and asynchronous dynamics
• Metastability of the three-state Potts model with general interactions

Cited by in corpus (3)

• Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics
• Thermodynamics of Innovation: A Statistical Mechanics Framework of Social Adoption
• Critical configurations of the hard-core model on square grid graphs