A study of the properties of mesoscopic consensus clusters that arise due to Ising dynamics on graphs - Sowdi Ravindra Bose, Karthik (2016)
We studied the impact of graph structure and temperature has on the sizes and lifetimes of crowds (mesoscopic clusters of consensus) that form when due to Ising Spin dynamics. Spin models have proven successful in modelling the formation of global consensus in opinion for large groups of people. We focus on the case where lots of small groups form consensuses but no global consensus forms. Our results can be interpreted in the light of consumer opinion and give an explanation for fads and brand reputation. Data was obtained by performing Monte Carlo simulations of the Ising Model on various graphs, our key control parameter was the inverse temperature β =1/kBT. In the Ising Model each vertex chooses an opinion −1 or +1 each time step based on the opinions of its neighbours. Low β ≪ 1 exhibit as vertices behaving noisily (flipping opinion often), while β → 1 tend to be quieter (flipping rarely). Interactions between neighbours creates feedback loops which determine crowd properties. The simulations we perform start with β_q = 0 for 0 ≤ t < 500, then the temperature is sharply quenched to a value βq ≥ 0 and kept constant for t ∈ (500, 15000]. We analyse the data in the time period 13000 ≤ t ≤ 15000. The simulations were performed on the 2-d rectangular lattice (RL), K-degree 1-d circular lattice (CL), Erdős-Rényi (ER), and 1-d Watts-Strogatz (WS) graphs. We conclude that crowds with multiple unique feedback loops are more robust against noisy vertices and crowd-crowd interactions. On the CL many feedback loops use the same paths which makes it easy for noisy vertices to disrupt the loops simultaneously. However 1-d structure limits crowd-crowd interactions making crowds strong against absorption by their neighbours. On the RL there are many paths between vertices, thus feedback loops can be easily rerouted around noisy vertices and crowds are robust against noise. On weakly connected ER graphs many feedback loops share the same paths and thus crowds are extremely susceptible to noisy vertices. Disorder in the 1-d lattice structure in WS graphs reduces the number of feedback loops in certain regions, this makes some crowds weak against both noisy vertices and neighbouring crowds. On the RL, CL and WS graphs we observe that size of crowds changes like an anomalous diffusion process. The anomalous diffusion exponent α was determined by plotting the root mean square change in crowd size sqrt(<ΔP²>) as a function of the crowd lifetime L. On the RL changes in crowd size were super-diffusive for 0.35 ≲ β ≲ 0.65; crowds grew very large in short periods of time, however such crowds also had short lifetimes as they were constantly replaced by new crowds. This behaviour is similar to the coming and going of fads. For 0.65 ≲ β ≲ 1 crowd sizes changed sub-diffusively, crowds grew in very small amounts to very large sizes over long time periods, this is analogous to the spread of reputation through word of mouth.