Effect of modularity on the Glauber dynamics of the dilute spin glass model

Abstract

We study the Glauber dynamics of the dilute, infinite-ranged spin glass model, the so-called dilute Sherrington-Kirkpatrick (dSK) model. The dSK model has sparse couplings and can be classified by the modularity (M) of the coupling matrix. We investigate the effect of the modularity on the relaxation dynamics starting from a random initial state. By using the Glauber dynamics and the replica method, we derive the relaxation dynamics equations for the magnetization (m) and the energy per spin (r), in addition to the equation for the spin glass order parameter (q αβ ). In the replica symmetric (RS) analysis, we find that there are two solutions for the RS spin glass order parameter (q): q = 0which is stable for r 1/2 in the non-modular system and q = 0 which is stable for r 1/\(\sqrt 8 \) in the completely modular system. By substituting the proper q values into the equations for r, we find that the relaxation dynamics of r depends on the modularity, M. These results suggest that, in the context of evolutionary theory, the modularity may emerge spontaneously in the point-mutation-only framework (Glauber dynamics) under a changing environment.