On the Universality of the Energy Response Function in the Long-Range Spin Glass Model with Sparse, Modular Couplings

TL;DR

This study investigates how modularity in the coupling matrix affects energy relaxation in a long-range spin glass model, revealing size-dependent effects on the relaxation dynamics and energy states.

Contribution

It is the first to analyze the impact of modularity on energy relaxation in the dilute SK model, highlighting size-dependent behaviors.

Findings

Modularity has negligible effect on relaxation in the infinite size limit.

Finite systems show less favorable energy states with increased modularity.

Large finite systems may benefit from modularity at intermediate times.

Abstract

We consider energy relaxation of the long-range spin glass model with sparse couplings, the so-called dilute Sherrington-Kirkpatrick (SK) model, starting from a random initial state. We consider the effect that modularity of the coupling matrix has on this relaxation dynamics. In the absence of finite size effects, the relaxation dynamics appears independent of modularity. For finite sizes, a more modular system reaches a less favorable energy at long times. For small sizes, a more modular system also has a less favorable energy at short times. For large sizes, modularity appears to lead to slightly more favorable energies at intermediate times. We discuss these results in the context of evolutionary theory, where horizontal gene transfer, absent in the Glauber equilibration dynamics of the SK model studied here, endows modular organisms with larger response functions at short times.