Is directed percolation class for synchronization transition robust with multi-site interactions?

Abstract

Coupled map lattice with pairwise local interactions is a well-studied system. However, in several situations, such as neuronal or social networks, multi-site interactions are possible. In this work, we study the coupled Gauss map in one dimension with 2-site, 3-site, 4-site and 5-site interaction. This coupling cannot be decomposed in pairwise interactions. We coarse-grain the variable values by labeling the sites above \(x^{\star }\) as up spin (+ 1) and the rest as down spin (– 1) where \(x^{\star }\) is the fixed point. We define flip rate F(t) as the fraction of sites i such that \(s_{i}(t-1) \ne s_{i}(t)\) and persistence P(t) as the fraction of sites i such that \(s_{i}(t')=s_{i}(0)\) for all \(t' \le t\). The dynamic phase transitions to a synchronized state is studied above quantifiers. For 3 and 5 sites interaction, we find that at the critical point, \(F(t) \sim t^{-\delta }\) with \(\delta =0.159\) and \(P(t) \sim t^{-\theta }\) with \(\theta =1.5\). They match the directed percolation (DP) class. Finite-size and off-critical scaling is consistent with DP class. For 2 and 4 site interactions, the exponent \(\delta \) and behavior of P(t) at critical point changes. Furthermore, we observe logarithmic oscillations over and above power-law decay at the critical point for 4-site coupling. Thus multi-site interactions can lead to new universality class(es).