Universality of phase and synchronization transitions on a complex network

Natural systems are often modeled by regular or disordered lattices or, in general, graphs or complex networks. The dynamics occurring on a network gets affected by its underlying structures. In regular lattices, the Euclidean dimension is the only relevant parameter that determines the universality of the emergent phenomena occurring in the dynamics. However, due to its structural heterogeneity, a complex network generally acts as a 
disordered system. In this talk, I will discuss the role of the spectral dimension (ds) of a network in determining the universality of dynamics occurring on it. Specifically, I will talk about synchronization and entrainment transitions in the nonequilibrium dynamics of the Kuramoto model and the phase transition in the equilibrium dynamics of the classical XY model, thereby covering a broad spectrum from nonlinear dynamics to statistical and condensed matter physics. Using linear theory, I will show, in general, how the dynamics is related to the underlying network properties. This yields the lower critical spectral dimension of the phase synchronization and entrainment transitions in the Kuramoto model as ds = 4 and ds = 2 respectively, whereas, for the phase transition in the XY model, it is ds = 2. We test our theoretical hypotheses on a network where two nodes are connected with a probability proportional to a power law of the distance between the nodes. One can realize any desired ds ∈ [1, ∞) on such a network. We found by a detailed numerical investigation that the network disorder in the region 2 ≤ ds ≲ 3 seems to be relevant, affecting the dynamics profoundly

*Reference: *M. Sarkar, T. Enss, and N. Defenu, arXiv:2401.00092 (2023)