We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdős-Rényi graph with parameter pn ∈ (0, 1], where n is the size of the graph (i.e., the number of particles). If pn ≡ 1 the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as n → ∞ to the solution of a PDE: a McKean-Vlasov (or Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdős-Rényi graphs with limn pnn = ∞, and properly rescaling the interaction to account for the dilution introduced by pn. However, these results have been proven under strong assumptions on the initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results -Law of Large Numbers and Large Deviation Principle -assuming only the convergence of the empirical measure of the initial condition.

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. Using the finite gap method, two of us (PGG and PMS) have recently solved, to leading order and in terms of elementary functions of the initial data, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number of unstable modes. In this paper, concentrating on the simplest case of a single unstable mode, we study the periodic Cauchy problem of the AWs for the NLS equation perturbed by a linear loss or gain term. Using the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic model describing quantitatively how the solution evolves, after a suitable transient, into slowly varying lower dimensional patterns (attractors) in the (x, t) plane, characterized by ∆X = L/2 in the case of loss, and by ∆X = 0 in the case of gain, where ∆X is the x-shift of the position of the AW during the recurrence, and L is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces O(1) effects on the periodic AW dynamics, generating the slowly varying loss/gain attractors analytically described in this paper, we expect that these attractors, together with their generalizations corresponding to more unstable modes, will play a basic role in the theory of periodic AWs in nature.

The long time dynamics of the stochastic Kuramoto model defined on a graph is analyzed in the subcritical regime. The emphasis is posed on the relationship between the mean field behavior and the connectivity of the underlying graph: we give an explicit deterministic condition on the sequence of graphs such that, for any initial condition, even dependent on the network, the system approaches the unique stable stationary solution and it remains close to it, up to almost exponential times. The condition on the sequence of graphs is expressed through a concentration in ℓ∞ → ℓ1 norm and it is shown to be satisfied by a large class of graphs, random and deterministic, provided that the number of neighbors per site diverges, as the size of the system tends to infinity.

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We consider a class of weakly interacting particle systems of meanfield type. The interactions between the particles are encoded in a graph sequence, i.e., two particles are interacting if and only if they are connected in the underlying graph. We establish a Law of Large Numbers for the empirical measure of the system that holds whenever the graph sequence is convergent in the sense of graph limits theory. The limit is shown to be the solution to a non-linear Fokker-Planck equation weighted by the (possibly random) graph limit. No regularity assumptions are made on the graphon limit so that our analysis allows for very general graph sequences, such as exchangeable random graphs. For these, we also prove a propagation of chaos result. Finally, we fully characterize the graph sequences for which the associated empirical measure converges to the mean-field limit, i.e., to the solution of the classical McKean-Vlasov equation.