We study the equilibrium phases of a generalized Lotka-Volterra model characterized by a species interaction matrix which is random, sparse and symmetric. Dynamical fluctuations are modeled by a demographic noise with amplitude proportional to the effective temperature T. The equilibrium distribution of species abundances is obtained by means of the cavity method and the Belief Propagation equations, which allow for an exact solution on sparse networks. Our results reveal a rich and non-trivial phenomenology that deviates significantly from the predictions of fully connected models. Consistently with data from real ecosystems, which are characterized by sparse rather than dense interaction networks, we find strong deviations from Gaussianity in the distribution of abundances. In addition to the study of these deviations from Gaussianity, which are not related to multiple-equilibria, we also identified a novel topological glass phase, present at both finite temperature, as shown here, and at T=0, as previously suggested in the literature. The peculiarity of this phase, which differs from the multiple-equilibria phase of fully-connected networks, is its strong dependence on the presence of extinctions. These findings provide new insights into how network topology and disorder influence ecological networks, particularly emphasizing that sparsity is a crucial feature for accurately modeling real-world ecological phenomena.
Tonolo et al. (Wed,) studied this question.