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We study the dynamics of the generalized Lotka-Volterra model with a network structure. Performing a high connectivity expansion for graphs, we write down a mean-field dynamical theory that incorporates degree heterogeneity. This allows us to describe the fixed points of the model in terms of a few simple order parameters. We extend the analysis even for diverging abundances, using a mapping to the replicator model. With this we present a unified approach for both cooperative and competitive systems that display complementary behaviors. In particular we show the central role of an order parameter called the critical degree, $g_c$. In the competitive regime $g_c$ serves to distinguish high degree nodes that are more likely to go extinct, while in the cooperative regime it has the reverse role, it will determine the low degree nodes that tend to go relatively extinct., Comment: 20 pages, 6 figures

Random features models play a distinguished role in the theory of deep learning, describing the behavior of neural networks close to their infinite-width limit. In this work, we present a thorough analysis of the generalization performance of random features models for generic supervised learning problems with Gaussian data. Our approach, built with tools from the statistical mechanics of disordered systems, maps the random features model to an equivalent polynomial model, and allows us to plot average generalization curves as functions of the two main control parameters of the problem: the number of random features $N$ and the size $P$ of the training set, both assumed to scale as powers in the input dimension $D$. Our results extend the case of proportional scaling between $N$, $P$ and $D$. They are in accordance with rigorous bounds known for certain particular learning tasks and are in quantitative agreement with numerical experiments performed over many order of magnitudes of $N$ and $P$. We find good agreement also far from the asymptotic limits where $D\to \infty$ and at least one between $P/D^K$, $N/D^L$ remains finite., Comment: 11 pages + appendix, 4 figures. Comments are welcome

There are two key paradigms for non-equilibrium dynamics: on the one hand, aging towards an equilibrium state that cannot be reached on reasonable timescales; on the other, external driving that can lead to non-equilibrium steady states. We explore how these two mechanisms interact, by studying the behaviour of trap models, which are paradigmatic descriptions of slow glassy dynamics, when driven by trajectory bias towards high or low activity. To diagnose whether the driven systems continue to age, we establish a framework for mapping the biased dynamics to a Markovian time evolution with time-dependent transition rates. We find that the original aging dynamics reacts in two qualitatively distinct ways to the driving: it can be destroyed by driving of any nonzero strength (fragile aging), whereby the dynamics either reaches an active steady state or effectively freezes; or it can persist within a finite range of driving strengths around the undriven case (robust aging). This classification into fragile and robust aging could form the basis for distinguishing different universality classes of aging dynamics., Comment: 26 pages, 9 figures

Economic and ecological models can be extremely complex, with a large number of agents/species each featuring multiple interacting dynamical quantities. In an attempt to understand the generic stability properties of such systems, we define and study an interesting new matrix ensemble with extensive correlations, generalising the elliptic ensemble. We determine analytically the boundary of its eigenvalue spectrum in the complex plane, as a function of the correlations determined by the model at hand. We solve numerically our equations in several cases of interest, and show that the resulting spectra can take a surprisingly wide variety of shapes., Comment: 19 pages, 6 figures

Social media provides an essential platform for shaping and sharing opinions and consuming information in a decentralized way. However, users often interact with and are exposed to information mostly aligned with their beliefs, creating a positive feedback mechanism that reinforces those beliefs and excludes contrasting ones. In this paper, we study such mechanisms by analyzing the social network dynamics of controversial Twitter discussions using unsupervised methods that demand little computational power. Specifically, we focus on the retweet networks of the climate change conversation during 2019, when important climate social movements flourished. We find echo chambers of climate believers and climate skeptics that we identify based solely on the retweeting patterns of Twitter users. In particular, we study the information sources, or chambers, consumed by the audience of the leading users of the conversation. Users with similar (contrasting) ideological positions show significantly high (low)-overlapping chambers, resulting in a bimodal overlap distribution. Further, we uncover the ideological position of previously unobserved high-impact users based on how many audience members fall into either echo chamber. We uncover the ideology of more than half of the retweeting population as either climate believers or skeptics and find that the cross-group communication is small. Moreover, we find that, while the echo chamber structures are consistent throughout the year, most users inside the echo chambers change from one week to the next, suggesting that they are a stable emergent property of the Twittersphere. Interestingly, we detect a significant decrease in polarization and a significant increase on the number of climate skeptics that coincides with the main #FridaysForFuture strikes during September, 2019., Comment: 25 pages (17 of main text + 8 of appendices), 11 figures

Asymmetry in the synaptic interactions between neurons plays a crucial role in determining the memory storage and retrieval properties of recurrent neural networks. In this work, we analyze the problem of storing random memories in a network of neurons connected by a synaptic matrix with a definite degree of asymmetry. We study the corresponding satisfiability and clustering transitions in the space of solutions of the constraint satisfaction problem associated with finding synaptic matrices given the memories. We find, besides the usual SAT/UNSAT transition at a critical number of memories to store in the network, an additional transition for very asymmetric matrices, where the competing constraints (definite asymmetry vs. memories storage) induce enough frustration in the problem to make it impossible to solve. This finding is particularly striking in the case of a single memory to store, where no quenched disorder is present in the system., Comment: 24 pages + appendix, 7 figures

It is clear that conventional statistical inference protocols need to be revised to deal correctly with the high-dimensional data that are now common. Most recent studies aimed at achieving this revision rely on powerful approximation techniques, that call for rigorous results against which they can be tested. In this context, the simplest case of high-dimensional linear regression has acquired significant new relevance and attention. In this paper we use the statistical physics perspective on inference to derive a number of new exact results for linear regression in the high-dimensional regime., Comment: Most recent version accepted for publication in Physical Review E