arXiv subjects

Tobias Galla

Publications and source records attributed to Tobias Galla.

At least 19 recordsLinked to original sources

arXiv2026

Complex dynamics in the Sherrington-Kirkpatrick game

We study the outcome of adaptive learning of a large number of players engaging in sets of two-strategy two-player games. We are interested in typical games, and generate the payoff matrices at random at the beginning. The payoff matrices then remain fixed during the learning process. This provides a game theoretic foundation for the Sherrington-Kirkpatrick (SK) game, recently introduced by Garnier-Brun, Benzaquen and Bouchaud. The original model by these authors is a special case, with no bias towards any strategy. We here determine stability of learning for SK games with general random bias, and find that the nature of the stable state is affected by random fields. We also introduce a grand-canonical version of the SK game, in which players can choose to abstain. We determine the stability of learning for this game. Our analysis confirms that complex situations involving many players are frequently unlearnable, even if each player only chooses between two different actions. The rate with which players lose memory of past payoffs and the competitiveness of the game emerge as key parameters determining whether learning converges to a unique fixed point, whether there are many fixed points, or if the dynamics remains persistently volatile.

cond-mat.dis-nn↗

arXiv2026

Opinion dynamics: Statistical physics and beyond

Opinion dynamics, the study of how individual beliefs and collective public opinion evolve, is a fertile domain for applying statistical physics to complex social phenomena. Like physical systems, societies exhibit macroscopic regularities from localized interactions, leading to outcomes such as consensus or fragmentation. This field has grown significantly, attracting interdisciplinary methods and driven by a surge in large-scale behavioral data. This review covers its rapid progress, bridging the literature dispersion. We begin with essential concepts and definitions, encompassing the nature of opinions, microscopic and macroscopic dynamics. This foundation leads to an overview of empirical research, from lab experiments to large-scale data analysis, which informs and validates models of opinion dynamics. We then present individual-based models, categorized by their macroscopic phenomena (e.g., consensus, polarization, echo chambers) and microscopic mechanisms (e.g., homophily, assimilation). Furthermore, the review covers common analytical and computational tools, including stochastic processes, treatments, simulations, and optimization. Finally, we explore emerging frontiers, such as connecting empirical data to models and using AI agents as testbeds for novel social phenomena. By systematizing terminology and emphasizing analogies with traditional physics, this review aims to consolidate knowledge, provide a robust theoretical foundation, and shape future research in opinion dynamics.

physics.soc-ph↗

arXiv2026

Numerical methods for quasi-stationary distributions

In stochastic processes with absorbing states, the quasi-stationary distribution provides valuable insights into the long-term behaviour prior to absorption. In this work, we revisit two well-established numerical methods for its computation. The first is an iterative algorithm for solving the non-linear equation that defines the quasi-stationary distribution. We generalise this technique to accommodate general Markov stochastic processes, either with discrete or continuous state space, and with multiple absorbing states. The second is a Monte Carlo method with resetting, for which we propose a novel single-trajectory approach that uses the trajectory's own history to perform resets after absorption. In addition to these methodological contributions, we provide a detailed analysis of implementation aspects for both methods. We also compare their accuracy and efficiency across a range of examples. The results indicate that the iterative algorithm is generally the preferred choice for problems with simple boundaries, while the Monte Carlo approach is more suitable for problems with complex boundaries, where the implementation of the iterative algorithm is a challenging task.

cond-mat.stat-mech↗

arXiv2026

Anticipated synchronization in systems with distributed delay

Anticipated synchronisation occurs when a driven dynamical system synchronises with the future state of the driver system to which it is unidirectionally coupled. Previous theoretical and experimental studies have focused on setups with a single delay time in the coupling term, for which exact anticipation can arise as a solution. Here we extend this framework to configurations with distributed delay times. Our main result is that, for a given delay distribution, approximate anticipated synchronisation can emerge over a range of coupling strengths. We analyse this phenomenon analytically for systems of linear oscillators, where we identify simple cases exhibiting exact synchronisation--up to a constant amplitude factor. Numerical simulations of nonlinear chaotic systems reveal stable forms of approximate anticipated synchronisation.

nlin.CD↗

arXiv2026

A diffusion approximation for systems with frequent weak resetting

We develop a diffusion approximation for systems subject to fast random resetting by small amplitudes. Equivalently, this describes systems with frequent but small catastrophes. We demonstrate the validity of the approximation by computing the stationary distribution and mean first-passage times of simple one-dimensional systems. The approximation captures dynamically induced correlations in multi-particle systems, and it can be used to generalise the conditionally independent and identically distributed structure recently found in systems with full resetting. Finally, we show that resetting can induce cycles and patterns, which can be characterised using the diffusion approximation.

cond-mat.stat-mech↗

arXiv2026

Forecasting emergency department visits in the reference hospital of the Balearic Islands: the role of tourist and weather data

Accurate forecasting of patient arrivals at emergency departments (EDs) is vital for efficient resource allocation and high-quality patient care. In this study we investigate the relevance of exogenous variables, namely tourism, weather, calendar and demographic variables, in forecasting ED visits in the reference hospital in Palma de Mallorca, a city with significant seasonal population fluctuations due to tourism. Using a machine learning approach, we develop a model that predicts ED visits based solely on these exogenous variables. We test different machine learning algorithms (random forests, support vector machines, and feedforward neural networks) with different combinations of input variables and compare their symmetric mean average percentage errors (SMAPEs). Our findings reveal that calendar information, resident, and tourist population data are statistically significant for the accuracy of the predictions, while the addition of weather data does not provide any further improvement. Comparison of non-time-series with time-series prediction models reveals that the latter provide better accuracy for short prediction horizons (e.g. shorter than a week). Furthermore, time-series models become less or equally accurate to models relying only on exogenous variables for long prediction horizons (e.g. fortnight or month). Our study highlights the importance of carefully selecting predictive variables to ensure short- and long-term, robust and reliable forecasts. This demonstrates that, despite their lower complexity, non-time-series models with well-chosen input variables can be as effective as time-series models when predicting for long time horizons.

physics.soc-ph↗

arXiv2025

Large scale statistically validated comorbidity networks

We obtain comorbidity networks starting from medical information stored in electronic health records collected by the Wellbeing Services County of Southwest Finland (Varha). Based on the data, we associate each patient to one or more diseases and construct complex comorbidity networks associated with large patient cohorts characterized by an age interval and sex. The information about diseases in electronic health records is coded using the highest granularity present in the international classification of diseases (ICD codes) provided by the World Health Organization. We statistically validate links in each cohort comorbidity network and furthermore partition the networks into communities of diseases. These are characterized by the over-expression of a few disease categories, and communities from different age or sex cohorts show various similarities in terms of these disease classes. Moreover, all the detected communities for all the cohorts can be organized into a hierarchical tree. This allows us to observe a number of clusters of communities, originating from diverse age and sex cohorts, that group together communities characterized by the same disease classes. We also perform a dismantling procedure of statistically validated comorbidity networks to highlight those categories of diseases that are most responsible for the compactedness of the comorbidity networks for a given cohort of patients.

physics.soc-ph↗

arXiv2025

Breaking coexistence: Zealotry vs. nonlinear social impact

We study how zealotry and nonlinear social impact affect consensus formation in the nonlinear voter model, evolutionary games, and the partisan voter model. In all three models, consensus is an absorbing state in finite populations, while coexistence is a possible outcome of the deterministic dynamics. We show that sufficiently strong zealotry, i.e. the presence of agents who never change state, can drive infinite populations to consensus in all three models. However, while evolutionary games and the partisan voter model permit zealotry-induced consensus for all values of their model parameters, the nonlinear voter model does not. Central to this difference is the shape of the social impact function, which quantifies how the influence of a group scales with size, and is therefore a measure of majority and minority effects. We derive general conditions relating the slope of this function at small group sizes to the local stability of consensus. Sublinear impact favours minorities and can override zealotry to prevent consensus, whereas superlinear impact promotes majorities and therefore facilitates consensus. We extend the analysis to finite populations, exploring the time-to-consensus, and the shape of quasi-stationary distributions.

physics.soc-ph↗

arXiv2025

Analysis of a voter model with an evolving number of opinion states

In traditional voter models, opinion dynamics are driven by interactions between individuals, where an individual adopts the opinion of a randomly chosen neighbor. However, these models often fail to capture the emergence of entirely new opinions, which can arise spontaneously in real-world scenarios. Our study introduces a novel element to the classic voter model: the concept of innovation, where individuals have a certain probability of generating new opinions independently of their neighbors' states. This innovation process allows for a more realistic representation of social dynamics, where new opinions can emerge and old ones may fade over time. Through analytical and numerical analysis, we find that the balance between innovation and extinction shapes the number of opinions in the steady state. Specifically, for low innovation rates, the system tends toward near-consensus, while higher innovation rates lead to greater opinion diversity. We also show that network structure influences opinion dynamics, with greater degree heterogeneity reducing the number of opinions in the system.

physics.soc-ph↗

arXiv2025

Efficient approximations of transcriptional bursting effects on the dynamics of a gene regulatory network

Mathematical models of gene regulatory networks are widely used to study cell fate changes and transcriptional regulation. When designing such models, it is important to accurately account for sources of stochasticity. However, doing so can be computationally expensive and analytically untractable, posing limits on the extent of our explorations and on parameter inference. Here, we explore this challenge using the example of a simple auto-negative feedback motif, in which we incorporate stochastic variation due to transcriptional bursting and noise from finite copy numbers. We find that transcriptional bursting may change the qualitative dynamics of the system by inducing oscillations when they would not otherwise be present, or by magnifying existing oscillations. We describe multiple levels of approximation for the model in the form of differential equations, piecewise deterministic processes, and stochastic differential equations. Importantly, we derive how the classical chemical Langevin equation can be extended to include a noise term representing transcriptional bursting. This approximation drastically decreases computation times and allows us to analytically calculate properties of the dynamics, such as their power spectrum. We explore when these approximations break down and provide recommendations for their use. Our analysis illustrates the importance of accounting for transcriptional bursting when simulating gene regulatory network dynamics and provides recommendations to do so with computationally efficient methods.

q-bio.MN↗

arXiv2025

Proliferating active disks with game dynamical interaction

We study a system of self-propelled, proliferating finite-size disks with game-theoretical interactions, where growth rates depend on local population composition. We analyze how these interactions influence spatial distribution, coexistence, and extinction. Three scenarios emerge: (i) stable coexistence with well-mixed distributions, (ii) bistability, where the outcome depends on initial conditions, and (iii) dominance, where one species always outcompetes the other. We further show that activity enhances spatial mixing in coexistence and reduces extinction times in both dominance and bistability. Additionally, stronger interactions promote coexistence while accelerating extinction in the latter two regimes. By varying diffusivity, activity, and interaction strength, we show how movement and spatial constraints affect population dynamics.

q-bio.PE↗

arXiv2025

Predictability of temporal network dynamics in normal ageing and brain pathology

Spontaneous brain activity generically displays transient spatiotemporal coherent structures, which can selectively be affected in various neurological and psychiatric pathologies. Here we model the full brain's electroencephalographic activity as a high-dimensional functional network performing a trajectory in a latent graph phase space. This approach allows us to investigate the orbital stability of brain's activity and in particular its short-term predictability. We do this by constructing a non-parametric statistic quantifying the expansion of initially close functional network trajectories. We apply the method to cohorts of healthy ageing individuals, and patients previously diagnosed with Parkinson's or Alzheimer's disease. Results not only characterise brain dynamics from a new angle, but further show that functional network predictability varies in a marked scale-dependent way across healthy controls and patient groups. The path towards both pathologies is markedly different. Furthermore, healthy ageing's predictability appears to strongly differ from that of Parkinson's disease, but much less from that of patients with Alzheimer's disease.

q-bio.NC↗

arXiv2025

Breakdown of random-matrix universality in persistent Lotka--Volterra communities

The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this letter, we offer a pertinent counterexample in the context of the generalised Lotka--Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. We thus show that the eigenvalue spectra of random matrices can be used to deduce the stability of `feasible' ecological communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account.

cond-mat.dis-nn↗

arXiv2025

Dispersal-induced instability in complex ecosystems

In his seminal work in the 1970s, Robert May suggested that there is an upper limit to the number of species that can be sustained in stable equilibrium by an ecosystem. This deduction was at odds with both intuition and the observed complexity of many natural ecosystems. The so-called stability-diversity debate ensued, and the discussion about the factors contributing to ecosystem stability or instability continues to this day. We show in this work that dispersal can be a destabilising influence. To do this, we combine ideas from Alan Turing's work on pattern formation with May's random-matrix approach. We demonstrate how a stable equilibrium in a complex ecosystem with trophic structure can become unstable with the introduction of dispersal in space, and we discuss the factors which contribute to this effect. Our work highlights that adding more details to the model of May can give rise to more ways for an ecosystem to become unstable. Making May's simple model more realistic is therefore unlikely to entirely remove the upper bound on complexity.

q-bio.PE↗

arXiv2025

Eigenvalues of random matrices with generalised correlations: a path integral approach

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalised correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.

cond-mat.dis-nn↗

arXiv2025

Geospatial distributions reflect rates of evolution of features of language

Quantifying the speed of linguistic change is challenging due to the fact that the historical evolution of languages is sparsely documented. Consequently, traditional methods rely on phylogenetic reconstruction. In this paper, we propose a model-based approach to the problem through the analysis of language change as a stochastic process combining vertical descent, spatial interactions, and mutations in both dimensions. A notion of linguistic temperature emerges naturally from this analysis as a dimensionless measure of the propensity of a linguistic feature to undergo change. We demonstrate how temperatures of linguistic features can be inferred from their present-day geospatial distributions, without recourse to information about their phylogenies. Thus the evolutionary dynamics of language, operating across thousands of years, leaves a measurable geospatial signature. This signature licenses inferences about the historical evolution of languages even in the absence of longitudinal data.

physics.soc-ph↗

arXiv2024

Characterising the dynamics of unlabelled temporal networks

Networks model the architecture backbone of complex systems. The backbone itself can change over time leading to what is called `temporal networks'. Interpreting temporal networks as trajectories in graph space of a latent graph dynamics has recently enabled the extension of concepts and tools from dynamical systems and time series to networks. Here we address temporal networks with unlabelled nodes, a case that has received relatively little attention so far. Situations in which node labelling cannot be tracked over time often emerge in practice due to technical challenges, or privacy constraints. In unlabelled temporal networks there is no one-to-one matching between a network snapshot and its adjacency matrix. Characterizing the dynamical properties of such unlabelled network trajectories is nontrivial. We here exploit graph invariants to extend to the unlabelled setting network-dynamical quantifiers of linear correlations and dynamical instability. In particular, we focus on autocorrelation functions and the sensitive dependence on initial conditions. We show with synthetic graph dynamics that the measures are capable of recovering and estimating these dynamical fingerprints even when node labels are unavailable. We also validate the methods for some empirical temporal networks with removed node labels.

physics.soc-ph↗

arXiv2024

Asymptotic Extinction in Large Coordination Games

We study the exploration-exploitation trade-off for large multiplayer coordination games where players strategise via Q-Learning, a common learning framework in multi-agent reinforcement learning. Q-Learning is known to have two shortcomings, namely non-convergence and potential equilibrium selection problems, when there are multiple fixed points, called Quantal Response Equilibria (QRE). Furthermore, whilst QRE have full support for finite games, it is not clear how Q-Learning behaves as the game becomes large. In this paper, we characterise the critical exploration rate that guarantees convergence to a unique fixed point, addressing the two shortcomings above. Using a generating-functional method, we show that this rate increases with the number of players and the alignment of their payoffs. For many-player coordination games with perfectly aligned payoffs, this exploration rate is roughly twice that of $p$-player zero-sum games. As for large games, we provide a structural result for QRE, which suggests that as the game size increases, Q-Learning converges to a QRE near the boundary of the simplex of the action space, a phenomenon we term asymptotic extinction, where a constant fraction of the actions are played with zero probability at a rate $o(1/N)$ for an $N$-action game.

cs.GT↗