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Strategies aimed at reducing the negative effects of long-term uncertainty and risk are common in biology, game theory, and finance, even if they entail a cost in terms of mean benefit. Here, we focus on the single mutant's invasion of a finite resident population subject to fluctuating environmental conditions. Thus, the game-theoretical model we analyze integrates environmental and demographic randomness, i.e., the two leading sources of stochasticity and uncertainty. We use simulations and mathematical analysis to study if and when strategies that either increase or reduce payoff variance across environmental states can enhance the mutant fixation probability. Variance aversion implies that the mutant pays insurance in terms of mean payoff to avoid worst-case scenarios. Variance-prone or gambling strategies, on the other hand, entail specialization, allowing the mutant to capitalize on transient favorable conditions, leading to a series of ``boom-and-bust'' cycles. Our analyses elucidate how the rate of change of environmental conditions and the shape of the probability distribution of possible states affect the possible most convenient strategies. We discuss how our results relate to the bet-hedging theory, which aims to reduce fitness variance rather than payoff variance. We also describe the analogies and differences between these similar yet distinct approaches.

In the study of disordered systems, one often chooses a matrix of independent identically distributed interaction coefficients to represent the quenched random couplings between components, perhaps with some symmetry constraint or correlations between diagonally opposite pairs of elements. However, a more general set of couplings, which still preserves the statistical interchangeability of the components, could involve correlations between interaction coefficients sharing only a single row or column index. These correlations have been shown to arise naturally in systems such as the generalised Lotka-Volterra equations (gLVEs). In this work, we perform a dynamic mean-field analysis to understand how single-index correlations affect the dynamics and stability of disordered systems, taking the gLVEs as our example. We show that in-row correlations raise the level of noise in the mean field process, even when the overall variance of the interaction coefficients is held constant. We also see that correlations between transpose pairs of rows and columns can either enhance or suppress feedback effects, depending on the sign of the correlation coefficient. In the context of the gLVEs, in-row and transpose row/column correlations thus affect both the species survival rate and the stability of ecological equilibria., Comment: 19 pages, 16 figures

We use generating functionals to derive a dynamic mean-field description for generalised Lotka-Volterra systems with higher-order quenched random interactions. We use the resulting single effective species process to determine the stability diagram in the space of parameters specifying the statistics of interactions, and to calculate the properties of the surviving community in the stable phase. We find that the behaviour as a function of the model parameters is often similar to the pairwise model. For example, the presence of more exploitative interactions increases stability. However we also find differences. For instance, we confirm in more general settings an observation made previously in model with third-order interactions that more competition between species can increase linear stability, and the diversity in the community, an effect not seen in the pairwise model. The phase diagram of the model with higher-order interactions is more complex than that of the model with pairwise interactions. We identify a new mathematical condition for a sudden onset of diverging abundances., Comment: 31+30 pages, 25+12 figures

We consider evolutionary games in which the agent selected for update compares their payoff to q neighbours, rather than a single neighbour as in standard evolutionary game theory. Through studying fixed point stability and fixation times for 2x2 games with all-to-all interactions, we find that the flow changes significantly as a function of q. Further, we investigate the effects of changing the underlying topology from an all-to-all interacting system to an uncorrelated graph via the pair approximation. We also develop the framework for studying games with more than two strategies, such as the rock-paper-scissors game where we show that changing q leads to the emergence of new types of flow., Comment: 12 + 21 pages, 10 + 8 figures

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.

This paper provides what is hopefully a self-contained set of notes describing the detailed steps of a generating-functional analysis of systems of generalised Lotka-Volterra equations with random interaction coefficients. Nothing in these notes is original, instead the generating-functional method (also known as the Martin-Siggia-Rose-DeDominic-Janssen formalism) and the resulting dynamic mean field theories have been used for the study of disordered systems and spin glasses for decades. But it is hard to find unifying sources which would allow a beginner to learn step-by-step how these methods can be used. My aim is to provide such a source. Most of the calculations are specific to generalised Lotka-Volterra systems, but much can be transferred to disordered systems in more general., Comment: 81 pages, 8 figures

In this paper we study networks of nodes characterised by binary traits that change both endogenously and through nearest-neighbour interaction. Our analytical results show that those traits can be ranked according to the noisiness of their transmission using only measures of order in the stationary state. Crucially, this ranking is independent of network topology. As an example, we explain why, in line with a long-standing hypothesis, the relative stability of the structural traits of languages can be estimated from their geospatial distribution. We conjecture that similar inferences may be possible in a more general class of Markovian systems. Consequently, in many empirical domains where longitudinal information is not easily available the propensities of traits to change could be estimated from spatial data alone., Comment: 6+78 pages, 4+23 figures

A central concern of community ecology is the interdependence between interaction strengths and the underlying structure of the network upon which species interact. In this work we present a solvable example of such a feedback mechanism in a generalised Lotka-Volterra dynamical system. Beginning with a community of species interacting on a network with arbitrary degree distribution, we provide an analytical framework from which properties of the eventual `surviving community' can be derived. We find that highly-connected species are less likely to survive than their poorly connected counterparts, which skews the eventual degree distribution towards a preponderance of species with low degree, a pattern commonly observed in real ecosystems. Further, the average abundance of the neighbours of a species in the surviving community is lower than the community average (reminiscent of the famed friendship paradox). Finally, we show that correlations emerge between the connectivity of a species and its interactions with its neighbours. More precisely, we find that highly-connected species tend to benefit from their neighbours more than their neighbours benefit from them. These correlations are not present in the initial pool of species and are a result of the dynamics., Comment: 10 pages, 6 figures plus appendix

A binary mixture of two-different-sizes proliferating motile disks is studied. As growth is space-limited, we focus on the conditions such that there is coexistence of both large and small disks, or dominance of the larger disks. The study involves systematically varying some system parameters, such as diffusivities, growth rates, and self-propulsion velocities. In particular, we demonstrate that diffusing faster confers a competitive advantage, so that larger disks can in the long time coexist or even dominate to the smaller ones. In the case of self-propelled disks a coexistence regime is induced by the activity where the two types of disks show the same spatial distribution: both phase separated or both homogenously distributed in the whole system.

Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are applicable in a myriad of different contexts. However, it is often assumed that all components of the complex system in question are statistically equivalent, which is unrealistic in many applications. Here, we introduce the concept of a finely-structured random matrix. These are random matrices with element-specific statistics, which can be used to model systems in which the individual components are statistically distinct. By supposing that the degree of `fine structure' in the matrix is small, we arrive at a succinct `modified' elliptical law. We demonstrate the direct applicability of our results to the niche and cascade models in theoretical ecology, as well as a model of a neural network, and a directed network with arbitrary degree distribution. The simple closed form of our central results allow us to draw broad qualitative conclusions about the effect of fine structure on stability., Comment: 13 pages, 5 figures, plus supplementary material

We study the ordering dynamics of nonlinear voter models with multiple states, also providing a discussion of the two-state model. The rate with which an individual adopts an opinion scales as the $q$-th power of the number of the individual's neighbours in that state. For $q>1$ the dynamics favor the opinion held by the most agents. The ordering to consensus is driven by deterministic drift, and noise only plays a minor role. For $q<1$ the dynamics favors minority opinions, and for multistate models the ordering proceeds through a noise-driven succession of metastable states. Unlike linear multi-state systems, the nonlinear model cannot be reduced to an effective two-state model. We find that the average density of active interfaces in the model with multiple opinion states does not show a single exponential decay in time for $q<1$, again at variance with the linear model. This highlights the special character of the conventional (linear) voter model, in which deterministic drift is absent. As part of our analysis, we develop a pair approximation for the multi-state model on graphs, valid for any positive real value of $q$, improving on previous approximations for nonlinear two-state voter models.

Quantifying the contributions, or weights, of comparisons or single studies to the estimates in a network meta-analysis (NMA) is an active area of research. We extend this to the contributions of paths to NMA estimates. We present a general framework, based on the path-design matrix, that describes the problem of finding path contributions as a linear equation. The resulting solutions may have negative coefficients. We show that two known approaches, called shortestpath and randomwalk, are special solutions of this equation, and both meet an optimization criterion, as they minimize the sum of absolute path contributions. In general, there is an infinite space of solutions, which can be identified using the generalized inverse (Moore-Penrose pseudoinverse). We consider two further special approaches. For complex networks we find that shortestpath is superior with respect to run time and variability, compared to the other approaches, and is thus recommended in practice. The path-weights framework also has the potential to answer more general research questions in network meta-analysis., Comment: Preprint, 17 pages, 7 figures. Submitted to Statistics in Medicine

We study the stationary states of variants of the noisy voter model, subject to fluctuating parameters or external environments. Specifically, we consider scenarios in which the herding-to-noise ratio switches randomly and on different time scales between two values. We show that this can lead to a phase in which polarised and heterogeneous states exist. Secondly, we analyse a population of noisy voters subject to groups of external influencers, and show how multi-peak stationary distributions emerge. Our work is based on a combination of individual-based simulations, analytical approximations in terms of a piecewise-deterministic Markov processes (PDMP), and on corrections to this process capturing intrinsic stochasticity in the linear-noise approximation. We also propose a numerical scheme to obtain the stationary distribution of PDMPs with three environmental states and linear velocity fields., Comment: 20 pages, 15 figures

By interpreting a temporal network as a trajectory of a latent graph dynamical system, we introduce the concept of dynamical instability of a temporal network, and construct a measure to estimate the network Maximum Lyapunov Exponent (nMLE) of a temporal network trajectory. Extending conventional algorithmic methods from nonlinear time-series analysis to networks, we show how to quantify sensitive dependence on initial conditions, and estimate the nMLE directly from a single network trajectory. We validate our method for a range of synthetic generative network models displaying low and high dimensional chaos, and finally discuss potential applications.

We study communities emerging from generalised random Lotka--Volterra dynamics with a large number of species and with competitive exclusion. Each species is endowed with a number of traits, and competition between pairs of species increases with their similarity in trait space. This leads to a model with random Hebbian interactions. We use tools from the theory of disordered systems, notably dynamic mean field theory, to characterise the statistics of the resulting communities at stable fixed points and determine analytically when stability breaks down. Two distinct types of transition are identified in this way, both marked by diverging abundances, but differing in the behaviour of the integrated response function. At fixed points only a fraction of the initial pool of species survives. We numerically study the eigenvalue spectra of the interaction matrix between extant species. We find evidence that the two types of dynamical transition are, respectively, associated with the bulk spectrum or an outlier eigenvalue crossing into the right half of the complex plane., Comment: 14 pages, 9 figures + Supplement

The conventional voter model is modified so that an agent's switching rate depends on the `age' of the agent, that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. Lewis' thinning algorithm can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

We describe the design, construction, and operation of an apparatus utilizing a piezoelectric transducer for in-vacuum loading of nanoparticles into an optical trap for use in levitated optomechanics experiments. In contrast to commonly used nebulizer-based trap-loading methods which generate aerosolized liquid droplets containing nanoparticles, the method produces dry aerosols of both spherical and high-aspect ratio particles ranging in size by approximately two orders of mangitude. The device has been shown to generate accelerations of order $10^7$ $g$, which is sufficient to overcome stiction forces between glass nanoparticles and a glass substrate for particles as small as $170$ nm diameter. Particles with sizes ranging from $170$ nm to $\sim 10$ $\mu$m have been successfully loaded into optical traps at pressures ranging from $1$ bar to $0.6$ mbar. We report the velocity distribution of the particles launched from the substrate and our results indicate promise for direct loading into ultra-high-vacuum with sufficient laser feedback cooling. This loading technique could be useful for the development of compact fieldable sensors based on optically levitated nanoparticles as well as matter-wave interference experiments with ultra-cold nano-objects which rely on multiple repeated free-fall measurements and thus require rapid trap re-loading in high vacuum conditions., Comment: 9 pages, 10 figures

Optically-levitated dielectric particles can serve as ultra-sensitive detectors of feeble forces and torques, as tools for use in quantum information science, and as a testbed for quantum coherence in macroscopic systems. Knowledge of the structural and optical properties of the particles is important for calibrating the sensitivity of such experiments. Here we report the results of nanomechanical testing of silica nanospheres and investigate an annealing approach which can produce closer to bulk-like behavior in the samples in terms of their elastic moduli. These results, combined with our experimental investigations of optical trap lifetimes in high vacuum at high trapping-laser intensity for both annealed and as-grown nanospheres, were used to provide a theoretical analysis of the effects of porosity and non-sphericity in the samples, identifying possible mechanisms of trapping instabilities for nanospheres with non-bulk-silica-like properties., Comment: 10 pages, 7 figures

In the analysis of complex ecosystems it is common to use random interaction coefficients, often assumed to be such that all species are statistically equivalent. In this work we relax this assumption by choosing interactions according to the cascade model, which we incorporate into a generalised Lotka-Volterra dynamical system. These interactions impose a hierarchy in the community. Species benefit more, on average, from interactions with species further below them in the hierarchy than from interactions with those above. Using dynamic mean-field theory, we demonstrate that a strong hierarchical structure is stabilising, but that it reduces the number of species in the surviving community, as well as their abundances. Additionally, we show that increased heterogeneity in the variances of the interaction coefficients across positions in the hierarchy is destabilising. We also comment on the structure of the surviving community and demonstrate that the abundance and probability of survival of a species is dependent on its position in the hierarchy., Comment: 10 pages, 6 Figures

We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multi-state voter models with all-to-all interaction and on uncorrelated networks. Individual realisations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a meta-stable state. The density of active links and the entropy in these states varies from realisation to realisation. Making some simple assumptions we are able to analytically calculate the average density of active links and the average entropy in each of these states. We also show that, averaged over realisations, the density of active links decays exponentially, with a time scale set by the size and geometry of the graph, but independent of the initial number of opinion states. The decay of the average entropy is exponential only at long times when there are at most two opinions left in the population. Finally, we show how meta-stable states comprised of only a subset of opinions can be artificially engineered by introducing precisely one zealot in each of the prevailing opinions., Comment: 18 pages, 18 figures

We introduce an individual-based model of a complex ecological community with random interactions. The model contains a large number of species, each with a finite population of individuals, subject to discrete reproduction and death events. The interaction coefficients determining the rates of these events is chosen from an ensemble of random matrices, and is kept fixed in time. The set-up is such that the model reduces to the known generalised Lotka-Volterra equations with random interaction coefficients in the limit of an infinite population for each species. Demographic noise in the individual-based model means that species which would survive in the Lotka-Volterra model can become extinct. These noise-driven extinctions are the focus of the paper. We find that, for increasing complexity of interactions, ecological communities generally become less prone to extinctions induced by demographic noise. An exception are systems composed entirely of predator-prey pairs. These systems are known to be stable in deterministic Lotka-Volterra models with random interactions, but, as we show, they are nevertheless particularly vulnerable to fluctuations., Comment: 20 pages, 10 figures

We study the effects of stochastic birth and death processes on the structural phases of systems of active and passive Brownian particles subject to volume exclusion. The total number of particles in the system is a fluctuating quantity, determined by the birth and death parameters, and on the activity of the particles. As the birth and death parameters are varied we find liquid, hexatic and solid phases. For passive particles these phases are found to be spatially homogeneous. For active particles motility-induced phase separation (co-existing hexatic and liquid phases) occurs for large activity and sufficiently small birth rates. We also observe a re-entrant transition to the hexatic phase when the birth rate is increased. This results from a balance of an increasing number of particles filling the system, and a larger number of defects resulting from the birth and death dynamics.

Network meta-analysis (NMA) is a technique used in medical statistics to combine evidence from multiple medical trials. NMA defines an inference and information processing problem on a network of treatment options and trials connecting the treatments. We believe that statistical physics can offer useful ideas and tools for this area, including from the theory of complex networks, stochastic modelling and simulation techniques. The lack of a unique source that would allow physicists to learn about NMA effectively is a barrier to this. In this article we aim to present the `NMA problem' and existing approaches to it coherently and in a language accessible to statistical physicists. We also summarise existing points of contact between statistical physics and NMA, and describe our ideas of how physics might make a difference for NMA in the future. The overall goal of the article is to attract physicists to this interesting, timely and worthwhile field of research., Comment: 73 pages, 13 figures

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. Our findings connect two previously disparate ways of modelling complex ecosystems: Robert May's random matrix approach and the generalised Lotka-Volterra equations. We show that the eigenvalue spectra of random matrices can be used to deduce the stability of dynamically constructed (or `feasible') communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account., Comment: 27 pages, 7 figures

The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rate trajectories. Further, we also compare the stochastic bridges produced using our method to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the paths produced using our method, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that our method can be used to judge the accuracy of the WKB approximation at finite levels of noise.