Showing papers in "Journal of physics in 2022"

Reciprocity, community detection, and link prediction in dynamic networks

Abstract: Abstract Many complex systems change their structure over time, in these cases dynamic networks can provide a richer representation of such phenomena. As a consequence, many inference methods have been generalized to the dynamic case with the aim to model dynamic interactions. Particular interest has been devoted to extend the stochastic block model and its variant, to capture community structure as the network changes in time. While these models assume that edge formation depends only on the community memberships, recent work for static networks show the importance to include additional parameters capturing structural properties, as reciprocity for instance. Remarkably, these models are capable of generating more realistic network representations than those that only consider community membership. To this aim, we present a probabilistic generative model with hidden variables that integrates reciprocity and communities as structural information of networks that evolve in time. The model assumes a fundamental order in observing reciprocal data, that is an edge is observed, conditional on its reciprocated edge in the past. We deploy a Markovian approach to construct the network’s transition matrix between time steps and parameters’ inference is performed with an expectation-maximization algorithm that leads to high computational efficiency because it exploits the sparsity of the dataset. We test the performance of the model on synthetic dynamical networks, as well as on real networks of citations and email datasets. We show that our model captures the reciprocity of real networks better than standard models with only community structure, while performing well at link prediction tasks.

Entropy, free energy, symmetry and dynamics in the brain

Abstract: Abstract Neuroscience is home to concepts and theories with roots in a variety of domains including information theory, dynamical systems theory, and cognitive psychology. Not all of those can be coherently linked, some concepts are incommensurable, and domain-specific language poses an obstacle to integration. Still, conceptual integration is a form of understanding that provides intuition and consolidation, without which progress remains unguided. This paper is concerned with the integration of deterministic and stochastic processes within an information theoretic framework, linking information entropy and free energy to mechanisms of emergent dynamics and self-organization in brain networks. We identify basic properties of neuronal populations leading to an equivariant matrix in a network, in which complex behaviors can naturally be represented through structured flows on manifolds establishing the internal model relevant to theories of brain function. We propose a neural mechanism for the generation of internal models from symmetry breaking in the connectivity of brain networks. The emergent perspective illustrates how free energy can be linked to internal models and how they arise from the neural substrate.

Coincidence complex networks

Abstract: Abstract Complex networks, which constitute the main subject of network science, have been wide and extensively adopted for representing, characterizing, and modeling an ample range of structures and phenomena from both theoretical and applied perspectives. The present work describes the application of the real-valued Jaccard and real-valued coincidence similarity indices for translating generic datasets into networks. More specifically, two data elements are linked whenever the similarity between their respective features, gauged by some similarity index, is greater than a given threshold. Weighted networks can also be obtained by taking these indices as weights. It is shown that the two proposed real-valued approaches can lead to enhanced performance when compared to cosine and Pearson correlation approaches, yielding a detailed description of the specific patterns of connectivity between the nodes, with enhanced modularity. In addition, a parameter α is introduced that can be used to control the contribution of positive and negative joint variations between the considered features, catering for enhanced flexibility while obtaining networks. The ability of the proposed methodology to capture detailed interconnections and emphasize the modular structure of networks is illustrated and quantified respectively to real-world networks, including handwritten letters and raisin datasets, as well as the Caenorhabditis elegans neuronal network. The reported methodology and results pave the way to a significant number of theoretical and applied developments.

Statistical physics of network structure and information dynamics

Abstract: Abstract In the last two decades, network science has proven to be an invaluable tool for the analysis of empirical systems across a wide spectrum of disciplines, with applications to data structures admitting a representation in terms of complex networks. On the one hand, especially in the last decade, an increasing number of applications based on geometric deep learning have been developed to exploit, at the same time, the rich information content of a complex network and the learning power of deep architectures, highlighting the potential of techniques at the edge between applied math and computer science. On the other hand, studies at the edge of network science and quantum physics are gaining increasing attention, e.g., because of the potential applications to quantum networks for communications, such as the quantum Internet. In this work, we briefly review a novel framework grounded on statistical physics and techniques inspired by quantum statistical mechanics which have been successfully used for the analysis of a variety of complex systems. The advantage of this framework is that it allows one to define a set of information-theoretic tools which find widely used counterparts in machine learning and quantum information science, while providing a grounded physical interpretation in terms of a statistical field theory of information dynamics. We discuss the most salient theoretical features of this framework and selected applications to protein–protein interaction networks, neuronal systems, social and transportation networks, as well as potential novel applications for quantum network science and machine learning.

Discrete curvature on graphs from the effective resistance*

Abstract: This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to calculate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on the theory of discrete curvature and its many applications in mathematics, network science, data science and physics.

EEG functional connectivity and deep learning for automatic diagnosis of brain disorders: Alzheimer’s disease and schizophrenia

Abstract: Abstract Mental disorders are among the leading causes of disability worldwide. The first step in treating these conditions is to obtain an accurate diagnosis. Machine learning algorithms can provide a possible solution to this problem, as we describe in this work. We present a method for the automatic diagnosis of mental disorders based on the matrix of connections obtained from EEG time series and deep learning. We show that our approach can classify patients with Alzheimer’s disease and schizophrenia with a high level of accuracy. The comparison with the traditional cases, that use raw EEG time series, shows that our method provides the highest precision. Therefore, the application of deep neural networks on data from brain connections is a very promising method for the diagnosis of neurological disorders.

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Abstract: Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington-Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree $c$ scales as a power $c \propto N^{b}$ ($b < 1$) of the total number $N \gg 1$ of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

Explosive transitions in epidemic dynamics

Abstract: Abstract Standard epidemic models exhibit one continuous, second order phase transition to macroscopic outbreaks. However, interventions to control outbreaks may fundamentally alter epidemic dynamics. Here we reveal how such interventions modify the type of phase transition. In particular, we uncover three distinct types of explosive phase transitions for epidemic dynamics with capacity-limited interventions. Depending on the capacity limit, interventions may (i) leave the standard second order phase transition unchanged but exponentially suppress the probability of large outbreaks, (ii) induce a first-order discontinuous transition to macroscopic outbreaks, or (iii) cause a secondary explosive yet continuous third-order transition. These insights highlight inherent limitations in predicting and containing epidemic outbreaks. More generally our study offers a cornerstone example of a third-order explosive phase transition in complex systems.

Exact time-dependent dynamics of discrete binary choice models

Abstract: Abstract We provide a generic method to find full dynamical solutions to binary decision models with interactions. In these models, agents follow a stochastic evolution where they must choose between two possible choices by taking into account the choices of their peers. We illustrate our method by solving Kirman and Föllmer’s ant recruitment model for any number N of discrete agents and for any choice of parameters, recovering past results found in the limit N → ∞. We then solve extensions of the ant recruitment model for increasing asymmetry between the two choices. Finally, we provide an analytical time-dependent solution to the standard voter model and a semi-analytical solution to the vacillating voter model. Our results show that exact analytical time-dependent solutions can be achieved for discrete choice models without invoking that the number of agents N are continuous or that both choices are symmetric, and additionally show how to practically use the analytics for fast evaluation of the resulting probability distributions.

Which will be your firm’s next technology? Comparison between machine learning and network-based algorithms

Abstract: Abstract We reconstruct the innovation dynamics of about two hundred thousand companies by following their patenting activity for about ten years. We define the technology portfolios of these companies as the set of the technological sectors present in the patents they submit. By assuming that companies move more frequently towards related sectors, we leverage their past activity to build network-based and machine learning algorithms to forecast the future submissions of patents in new sectors. We compare different prediction methodologies using suitable evaluation metrics, showing that tree-based machine learning algorithms outperform the standard methods based on networks of co-occurrences. This methodology can be applied by firms and policymakers to disentangle, given the present innovation activity, the feasible technological sectors from those that are out of reach.

Adaptive survival movement strategy to local epidemic outbreaks in cyclic models

Abstract: Abstract We study the generalised rock-paper-scissors game with five species whose organisms face local epidemic outbreaks. As an evolutionary behavioural survival strategy, organisms of one out of the species move in the direction with more enemies of their enemies to benefit from protection against selection. We consider that each organism scans the environment, performing social distancing instead of agglomerating when perceiving that the density of sick organisms is higher than a tolerable threshold. Running stochastic simulations, we study the interference of the adaptive movement survival strategy in spatial pattern formation, calculating the characteristic length scale of the typical spatial domains inhabited by organisms of each species. We compute how social distancing trigger impacts the chances of an individual being killed in the cyclic game and contaminated by the disease. The outcomes show that the species predominates in the cyclic game because of the organisms’ local adaptation. The territory occupied by the species grows with the proportion of individuals learning to trigger the social distancing tactic. We also show that organisms that perceive large distances more properly execute the adaptive strategy, promptly triggering the social distancing tactic and choosing the correct direction to move. Our findings may contribute to understanding the role of adaptive behaviour when environmental changes threaten biodiversity.

Collective effects and synchronization of demand in real-time demand response

Abstract: Abstract Future energy systems will be dominated by variable renewable power generation and interconnected sectors, leading to rapidly growing complexity. Flexible elements are required to balance the variability of renewable power sources, including backup generators and storage devices, but also flexible consumers. Demand response (DR) aims to adapt the demand to the variable generation, in particular by shifting the load in time. In this article, we provide a detailed statistic analysis of the collective operation of many DR units. We establish and simulate a model for load shifting in response to real-time electricity pricing using local storage systems. We show that DR drives load shifting as desired but also induces strong collective effects that may threaten system stability. The load of individual households synchronizes, leading to extreme demand peaks. We provide a detailed statistical analysis of the grid load and quantify both the likelihood and extent of extreme demand peaks.

Clustering matrices through optimal permutations

Abstract: Abstract Matrices are two-dimensional data structures allowing one to conceptually organize information. For example, adjacency matrices are useful to store the links of a network; correlation matrices are simple ways to arrange gene co-expression data or correlations of neuronal activities. Clustering matrix entries into geometric patterns that are easy to interpret helps us to understand and explain the functional and structural organization of the system components described by matrix entries. Here we introduce a theoretical framework to cluster a matrix into a desired pattern by performing a similarity transformation obtained by solving an optimization problem named optimal permutation problem. On the numerical side, we present an efficient clustering algorithm that can be applied to any type of matrix, including non-normal and singular matrices. We apply our algorithm to the neuronal correlation matrix and the synaptic adjacency matrix of the Caenorhabditis elegans nervous system by performing different types of clustering, using block-diagonal, nested, banded, and triangular patterns. Some of these clustering patterns show their biological significance in that they separate matrix entries into groups that match the experimentally known classification of C. elegans neurons into four broad categories made up of interneurons, motor, sensory, and polymodal neurons.

Reward versus punishment: averting the tragedy of the commons in eco-evolutionary dynamics

Abstract: Abstract We consider an unstructured population of individuals who are randomly matched in an underlying population game in which the payoffs depend on the evolving state of the common resource exploited by the population. There are many known mechanisms for averting the overexploitation (tragedy) of the (common) resource. Probably one of the most common mechanism is reinforcing cooperation through rewards and punishments. Additionally, the depleting resource can also provide feedback that reinforces cooperation. Thus, it is an interesting question that how reward and punishment comparatively fare in averting the tragedy of the common (TOC) in the game-resource feedback evolutionary dynamics. Our main finding is that, while averting the TOC completely, rewarding cooperators cannot get rid of all the defectors, unlike what happens when defectors are punished; and as a consequence, in the completely replete resource state, the outcome of the population game can be socially optimal in the presence of the punishment but not so in the presence of the reward.

The curvature effect in Gaussian random fields

Abstract: Abstract Random field models are mathematical structures used in the study of stochastic complex systems. In this paper, we compute the shape operator of Gaussian random field manifolds using the first and second fundamental forms (Fisher information matrices). Using Markov chain Monte Carlo techniques, we simulate the dynamics of these random fields and compute the Gaussian, mean and principal curvatures of the parametric space, analyzing how these quantities change along dynamics exhibiting phase transitions. During the simulations, we have observed an unexpected phenomenon that we called the curvature effect , which indicates that a highly asymmetric geometric deformation happens in the underlying parametric space when there are significant increase/decrease in the system’s entropy. When the system undergoes a phase transition from randomness to clustered behavior the curvature is smaller than that observed in the reverse phase transition. This asymmetric pattern relates to the emergence of hysteresis phenomenon, leading to an intrinsic arrow of time along the random field dynamics.

A random matrix theory approach to damping in deep learning

Abstract: Abstract We conjecture that the inherent difference in generalisation between adaptive and non-adaptive gradient methods in deep learning stems from the increased estimation noise in the flattest directions of the true loss surface. We demonstrate that typical schedules used for adaptive methods (with low numerical stability or damping constants) serve to bias relative movement towards flat directions relative to sharp directions, effectively amplifying the noise-to-signal ratio and harming generalisation. We further demonstrate that the numerical damping constant used in these methods can be decomposed into a learning rate reduction and linear shrinkage of the estimated curvature matrix. We then demonstrate significant generalisation improvements by increasing the shrinkage coefficient, closing the generalisation gap entirely in both logistic regression and several deep neural network experiments. Extending this line further, we develop a novel random matrix theory based damping learner for second order optimisers inspired by linear shrinkage estimation. We experimentally demonstrate our learner to be very insensitive to the initialised value and to allow for extremely fast convergence in conjunction with continued stable training and competitive generalisation. We also find that our derived method works well with adaptive gradient methods such as Adam.

Atlas of urban scaling laws

Abstract: Abstract Accurate estimates of the urban fractal dimension D f are obtained by implementing the detrended moving average algorithm on high-resolution multi-spectral satellite images from the WorldView2 (WV2) database covering the largest European cities. Fractal dimension D f varies between 1.65 and 1.90 with high values for highly urbanised urban sectors and low ones for suburban and peripheral ones. Based on recently proposed models, the values of the fractal dimension D f are checked against the exponents β s and β i of the scaling law Y ∼ N β , respectively for socio-economic and infrastructural variables Y , with N the population size. The exponents β s and β i are traditionally derived as if cities were zero-dimensional objects, with the relevant feature Y related to a single homogeneous population value N , thus neglecting the microscopic heterogeneity of the urban structure. Our findings go beyond this limit. High sensitive and repeatable satellite records yield robust local estimates of the urban scaling exponents. Furthermore, the work discusses how to discriminate among different scaling theories, shedding light on the debated issue of scaling phenomena contradictory perspectives and pave paths to a more systematic adoption of the complex system science methods to urban landscape analysis.

Criticality and network structure drive emergent oscillations in a stochastic whole-brain model

Abstract: Abstract Understanding the relation between the structure of brain networks and their functions is a fundamental open question. Simple models of neural activity based on real anatomical networks have proven to be effective in describing features of whole-brain spontaneous activity when tuned at their critical point. In this work, we show that structural networks are indeed a crucial ingredient in the emergence of collective oscillations in a whole-brain stochastic model at criticality. We study analytically a stochastic Greenberg–Hastings cellular automaton in the mean-field limit, showing that it undergoes an abrupt phase transition with a bistable region. In particular, no global oscillations emerge in this limit. Then, we show that by introducing a network structure in the homeostatic normalization regime, the bistability may be disrupted, and the transition may become smooth. Concomitantly, through an interplay between network topology and weights, a large peak in the power spectrum appears around the transition point, signaling the emergence of collective oscillations. Hence, both the structure of brain networks and criticality are fundamental in driving the collective responses of whole-brain stochastic models.

Subduing always defecting mutants by multiplayer reactive strategies: non-reciprocity versus generosity

Abstract: Abstract A completely non-generous and reciprocal population of players can create a robust cooperating state that cannot be invaded by always defecting free riders if the interactions among players are repeated for long enough. However, strict non-generosity and strict reciprocity are ideal concepts, and may not even be desirable sometimes. Therefore, to what extent generosity or non-reciprocity can be allowed while still not be swamped by the mutants, is a natural question. In this paper, we not only ask this question but furthermore ask how generosity comparatively fares against non-reciprocity in this context. For mathematical concreteness, we work within the framework of multiplayer repeated prisoner’s dilemma game with reactive strategies in a finite and an infinite population; and explore the aforementioned questions through the effects of the benefit to cost ratio, the interaction group size, and the population size.

Influence maximization under limited network information: seeding high-degree neighbors

Abstract: Abstract The diffusion of information, norms, and practices across a social network can be initiated by compelling a small number of seed individuals to adopt first. Strategies proposed in previous work either assume full network information or a large degree of control over what information is collected. However, privacy settings on the Internet and high non-response in surveys often severely limit available connectivity information. Here we propose a seeding strategy for scenarios with limited network information: Only the degrees and connections of some random nodes are known. This new strategy is a modification of ‘random neighbor sampling’ (or ‘one-hop’) and seeds the highest-degree neighbors of randomly selected nodes. Simulating a fractional threshold model, we find that this new strategy excels in networks with heavy tailed degree distributions such as scale-free networks and large online social networks. It outperforms the conventional one-hop strategy even though the latter can seed 50% more nodes, and other seeding possibilities including pure high-degree seeding and clustered seeding.

Simple mechanism rules the dynamics of volleyball

Abstract: Abstract In volleyball games, we define a rally as the succession of events observed since the ball is served until one of the two teams on the court scores the point. In this process, athletes evolve in response to physical and information constraints, spanning several spatiotemporal scales and interplaying co-adaptively with the environment. Aiming to study the emergence of complexity in this system, we carried out a study focused on three steps: data collection, data analysis, and modeling. First, we collected data from 20 high-level professional volleyball games. Then we conducted a data-driven analysis from where we identified fundamental insights that we used to define a parsimonious stochastic model for the dynamics of the game. On these bases, we show that it is possible to give a closed-form expression for the probability that the players perform n hits in a rally using only two stochastic variables. Our results fully agree with the empirical observations and represent a new advance in the comprehension of team-sports competition complexity and dynamics.

Complex systems for the most vulnerable

Abstract: Abstract In a rapidly changing world, facing an increasing number of socioeconomic, health and environmental crises, complexity science can help us to assess and quantify vulnerabilities, and to monitor and achieve the UN sustainable development goals. In this perspective, we provide three exemplary use cases where complexity science has shown its potential: poverty and socioeconomic inequalities, collective action for representative democracy, and computational epidemic modeling. We then review the challenges and limitations related to data, methods, capacity building, and, as a result, research operationalization. We finally conclude with some suggestions for future directions, urging the complex systems community to engage in applied and methodological research addressing the needs of the most vulnerable.

Social network heterogeneity benefits individuals at the expense of groups in the creation of innovation

Abstract: Abstract Innovation is fundamental for development and provides a competitive advantage for societies. It is the process of creating more complex technologies, ideas, or protocols from existing ones. While innovation may be created by single agents (i.e. individuals or organisations), it is often a result of social interactions between agents exchanging and combining complementary expertise and perspectives. The structure of social networks impacts this knowledge exchange process. To study the role of social network structures on the creation of new technologies, we design an evolutionary mechanistic model combining self-creation and social learning. We find that social heterogeneity allows agents to leverage the benefits of diversity and to develop technologies of higher complexity. Social heterogeneity, however, reduces the group ability to innovate. Not only the social structure but also the openness of agents to collaborate affect innovation. We find that interdisciplinary interactions lead to more complex technologies benefiting the entire group but also increase the inequality in the innovation output. Lower openness to interdisciplinary collaborations may be compensated by a higher ability to collaborate with multiple peers, but low openness also neutralises the intrinsic benefits of network heterogeneity. Our findings indicate that social network heterogeneity has contrasting effects on microscopic (local) and macroscopic (group) levels, suggesting that the emergence of innovation leaders may suppress the overall group performance.

Visibility graphs of animal foraging trajectories

Abstract: Abstract The study of self-propelled particles is a fast growing research topic where biological inspired movement is increasingly becoming of much interest. A relevant example is the collective motion of social insects, whose variety and complexity offer fertile grounds for theoretical abstractions. It has been demonstrated that the collective motion involved in the searching behaviour of termites is consistent with self-similarity, anomalous diffusion and Lévy walks. In this work we use visibility graphs—a method that maps time series into graphs and quantifies the signal complexity via graph topological metrics—in the context of social insects foraging trajectories extracted from experiments. Our analysis indicates that the patterns observed for isolated termites change qualitatively when the termite density is increased, and such change cannot be explained by jamming effects only, pointing to collective effects emerging due to non-trivial foraging interactions between insects as the cause. Moreover, we find that such onset of complexity is maximised for intermediate termite densities.

Multiplex mobility network and metapopulation epidemic simulations of Italy based on open data

Abstract: The patterns of human mobility play a key role in the spreading of infectious diseases and thus represent a key ingredient of epidemic modeling and forecasting. Unfortunately, as the Covid-19 pandemic has dramatically highlighted, for the vast majority of countries there is no availability of granular mobility data. This hinders the possibility of developing computational frameworks to monitor the evolution of the disease and to adopt timely and adequate prevention policies. Here we show how this problem can be addressed in the case study of Italy. We build a multiplex mobility network based solely on open data, and implement a SIR metapopulation model that allows scenario analysis through data-driven stochastic simulations. The mobility flows that we estimate are in agreement with real-time proprietary data from smartphones. Our modeling approach can thus be useful in contexts where high-resolution mobility data is not available.

Local multifractality in urban systems—the case study of housing prices in the greater Paris region

Abstract: Abstract Even though the study of fractal and multifractal properties has now become an established approach for statistical urban data analysis, the accurate multifractal characterisation of smaller, district-scale spatial units is still a somewhat challenging task. The latter issue is key for understanding complex spatial correlations within urban regions while the methodological challenge can be mainly attributed to inhomogeneous data availability over their territories. We demonstrate how the approach proposed here for the multifractal analysis of irregular marked point processes is able to estimate local self-similarity and intermittency exponents in a satisfactory manner via combining methods from classical multifractal and geographical analysis. With the aim of emphasizing general applicability, we first introduce the procedure on synthetic data using a multifractal random field as mark superposed on two distinct spatial distributions. We go on to illustrate the methodology on the example of home prices in the greater Paris region, France. In the context of complex urban systems, our findings proclaim the need for separately tackling processes on the geolocation (support) and any attached value (mark, e.g. home prices) of geospatial data points in an attempt to fully describe the phenomenon under observation. In particular, the results are indicators of the strength of global and local spatial dependency in the housing price structure and how these build distinct layered patterns within and outside of the municipal boundary. The derived properties are of potential urban policy and strategic planning relevance for the timely identification of local vulnerabilities while they are also intended to be combinable with existing price indices in the regional economics context.

Spectral properties of the generalized diluted Wishart ensemble

Abstract: Abstract The celebrated Marčenko–Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as F ≡ 1 2 d X Y T + Y X T , where X and Y are diluted N × P rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution P ( x i μ , y i μ ) = d N ϱ ( x i μ , y i μ ) + 1 − d N δ x i μ , 0 δ y i μ , 0 , with the probability density ϱ ( x , y ) controlling the correlation between the matrices entries of X and Y . Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph d , the rectangularity of the matrices α = N / P , and the degree of correlation of the matrix entries via the density ϱ ( x , y ). Finally, we compare our findings to numerical diagonalisation showing excellent agreement.

The training response law explains how deep neural networks learn

Abstract: Abstract Deep neural network is the widely applied technology in this decade. In spite of the fruitful applications, the mechanism behind that is still to be elucidated. We study the learning process with a very simple supervised learning encoding problem. As a result, we found a simple law, in the training response, which describes neural tangent kernel. The response consists of a power law like decay multiplied by a simple response kernel. We can construct a simple mean-field dynamical model with the law, which explains how the network learns. In the learning, the input space is split into sub-spaces along competition between the kernels. With the iterated splits and the aging, the network gets more complexity, but finally loses its plasticity.

Coexistence of coordination and anticoordination in nonlinear public goods game

Abstract: Abstract There is a plethora of instances of interactions between players, in both biological and socio-economical context, that can be modeled as the paradigmatic PGG. However, in such interactions, arguably the PGG is often nonlinear in nature. This is because the increment in benefit generated, owing to additional cost contributed by the players, is realistically seldom linear. Furthermore, sometimes a social good is created due to interspecific interactions, e.g. in cooperative hunting by animals of two different species. In this paper, we study the evolutionary dynamics of a heterogenous population of cooperators and defectors playing nonlinear PGG; here we define heterogenous population as the one composed of distinct subpopulations with interactions among them. We employ the replicator equations for this investigation, and present the non-trivial effects of nonlinearity and size of the groups involved in the game. We report the possibility of discoordination, and coexistence of coordination and anti-coordination in such nonlinear PGG.

Klaus Hasselmann and Economics*

Abstract: Abstract Klaus Hasselmann has earned the 2021 Nobel Prize in physics for his breakthroughs in analysing the climate system as a complex physical system. Since decades, as a leading climate scientist he is aware of the need for creative cooperation between climate scientists and researchers from other fields, especially economics. To facilitate such cooperation, he has designed a productive research program for economic analysis in view of climate change. Without blurring the differences between economics and physics, the Hasselmann program stresses the complexities of today’s economy. This includes the importance of heterogeneous actors and different time scales, of making major uncertainties explicit and bringing researchers and practitioners in close interaction. The program has triggered decades of collaborative research, especially in the network of the Global Climate Forum, that he has founded for this purpose. Research inspired by Hasselmann’s innovative ideas has led to a farewell to outdated economic approaches: single-equilibrium models, a single constant discount rate, framing the climate challenge as a kind of prisoner’s dilemma and framing it as a problem of scarcity requiring sacrifices from the majority of today’s population. Instead of presenting the climate problem as the ultimate apocalyptic narrative, he sees it as a challenge to be mastered. To meet this challenge requires careful research in order to identify underutilisation of human, technical and social capacities that offer the keys to a climate friendly world economy. Climate neutrality may then be achieved by activating these capacities through investment-oriented climate strategies, designed and implemented by different actors both in industrialised and developing countries. The difficulties to bring global greenhouse gas emissions down to net zero are enormous; the Hasselmann program holds promise of significant advances in this endeavour.