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The management of common-pool resources is a complex challenge due to the risk of overexploitation and the tragedy of the commons. A novel framework has been introduced to address this issue, focusing on the coevolutionary relationship between human behavior and common-pool resources within a human-environment system. However, the impact of the Allee effect on the coevolution and its resource sustainability is still unexplored. The Allee effect, a biological phenomenon characterized by a correlation between resource availability and growth rate, is a fundamental attribute of numerous natural resources. In this paper, we introduce two coevolutionary models of resource and strategy under replicator dynamics and knowledge feedback by applying the Allee effect to the common-pool resources within human-environment system. These models encapsulate various facets of resource dynamics and the players' behavior, such as resource growth function, the extraction rates, and the strategy update rules. We find that the Allee effect can induce bi-stability and critical transition, leading to either sustainable or unsustainable outcomes depending on the initial condition and parameter configuration. We demonstrate that knowledge feedback enhances the resilience and sustainability of the coevolving system, and these results advances the understanding of human-environment system and management of common-pool resources.

Games with environmental feedback have become a crucial area of study across various scientific domains, modelling the dynamic interplay between human decisions and environmental changes, and highlighting the consequences of our choices on natural resources and biodiversity. In this work, we propose a co-evolutionary model for human-environment systems that incorporates the effects of knowledge feedback and social interaction on the sustainability of common pool resources. The model represents consumers as agents who adjust their resource extraction based on the resource's state. These agents are connected through social networks, where links symbolize either affinity or aversion among them. The interplay between social dynamics and resource dynamics is explored, with the system's evolution analyzed across various network topologies and initial conditions. We find that knowledge feedback can independently sustain common pool resources. However, the impact of social interactions on sustainability is dual-faceted: it can either support or impede sustainability, influenced by the network's connectivity and heterogeneity. A notable finding is the identification of a critical network mean degree, beyond which a depletion/repletion transition parallels an absorbing/active state transition in social dynamics, i.e., individual agents and their connections are/are not prone to being frozen in their social states. Furthermore, the study examines the evolution of the social network, revealing the emergence of two polarized groups where agents within each community have the same affinity. Comparative analyses using Monte-Carlo simulations and rate equations are employed, along with analytical arguments, to reinforce the study's findings. The model successfully captures how information spread and social dynamics may impact the sustanebility of common pool resource.

Common-pool resource governance requires users to cooperate and avoid overexploitation, but defection and free-riding often undermine cooperation. We model a human-environmental system that integrates dynamics of resource and users' strategies. The resource follows a logistic function that depends on natural growth rate, carrying capacity, and extraction rates of cooperators and defectors. The users' strategies evolve according to different processes that capture effects of payoff, resource, and noise. We analyze the feedback between resource availability and strategic adaptation, and explores the conditions for the emergence and maintenance of cooperation. We find different processes lead to different regimes of equilibrium solutions and resource levels depending on the parameter configuration and initial conditions. We also show that some processes can enhance the sustainability of the resource by making the users more responsive to the resource scarcity. The paper advances the understanding of human-environmental system and offers insights for resource governance policies and interventions.

Sustainability of common-pool resources hinges on the interplay between human and environmental systems. However, there is still a lack of a novel and comprehensive framework for modelling extraction of common-pool resources and cooperation of human agents that can account for different factors that shape the system behavior and outcomes. In particular, we still lack a critical value for ensuring resource sustainability under different scenarios. In this paper, we present a novel framework for studying resource extraction and cooperation in human-environmental systems for common-pool resources. We explore how different factors, such as resource availability and conformity effect, influence the players' decisions and the resource outcomes. We identify critical values for ensuring resource sustainability under various scenarios. We demonstrate the observed phenomena are robust to the complexity and assumptions of the models and discuss implications of our study for policy and practice, as well as the limitations and directions for future research.

Complex systems are ubiquitous in nature and engineering, but their analysis and control are hampered by their high dimensionality and the influence of various factors on their dynamics. Dimensionality reduction aims to find a low-dimensional representation of the complex system that preserves its essential features and reveals its underlying mechanisms and long-term dynamics. However, most existing methods for dimensionality reduction assume deterministic systems, while many real-world systems exhibit stochasticity. Here, we develop a general analytical framework for dimensionality reduction of stochastic complex dynamical networks that can map a high-dimensional system with stochastic terms to a low-dimensional effective system with a single effective state variable and few effective parameters. The effective parameters are those that determine the network's dynamical behavior and are associated with specific system states. The effective equation is a low-dimensional representation of the original stochastic complex dynamical network that preserves its essential dynamical features. The framework also allows us to analyze the dynamic behavior and potential convergence of the stochastic complex dynamical network by using the standard deviation of the effective equation., Comment: 11 pages, 2 figures

One of the outstanding problems in complexity science and dynamical system theory is understanding the dynamic behavior of high-dimensional networked systems and their susceptibility to transitions to undesired states. Because of varied interactions, large number of parameters and different initial conditions, the study is extremely difficult and existing methods can be applied only to continuous-time systems. Here we propose an analytical framework for collapsing N-dimensional discrete-time systems into a S+1-dimensional manifold as a function of S effective parameters with S << N. Specifically, we provide a quantitative prediction of the quality of the low-dimensional collapse. We test our framework on a variety of real-world complex systems showing its good performance and correctly identify the regions in the parameter space corresponding to the system's transitions. Our work offers an analytical tool to reduce dimensionality of discrete-time networked systems that can be applied to a broader set of systems and dynamics.

The sustainable use of common-pool resources (CPRs) is a major environmental governance challenge because of their possible over-exploitation. Research in this field has overlooked the feedback between user decisions and resource dynamics. Here we develop an online game to perform a set of experiments in which users of the same CPR decide on their individual harvesting rates, which in turn depend on the resource dynamics. We show that, if users share common goals, a high level of self-organized cooperation emerges, leading to long-term resource sustainability. Otherwise, selfish/individualistic behaviors lead to resource depletion ("Tragedy of the Commons"). To explain these results, we develop an analytical model of coupled resource-decision dynamics based on optimal control theory and show how this framework reproduces the empirical results.

In our recent article (Tu et al., Warnings and caveats in brain controllability, arXiv:1705.08261) we provided quantitative evidence to show that there are warnings and caveats in the way Gu and collaborators (Gu et al. Controllability of structural brain networks. Nature communications 6 (2015): 8414) define brain controllability. The comment by Pasqualetti et al. (Pasqualetti et al. RE: Warnings and Caveats in Brain Controllability. NeuroImage 297 (2019), 586-588) confirms the need to go beyond the methodology and approach presented in Gu et al. original work. In fact, they recognize that the source of confusion is due to the fact that assessing controllability via numerical analysis typically leads to ill-conditioned problems, and thus often generates results that are difficult to interpret. This is indeed the first warning we discussed: our work was not meant to prove that brain networks are not controllable from one node, rather we wished to highlight that the one node controllability framework and all consequent results were not properly justified based on the methodology presented in Gu et al. We used in our work the same method of Gu et al. not because we believe it is the best methodology, but because we extensively investigated it with the aim of replicating, testing and extending their results. And the warning and caveats we have proposed are the results of this investigation., Comment: 3 pages, 1 figure

The year 2017 saw the rise and fall of the crypto-currency market, followed by high variability in the price of all crypto-currencies. In this work, we study the abrupt transition in crypto-currency residuals, which is associated with the critical transition (the phenomenon of critical slowing down) or the stochastic transition phenomena. We find that, regardless of the specific crypto-currency or rolling window size, the autocorrelation always fluctuates around a high value, while the standard deviation increases monotonically. Therefore, while the autocorrelation does not display signals of critical slowing down, the standard deviation can be used to anticipate critical or stochastic transitions. In particular, we have detected two sudden jumps in the standard deviation, in the second quarter of 2017 and at the beginning of 2018, which could have served as early warning signals of two majors price collapses that have happened in the following periods. We finally propose a mean-field phenomenological model for the price of crypto-currency to show how the use of the standard deviation of the residuals is a better leading indicator of the collapse in price than the time series' autocorrelation. Our findings represent a first step towards a better diagnostic of the risk of critical transition in the price and/or volume of crypto-currencies., Comment: 14 pages, 5 figures, 1 table

Empirical observations show that ecological communities can have a huge number of coexisting species, also with few or limited number of resources. These ecosystems are characterized by multiple type of interactions, in particular displaying cooperative behaviors. However, standard modeling of population dynamics based on Lotka-Volterra type of equations predicts that ecosystem stability should decrease as the number of species in the community increases and that cooperative systems are less stable than communities with only competitive and/or exploitative interactions. Here we propose a stochastic model of population dynamics, which includes exploitative interactions as well as cooperative interactions induced by cross-feeding. The model is exactly solved and we obtain results for relevant macro-ecological patterns, such as species abundance distributions and correlation functions. In the large system size limit, any number of species can coexist for a very general class of interaction networks and stability increases as the number of species grows. For pure mutualistic/commensalistic interactions we determine the topological properties of the network that guarantee species coexistence. We also show that the stationary state is globally stable and that inferring species interactions through species abundance correlation analysis may be misleading. Our theoretical approach thus show that appropriate models of cooperation naturally leads to a solution of the long-standing question about complexity-stability paradox and on how highly biodiverse communities can coexist., Comment: Incorrectdly posted twice. Current version arXiv:1708.03154

Empirical observations show that ecological communities can have a huge number of coexisting species, also with few or limited number of resources. These ecosystems are characterized by multiple type of interactions, in particular displaying cooperative behaviors. However, standard modeling of population dynamics based on Lotka-Volterra type of equations predicts that ecosystem stability should decrease as the number of species in the community increases and that cooperative systems are less stable than communities with only competitive and/or exploitative interactions. Here we propose a stochastic model of population dynamics, which includes exploitative interactions as well as cooperative interactions induced by cross-feeding. The model is exactly solved and we obtain results for relevant macro-ecological patterns, such as species abundance distributions and correlation functions. In the large system size limit, any number of species can coexist for a very general class of interaction networks and stability increases as the number of species grows. For pure mutualistic/commensalistic interactions we determine the topological properties of the network that guarantee species coexistence. We also show that the stationary state is globally stable and that inferring species interactions through species abundance correlation analysis may be misleading. Our theoretical approach thus show that appropriate models of cooperation naturally leads to a solution of the long-standing question about complexity-stability paradox and on how highly biodiverse communities can coexist., Comment: 25 pages, 10 figures

In this work we challenge the main conclusions of Gu et al work (Controllability of structural brain networks. Nature communications 6, 8414, doi:10.1038/ncomms9414, 2015) on brain controllability. Using the same methods and analyses on four datasets we find that the minimum set of nodes to control brain networks is always larger than one. We also find that the relationships between the average/modal controllability and weighted degrees also hold for randomized data and the there are not specific roles played by Resting State Networks in controlling the brain. In conclusion, we show that there is no evidence that topology plays specific and unique roles in the controllability of brain networks. Accordingly, Gu et al. interpretation of their results, in particular in terms of translational applications (e.g. using single node controllability properties to define target region(s) for neurostimulation) should be revisited. Though theoretically intriguing, our understanding of the relationship between controllability and structural brain network remains elusive., Comment: 9 pages, 1 Figure, 1 Table

Recently, a theoretical framework aimed at separating the roles of dynamics and topology in multi-dimensional systems has been developed (Gao et al, \textit{Nature}, Vol 530:307 (2016)). The validity of their method is assumed to hold depending on two main hypothesis: $(i)$ The network determined by the the interaction between pairs of nodes has negligible degree correlations; $(ii)$ The node activities are uniform across nodes on both the drift and pair-wise interaction functions. Moreover, the authors consider only positive (mutualistic) interactions. Here we show the conditions proposed by Gao and collaborators are neither sufficient nor necessary to guarantee that their method works in general, and validity of their results are not independent of the model chosen within the class of dynamics they considered. Indeed we find that a new condition poses effective limitations to their framework and we provide quantitative predictions of the quality of the one dimensional collapse as a function of the properties of interaction networks and stable dynamics using results from random matrix theory. We also find that multi-dimensional reduction may work also for interaction matrix with a mixture of positive and negative signs, opening up application of the framework to food-webs, neuronal networks and social/economic interactions.

The easy access to large data sets has allowed for leveraging methodology in network physics and complexity science to disentangle patterns and processes directly from the data, leading to key insights in the behavior of systems. Here we use to country specific food production data to study binary and weighted topological properties of the bipartite country-food production matrix. This country-food production matrix can be: 1) transformed into overlap matrices which embed information regarding shared production of products among countries, and or shared countries for individual products, 2) identify subsets of countries which produce similar commodities or subsets of commodities shared by a given country allowing for visualization of correlations in large networks, and 3) used to rank country's fitness (the ability to produce a diverse array of products weighted on the type of food commodities) and food specialization (quantified on the number of countries producing that food product weighted on their fitness). Our results show that, on average, countries with high fitness producing highly specialized food commodities also produce low specialization goods, while nations with low fitness producing a small basket of diverse food products, typically produce low specialized food commodities.