Wealth condensation in a multiplicative random asset exchange model

Random Asset Exchange (RAE) models, despite a number of simplifying assumptions, serve the purpose of establishing direct relationships between microscopic exchange mechanisms and observed economical data. In this work a conservative multiplicative RAE model is discussed in which, at each timestep, two agents “bet” for a fraction f of the poorest agent's wealth. When the poorest agent wins the bet with probability p, we show that, in a well defined region of the (p,f) phase space, there is wealth condensation. This means that all wealth ends up owned by only one agent, in the long run. We derive the condensation conditions analytically by two different procedures, and find results in accordance with previous numerical estimates. In the non-condensed phase, the equilibrium wealth distribution is a power law for small wealths. The associated exponent is derived analytically and it is found that it tends to -1 on the condensation interface. I turns out that wealth condensation happens also for values of p much larger than 0.5, that is under microscopic exchange rules that, apparently, favor the poor. We argue that the observed “rich get richer” effect is enhanced by the multiplicative character of the dynamics.     

Statistical mechanics of money

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Boltzmann-Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Boltzmann-Gibbs law does not hold. The instantaneous distribution of money among the agents of a system should not be confused with the distribution of wealth. The latter also includes material wealth, which is not conserved, and thus may have a different (e.g. power-law) distribution. Received 22 June 2000     

Class formation in a social network with asset exchange

We study two kinds of economic exchanges, additive and multiplicative, in a system of N agents. The work is divided into two parts. In the first one, the agents are free to interact with each other. The system evolves to a Boltzmann-Gibbs distribution with additive exchange and condenses with a multiplicative one. If bankruptcy is introduced, both types of exchange lead to condensation. Condensation times have been studied. In the second part, the agents are placed in a social network. We analyze the behavior of wealth distributions in time, and the formation of economic classes is observed for certain values of network connectivity.     

Laplace transform analysis of a multiplicative asset transfer model

We analyze a simple asset transfer model in which the transfer amount is a fixed fraction f of the giver’s wealth. The model is analyzed in a new way by Laplace transforming the master equation, solving it analytically and numerically for the steady-state distribution, and exploring the solutions for various values of f∈(0,1). The Laplace transform analysis is superior to agent-based simulations as it does not depend on the number of agents, enabling us to study entropy and inequality in regimes that are costly to address with simulations. We demonstrate that Boltzmann entropy is not a suitable (e.g. non-monotonic) measure of disorder in a multiplicative asset transfer system and suggest an asymmetric stochastic process that is equivalent to the asset transfer model.     

A self-adapting herding model: The agent judge-abilities influence the dynamic behaviors

We propose a self-adapting herding model, in which the financial markets consist of agent clusters with different sizes and market desires. The ratio of successful exchange and merger depends on the volatility of the market and the market desires of the agent clusters. The desires are assigned in term of the wealth of the agent clusters when they merge. After an exchange, the beneficial cluster’s desire keeps on the same, the losing one’s desire is altered which is correlative with the agent judge-ability. A parameter R is given to all agents to denote the judge-ability. The numerical calculation shows that the dynamic behaviors of the market are influenced distinctly by R, which includes the exponential magnitudes of the probability distribution of sizes of the agent clusters and the volatility autocorrelation of the returns, the intensity and frequency of the volatility.     

Motif structure and cooperation in real-world complex networks

Networks of dynamical nodes serve as generic models for real-world systems in many branches of science ranging from mathematics to physics, technology, sociology and biology. Collective behavior of agents interacting over complex networks is important in many applications. The cooperation between selfish individuals is one of the most interesting collective phenomena. In this paper we address the interplay between the motifs’ cooperation properties and their abundance in a number of real-world networks including yeast protein-protein interaction, human brain, protein structure, email communication, dolphins’ social interaction, Zachary karate club and Net-science coauthorship networks. First, the amount of cooperativity for all possible undirected subgraphs with three to six nodes is calculated. To this end, the evolutionary dynamics of the Prisoner’s Dilemma game is considered and the cooperativity of each subgraph is calculated as the percentage of cooperating agents at the end of the simulation time. Then, the three- to six-node motifs are extracted for each network. The significance of the abundance of a motif, represented by a Z-value, is obtained by comparing them with some properly randomized versions of the original network. We found that there is always a group of motifs showing a significant inverse correlation between their cooperativity amount and Z-value, i.e. the more the Z-value the less the amount of cooperativity. This suggests that networks composed of well-structured units do not have good cooperativity properties.     

Quasi-particle excitations around a pair of half-quantum vortices in p- wave superconductors

Triplet superconductors such as Sr₂RuO₄ and Na_xCoO₂⋅yH₂O are now found to be p-wave (p_x ± ip_y) or f-wave ((p_x ± ip_y) cos cp_z) superconductors. It was phenomenologically suggested that in these p-wave or f-wave superconductors, a pair of half-quantum vortices (HQVs) becomes stable. Using the Bogoliubov-de Gennes equation, previously we have analyzed quasi-particle excitations around an HQV at one end of a d-soliton for simplicity. In next study, we will investigate the stability of the pair of HQV’s, which are connected by the d-soliton. For this purpose, we have developed a new numerical method to solve the Bogoliubov-de Gennes equation for two vortices state, using Mathieu functions.     

Dynamics of bloggers’ communities: Bipartite networks from empirical data and agent-based modeling

We present an analysis of the empirical data and the agent-based modeling of the emotional behavior of users on the Web portals where the user interaction is mediated by posted comments, like Blogs and Diggs. We consider the dataset of discussion-driven popular Diggs, in which all comments are screened by machine-learning emotion detection in the text, to determine positive and negative valence (attractiveness and aversiveness) of each comment. By mapping the data onto a suitable bipartite network, we perform an analysis of the network topology and the related time-series of the emotional comments. The agent-based model is then introduced to simulate the dynamics and to capture the emergence of the emotional behaviors and communities. The agents are linked to posts on a bipartite network, whose structure evolves through their actions on the posts. The emotional states (arousal and valence) of each agent fluctuate in time, subject to the current contents of the posts to which the agent is exposed. By an agent’s action on a post its current emotions are transferred to the post. The model rules and the key parameters are inferred from the considered empirical data to ensure their realistic values and mutual consistency. The model assumes that the emotional arousal over posts drives the agent’s action. The simulations are preformed for the case of constant flux of agents and the results are analyzed in full analogy with the empirical data. The main conclusions are that the emotion-driven dynamics leads to long-range temporal correlations and emergent networks with community structure, that are comparable with the ones in the empirical system of popular posts. In view of pure emotion-driven agents actions, this type of comparisons provide a quantitative measure for the role of emotions in the dynamics on real blogs. Furthermore, the model reveals the underlying mechanisms which relate the post popularity with the emotion dynamics and the prevalence of negative emotions (critique). We also demonstrate how the community structure is tuned by varying a relevant parameter in the model. All data used in these works are fully anonymized.     

A multi-agent dynamic model based on different kinds of bequests

We investigate how wealth transfer that happens at the end of an agent’s life affects its final distribution based on a multi-agent dynamic model. We discuss two kinds of wealth transfers: to a single agent and to charities. The first kind of bequest is common in our realistic world and is always regarded by the public as unequal inheritance. The bequests to charities will be gathered and then equally redistributed among the survivors in our model. We find that when all the decedents choose the second kind of bequest, the final distribution is the Gibbs exponential function. When all the decedents choose the first kind of bequest, the result is condensation that a single individual accumulates all the available wealth. When an increasing portion of decedents choose the one-heir bequests, the exponential distribution evolves towards a power law shape (accompanied by deteriorating inequality). This shape firstly appears from the intermediate range of wealth and extends towards the top end of the simulated distribution, while the distribution remains exponential for high values of the wealth. At the same time, the Gini coefficient increases and the wealth accumulation becomes serious. At last, we analyze the source of the inequality. We find that not only unequal inheritances, but also equal division of the charity’s wealth can relatively contribute to an inequality of wealth distribution.     

Model of wealth and goods dynamics in a closed market

A simple computer simulation model of a closed market on a fixed network with free flow of goods and money is introduced. The model contains only two variables: the amount of goods and money beside the size of the system. An initially flat distribution of both variables is presupposed. We show that under completely random rules, i.e. through the choice of interacting agent pairs on the network and of the exchange rules that the market stabilizes in time and shows diversification of money and goods. We also indicate that the difference between poor and rich agents increases for small markets, as well as for systems in which money is steadily deduced from the market through taxation. It is also found that the price of goods decreases when taxes are introduced, likely due to the less availability of money.     

Emergence of power-law in a market with mixed models

We investigate the problem of wealth distribution from the viewpoint of asset exchange. Robust nature of Pareto's law across economies, ideologies and nations suggests that this could be an outcome of trading strategies. However, the simple asset exchange models fail to reproduce this feature. A Yardsale (YS) model in which amount put on the bet is a fraction of minimum of the two players leads to condensation of wealth in hands of some agent while theft and fraud (TF) model in which the amount to be exchanged is a fraction of loser's wealth leads to an exponential distribution of wealth. We show that if we allow few agents to follow a different model than others, i.e., there are some agents following TF model while rest follow YS model, it leads to distribution with power-law tails. Similar effect is observed when one carries out transactions for a fraction of one's wealth using TF model and for the rest YS model is used. We also observe a power-law tail in wealth distribution if we allow the agents to follow either of the models with some probability.     

Adaptive cyclically dominating game on co-evolving networks: numerical and analytic results

A co-evolving and adaptive Rock (R)–Paper (P)–Scissors (S) game (ARPS) in which an agent uses one of three cyclically dominating strategies is proposed and studied numerically and analytically. An agent takes adaptive actions to achieve a neighborhood to his advantage by rewiring a dissatisfying link with a probability p or switching strategy with a probability 1 - p. Numerical results revealed two phases in the steady state. An active phase for p < p _c has one connected network of agents using different strategies who are continually interacting and taking adaptive actions. A frozen phase for p > p _c has three separate clusters of agents using only R, P, and S, respectively with terminated adaptive actions. A mean-field theory based on the link densities in co-evolving network is formulated and the trinomial closure scheme is applied to obtain analytical solutions. The analytic results agree with simulation results on ARPS well. In addition, the different probabilities of winning, losing, and drawing a game among the agents are identified as the origin of the small discrepancy between analytic and simulation results. As a result of the adaptive actions, agents of higher degrees are often those being taken advantage of. Agents with a smaller (larger) degree than the mean degree have a higher (smaller) probability of winning than losing. The results are informative for future attempts on formulating more accurate theories.     

Centralized versus decentralized control—A solvable stylized model in transportation

Today’s supply networks consist of a certain amount of logistics objects that are enabled to interact with each other and to decide autonomously upon their next steps; in other words, they exhibit a certain degree of autonomous cooperation. Therefore, modern logistics research regards them as complex adaptive logistics systems. In order to analyze evolving dynamics and underlying implications for the respective systems’ behavior as well as the potential outcomes resulting from the interaction between autonomous decision-making “smart parts”, we propose in this contribution a fully solvable stylized model. We consider a population of homogeneous, autonomous interacting agents traveling on R with a given velocity that is itself corrupted by White Gaussian Noise. Based on real time observations of the positions of his neighbors, each agent is allowed to adapt his traveling velocity. These agent interactions are restricted to neighboring entities confined in finite spatial clusters (i.e. we have range-limited interactions). In the limit of a large population of neighboring agents, a mean-field dynamics can be derived and, for small interaction range, the resulting dynamics coincides with the exactly solvable Burgers’ nonlinear field equation. Explicit Burgers’ solution enables to explicitly appreciate the emergent structure due to the local and individual agent interactions. In particular, for strongly interactive regimes in the present model, the resulting spatial distribution of agents converges to a shock wave pattern. To compare performances of centralized versus decentralized organization, we assign cost functions incurred when velocity adaptations are triggered either by multi-agent interactions or by central control. The multi-agent cumulative costs are then compared with the costs that would be incurred by implementing an effective optimal central controller able, for a given time horizon, to reproduce an identical spatial probability distribution of agents. The resulting optimal control problem can be solved exactly and the corresponding costs can be expressed as the Kullback-Leibler relative entropy between the free and the controlled probability measures. This enables one to conclude that for time horizons shorter than a critical value, multi-agent interactions generate smaller cumulative costs than an optimal effective central controller.     

Modélisation d'un échangeur de chaleur compact à courants croisés séparés par des couches de matériau à changement de phase

A mathematical model was developed to predict the thermal behaviour of a crossflow compact heat exchanger with layers of phase change material (PCM) operating under winter conditions. The model is validated and a parametric study is conducted to determine the design and operating conditions that promote condensation and may lead eventually to the frosting of the exchanger. The transient and steady-state thermal efficiency of the exchanger are also predicted as a function of the number of thermal units NTU_f, of the ratio of the products of the mass flow rate by the heat capacity on the cold and hot sides C and of the Biot number Bi. Results indicate that for Bi = 0.6 (which corresponds to a 3 mm thick PCM layer), condensation occurs when NTU_f ≥ 2.0 for C = 1.0 and when NTU_f ≥ 5.0 for C = 0.5. If the thickness of the PCM layer increases, condensation is avoided and the duration of the heat recovery period is prolonged. At the same time however, the steady state thermal efficiency diminishes. A heat exchanger for which Bi = 0.6, C = 0.5 and NTU_f = 4.0 appears to be a good compromise for acceptable heat recovery and efficiency. In this case, the heat recovery period lasts 1.6 h and the steady state thermal efficiency levels off at 64 %.     

Synchronization of mobile chaotic agents on connected networks

This work studies the synchronization of a number of mobile agents on a substrate network. Each agent carries a chaotic map and randomly walks on a connected network. The collection of agents consists of another time-varying network derived from the substrate network. It is found that the synchronization conditions of this agent network depend on the average degree of the substrate network’s connectivity, the coupling strength between interacting agents, and the agent density in the network. Synchronization of the agent network on scale-free and ER networks is considered here, and it is found that the scale-free topology is more applicable to synchronize mobile chaotic agents. To get analytical insights, the star graph is taken and considered as a substrate network.     

Magnetic field dependence of the Peierls instability in one-dimensional conductors

We show that an applied magnetic field will suppress the Peierls instability in one-dimensional conductors. The effect is big enough to be observable. In sufficiently strong fields the transition between the metallic and insulating state is due to the condensation of a phonon with wave vectorq≠2p _F, but close to 2p _F. The role of spinorbit interactions is also discussed.     

How Big Is Too Big? Critical Shocks for Systemic Failure Cascades

External or internal shocks may lead to the collapse of a system consisting of many agents. If the shock hits only one agent initially and causes it to fail, this can induce a cascade of failures among neighboring agents. Several critical constellations determine whether this cascade affects the system in part or as a whole which, in the second case, leads to systemic risk. We investigate the critical parameters for such cascades in a simple model, where agents are characterized by an individual threshold θ _i determining their capacity to handle a load αθ _i with 1?α being their safety margin. If agents fail, they redistribute their load equally to K neighboring agents in a regular network. For three different threshold distributions P(θ), we derive analytical results for the size of the cascade, X(t), which is regarded as a measure of systemic risk, and the time when it stops. We focus on two different regimes, (i) EEE, an external extreme event where the size of the shock is of the order of the total capacity of the network, and (ii) RIE, a random internal event where the size of the shock is of the order of the capacity of an agent. We find that even for large extreme events that exceed the capacity of the network finite cascades are still possible, if a power-law threshold distribution is assumed. On the other hand, even small random fluctuations may lead to full cascades if critical conditions are met. Most importantly, we demonstrate that the size of the “big” shock is not the problem, as the systemic risk only varies slightly for changes in the number of initially failing agents, the safety margin and the threshold distribution, which further gives hints on how to reduce systemic risk.     

Effect of Savings on a Gas-Like Model Economy with Credit and Debt

In kinetic exchange models, agents make transactions based on well-established microscopic rules that give rise to macroscopic variables in analogy to statistical physics. These models have been applied to study processes such as income and wealth distribution, economic inequality sources, economic growth, etc., recovering well-known concepts in the economic literature. In this work, we apply ensemble formalism to a geometric agents model to study the effect of saving propensity in a system with money, credit, and debt. We calculate the partition function to obtain the total money of the system, with which we give an interpretation of the economic temperature in terms of the different payment methods available to the agents. We observe an interplay between the fraction of money that agents can save and their maximum debt. The system’s entropy increases as a function of the saved proportion, and increases even more when there is debt.     

Networking effects on evolutionary snowdrift game in networks with fixed degrees

We study the effects of spatial structures other than the degree distribution on the extent of the emergence of cooperation in an evolutionary snowdrift game. By swapping the links in three different types of regular lattices with a fixed degree k, we study how the frequency of cooperator f_C changes as the clustering coefficient (CC), which signifies how the nearest neighbors of a vertex are connected, and the sharing coefficient (SC), which signifies how the next-nearest neighbors of a vertex are shared by the nearest neighbors, are varied. For small k, a non-vanishing CC tends to suppress f_C. A non-vanishing SC also leads to a suppressed f_C for the networks studied. As the degree increases, the sensitivity of f_C to the network properties is found to become increasingly weak. The result is discussed within the context of the ranking patterns of average payoffs as k changes. An approximation for f_C, which is based on the idea of a finite fully connected network and gives results in good agreement with numerical results, is derived in the limit of large k.