Heat-Bath and Metropolis Dynamics in Ising-like Models on Directed Regular Random Graphs.

Using a single-site mean-field approximation (MFA) and Monte Carlo simulations, we examine Ising-like models on directed regular random graphs. The models are directed-network implementations of the Ising model, Ising model with absorbing states, and majority voter models. When these nonequilibrium models are driven by the heat-bath dynamics, their stationary characteristics, such as magnetization, are correctly reproduced by MFA as confirmed by Monte Carlo simulations. It turns out that MFA reproduces the same result as the generating functional analysis that is expected to provide the exact description of such models. We argue that on directed regular random graphs, the neighbors of a given vertex are typically uncorrelated, and that is why MFA for models with heat-bath dynamics provides their exact description. For models with Metropolis dynamics, certain additional correlations become relevant, and MFA, which neglects these correlations, is less accurate. Models with heat-bath dynamics undergo continuous phase transition, and at the critical point, the power-law time decay of the order parameter exhibits the behavior of the Ising mean-field universality class. Analogous phase transitions for models with Metropolis dynamics are discontinuous.

Equilibrium behavior in a nonequilibrium system: Ising-doped voter model on complete graph.

While the Ising model belongs to the realm of equilibrium statistical mechanics, the voter model is an example of a nonequilibrium system. We examine an opinion formation model, which is a mixture of Ising and voter agents with concentrations p and 1-p, respectively. Although in our model for p<1 a detailed balance is violated, on a complete graph the average magnetization in the stationary state for any p>0 is shown to satisfy the same equation as for the pure Ising model (p=1). Numerical simulations confirm such a behavior. Variance of magnetization and susceptibility in our model increase for decreasing p and diverge at the temperature at which magnetization vanishes. Simulations on a random graph also show that a small concentration of Ising agents is sufficient to induce a ferromagnetic ordering.

Stability and Evolution of Synonyms and Homonyms in Signaling Game.

Synonyms and homonyms appear in all natural languages. We analyze their evolution within the framework of the signaling game. Agents in our model use reinforcement learning, where probabilities of selection of a communicated word or of its interpretation depend on weights equal to the number of accumulated successful communications. When the probabilities increase linearly with weights, synonyms appear to be very stable and homonyms decline relatively fast. Such behavior seems to be at odds with linguistic observations. A better agreement is obtained when probabilities increase faster than linearly with weights. Our results may suggest that a certain positive feedback, the so-called Metcalfe's Law, possibly drives some linguistic processes. Evolution of synonyms and homonyms in our model can be approximately described using a certain nonlinear urn model.

Evolution towards Linguistic Coherence in Naming Game With Migrating Agents.

As an integral part of our culture and way of life, language is intricately related to the migrations of people. To understand whether and how migration shapes language formation processes, we examine the dynamics of the naming game with migrating agents. (i) When all agents may migrate, the dynamics generates effective surface tension that drives the coarsening. Such behaviour is very robust and appears for a wide range of densities of agents and their migration rates. (ii) However, when only multilingual agents are allowed to migrate, monolingual islands are typically formed. In such a case, when the migration rate is sufficiently large, the majority of agents acquire a common language that spontaneously emerges with no indication of surface-tension-driven coarsening. Relatively slow coarsening that takes place in a dense static population is very fragile, and an arbitrarily small migration rate can most likely divert the system towards the quick formation of monolingual islands. Our work shows that migration influences language formation processes, but additional details such as density or mobility of agents are needed to more precisely specify this influence.

Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds.

Using simulated annealing, we examine a bipartitioning of small worlds obtained by adding a fraction of randomly chosen links to a one-dimensional chain or a square lattice. Models defined on small worlds typically exhibit a mean-field behavior, regardless of the underlying lattice. Our work demonstrates that the bipartitioning of small worlds does depend on the underlying lattice. Simulations show that for one-dimensional small worlds, optimal partitions are finite size clusters for any fraction of additional links. In the two-dimensional case, we observe two regimes: when the fraction of additional links is sufficiently small, the optimal partitions have a stripe-like shape, which is lost for a larger number of additional links as optimal partitions become disordered. Some arguments, which interpret additional links as thermal excitations and refer to the thermodynamics of Ising models, suggest a qualitative explanation of such a behavior. The histogram of overlaps suggests that a replica symmetry is broken in a one-dimensional small world. In the two-dimensional case, the replica symmetry seems to hold, but with some additional degeneracy of stripe-like partitions.

Entropic long-range ordering in an adsorption-desorption model.

We examine a two-dimensional nonequilibrium lattice model where particles adsorb at empty sites and desorb when the number of neighboring particles is greater than a given threshold. In a certain range of parameters the model exhibits entropic ordering similar to some hard-core systems. However, contrary to hard-core systems, on increasing the density of particles the ordering is destroyed. In the heterogenous version of our model, a regime with slow dynamics appears that might indicate formation of some kind of glassy structures.

Emergence of linguistic conventions in multi-agent reinforcement learning.

Recently, emergence of signaling conventions, among which language is a prime example, draws a considerable interdisciplinary interest ranging from game theory, to robotics to evolutionary linguistics. Such a wide spectrum of research is based on much different assumptions and methodologies, but complexity of the problem precludes formulation of a unifying and commonly accepted explanation. We examine formation of signaling conventions in a framework of a multi-agent reinforcement learning model. When the network of interactions between agents is a complete graph or a sufficiently dense random graph, a global consensus is typically reached with the emerging language being a nearly unique object-word mapping or containing some synonyms and homonyms. On finite-dimensional lattices, the model gets trapped in disordered configurations with a local consensus only. Such a trapping can be avoided by introducing a population renewal, which in the presence of superlinear reinforcement restores an ordinary surface-tension driven coarsening and considerably enhances formation of efficient signaling.

Language competition in a population of migrating agents.

Influencing various aspects of human activity, migration is associated also with language formation. To examine the mutual interaction of these processes, we study a Naming Game with migrating agents. The dynamics of the model leads to formation of low-mobility clusters, which turns out to break the symmetry of the model: although the Naming Game remains symmetric, low-mobility languages are favored. High-mobility languages are gradually eliminated from the system, and the dynamics of language formation considerably slows down. Our model is too simple to explain in detail language competition of migrating human communities, but it certainly shows that languages of settlers are favored over nomadic ones.

Phase transition and power-law coarsening in an Ising-doped voter model.

We examine an opinion formation model, which is a mixture of Voter and Ising agents. Numerical simulations show that even a very small fraction (∼1%) of the Ising agents drastically changes the behavior of the Voter model. The Voter agents act as a medium, which correlates sparsely dispersed Ising agents, and the resulting ferromagnetic ordering persists up to a certain temperature. Upon addition of the Ising agents, a logarithmically slow coarsening of the Voter model (d=2), or its active steady state (d=3), change into an Ising-type power-law coarsening.

Phase transitions in Ising models on directed networks.

We examine Ising models with heat-bath dynamics on directed networks. Our simulations show that Ising models on directed triangular and simple cubic lattices undergo a phase transition that most likely belongs to the Ising universality class. On the directed square lattice the model remains paramagnetic at any positive temperature as already reported in some previous studies. We also examine random directed graphs and show that contrary to undirected ones, percolation of directed bonds does not guarantee ferromagnetic ordering. Only above a certain threshold can a random directed graph support finite-temperature ferromagnetic ordering. Such behavior is found also for out-homogeneous random graphs, but in this case the analysis of magnetic and percolative properties can be done exactly. Directed random graphs also differ from undirected ones with respect to zero-temperature freezing. Only at low connectivity do they remain trapped in a disordered configuration. Above a certain threshold, however, the zero-temperature dynamics quickly drives the model toward a broken symmetry (magnetized) state. Only above this threshold, which is almost twice as large as the percolation threshold, do we expect the Ising model to have a positive critical temperature. With a very good accuracy, the behavior on directed random graphs is reproduced within a certain approximate scheme.

Robust criticality of an Ising model on rewired directed networks.

We show that preferential rewiring, which is supposed to mimic the behavior of financial agents, changes a directed-network Ising ferromagnet with a single critical point into a model with robust critical behavior. For the nonrewired random graph version, due to a constant number of out-links for each site, we write a simple mean-field-like equation describing the behavior of magnetization; we argue that it is exact and support the claim with extensive Monte Carlo simulations. For the rewired version, this equation is obeyed only at low temperatures. At higher temperatures, rewiring leads to strong heterogeneities, which apparently invalidates mean-field arguments and induces large fluctuations and divergent susceptibility. Such behavior is traced back to the formation of a relatively small core of agents that influence the entire system.

Generic criticality of community structure in random graphs.

We examine a community structure in random graphs of size n and link probability p/n determined with the Newman greedy optimization of modularity. Calculations show that for p<1 communities are nearly identical with clusters. For p=1 the average sizes of a community s(av) and of the giant community s(g) show a power-law increase s(av)∼n(α') and s(g)∼n(α). From numerical results we estimate α'≈0.26(1) and α≈0.50(1) and using the probability distribution of sizes of communities we suggest that α'=α/2 should hold. For p>1 the community structure remains critical: (i) s(av) and s(g) have a power-law increase with α'≈α<1 and (ii) the probability distribution of sizes of communities is very broad and nearly flat for all sizes up to s(g). For large p the modularity Q decays as Q∼p(-0.55), which is intermediate between some previous estimations. To check the validity of the results, we also determine the community structure using another method, namely, a nongreedy optimization of modularity. Tests with some benchmark networks show that the method outperforms the greedy version. For random graphs, however, the characteristics of the community structure determined using both greedy and nongreedy optimizations are, within small statistical fluctuations, the same.

Emergence of social structures via preferential selection.

We examine a weighted-network multiagent model with preferential selection such that agents choose partners with probability p(w), where w is the number of their past selections. When p(w) increases sublinearly with the number of past selections [p(w)∼w(α),α<1], agents develop a uniform preference for all other agents. At α=1, this state loses stability and more complex structures form. For a superlinear increase (α>1), strong heterogeneities emerge and agents make selections mainly within small and sometimes asymmetric clusters. Even in a few-agent case, the formation of such clusters resembles phase transitions with spontaneous symmetry breaking.

Naming game on adaptive weighted networks.

We examine a naming game on an adaptive weighted network. A weight of connection for a given pair of agents depends on their communication success rate and determines the probability with which the agents communicate. In some cases, depending on the parameters of the model, the preference toward successfully communicating agents is essentially negligible and the model behaves similarly to the naming game on a complete graph. In particular, it quickly reaches a single-language state, albeit some details of the dynamics are different from the complete-graph version. In some other cases, the preference toward successfully communicating agents becomes much more important and the model gets trapped in a multi-language regime. In this case gradual coarsening and extinction of languages lead to the emergence of a dominant language, albeit with some other languages still present. A comparison of distribution of languages in our model and in the human population is discussed.

Statistical mechanics model of angiogenic tumor growth.

We examine a lattice model of tumor growth where the survival of tumor cells depends on the supplied nutrients. When such a supply is random, the extinction of tumors belongs to the directed percolation universality class. However, when the supply is correlated with the distribution of tumor cells, which as we suggest might mimic the angiogenic growth, the extinction shows different critical behavior. Such a correlation affects also the morphology of the growing tumors and drastically raises tumor-survival probability.

Diffusive behavior of a greedy traveling salesman.

Using Monte Carlo simulations we examine the diffusive properties of the greedy algorithm in the d-dimensional traveling salesman problem. Our results show that for d=3 and 4 the average squared distance from the origin (r(2)) is proportional to the number of steps t. In the d=2 case such a scaling is modified with some logarithmic corrections, which might suggest that d=2 is the critical dimension of the problem. The distribution of lengths also shows marked differences between d=2 and d>2 versions. A simple strategy adopted by the salesman might resemble strategies chosen by some foraging and hunting animals, for which anomalous diffusive behavior has recently been reported and interpreted in terms of Lévy flights. Our results suggest that broad and Lévy-like distributions in such systems might appear due to dimension-dependent properties of a search space.

Slow dynamics in a driven two-lane particle system.

We study a two-lane model of two species of particles that perform biased diffusion. Extensive numerical simulations show that when bias q is strong enough, oppositely drifting particles form some clusters that block each other. Coarsening of such clusters is very slow and their size increases logarithmically in time. For smaller q, particles collapse essentially on a single cluster whose size seems to diverge at a certain value of q=qc. Simulations show that despite slow coarsening, the model has rather large power-law cooling-rate effects. It makes its dynamics different from glassy systems but similar to some three-dimensional Ising-type models (gonihedric models).

Language structure in the n -object naming game.

We examine a naming game with two agents trying to establish a common vocabulary for n objects. Such efforts lead to the emergence of language that allows for an efficient communication and exhibits some degree of homonymy and synonymy. Although homonymy reduces the communication efficiency, it seems to be a dynamical trap that persists for a long, and perhaps indefinite, time. On the other hand, synonymy does not reduce the efficiency of communication but appears to be only a transient feature of the language. Thus, in our model the role of synonymy decreases and in the long-time limit it becomes negligible. A similar rareness of synonymy is observed in present natural languages. The role of noise, that distorts the communicated words, is also examined. Although, in general, the noise reduces the communication efficiency, it also regroups the words so that they are more evenly distributed within the available "verbal" space.

Heat conduction and diffusion of hard disks in a narrow channel.

Using molecular dynamics we study heat conduction and diffusion of hard disks in one-dimensional narrow channels. When collisions preserve momentum the heat conduction kappa diverges with the number of disks N as kappa approximately N alpha (alpha approximately 1/3) . Such a behavior is seen both when the ordering of disks is fixed ("pen-case" model), and when they can exchange their positions. Momentum conservation results also in sound-wave effects that enhance diffusive behavior and on an intermediate time scale (that diverges in the thermodynamic limit) normal diffusion takes place even in the "pen-case" model. When collisions do not preserve momentum, kappa remains finite and sound-wave effects are absent.

Long-term evolution of an ecosystem with spontaneous periodicity of mass extinctions.

Twenty years ago, after analysing palaeontological data, Raup and Sepkoski suggested that mass extinctions on Earth appear cyclically in time with a period of approximately 26 million years (My). To explain the 26My period, a number of proposals were made involving, e.g., astronomical effects, increased volcanic activity, or the Earth's magnetic field reversal, none of which, however, has been confirmed. Here we study a spatially extended discrete model of an ecosystem and show that the periodicity of mass extinctions might be a natural feature of the ecosystem's dynamics and not the result of a periodic external perturbation. In our model, periodic changes of the diversity of an ecosystem and some of its other characteristics are induced by the coevolution of species. In agreement with some palaeontological data, our results show that the longevity of a species depends on the evolutionary stage at which the species is created. Possible further tests of our model are also discussed.