Spatial evolution character of multi-objective evolutionary algorithm based on self-organized criticality theory

This paper analyzes the spatial evolution character of multi-objective evolutionary algorithms using self-organized criticality theory. The spatial evolution character is modeled by the statistical property of crowding distance, which displays a scale-free feature and a power-law distribution. We propose that the evolutional rule of multi-objective optimization algorithms is a self-organized state transition from an initial scale-free state to a final scale-free state. The target is to get close to a critical state representing the true Pareto-optimal front. Besides, the anti-Matthew effect is the internal incentive factor of most strategies. The final scale-free state reflects the quality of the final Pareto-optimal front. The speed of the state transition reflects the efficiency of the algorithm. We simulate the spatial evolution characters of three typical multi-objective evolutionary algorithms representing three fields, i.e., Genetic Algorithm, Differential Evolution and the Artificial Immune System algorithm. The results prove that the model and the explanation are effective for analyzing the evolutional rule of multi-objective evolutionary algorithms.     

The origins of dragon-kings and their occurrence in society

A society is a medium with a complex structure of one-to-one relations between people. Those could be relations between friends, wife–husband relationships, relations between business partners, and so on. At a certain level of analysis, a society can be regarded as a gigantic maze constituted of one-to-one relationships between people. From a physical standpoint it can be considered as a highly porous medium. Such media are widely known for their outstanding properties and effects like self-organized criticality, percolation, power-law distribution of network cluster sizes, etc. In these media supercritical events, referred to as dragon-kings, may occur in two cases: when increasing stress is applied to a system (self-organized criticality scenario) or when increasing conductivity of a system is observed (percolation scenario). In social applications the first scenario is typical for negative effects: crises, wars, revolutions, financial breakdowns, state collapses, etc. The second scenario is more typical for positive effects like emergence of cities, growth of firms, population blow-ups, economic miracles, technology diffusion, social network formation, etc. If both conditions (increasing stress and increasing conductivity) are observed together, then absolutely miraculous dragon-king effects can occur that involve most human society. Historical examples of this effect are the emergence of the Mongol Empire, world religions, World War II, and the explosive proliferation of global internet services. This article describes these two scenarios in detail beginning with an overview of historical dragon-king events and phenomena starting from the early human history till the last decades and concluding with an analysis of their possible near future consequences on our global society. Thus we demonstrate that in social systems dragon-king is not a random outlier unexplainable by power-law statistics, but a natural effect. It is a very large cluster in a porous percolation medium. It occurs as a result of changes in external conditions, such as supercritical load, increase in system elements’ sensitivity, or system connectivity growth.     

Scale-free random graphs and Potts model

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertexi has a prescribed weight P_i ∝ i^-μ (0 < μ< 1) and an edge can connect verticesi andj with rateP _i P _j . Corresponding equilibrium ensemble is identified and the problem is solved by theq → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, whereλ = 1 +μ ^-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finiteN shows double peaks.     

Landslides on sandpiles: Some moment relations in one dimension

Several relations between the structure of stable recurrent states and the statistics of avalanches in a one-dimensional sandpile automaton are derived and numerically verified. In particular, it is shown that the average avalanche size is determined by the second rather than the first moment of the distribution of trough distances. The two moments scale differently with system size, which implies multiscaling for the distribution. Moreover, the scaling of edge events (avalanches which fall off the pile) is shown to differ from that of bulk events (avalanches which remain on the pile).     

Scale-free networks via attaching to random neighbors

Preferential attachment is considered one of the key factors in the formation of scale-free networks. However, complete random attachment without a preferential mechanism can also generate scale-free networks in nature, such as protein interaction networks in cells. This article presents a new scale-free network model that applies the following general mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach to random neighbors of random vertices that are already well connected. The proposed model does not require global-based preferential strategies and utilizes only the random attachment method. Theoretical analysis and numerical simulation results denote that the proposed model has steady scale-free network characteristics, and random attachment without a preferential mechanism may generate scale-free networks.     

Evolution of species from Darwin theory: A simple model

Evolution of species is a complex phenomenon. Some theoretical models take into account evolution of species, like the Bak–Sneppen model that obtain punctuated equilibrium from self-organized criticality and the Penna model for biological aging that consists in a bit-string model subjected to aging, reproduction and death. In this work we propose a simple model to study different scenarios used to simulate the evolution of species. This model is based on Darwin’s ideas of evolution. The present findings show that punctuated equilibria and stasis seem to be obtained directly from the mutation, selection of parents and the genetic crossover, and are very close to the fossil data analysis.     

Comments on “Scale-free networks without growth”

In [Y.-B. Xie, T. Zhou, B.-H. Wang, Scale-free networks without growth, Physica A 387 (2008) 1683-1688], a nongrowing scale-free network model has been introduced, which shows that the degree distribution of the model varies from the power-law form to the Poisson form as the free parameter α increases, and indicates that the growth may not be necessary for a scale-free network structure to emerge. However, the model implicitly assumes that self-loops and multiple-links are allowed in the model and counted in the degree distribution. In many real-life networks, such an assumption may not be reasonable. We showed here that the degree distribution of the emergent network does not obey a power-law form if self-loops and multiple-links are allowed in the model but not counted in the degree distribution. We also observed the same result when self-loops and multiple-links are not allowed in the model. Furthermore, we showed that the effect of self-loops and multiple-links on the degree distribution weakens as α increases and even becomes negligible when α is sufficiently large.     

Coherent noise, scale invariance and intermittency in large systems

We introduce a new class of models in which a large number of “agents” organize under the influence of an externally imposed coherent noise. The model shows reorganization events whose size distribution closely follows a power law over many decades, even in the case where the agents do not interact with each other. In addition, the system displays “aftershock” events in which large disturbances are followed by a string of others at times which are distributed according to a t⁻¹ law. We also find that the lifetimes of the agents in the system possess a power-law distribution. We explain all these results using an approximate analytic treatment of the dynamics and discuss a number of variations on the basic model relevant to the study of particular physical systems.     

Steady State of Stochastic Sandpile Models

We study the steady state of the Abelian sandpile models with stochastic toppling rules. The particle addition operators commute with each other, but in general these operators need not be diagonalizable. We use their Abelian algebra to determine their eigenvalues, and the Jordan block structure. These are then used to determine the probability of different configurations in the steady state. We illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example, for systems of size ≤12, and also study the density profile in the steady state.     

The impact of connection density on scale-free distribution in random networks

Preferential attachment is considered as a fundamental mechanism that contributes to the scale-free characteristics of random networks, which include growth and non-growth networks. There exist some situations of non-growth random networks, particularly for very sparse or dense networks, where preferential attachments cannot consequentially result in true scale-free features, but only in scale-free-like appearances. This phenomenon implies that, a close relationship exists between the connection density p$p$ and the scaling. In this study, we propose a self-organized model with constant network size to study the phenomenon. We show analytically and numerically that there exists a certain critical point _pc$p_c$. Only when p=_pc$p=p_c$, the random network evolves into steady scale-free state. Otherwise, the network exhibits a steady scale-free-like state. The closer the p$p$ approximates _pc$p_c$, the closer the scale-free-like distribution approximates the true scale-free distribution. Our results show that, in random network lack of growth, a preferential scheme does not necessarily lead to a scale-free state, and a formation of scale-free is a consequence of two mechanisms: (i) a preferential scheme and (ii) appropriate connection density.     

Scale-free properties of information flux networks in genetic algorithms

In this study, we present empirical analysis of statistical properties of mating networks in genetic algorithms (GAs). Under the framework of GAs, we study a class of interaction network model—information flux network (IFN), which describes the information flow among generations during evolution process. The IFNs are found to be scale-free when the selection operator uses a preferential strategy rather than a random. The topology structure of IFN is remarkably affected by operations used in genetic algorithms. The experimental results suggest that the scaling exponent of the power-law degree distribution is shown to decrease when crossover rate increases, but increase when mutation rate increases, and the reason may be that high crossover rate leads to more edges that are shared between nodes and high mutation rate leads to many individuals in a generation possessing low fitness. The magnitude of the out-degree exponent is always more than the in-degree exponent for the systems tested. These results may provide a new viewpoint with which to view GAs and guide the dissemination process of genetic information throughout a population.     

Critical brain networks

Highly correlated brain dynamics produces synchronized states with no behavioral value, while weakly correlated dynamics prevents information flow. We discuss the idea put forward by Per Bak that the working brain stays at an intermediate (critical) regime characterized by power-law correlations.     

Return distributions in dog-flea model revisited

A recent study of coherent noise model for the system size independent case provides an exact relation between the exponent τ of avalanche size distribution and the q value of the appropriate q-Gaussian that fits the return distribution of the model. This relation is applied to Ehrenfest’s historical dog-flea model by treating the fluctuations around the thermal equilibrium as avalanches. We provide a clear numerical evidence that the relation between the exponent τ of fluctuation length distribution and the q value of the appropriate q-Gaussian obeys this exact relation when the system size is large enough. This allows us to determine the value of the q-parameter a priori from one of the well known exponents of such dynamical systems. Furthermore, it is shown that the return distribution in dog-flea model gradually approaches q-Gaussian as the system size increases and this tendency can be analyzed by a well defined analytical expression.     

Stretched exponential distribution of recurrent time of wars in China

As a killing machine and a decisive factor of history, wars play an important role in social system. In this paper, we present an empirical exploration of the distribution of recurrent time of wars in ancient China and find that it obeys a stretched exponential form. The pattern we found implies that there are undetected mechanisms that underlie the dynamics of wars. In order to explain the origin of this form, a model mainly based on the correlation between two consecutive wars is constructed, which is somewhat similar to the Bak-Sneppen model. The simulation results of the model are in agreement with the empirical statistics and suggest that the dynamics of wars could relate with self-organized criticality.     

Self-organized criticality in MHD driven plasma edge turbulence

We analyze long-range time correlations and self-similar characteristics of the electrostatic turbulence at the plasma edge and scrape-off layer in the Tokamak Chauffage Alfvén Brésillien (TCABR), with low and high Magnetohydrodynamics (MHD) activity. We find evidence of self-organized criticality (SOC), mainly in the region near the tokamak limiter. Comparative analyses of data before and during the MHD activity reveals that during the high MHD activity the Hurst parameter decreases. Finally, we present a cellular automaton whose parameters are adjusted to simulate the analyzed turbulence SOC change with the MHD activity variation.     

Spatial correlations in permeability distributions due to extreme dynamics restructuring of unconsolidated sandstone

We propose a model for porous sandstone formation from unconsolidated sand based on a series of restructuring events where the local pressure difference due to flow in the sand is the largest. We investigate the local and global permeability distributions after steady state has been reached. Whereas we find no spatial correlations in the local permeability distribution, the distribution of inverse permeability shows spatial correlations consistent with a fractional Brownian noise characterized by a Hurst exponent of 0.88(9). The global permeability of the system shows time fluctuations as restructuring proceeds consistent with self-affinity characterized by a Hurst exponent of 0.25(3), crossing over to white noise at larger time scales.     

The Artic-sea-ice cover: Problem of forecasting

The ice-research station “North Pole 32” (NP 32) was established on the ice pack in the Arctic Ocean in 2003 and drifted up to February 2004, when a “global” perturbation in the sea-ice-cover caused the intensive ice fragmentation around the actual position of the NP 32. As a result, the station ceased its activity and was abandoned in March 2004. The statistical characterization of the sea-ice cover fragmentation and the drift dynamics during the final weeks of the work of the NP 32 are under consideration in this communication. The work is an attempt to reveal some features in the prehistory of this large-scale event which could serve for forecasting the substantial perturbations in the sea-ice cover. The most prominent feature that indicated the approach of the sea-ice fragmentation over the area of ∼10⁵ km² was the break of the correlated motion of ice-fields observed 2 days before the “catastrophic” event.     

On the relation between Vicsek and Kuramoto models of spontaneous synchronization

The Vicsek model for self-propelling particles in 2D is investigated with respect to the addition of the stochastic perturbation of dynamic equations. We show that this model represents in essence the same type of bifurcations under a different type of noise as the celebrated Kuramoto model of spontaneous synchronization. These models demonstrate similar behavior at least within the mean-field approach. To prove this we consider two types of noise for the Vicsek model which are commonly considered in the literature: the intrinsic and the extrinsic ones (according to the terminology of Pimentel et al. [J.A. Pimentel, M. Aldana, C. Huepe, H. Larralde, Intrinsic and extrinsic noise effects on phase transitions of network models with applications to swarming systems, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 77 (6) (2008) doi:10.1103/PhysRevE.77.061138. URL: http://dx.doi.org/10.1103/PhysRevE.77.061138]). The qualitative correspondence with the bifurcation of stationary states in the Kuramoto model is stated. A new type of stochastic perturbation—the “mixed” noise is proposed. It is constructed as the weighted superposition of the intrinsic and the extrinsic noises. The corresponding phase diagram “noise amplitude vs. interaction strength” is obtained. The possibility of the tricritical behavior for the Vicsek model is predicted.