Universality of random-site percolation thresholds for two-dimensional complex noncompact neighborhoods

The phenomenon of percolation is one of the core topics in statistical mechanics. It allows one to study the phase transition known in real physical systems only in a purely geometrical way. In this paper, we determine thresholds p_{c} for random-site percolation in triangular and honeycomb lattices for all available neighborhoods containing sites from the sixth coordination zone. The results obtained (together with the percolation thresholds gathered from the literature also for other complex neighborhoods and also for a square lattice) show the power-law dependence p_{c}∝(ζ/K)^{-γ} with γ=0.526(11), 0.5439(63), and 0.5932(47), for a honeycomb, square, and triangular lattice, respectively, and p_{c}∝ζ^{-γ} with γ=0.5546(67) independently on the underlying lattice. The index ζ=∑_{i}z_{i}r_{i} stands for an average coordination number weighted by distance, that is, depending on the coordination zone number i, the neighborhood coordination number z_{i}, and the distance r_{i} to sites in the ith coordination zone from the central site. The number K indicates lattice connectivity, that is, K=3, 4, and 6 for the honeycomb, square, and triangular lattice, respectively.

Recovering Zipf's law in intercontinental scientific cooperation

Scientific cooperation on an international level has been well studied in the literature. However, much less is known about this cooperation on an intercontinental level. In this paper, we address this issue by creating a collection of approximately 13.8×106 publications around the papers by one of the highly cited authors working in complex networks and their applications. The obtained rank-frequency distribution of the probability of sequences describing continents and the number of countries-with which authors of papers are affiliated-follows the power law with an exponent -1.9108(15). Such dependence is known in the literature as Zipf's law, and it has been originally observed in linguistics; later, it turned out that it is very commonly observed in various fields. The number of distinct "continent (number of countries)" sequences in a function of the number of analyzed papers grows according to power law with exponent 0.527(14); i.e., it follows Heap's law.

Thermal properties of structurally balanced systems on classical random graphs

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) under the influence of the temperature modeling the social noise level are discussed for interacting actors occupying nodes of classical random graphs. Depending on the graph density D, either a smooth crossover or a first-order phase transition from a balanced to an imbalanced state of the system is observed with an increase in the thermal noise level. The minimal graph density Dmin for which the first-order phase transition can be observed decreases with the system size N as Dmin∝N-0.58(1). For graph densities D>Dmin, the reduced critical temperature Tc⋆=Tc/Tc(D=1) increases with the graph density as Tc⋆∝D1.719(6) independently of the system size N.

Vanishing Opinions in Latané Model of Opinion Formation

In this paper, the results of computer simulations based on the Nowak-Szamrej-Latané model with multiple (from two to five) opinions available in the system are presented. We introduce the noise discrimination level (which says how small the clusters of agents could be considered negligible) as a quite useful quantity that allows qualitative characterization of the system. We show that depending on the introduced noise discrimination level, the range of actors' interactions (controlled indirectly by an exponent in the distance scaling function, the larger the exponent, the more influential the nearest neighbors are) and the information noise level (modeled as social temperature, which increases results in the increase in randomness in taking the opinion by the agents), the ultimate number of the opinions (measured as the number of clusters of actors sharing the same opinion in clusters greater than the noise discrimination level) may be smaller than the number of opinions available in the system. These are observed in small and large information noise limits but result in either unanimity, or polarization, or randomization of opinions.

Mean-field approximation for structural balance dynamics in heat bath

A critical temperature for a complete signed graph of N agents where the time-dependent polarization of links tends towards the Heider (structural) balance is found analytically using the heat-bath approach and the mean-field approximation as T^{c}=(N-2)/a^{c}, where a^{c}≈1.71649. The result is in perfect agreement with numerical simulations starting from the paradise state where all links are positively polarized as well as with the estimation of this temperature received earlier with much more sophisticated methods. When heating the system, one observes a discontinuous and irreversible phase transition at T^{c} from a nearly balanced state when the mean link polarization is about x_{c}=0.796388 to a disordered and unbalanced state where the polarization vanishes. When the initial conditions for the polarization of links are random, then at low temperatures a balanced bipolar state of two mutually hostile cliques exists that decays towards the disorder and there is a discontinuous phase transition at a temperature T^{d} that is lower than T^{c}. The system phase diagram corresponds to the so-called fold catastrophe when a stable solution of the mean-field equation collides with a separatrix, and as a result a hysteresislike loop is observed.

Random site percolation on honeycomb lattices with complex neighborhoods

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with a radius ranging from one to three and containing 3-24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number z of sites in the neighborhoods, the site percolation thresholds _p follow the dependency _p ∝ 1 / z, as recently shown by Xun et al. [Phys. Rev. E 105, 024105 (2022)]. On the contrary, non-compact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of _p corresponding to the same number z of sites in the neighborhoods). An example of a single-value index ζ = ∑ i _{z r}-where _z and _r are the number of sites and radius of the ith coordination zone, respectively-characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence _p ∝ 1 / ζ. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Phys. Rev. E 64, 016706 (2001)] are also presented. The latter may be useful for computer physicists dealing with solid-state physics and interdisciplinary statistical physics applications, where the honeycomb lattice is the underlying network topology.

Does Social Distancing Matter for Infectious Disease Propagation? An SEIR Model and Gompertz Law Based Cellular Automaton

In this paper, we present stochastic synchronous cellular automaton defined on a square lattice. The automaton rules are based on the SEIR (susceptible → exposed → infected → recovered) model with probabilistic parameters gathered from real-world data on human mortality and the characteristics of the SARS-CoV-2 disease. With computer simulations, we show the influence of the radius of the neighborhood on the number of infected and deceased agents in the artificial population. The increase in the radius of the neighborhood favors the spread of the pandemic. However, for a large range of interactions of exposed agents (who neither have symptoms of the disease nor have been diagnosed by appropriate tests), even isolation of infected agents cannot prevent successful disease propagation. This supports aggressive testing against disease as one of the useful strategies to prevent large peaks of infection in the spread of SARS-CoV-2-like diseases.

Thermal properties of structurally balanced systems on diluted and densified triangulations

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) are discussed for various networks of interacting actors under the influence of the temperature modeling the social noise level. For that purpose, two main types of lattices are considered. The first is created by removing some links from a regular triangular lattice to produce a diluted triangular lattice, and the second by adding more links to create an enhanced triangular lattice. In both those cases, the full range of possible graph densities is discussed, limited by the extreme cases of networks which consist of a small number of separated triads and fully connected networks. It is shown that the existence of the balanced state is not possible if the average node degree is too close to the value characterizing the regular triangular lattice. Otherwise, both balanced (or partially balanced) and imbalanced states are possible, depending on the temperature. However, only for graphs which are dense enough a phase transition of the first kind is observed, while less enhanced networks (and all diluted) indicate a smooth crossover between the two states. The crossover temperatures are size independent only for the diluted triangular lattices and depend on the size of the system for the enhanced triangular lattices, as is the case also for the critical temperatures of the phase transition observed in denser enhanced lattices.

Wealth Rheology

We study wealth rank correlations in a simple model of macroeconomy. To quantify rank correlations between wealth rankings at different times, we use Kendall's τ and Spearman's ρ, Goodman-Kruskal's γ, and the lists' overlap ratio. We show that the dynamics of wealth flow and the speed of reshuffling in the ranking list depend on parameters of the model controlling the wealth exchange rate and the wealth growth volatility. As an example of the rheology of wealth in real data, we analyze the lists of the richest people in Poland, Germany, the USA and the world.

Comment on "Phase transition in a network model of social balance with Glauber dynamics"

In a recent work [R. Shojaei et al., Phys. Rev. E 100, 022303 (2019)2470-004510.1103/PhysRevE.100.022303] the authors calculate numerically the critical temperature T_{c} of the balanced-imbalanced phase transition in a fully connected graph. According to their findings, T_{c} decreases with the number of nodes N. Here we calculate the same critical temperature using the heat-bath algorithm. We show that T_{c} increases with N as N^{γ}, with γ close to 0.5 or 1.0. This value depends on the initial fraction of positive bonds.

Percolation thresholds on a triangular lattice for neighborhoods containing sites up to the fifth coordination zone

We determine thresholds p_{c} for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighborhoods. The dependence of the value of the percolation thresholds p_{c} on the coordination number z are tested against various theoretical predictions. The proposed single scalar index ξ=∑_{i}z_{i}r_{i}^{2}/i (depending on the coordination zone number i, the neighborhood coordination number z, and the square distance r^{2} to sites in ith coordination zone from the central site) allows one to differentiate among various neighborhoods and relate p_{c} to ξ. The thresholds roughly follow a power law p_{c}∝ξ^{-γ} with γ≈0.710(19).

Expulsion from structurally balanced paradise

We perform simulations of structural balance evolution on a triangular lattice using the heat-bath algorithm. In contrast to similar approaches-but applied to the analysis of complete graphs-the triangular lattice topology successfully prevents the occurrence of even partial Heider balance. Starting with the state of Heider's paradise, it is just a matter of time when the evolution of the system leads to an unbalanced and disordered state. The time of the system relaxation does not depend on the system size. The lack of any signs of a balanced state was not observed in earlier investigated systems dealing with the structural balance.

Site percolation thresholds on triangular lattice with complex neighborhoods

We determine thresholds p_c for random site percolation on a triangular lattice for neighborhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN), and next-next-next-next-nearest (5NN) neighbors, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, and 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [Phys. Rev. E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [Phys. Rev. E 90, 062101 (2014)], for estimating thresholds from low statistics data. The estimated values of percolation thresholds are p_c(4NN)=0.192410(43), p_c(3NN+2NN)=0.232008(38), p_c(5NN+4NN)=0.140286(5), p_c(3NN+2NN+NN)=0.215484(19), p_c(5NN+4NN+NN)=0.131792(58), p_c(5NN+4NN+3NN+2NN)=0.117579(41), and p_c(5NN+4NN+3NN+2NN+NN)=0.115847(21). The method is tested on the standard case of site percolation on the triangular lattice, where p_c(NN)=p_c(2NN)=p_c(3NN)=p_c(5NN)=12 is recovered with five digits accuracy p_c(NN)=0.500029(46) by averaging over one thousand lattice realizations only.

Ageing of complex networks

Many real-world complex networks arise as a result of a competition between growth and rewiring processes. Usually the initial part of the evolution is dominated by growth while the later one rather by rewiring. The initial growth allows the network to reach a certain size while rewiring to optimize its function and topology. As a model example we consider tree networks which first grow in a stochastic process of node attachment and then age in a stochastic process of local topology changes. The ageing is implemented as a Markov process that preserves the node-degree distribution. We quantify differences between the initial and aged network topologies and study the dynamics of the evolution. We implement two versions of the ageing dynamics. One is based on reshuffling of leaves and the other on reshuffling of branches. The latter one generates much faster ageing due to nonlocal nature of changes.

Noise induced unanimity and disorder in opinion formation

We propose an opinion dynamics model based on Latané’s social impact theory. Actors in this model are heterogeneous and, in addition to opinions, are characterised by their varying levels of persuasion and support. The model is tested for two and three initial opinions randomly distributed among actors. We examine how the noise (randomness of behaviour) and the flow of information among actors affect the formation and spread of opinions. Our main research involves the process of opinion formation and finding phases of the system in terms of parameters describing noise and flow of the information for two and three opinions available in the system. The results show that opinion formation and spread are influenced by both (i) flow of information among actors (effective range of interactions among actors) and (ii) noise (randomness in adopting opinions). The noise not only leads to opinions disorder but also it promotes consensus under certain conditions. In disordered phase and when the exchange of information is spatially effectively limited, various faces of disorder are observed, including system states, where the signatures of self-organised criticality manifest themselves as scale-free probability distribution function of cluster sizes. Then increase of noise level leads system to disordered random state. The critical noise level above which histograms of opinion clusters’ sizes loose their scale-free character increases with increase of the easy of information flow.

Impact of temperature on the physicochemical, structural and biological features of copper-silica nanocomposites

Classical wet chemical synthesis was used to fabricate a hybrid composite that contained copper nanoparticles (average size ∼1 nm), which were embedded into a silicon oxide carrier. The structural and chemical alternations in the copper-functionalized silica were investigated in systems that were sintered at 573 K, 873 K, 1173 K, and 1473 K. A general trend, which was associated with the transformation of metallic copper with a cubic structure into copper(II) oxide with a monoclinic structure in the heat-treated systems, was found. XPS and FTIR spectroscopies also revealed the presence of copper(I) oxide, which formed a shell around the CuO. SEM and TEM showed gradual densification of the hybrid system at ever higher sintering temperatures, which corresponded with the gradual copper agglomeration. A temperature of 873 K was determined to be the temperature at which amorphous silica was transformed into cristoballite and tridymite, as well as the formation of a bulk-like copper structure. In relation to the physicochemical and structural data, high antimicrobial features that had a relatively low toxicity effect on the normal human fibroblasts (NHDF) below 250 mg/L was found for the initial copper-silica composite and the samples that were sintered at 573 K. In turn, a significant decrease in the biological impact was observed in the samples that were sintered at temperatures above 573 K. As a result, the paper discusses the model of structural modifications in copper-silica nanocomposite concerning their biological impact that was developed.

Physicochemical and structural features of heat treated silver-silica nanocomposite and their impact on biological properties

In the last few decades, many nanostructures with varying properties and possible applications have been developed. These materials have been intended to work in various environmental temperature conditions. In this context, the main challenge has been to comprehend the impact of synergic interaction between individual elements included in non-annealed materials in relation to systems subjected to temperature impact. Another problem has corresponded to the impact of thermal modification on organisms such as bacteria and human cells. Such problems can be solved by the fabrication of a nanocomposite with mono-dispersed 8 nm silver (Ag⁰ or Ag⁺) embedded into a silica carrier, followed by the analysis of the impact of heat treatment under various temperature conditions on its physicochemical features. Therefore, methodical studies reported in this text have shown an increase of silver particle size up to 170 nm, a decrease of its concentration, as well as the formation of sub-nanometer Ag⁺ and/or Ag²⁺ clusters as the temperature rises to 1173 K. In turn, the structurally disordered silica carrier had been entirely transformed to cristobalite and tridymite only at 1473 K as well as partial reduction of Ag²⁺ to Ag⁺. Simultaneously, inhibition of growth of Gram-positive and Gram-negative bacteria, as well as an increase in cytotoxicity towards human cells was observed as the temperature rose. As a final point, for the first time, a "pseudo" phase diagram of the structural alterations in the Ag/SiO₂ nanocomposite has been created, as well as a model of silver-silica transformation to biological systems has been developed.

Unique properties of silver and copper silica-based nanocomposites as antimicrobial agents

The paper reports a new route for the fabrication and determination of physicochemical properties and biological activity, of metallic silica-based nanostructure (Ag/SiO₂, Cu/SiO₂).

Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

In this paper, random-site percolation thresholds for a simple cubic (SC) lattice with site neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation (Bastas et al., arXiv:1411.5834) is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are p(C)(4NN)=0.31160(12),p(C)(4NN+NN)=0.15040(12),p(C)(4NN+2NN)=0.15950(12),p(C)(4NN+3NN)=0.20490(12),p(C)(4NN+2NN+NN)=0.11440(12),p(C)(4NN+3NN+NN)=0.11920(12),p(C)(4NN+3NN+2NN)=0.11330(12), and p(C)(4NN+3NN+2NN+NN)=0.10000(12), where 3NN, 2NN, and NN stand for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with a lattice constant that is twice as large, the percolation threshold p(C)(4NN) is exactly equal to p(C)(NN). The simplified method of Bastas et al. allows for uncertainty of the percolation threshold value p(C) to be reached, similar to that obtained with the classical method but ten times faster.

The risk of extinction - the mutational meltdown or the overpopulation

The phase diagrams survival-extinction for the Penna model with parameters: (mutations rate)-(birth rate), (mutation rate)-(harmful mutations threshold), (harmful mutation threshold)-(minimal reproduction age) are presented. The extinction phase may be caused by either mutational meltdown or overpopulation. When the Verhulst factor is responsible for removing only newly born babies and does not act on adults the overpopulation is avoided and only genetic factors may lead to species extinction.

Restoring site percolation on damaged square lattices

Restoring site percolation on a damaged square lattice with the nearest neighbor (N2) is investigated using two different strategies. In the first one, a density y of new sites are created on the empty sites with longer range links, either next-nearest neighbor (N3) or next-next-nearest neighbor (N4), but without N2. In the second one, new longer range links N3 or N4 are added to N2 but only for a fraction v of the remaining non-destroyed sites. Starting at p(c)(N2), with a density x of randomly destroyed sites, the values of y(c) and v(c), which restore site percolation, are calculated for both strategies with, respectively, N3 and N4 using Monte Carlo simulations. Results are obtained for the whole range 0 < or = x < or = p(c)(N2).

Magnetization reversal in spin patterns with complex geometry

We study field-driven dynamics of spins with antiferromagnetic interactions along the links of a complex substrate geometry, which is modeled by graphs of a controlled connectivity distribution. The magnetization reversal occurs in avalanches of spin flips, which are pinned by the topological constraints of the underlying graph. The hysteresis loop and avalanche sizes are analyzed and classified in terms of the graph's connectivity and clustering. The results are relevant for magnets with a hierarchical spatial inhomogeneity and for design of nanoscale magnetic devices.

Square-lattice site percolation at increasing ranges of neighbor bonds

We report site percolation thresholds for square lattice with neighbor bonds at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (NN), next-nearest neighbors (NNN), next-next-nearest neighbors (4N), and fifth-nearest neighbors (6N) yield the same pc = 0.592... . The fourth-nearest neighbors (5N) give pc = 0.298... . This equality is proved to be mathematically exact using symmetry argument. We then consider combinations of various kinds of neighborhoods with (NN+NNN), (NN+4N), (NN+NNN+4N), and (NN+5N). The calculated associated thresholds are respectively pc = 0.407..., 0.337..., 0.288..., and 0.234... . The existing Galam-Mauger universal formula for percolation thresholds does not reproduce the data showing dimension and coordination number are not sufficient to build a universal law which extends to complex lattices.