arXiv subjects

Krzysztof Malarz

Publications and source records attributed to Krzysztof Malarz.

At least 19 recordsLinked to original sources

arXiv2026

Heider balance in square versus triangular lattices under an external field

We discuss the Heider model in the presence of an external social field. This field breaks the possible symmetry between the probabilities of hostile and friendly relationships. We consider the system in the presence of fluctuations generated by thermal noise and present the results of a comparative study of two-dimensional triangular and square networks with periodic boundary conditions. The results were obtained using three different methods: exact calculations for small systems ($3 \times 3$ square lattice and $2 \times 2$ triangular lattice), Monte Carlo simulations of medium-sized systems ($16 \times 16$ lattices), and exact calculations in the thermodynamic limit (corresponding to infinite size) of certain limiting cases for which analytical solutions are possible. In particular, we exploit the recently discovered equivalence between structurally balanced systems and the Ising model to derive an exact form of the edge magnetization susceptibility for systems in Heider equilibrium.

cond-mat.stat-mech↗

arXiv2026

Extending the Biswas--Chatterjee--Sen model with nonconformists and inflexibles

Originally, the Biswas--Chatterjee--Sen model was shown to exhibit an order/disorder phase transition for a sufficiently large number of negative interactions among actors. In this paper, the model is extended by the existence of anticonformists and inflexibles. Anticonformists are actors who define themselves in opposition to the group and may intentionally reject what most people accept, while inflexibles are those who do not change their opinions at all. Both discrete and continuous opinions are considered. With direct Monte Carlo simulations and mean-field calculations, we check the influence of fractions of anticonformists and inflexibles on the mean opinion in the system. With the mean-field calculations, we identify ranges of fractions of anticonformists where an ordered phase of the system is available. The results of the mean-field calculations perfectly match the results of the Monte Carlo simulations. We consider inflexibles adhered: (i) to extreme opinions; (ii) to specific opinions, and (iii) chosen independently of their initial opinion. For inflexibles adhered to specific and extreme opinions, they play a role of an effective bias suppressing the disordered phase in the system. The qualitative results of introducing anticonformists (inflexibles) in various ways (discrete/continuous opinions and annealed/quenched disorder) are roughly the same. However, for the model extended by inflexibles, we can observe a systematic shift of the mean order parameter to its higher values for quenched disorder compared with annealed disorder. On the other hand, for anticonformists modeled with a continuous space of opinions, we can observe a systematic shift of the mean order parameter to its higher values compared with the discrete space of opinions.

physics.soc-ph↗

arXiv2026

Random site percolation with complex neighborhoods in five dimension

In this paper, the random site percolation problem in a five-dimensional space for complex neighborhoods is studied. The efficient C++ code (with ordinary work division) of the classical Newman--Ziff algorithm is presented. The obtained speed-up of computations reduces 2170 years of single-core computations -- necessary for obtaining the results presented in this paper -- much below the typical half-decay time of the scientist. For complex neighborhoods, it is for neighborhoods composed with sites taken from several coordination zones (up to the seventh coordination zone), the 127 percolation thresholds are calculated (with 120 among them being estimated for the first time). For seven extended (compact) neighborhoods, the fractal dimensions are also calculated. The mean value of these fractal dimensions, averaged over these seven compact neighborhoods, is estimated as $\langle d_f\rangle\approx 3.5581(70)$. The percentage errors of the values obtained for the fractal dimensions for compact neighborhoods vary from 0.26\% to 1.76\% with respect to the theoretically predicted value based on scaling relations and the most recent estimates of critical exponents for five-dimensional space. The universality of the percolation threshold as dependent on the weighted coordination number $ζ=\sum_i z_i r_i$ (where $z_i$ is the number of sites in the $i$-th coordination zone and $r_i$ is the Euclidean distance from the sites in the $i$-th coordination zone to the central site) is also verified. The latter manifests itself as the power law ($p_c\proptoζ^{-g}$) with $g\approx 0.7913(43)$

physics.comp-ph↗

arXiv2026

Relationship between Heider links and Ising spins

We show that the Heider model with an external field is equivalent, in the limit of structural balance, to the Ising model with nearest-neighbor interactions without an external field. More precisely, we claim that the signs of the Heider relations that maintain structural equilibrium in the system can be represented as nearest neighbor Ising spin products. We demonstrate this explicitly for a complete graph and provide a general argument for an arbitrary graph. A consequence of the equivalence is that the system of balanced Heider states undergoes a phase transition, inherited from the Ising model, at a critical value of the social field at which the fluctuations of edge magnetization are maximal.

cond-mat.stat-mech↗

arXiv2025

Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone

The site percolation problem is one of the core topics in statistical physics. Evaluation of the percolation threshold, which separates two phases (sometimes described as conducting and insulating), is useful for a range of problems from core condensed matter to interdisciplinary application of statistical physics in epidemiology or other transportation or connectivity problems. In this paper with Newman--Ziff fast Monte Carlo algorithm and finite-size scaling theory the random site percolation thresholds $p_c$ for a square lattice with complex neighborhoods containing sites from the sixth coordination zone are computed. Complex neighborhoods are those that contain sites from various coordination zones (which are not necessarily compact). We also present the source codes of the appropriate procedures (written in C) to be replaced in original Newman--Ziff code. Similar to results previously found for the honeycomb lattice, the percolation thresholds for complex neighborhoods on a square lattice follow the power law $p_c(ζ)\proptoζ^{-γ_2}$ with $γ_2=0.5454(60)$, where $ζ=\sum_i z_i r_i$ is the weighted distance of sites in complex neighborhoods ($r_i$ and $z_i$ are the distance from the central site and the number of sites in the coordination zone $i$, respectively).

cond-mat.stat-mech↗

arXiv2025

Fine structure of phase diagram for social impact theory

In this paper, the social impact theory introduced by Latané is reconsidered. A fully differentiated society is considered; that is, initially every actor has their own opinion. The equivalent of Muller's ratchet guards that -- even for the non-deterministic case (with a positive social temperature) -- any opinion once removed from the opinion space does not appear again. With computer simulation, we construct the phase diagram for Latané model based on the number of surviving opinions after various evolution times. The phase diagram is constructed on the two-dimensional plane of model control parameters responsible for the effective range of interaction among actors and the social temperature. Introducing the Muller's ratchet-like mechanism gives a non-zero chance for any opinion to be removed from the system. We believe that in such a case, for any positive temperature, ultimately a consensus is reached. However, even for a moderate system size, the time to reach consensus is very long. In contrast, for the deterministic case (without social temperature), the system may be frozen with clusters of actors having several different opinions, or even reach the cycle limit (with blinking structures).

physics.soc-ph↗

arXiv2025

Lower limit of percolation threshold on a square lattice with complex neighborhoods

In this paper, the 60-year-old concept of long-range interaction in percolation problems introduced by Dalton, Domb, and Sykes, is reconsidered. With Monte Carlo simulation -- based on Newman-Ziff algorithm and finite-size scaling hypothesis -- we estimate 64 percolation thresholds for random site percolation problem on a square lattice with neighborhoods that contain sites from the 7th coordination zone. The percolation thresholds obtained range from 0.27013 (for the neighborhood that contains only sites from the 7th coordination zone) to 0.11535 (for the neighborhood that contains all sites from the 1st to the 7th coordination zone). Similarly to neighborhoods with smaller ranges, the power-law dependence of the percolation threshold on the effective coordination number with exponent close to -1/2 is observed. Finally, we empirically determine the limit of the percolation threshold on square lattices with complex neighborhoods. This limit scales with the inverse square of the mean radius of the neighborhood. The boundary of this limit is touched for threshold values associated with extended (compact) neighborhoods.

cond-mat.stat-mech↗

arXiv2025

Heider balance on Archimedean lattices and cliques

We investigate the work function $U(T)$ for the Heider balance, driven by a thermal noise $T$, on several planar networks that contain separated triangles, pairs of triangles, chains of triangles and complex structures of triangles. In simulations, the heat-bath algorithm is applied. Two schemes of link values updating are considered: synchronous and asynchronous (sequential). The latter results are compared with analytical calculations for small cliques. We argue that the actual shape of $U(T)$ is a consequence of a local topology rather than of a macroscopic ordering. Finally, we present the mathematical proof that for any planar lattice, perfect structural (Heider) balance is unreachable at $T>0$.

physics.soc-ph↗

arXiv2024

Universality of opinions disappearing in sociophysical models of opinion dynamics: From initial multitude of opinions to ultimate consensus

Possibility of reaching a consensus in social systems with strong initial fragmentation is one of the most interesting issues in sociopysics. It is also intriguing what the dynamics of such processes is. To address those problems, we performed computer simulations using well-established models of social opinion formation, namely the voter, Sznajd, and Latané models. We investigated opinion dynamics in cases where the initial number of opinions is very large, equal to the number of actors (the voter and Latané models) or when every second actor has their own opinion (Sznajd model), with some variations on the update schemes, lattice topologies, effective ranges of interaction, and information noise levels. For all considered models, the number of opinions assumed by the actors is finally almost always reduced to only one. However, while the voter and Latané models exhibit a power-law time decrease in the number of opinions, the Sznajd model follows a complex three-stage behavior. We also demonstrated that the mean/median time of reaching the consensus scales with system size according to a power law for voter and Sznajd models, while for the Latané model this increase is even faster. Our results show that in the studied models the consensus is possible, provided that a long enough and model-dependent time to reach this state is available.

physics.soc-ph↗

arXiv2024

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 their radius ranging from one to three and containing from three to 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 $p_c$ follow the dependency $p_c\propto 1/z$, as recently shown by Xun, Hao and Ziff [Physical Review E 105, 024105 (2022)]. On the contrary, noncompact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of $p_c$ corresponding to the same number $z$ of sites in the neighborhoods). An example of a single-value index $ζ=\sum_i z_i r_i$ -- where $z_i$ and $r_i$ are the number of sites and radius of the $i$-th 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 $p_c\propto 1/\sqrtζ$. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Physical Review E 64, 016706 (2001)] are also presented.

cond-mat.stat-mech↗

arXiv2024

Universality of random-site percolation thresholds for two-dimensional complex non-compact 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 $p_c\propto(ζ/K)^{-γ}$ with $γ=0.526(11)$, $0.5439(63)$ and $0.5932(47)$, for honeycomb, square, and triangular lattice, respectively, and $p_c\proptoζ^{-γ}$ with $γ=0.5546(67)$ independently on the underlying lattice. The index $ζ=\sum_i z_i r_i$ 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 $i$-th 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.

cond-mat.stat-mech↗

arXiv2023

Recovering Zipf's law in intercontinental scientific collaboration

Scientific cooperation on an international level has been well studied in the literature. However, much less is known about this cooperation on the intercontinental level. In this paper, we address this issue by creating a collection of approximately 13.8 million publications around the papers by one of the highly cited author working in complex networks and their applications. The obtained rank-frequency distribution of the probability of sequences describing continents and number of countries -- with which authors of papers are affiliated -- follows the power law with an exponent $-1.9108(15)$. Such a 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.

cs.DL↗

arXiv2023

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 cross-over or a first-order phase transition from a balanced to an imbalanced state of the system is observed with an increase of the thermal noise level. The minimal graph density $D_\text{min}$ for which the first-order phase transition can be observed decreases with system size $N$ as $D_\text{min}\propto N^{-0.58(1)}$. For graph densities $D>D_\text{min}$ the reduced critical temperature $T_c^\star=T_c/T_c(D=1)$ increases with the graph density as $T_c^\star\propto D^{1.719(6)}$ independently of the system size $N$.

physics.soc-ph↗

arXiv2022

Vanishing opinions in Latané model of opinion formation

In this paper, the results of computer simulations based on 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 as 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 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 increase of 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.

physics.soc-ph↗

arXiv2022

Mean-field approximation for structural balance dynamics in heat-bath

A critical temperature for a complete signed graph of $N$ agents where time-dependent links polarization 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 $a^c\approx 1.71649$. 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 links polarization 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 hysteresis-like loop is observed.

physics.soc-ph↗

arXiv2022

Does social distancing matter for infectious disease propagation? A 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 $\to$ exposed $\to$ infected $\to$ 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 epidemic. 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 disease.

nlin.CG↗

arXiv2022

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 cross-over between the two states. The cross-over 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.

physics.soc-ph↗

arXiv2021

Wealth rheology

We study wealth rank correlations in a simple model of macro-economy. 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.

econ.TH↗