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Krzysztof Kulakowski

Publications and source records attributed to Krzysztof Kulakowski.

At least 19 recordsLinked to original sources
arXiv2022

Perfect cycles in the synchronous Heider dynamics in complete network

We discuss a cellular automaton simulating the process of reaching Heider balance in a fully connected network. The dynamics of the automaton is defined by a deterministic, synchronous and global update rule. The dynamics has a very rich spectrum of attractors including fixed points and limit cycles, the length and number of which change with the size of the system. In this paper we concentrate on a class of limit cycles that preserve energy spectrum of the consecutive states. We call such limit cycles perfect. Consecutive states in a perfect cycle are separated from each other by the same Hamming distance. Also the Hamming distance between any two states separated by $k$ steps in a perfect cycle is the same for all such pairs of states. The states of a perfect cycle form a very symmetric trajectory in the configuration space. We argue that the symmetry of the trajectories is rooted in the permutation symmetry of vertices of the network and a local symmetry of a certain energy function measuring the level of balance/frustration of triads.

physics.comp-ph↗
arXiv2021

Towards the Heider balance -- a cellular automaton with a global neighborhood

We study a simple deterministic map that leads a fully connected network to Heider balance. The map is realized by an algorithm that updates all links synchronously in a way depending on the state of the entire network. We observe that the probability of reaching a balanced state increases with the system size N. Jammed states become less frequent for larger N. The algorithm generates also limit cycles, mostly of length $2$, but also of length 3, 4, 6, 12 or 14. We give a simple argument to estimate the mean size of basins of attraction of balanced states and discuss symmetries of the system including the automorphism group as well as gauge invariance of triad configurations. We argue that both the symmetries play an essential role in the occurrence of cycles observed in the synchronous dynamics realized by the algorithm.

nlin.CG↗
arXiv2019

The Heider balance and the looking-glass self

We consider the dynamics of interpersonal relations which leads to balanced states in a fully connected network. Here this approach is applied to directed networks with asymmetric relations, and it is generalized to include self-evaluation of actors, according to the 'looking-glass self' theory. A new index of self-acceptance is proposed: the relation of an actor to him/herself is positive, if the majority of his/her positive relations to others are reciprocated. Sets of stable configurations of relations are obtained under the dynamics, where the self-evaluation of some actors is negative. Within each set all configurations have the same structure.

physics.soc-ph↗
arXiv2018

Separate or perish - the coevolving voter model

Recent generalization of the coevolving voter model (J. Toruniewska et al, PRE 96 (2017) 042306) is further generalized here, including spin-dependent probability of rewiring. Mean field results indicate that either the system splits into two separate networks with different spins, or one of spin orientation goes extinct. In both cases, the density of active links is equal to zero. The results are discussed in terms of homophily in social contacts.

physics.soc-ph↗
arXiv2018

Mean field model of a game for power

Our aim is to model a game for power as a dynamical process, where an excess of power possessed by a player allows him to gain even more power. Such a positive feedback is often termed as the Matthew effect. Analytical and numerical methods allow to identify a set of fixed points of the model dynamics. The positions of the unstable fixed points give an insight on the basins of attraction of the stable fixed points. The results are interpreted in terms of modeling of coercive power.

physics.soc-ph↗
arXiv2017

Paradox of integration - mean field approach

Recently a computational model has been proposed of the social integration, as described in sociological terms by Peter Blau. In this model, actors praise or critique each other, and these actions influence their social status and raise negative or positive emotions. The role of a self-deprecating strategy of actors with high social status has also been discussed there. Here we develop a mean field approach, where the active and passive roles (praising and being praised, etc.) are decoupled. The phase transition from friendly to hostile emotions has been reproduced, similarly to the previously applied purely computational approach. For both phases, we investigate the time dependence of the distribution of social status. There we observe a diffusive spread, which - after some transient time - appears to be limited from below or from above, depending on the phase. As a consequence, the mean status flows.

physics.soc-ph↗
arXiv2016

Consecutive partitions of social networks between rivaling leaders

A model algorithm is proposed to study subsequent partitions of complex networks describing social structures. The partitions are supposed to appear as actions of rivaling leaders corresponding to nodes with large degrees. The condition of a partition is that the distance between two leaders is at least three links. This ensures that the layer of nearest neighbours of each leader remains attached to him. As a rule, numerically calculated size distribution of fragments of scale-free Albert-Barabasi networks reveals one large fragment which contains the original leader (hub of the network), and a number of small fragments with opponents that are described by two Weibull distributions. Numerical simulations and mean-field theory reveal that size of the larger fragment scales as the square root of the initial network size. The algorithm is applied to the data on political blogs in U.S. (L. Adamic and N. Glance, Proc. WWW-2005). The obtained fragments are clearly polarized; either they belong to Democrats, or to the GOP.

physics.soc-ph↗
arXiv2016

Paradox of integration -- a computational model

The paradoxical aspect of integration of a social group has been highlighted by Peter Blau (Exchange and Power in Social Life, Wiley and Sons, 1964). During the integration process, the group members simultaneously compete for social status and play the role of the audience. Here we show that when the competition prevails over the desire of approval, a sharp transition breaks all friendly relations. However, as was described by Blau, people with high status are inclined to bother more with acceptance of others; this is achieved by praising others and revealing her/his own weak points. In our model, this action smooths the transition and improves interpersonal relations.

physics.soc-ph↗
arXiv2015

Inferring cultural regions from correlation networks of given baby names

We report investigations on the statistical characteristics of the baby names given between 1910 and 2010 in the United States of America. For each year, the 100 most frequent names in the USA are sorted out. For these names, the correlations between the names profiles are calculated for all pairs of states (minus Hawaii and Alaska). The correlations are used to form a weighted network which is found to vary mildly in time. In fact, the structure of communities in the network remains quite stable till about 1980. The goal is that the calculated structure approximately reproduces the usually accepted geopolitical regions: the North East, the South, and the "Midwest + West" as the third one. Furthermore, the dataset reveals that the name distribution satisfies the Zipf law, separately for each state and each year, i.e. the name frequency $f\propto r^{-α}$, where r is the name rank. Between 1920 and 1980, the exponent alpha is the largest one for the set of states classified as 'the South', but the smallest one for the set of states classified as "Midwest + West". Our interpretation is that the pool of selected names was quite narrow in the Southern states. The data is compared with some related statistics of names in Belgium, a country also with different regions, but having quite a different scale than the USA. There, the Zipf exponent is low for young people and for the Brussels citizens.

physics.soc-ph↗
arXiv2015

Emerging communities in networks - a flow of ties

Algorithms for search of communities in networks usually consist discrete variations of links. Here we discuss a flow method, driven by a set of differential equations. Two examples are demonstrated in detail. First is a partition of a signed graph into two parts, where the proposed equations are interpreted in terms of removal of a cognitive dissonance by agents placed in the network nodes. There, the signs and values of links refer to positive or negative interpersonal relationships of different strength. Second is an application of a method akin to the previous one, dedicated to communities identification, to the Sierpinski triangle of finite size. During the time evolution, the related graphs are weighted; yet at the end the discrete character of links is restored. In the case of the Sierpinski triangle, the method is supplemented by adding a small noise to the initial connectivity matrix. By breaking the symmetry of the network, this allows to a successful handling of overlapping nodes.

physics.soc-ph↗
arXiv2013

Size dependence of the largest distance between random points

A set of $N$ points is chosen randomly in a $D$-dimensional volume $V=a^D$, with periodic boundary conditions. For each point $i$, its distance $d_i$ is found to its nearest neighbour. Then, the maximal value is found, $d_{max}=max(d_i, i=1,...,N)$. Our numerical calculations indicate, that when the density $N/V$=const, $d_{max}$ scales with the linear system size as $d^2_{max}\propto a^ϕ$, with $ϕ=0.24\pm0.04$ for $D=1,2,3,4$.

physics.comp-ph↗
arXiv2013

How many parameters to model states of mind ?

A series of examples of computational models is provided, where the model aim is to interpret numerical results in terms of internal states of agents minds. Two opposite strategies or research can be distinguished in the literature. First is to reproduce the richness and complexity of real world as faithfully as possible, second is to apply simple assumptions and check the results in depth. As a rule, the results of the latter method agree only qualitatively with some stylized facts. The price we pay for more detailed predictions within the former method is that consequences of the rich set of underlying assumptions remain unchecked. Here we argue that for computational reasons, complex models with many parameters are less suitable.

physics.soc-ph↗
arXiv2013

Competing of Sznajd and voter dynamics in the Watts-Strogatz network

We investigate the Watts-Strogatz network with the clustering coefficient C dependent on the rewiring probability. The network is an area of two opposite contact processes, where nodes can be in two states, S or D. One of the processes is governed by the Sznajd dynamics: if there are two connected nodes in D-state, all their neighbors become D with probability p. For the opposite process it is sufficient to have only one neighbor in state S; this transition occurs with probability 1. The concentration of S-nodes changes abruptly at given value of the probability p. The result is that for small p, in clusterized networks the activation of S nodes prevails. This result is explained by a comparison of two limit cases: the Watts-Strogatz network without rewiring, where C=0.5, and the Bethe lattice where C=0.

physics.soc-ph↗
arXiv2012

Coarse-grained cellular automaton for traffic systems

A coarse-grained cellular automaton is proposed to simulate traffic systems. There, cells represent road sections. A cell can be in two states: jammed or passable. Numerical calculations are performed for a piece of square lattice with open boundary conditions, for the same piece with some cells removed and for a map of a small city. The results indicate the presence of a phase transition in the parameter space, between two macroscopic phases: passable and jammed. The results are supplemented by exact calculations of the stationary probabilities of states for the related Kripke structure constructed for the traffic system. There, the symmetry-based reduction of the state space allows to partially reduce the computational limitations of the numerical method.

nlin.CG↗
arXiv2012

Competing contact processes on homogeneous networks with tunable clusterization

We investigate two homogeneous networks: the Watts-Strogatz network and the random Erdos-Renyi network, the latter with tunable clustering coefficient $C$. The network is an area of two competing contact processes, where nodes can be in two states, S or D. A node S becomes D with probability 1 if at least two its mutually linked neighbours are D. A node D becomes S with a given probability $p$ if at least one of its neighbours is S. The competition between the processes is described by a phase diagram, where the critical probability $p_c$ depends on the clustering coefficient $C$. For $p>p_c$ the rate of state S increases in time, seemingly to dominate in the whole system. Below $p_c$, the contribution of D-nodes remains finite. The numerical results, supported by mean field approach, indicate that the transition is discontinuous.

physics.soc-ph↗
arXiv2011

Bounded confidence model: addressed information maintain diversity of opinions

A community of agents is subject to a stream of messages, which are represented as points on a plane of issues. Messages are sent by media and by agents themselves. Messages from media shape the public opinion. They are unbiased, i.e. positive and negative opinions on a given issue appear with equal frequencies. In our previous work, the only criterion to receive a message by an agent is if the distance between this message and the ones received earlier does not exceed the given value of the tolerance parameter. Here we introduce a possibility to address a message to a given neighbour. We show that this option reduces the unanimity effect, what improves the collective performance.

physics.soc-ph↗