arXiv subjects

Agata Fronczak

Publications and source records attributed to Agata Fronczak.

At least 19 recordsLinked to original sources

arXiv2026

A mini-review on combinatorial solutions to the Marcus-Lushnikov irreversible aggregation

Over the past decade, a combinatorial framework for discrete, finite, and irreversibly aggregating systems has emerged. This work reviews its progress, practical applications, and limitations. We outline the approach's assumptions and foundations, based on direct enumeration of system states, contrasting with classical Smoluchowski and Marcus-Lushnikov methods. Using the constant kernel as an example, we derive combinatorial expressions for the average number of clusters of a given size and their standard deviation, and present the complete probability distribution for cluster counts. The method is then extended to several kernels (additive, product, linear-chain, condensation) by explicitly enumerating ways to form clusters of a given size. For general kernels, approximate solutions are obtained via recursive expressions, enabling predictions without explicit solutions. Applications to aerosol growth and planetesimal formation are demonstrated, with comparisons to numerical results. We summarize issues of validity and precision and propose open problems. The appendix includes partial Bell polynomials, generating functions, Lagrange inversion, potential applications, and links between combinatorial and scaling solutions of the Smoluchowski equation.

cond-mat.stat-mech↗

arXiv2025

Exploring the dynamics of external and self-citations and their role in shaping scientific impact

Understanding the mechanisms driving the distribution of scientific citations is a key challenge in assessing the scientific impact of authors. We investigate the influence of the preferential attachment rule (PAR) in this process by analysing individual citation events from the DBLP dataset and two Scopus-based datasets, enabling us to estimate the probability of citations being assigned preferentially. Our findings reveal that, for the aggregated dataset, PAR dominates the citation distribution process, with approximately 70% of citations adhering to this mechanism. However, analysis at the individual level shows significant variability, with some authors experiencing a greater prevalence of preferential citations, particularly in the context of external citations. In contrast, self-citations exhibit notably different behaviour, with only 20% following PAR. We also demonstrate that the prominence of PAR increases with an author's citability (average citations per paper), suggesting that more citable authors are preferentially cited, while less-cited authors experience more random citation patterns. Furthermore, we show that self-citations may influence bibliometric indices, such as the h-index. Our results confirm the distinct dynamics of self-citations compared to external citations, raising questions about the mechanisms driving self-citation patterns. These findings provide new insights into citation behaviours and highlight the limitations of existing approaches.

cs.DL↗

arXiv2025

Beyond traditional box-covering: Determining the fractal dimension of complex networks using a fixed number of boxes of flexible diameter

In this article, we present a novel box-covering algorithm for analyzing the fractal properties of complex networks. Unlike traditional algorithms that impose a predetermined box size, our approach assigns nodes to boxes identified by their nearest local hubs without enforcing rigid distance constraints. This flexibility leads to a key methodological shift: instead of fixing the box size in advance, we first determine the number of boxes and then compute their average size. We argue that this procedure is fully consistent with the recently proposed scaling theory of fractal complex networks and closely related to the concept of hidden metric spaces in which network nodes are embedded. We demonstrate that our approach not only significantly reduces computational complexity compared to existing methods, but also (despite relaxing constraints on box diameter) covers networks using boxes of more similar sizes than, for instance, the classical greedy coloring (GC) algorithm. To evaluate the effectiveness of our method, we analyze nine complex networks (three model-based and six real-world) representing a broad spectrum: from networks with confirmed fractality, through those with initially uncertain, but here confirmed, fractal properties (such as the Internet at the level of autonomous systems), to large-scale networks that have so far remained beyond the reach of existing algorithms due to their size.

cond-mat.dis-nn↗

arXiv2025

Model-based reconstruction of real-world fractal complex networks

This paper presents a versatile model for generating fractal complex networks that closely mirror the properties of real-world systems. By combining features of reverse renormalization and evolving network models, the proposed approach introduces several tunable parameters, offering exceptional flexibility in capturing the diverse topologies and scaling behaviors found in both natural and man-made networks. The model effectively replicates their key characteristics such as fractal dimensions, power-law degree distributions, and scale-invariant properties of hierarchically nested boxes. Unlike traditional deterministic models, it incorporates stochasticity into the network growth process, overcoming limitations like discontinuities in degree distributions and rigid size constraints. The model's applicability is demonstrated through its ability to reproduce the structural features of real-world fractal networks, including the Internet, the World Wide Web, and co-authorship networks.

physics.soc-ph↗

arXiv2025

Universality classes in the time evolution of epidemic outbreaks on complex networks

We investigate the full temporal evolution of epidemic outbreaks in complex networks, focusing on the susceptible-infected (SI) model of disease transmission. Combining theoretical analysis with large-scale numerical simulations, we uncover two universal patterns of epidemic growth, determined by the structure of the underlying network. In small-world networks, the prevalence follows a Gompertz-like curve, while in fractal networks it evolves according to Avrami-type dynamics, which is typical of spatially constrained systems. These regimes define distinct universality classes that remain robust across arbitrary transmission rates. Notably, our approach provides explicit analytical formulas for the global epidemic prevalence and class-specific scaling relations capturing its dependence on the transmission rate. We show that the commonly assumed early exponential growth occurs only in small-world networks, where it corresponds to the short-time approximation of the Gompertz function. In contrast, this exponential phase is entirely absent in fractal networks, where spreading is markedly slower and governed by different mechanisms. Our approach clarifies the structural origins of these contrasting behaviors and offer a unified framework for understanding epidemic dynamics across diverse network topologies.

physics.soc-ph↗

arXiv2024

Interplay between tie strength and neighbourhood topology in complex networks: Granovetter's theory and beyond

Granovetter's weak ties theory is a very important sociological theory according to which a correlation between edge weight and the network's topology should exist. More specifically, the neighbourhood overlap of two nodes connected by an edge should be positively correlated with edge weight (tie strength). However, some real social networks exhibit a negative correlation - the most prominent example is the scientific collaboration network, for which overlap decreases with edge weight. It has been demonstrated that the aforementioned inconsistency with Granovetter's theory can be alleviated in the scientific collaboration network through the use of asymmetric measures. In this paper, we explain that while asymmetric measures are often necessary to describe complex networks and to confirm Granovetter's theory, their interpretation is not simple, and there are pitfalls that one must be wary of. The definitions of asymmetric weights and overlaps introduce structural correlations that must be filtered out. We show that correlation profiles can be used to overcome this problem. Using this technique, not only do we confirm Granovetter's theory in various real and artificial social networks, but we also show that Granovetter-like weight-topology correlations are present in other complex networks (e.g. metabolic and neural networks). Our results suggest that Granovetter's theory is a sociological manifestation of more general principles governing various types of complex networks.

physics.soc-ph↗

arXiv2024

Scaling theory of fractal complex networks

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach - grounded in both scaling theory of phase transitions and renormalization group theory - leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information - the World Wide Web, biological - the human brain, and social - scientific collaboration networks) and in several fractal network models.

physics.soc-ph↗

arXiv2023

Towards fractal origins of the community structure in complex networks: a model-based approach

In this paper, we pose a hypothesis that the structure of communities in complex networks may result from their latent fractal properties. This hypothesis is based not only on the general observation that many real networks have multilevel organization, which is reminiscent of the geometric self-similarity of classical fractals. Quantitative arguments supporting this hypothesis are: first, many non-fractal real complex networks that have a well-defined community structure reveal fractal properties when suitably diluted; second, the scale-free community size distributions observed in many real networks directly relate to scale-invariant box mass distributions, which have recently been described as a fundamental feature of fractal complex networks. We test this hypothesis in a general model of evolving network with community structure that exhibits dual scale invariance: at the level of node degrees and community sizes, respectively. We show that, at least in this model, the proposed hypothesis cannot be rejected. The argument for this is that a kind of fractal core can be identified in the networks studied, which appears as a macroscopic connected component when the edges between modules identified by the community detection algorithm are removed in a supervised manner.

physics.soc-ph↗

arXiv2023

Asymmetry of social interactions and its role in link predictability: the case of coauthorship networks

The paper provides important insights into understanding the factors that influence tie strength in social networks. Using local network measures that take into account asymmetry of social interactions we show that the observed tie strength is a kind of compromise, which depends on the relative strength of the tie as seen from its both ends. This statement is supported by the Granovetter-like, strongly positive weight-topology correlations, in the form of a power-law relationship between the asymmetric tie strength and asymmetric neighbourhood overlap, observed in three different real coauthorship networks and in a synthetic model of scientific collaboration. This observation is juxtaposed against the current misconception that coauthorship networks, being the proxy of scientific collaboration networks, contradict the Granovetter's strength of weak ties hypothesis, and the reasons for this misconception are explained. Finally, by testing various link similarity scores, it is shown that taking into account the asymmetry of social ties can remarkably increase the efficiency of link prediction methods. The perspective outlined also allows us to comment on the surprisingly high performance of the resource allocation index -- one of the most recognizable and effective local similarity scores -- which can be rationalized by the strong triadic closure property, assuming that the property takes into account the asymmetry of social ties.

cs.SI↗

arXiv2021

Scientific success from the perspective of the strength of weak ties

We present the first complete confirmation of Granovetter's theory of social networks using a massive dataset. For this purpose, we study a scientific collaboration network, which is considered one of the most important examples that contradicts the universality of this theory. We achieve this goal by rejecting the assumption of the symmetry of social ties. Our approach is grounded in well-established heterogeneous (degree-based) mean-field theory commonly used to study dynamical processes on complex networks. Granovetter's theory is based on two hypotheses that assign different roles to interpersonal, information-carrying connections. The first hypothesis states that strong ties carrying the majority of interaction events are located mainly within densely connected groups of people. The second hypothesis maintains that these groups are connected by sparse weak ties that are of vital importance for the diffusion of information - individuals who have access to weak ties have an advantage over those who do not. Given the scientific collaboration network, with strength of directed ties measured by the asymmetric fraction of joint publications, we show that scientific success is strongly correlated with the structure of a scientist's collaboration network. First, among two scientists, with analogous achievements, the one with weaker ties tends to have the higher h-index, and second, teams connected by such ties create more valuable publications.

physics.soc-ph↗

arXiv2021

Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework

For this work, we studied a finite system of discreet-size aggregating particles for two types of kernels with arbitrary parameters, a condensation (or branched-chain polymerization) kernel, $K(i,j)=(A+i)(A+j)$, and a linear combination of the constant and additive kernels, $K(i,j)=A+i+j$. They were solved under monodisperse initial conditions in the combinatorial approach where discreet time is counted as subsequent states of the system. A generating function method and Lagrange inversion were used for derivations. Expressions for an average number of clusters of a given size and its corresponding standard deviation were obtained and tested against numerical simulation. High precision of the theoretical predictions can be observed for a wide range of $A$ and coagulation stages, excepting post-gel phase in the case of the condensation kernel (a giant cluster presence is preserved). For appropriate $A$, these two kernels reproduced known results of the constant, additive and product kernels. Beside a previously solved linear-chain kernel, they extend the number of arbitrary-parameter kernels solved in the combinatorial approach.

cond-mat.stat-mech↗

arXiv2021

Exact combinatorial approach to finite coagulating systems through recursive equations

This work outlines an exact combinatorial approach to finite coagulating systems through recursive equations and use of generating function method. In the classic approach the mean-field Smoluchowski coagulation is used. However, the assumptions of the mean-field theory are rarely met in real systems which limits the accuracy of the solution. In our approach, cluster sizes and time are discrete, and the binary aggregation alone governs the time evolution of the systems. By considering the growth histories of all possible clusters and applying monodisperse initial conditions, the exact expression for the probability of finding a coagulating system with an arbitrary kernel in a given cluster configuration is derived. Then, the average number of such clusters and the standard deviation of these solutions can be calculated. In this work, recursive equations for all possible growth histories of clusters are introduced. The correctness of our expressions was proved based on the comparison with numerical results obtained for systems with constant, multiplicative and additive kernels. For the first time the exact solutions for the multiplicative and additive kernels were obtained with this framework. In addition, our results were compared with the results arising from the solutions to the mean-field Smoluchowski equation. Our theoretical predictions outperform the classic approach.

cond-mat.stat-mech↗

arXiv2020

Combinatorial solutions to generalized electrorheological kernel aggregation

For this paper, we studied the time evolution of a system of coagulating particles under a generalized electrorheological (ER) kernel with real power, $K\left(i,j\right) = \left( \frac{1}{i}+\frac{1}{j} \right)^α$, and monodisperse initial conditions. We used a combinatorial framework in which time and cluster sizes were discrete and the binary aggregation governed the time evolution of the system. We modified a previously-known solution for the constant kernel to cover the generalized ER kernel and used it in the framework to obtain the exact expression for the cluster size distribution (the average number of particles of a given size) and the standard deviation. Our theoretical solution is validated by a comparison to numerically simulated results for several values of $α$ and to the experimental data of coagulating polystyrene particles. Theoretical predictions were accurate for any time of the aggregation process and for a wide range of $α$.

cond-mat.stat-mech↗

arXiv2020

Bell polynomials in the series expansions of the Ising model

Through applying Bell polynomials to the integral representation of the free energy of the Ising model for the triangular and hexagonal lattices we obtain the exact combinatorial formulas for the number of spin configurations at a given energy (i.e. low-temperature series expansion of the partition function or, alternatively, the number of states). We also generalize this approach to the wider class of the (chequered) Utiyama graphs. Apart from the presented exact formulas, our technique allows one to establish the correspondence between the perfect gas of clusters and the Ising model on the lattices which have positive coefficients in the low-temperature expansion (e.g. square lattice, hexagonal lattice). However it is not always the case -- we present that for the triangular lattice the coefficients could be negative and the perfect gas of clusters interpretation is problematic.

math-ph↗

arXiv2020

A veritable zoology of successive phase transitions in the asymmetric $q$-voter model on multiplex networks

We analyze a nonlinear $q$-voter model with stochastic noise, interpreted in the social context as independence, on a duplex network. The size of the lobby $q$ (i.e., the pressure group) is a crucial parameter that changes the behavior of the system. The $q$-voter model has been applied on multiplex networks in a previous work [Phys. Rev E. 92. 052812. (2015)], and it has been shown that the character of the phase transition depends on the number of levels in the multiplex network as well as the value of $q$. Here we study phase transition character in the case when on each level of the network the lobby size is different, resulting in two parameters $q_1$ and $q_2$. We find evidence of successive phase transitions when a continuous phase transition is followed by a discontinuous one or two consecutive discontinuous phases appear, depending on the parameter. When analyzing this system, we even encounter mixed-order (or hybrid) phase transition. We perform simulations and obtain supporting analytical solutions on a simple multiplex case - a duplex clique, which consists of two fully overlapped complete graphs (cliques).

cond-mat.stat-mech↗

arXiv2019

Truncated Lévy Walks and Superdiffusion in Boltzmann-Gibbs Equilibrium of the Hamiltonian Mean-Field Model

The Hamiltonian Mean-Field (HMF) model belongs to a broad class of statistical physics models with non-additive Hamiltonians that reveal many non-trivial properties, such as non-equivalence of statistical ensembles, ergodicity breaking, and negative specific heat. With this paper, we add to this set another intriguing feature, which is that of super-diffusive equilibrium dynamics. Using molecular dynamics techniques, we compare the diffusive properties of the HMF model in the quasi-stationary metastable state (QSS) and in the Boltzmann-Gibbs (BG) regime. In contrast to the current state of knowledge, we show that Lévy walks underlying super-diffusion in QSS do not disappear when the system settles in the thermodynamic equilibrium. We demonstrate that it is extremely difficult to distinguish QSS from the BG regime, by only examining the statistics of Lévy walks in HMF particle trajectories. We construct a simple stochastic model based on the truncated Lévy walks with rests that quantitatively resembles diffusion behavior observed in both stages of the HMF dynamics.

cond-mat.stat-mech↗

arXiv2019

Partial equivalence of statistical ensembles in a simple spin model with discontinuous phase transitions

In this paper, we draw attention to the problem of phase transitions in systems with locally affine microcanonical entropy, in which partial equivalence of (microcanonical and canonical) ensembles is observed. We focus on a very simple spin model, that was shown to be an equilibrium statistical mechanics representation of the biased random walk. The model exhibits interesting discontinuous phase transitions that are simultaneously observed in the microcanonical, canonical, and grand canonical ensemble, although in each of these ensembles the transition occurs in a slightly different way. The differences are related to fluctuations accompanying the discontinuous change of the number of positive spins. In the microcanonical ensemble, there is no fluctuation at all. In the canonical ensemble, one observes power-law fluctuations, which are, however, size-dependent and disappear in the thermodynamic limit. Finally, in the grand canonical ensemble, the discontinuous transition is of mixed-order (hybrid) kind with diverging (critical-like) fluctuations. In general, this paper consists of many small results, which together make up an interesting example of phase transitions that are not covered by the known classifications of these phenomena.

cond-mat.stat-mech↗

arXiv2018

Coagulation with product kernel and arbitrary initial conditions: Exact kinetics within the Marcus-Lushnikov framework

The time evolution of a system of coagulating particles under the product kernel and arbitrary initial conditions is studied. Using the improved Marcus-Lushnikov approach, the master equation is solved for the probability $W(Q,t)$ to find the system in a given mass spectrum $Q=\{n_1,n_2,\dots,n_g\dots\}$, with $n_g$ being the number of particles of size $g$. The exact expression for the average number of particles, $\langle n_g(t)\rangle$, at arbitrary time $t$ is derived and its validity is confirmed in numerical simulations of several selected initial mass spectra.

cond-mat.stat-mech↗