arXiv subjects

Kordian Makulski

Publications and source records attributed to Kordian Makulski.

4 recordsLinked to original sources

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↗

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↗