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Robert Jankowski

Publications and source records attributed to Robert Jankowski.

15 recordsLinked to original sources
arXiv2026

Latent geometry organizes higher-order interactions

Higher-order structures offer a natural representation of complex systems that involve interactions between groups of different sizes. A widespread feature of their higher-order structure is nestedness, whereby interactions involving smaller groups are contained within larger ones. Yet, why interactions of different orders organise into nested structures remains largely unexplained. Here, we introduce an analytically tractable geometric model of higher-order networks in which a single latent geometric space couples interactions across orders, leading to the spontaneous emergence of nestedness. We show analytically and numerically that nestedness undergoes a transition between a nested geometric regime, where it remains finite in the thermodynamic limit, and a regime where it vanishes with system size. In this regime, we uncover a weakly geometric range characterized by an anomalously slow finite-size decay, allowing substantial nestedness to persist in finite systems even when its asymptotic value vanishes. Finally, with a single geometric coupling parameter, the model reproduces the nestedness profiles observed in real-world hypergraphs across different domains. Our results reveal latent geometry as a simple organising principle underlying the nested organisation of higher-order interactions.

physics.soc-ph↗
arXiv2026

Hyperbolic Graph Embedders for Link Prediction and Topology Reconstruction

Hyperbolic embeddings provide compact geometric representations of complex networks in hyperbolic spaces, but systematic comparisons of methods developed in machine learning, network science, and algorithmics remain rare. We benchmark 13 unsupervised hyperbolic graph embedders under a unified protocol for link prediction and topology reconstruction on synthetic and empirical networks. The protocol captures both missing-link recovery and the preservation of local and global network structure. Maximum-likelihood and representation-learning-based approaches, including hybrid variants, achieve the strongest overall performance, although no method dominates across all tasks and structural regimes. Performance is more strongly associated with embedding paradigm than with disciplinary origin. We identify the network regimes in which different paradigms succeed or fail and provide practical guidance for method selection in downstream applications.

cs.LG↗
arXiv2026

Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization

Graph neural networks (GNNs) can operate on large graphs but become infrastructure-sensitive at the scale of millions of nodes and typically require scalable training techniques for even larger graphs. This raises a central question: when can a model trained on a smaller, scaled-down replica of a graph be deployed on the full-resolution graph without retraining? We introduce a zero-shot transfer protocol in which a GNN is trained on a graph coarse-grained by geometric renormalization (GR), and the resulting weights are transferred directly to the original network. Across synthetic and real-world networks, training on GR scaled-down replicas preserves much of the original-scale predictive performance while significantly reducing training cost. We further find that learned representations and predictive trajectories remain aligned across scales. These findings suggest that structural similarity may be more important than network size in determining GNN transferability, opening a path toward scale-equivariant graph architectures.

cs.LG↗
arXiv2026

Motif-based filtrations for persistent homology: A framework for graph isomorphism and property prediction

Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being computationally expensive. We address this task using persistent homology built on motif-based filtrations of graphs, a method from topological data analysis that summarizes the shape of data by tracking the persistence of structural features along filtrations. Specifically, we use edge-weighting schemes based on the densities of triangles, chordless squares, and chordless pentagons, which have been shown to be effective for detecting network dimensionality. Our cycle-density filtrations distinguish non-isomorphic graphs perfectly or nearly perfectly across four demanding graph families, many of which exhibit symmetries. We outperform curvature-based, degree-based, and Vietoris--Rips filtrations, and match or exceed the accuracy of egonet-distance methods while incurring a lower computational cost. The expressive power of our filtrations goes beyond isomorphism testing: because they capture rich structural information from graphs, they consistently achieve top performance on property prediction tasks using real-world data, and exhibit high sensitivity to edge rewiring and removal. Together, these findings establish cycle-density filtrations as an effective and computationally tractable framework for graph comparison and characterization, bridging topological data analysis and network science.

math.AT↗
arXiv2025

Trade-offs between structural richness and perceptual robustness in music network representations

Music is a structured and perceptually rich sequence of sounds in time, whose perception is shaped by the interplay of expectation and uncertainty about what comes next. Yet the uncertainty we infer from music depends on how the musical piece is encoded as an event sequence. In this work, we use network representations, in which event types are nodes and observed transitions are directed edges, to compare how different feature encodings shape the transition structure we recover and how robust that structure is under modeled perceptual constraints. We systematically analyse eight encodings of piano music, from single-feature vocabularies to richer multi-feature combinations. These representational choices reorganize the state space and fundamentally reshape network topology, shifting how uncertainty is distributed across transitions. To connect these descriptive differences to perception, we adopt a perceptual-constraint model that captures imperfect access to transition statistics. Overall, compressed single-feature representations yield dense transition structures with higher entropy rates, corresponding to higher average uncertainty per step, yet low model error, indicating that the constrained estimate stays close to the corpus transitions. In contrast, richer multi-feature representations preserve finer distinctions but expand the state space, sharpen transition profiles, lower entropy rates, and increase model error. Finally, across representations, uncertainty concentrates in diffusion-central nodes while model error remains low there, suggesting an informational landscape in which predictable flow coexists with localized surprise. Overall, our results show that feature choice shapes not only the networks we reconstruct, but also which transition statistics are available and how vulnerable they are to distortion under perceptual constraints.

physics.soc-ph↗
arXiv2025

Shortest-path percolation on scale-free networks

The shortest-path percolation (SPP) model aims at describing the consumption and eventual exhaustion of a network's resources. Starting from a network containing a macroscopic connected component, random pairs of nodes are sequentially selected, and if the length of the shortest path connecting the node pairs is smaller than a tunable budget parameter, then all edges along such a path are removed from the network. As edges are progressively removed, the network eventually breaks into multiple microscopic components, undergoing a percolation-like transition. It is known that SPP transition on Erd\H{o}s-R\'enyi networks (ERNs) belongs to same universality class as of the ordinary bond percolation if the budget parameter is finite; for unbounded budget, instead, the SPP transition becomes more abrupt than the ordinary percolation transition. By means of large-scale numerical simulations and finite-size scaling analysis, here we study the SPP transition on random scale-free networks (SFNs) characterized by power-law degree distributions. We find, in contrast with ordinary percolation, that the transition is identical to the one observed on ERNs, denoting independence from the degree exponent. Still, we distinguish finite- and infinite-budget SPP universality classes. Our findings follow from the fact that the SPP process drastically homogenizes the heterogeneous structure of SFNs before the SPP transition takes place.

physics.soc-ph↗
arXiv2025

Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis

Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of communities, impact efficient network navigation, and fundamentally shape connectivity and system behavior. We introduce a topological data analysis weighting scheme for graphs based on chordless cycles to estimate network dimensionality in a data-driven way. We further show that the resulting descriptors can effectively estimate network dimensionality using a neural network architecture trained on a synthetic graph database constructed for this purpose, which requires no retraining to transfer effectively to real-world networks. Thus, by combining cycle-aware filtrations, algebraic topology, and machine learning, our approach provides a robust and effective method for uncovering the hidden geometry of complex networks and guiding accurate modeling and low-dimensional embedding.

physics.soc-ph↗
arXiv2025

Task complexity shapes internal representations and robustness in neural networks

Neural networks excel across a wide range of tasks, yet remain black boxes. In particular, how their internal representations are shaped by the complexity of the input data and the problems they solve remains obscure. In this work, we introduce a suite of five data-agnostic probes-pruning, binarization, noise injection, sign flipping, and bipartite network randomization-to quantify how task difficulty influences the topology and robustness of representations in multilayer perceptrons (MLPs). MLPs are represented as signed, weighted bipartite graphs from a network science perspective. We contrast easy and hard classification tasks on the MNIST and Fashion-MNIST datasets. We show that binarizing weights in hard-task models collapses accuracy to chance, whereas easy-task models remain robust. We also find that pruning low-magnitude edges in binarized hard-task models reveals a sharp phase-transition in performance. Moreover, moderate noise injection can enhance accuracy, resembling a stochastic-resonance effect linked to optimal sign flips of small-magnitude weights. Finally, preserving only the sign structure-instead of precise weight magnitudes-through bipartite network randomizations suffices to maintain high accuracy. These phenomena define a model- and modality-agnostic measure of task complexity: the performance gap between full-precision and binarized or shuffled neural network performance. Our findings highlight the crucial role of signed bipartite topology in learned representations and suggest practical strategies for model compression and interpretability that align with task complexity.

cs.LG↗
arXiv2025

Mapping bipartite networks into multidimensional hyperbolic spaces

Bipartite networks appear in many real-world contexts, linking entities across two distinct sets. They are often analyzed via one-mode projections, but such projections can introduce artificial correlations and inflated clustering, obscuring the true underlying structure. In this paper, we propose a geometric model for bipartite networks that leverages the high levels of bipartite four-cycles as a measure of clustering to place both node types in the same similarity space, where link probabilities decrease with distance. Additionally, we introduce B-Mercator, an algorithm that infers node positions from the bipartite structure. We evaluate its performance on diverse datasets, illustrating how the resulting embeddings improve downstream tasks such as node classification and distance-based link prediction in machine learning. These hyperbolic embeddings also enable the generation of synthetic networks with node features closely resembling real-world ones, thereby safeguarding sensitive information while allowing secure data sharing. In addition, we show how preserving bipartite structure avoids the pitfalls of projection-based techniques, yielding more accurate descriptions and better performance. Our method provides a robust framework for uncovering hidden geometry in complex bipartite systems.

physics.soc-ph↗
arXiv2024

Hyperbolic Benchmarking Unveils Network Topology-Feature Relationship in GNN Performance

Graph Neural Networks (GNNs) have excelled in predicting graph properties in various applications ranging from identifying trends in social networks to drug discovery and malware detection. With the abundance of new architectures and increased complexity, GNNs are becoming highly specialized when tested on a few well-known datasets. However, how the performance of GNNs depends on the topological and features properties of graphs is still an open question. In this work, we introduce a comprehensive benchmarking framework for graph machine learning, focusing on the performance of GNNs across varied network structures. Utilizing the geometric soft configuration model in hyperbolic space, we generate synthetic networks with realistic topological properties and node feature vectors. This approach enables us to assess the impact of network properties, such as topology-feature correlation, degree distributions, local density of triangles (or clustering), and homophily, on the effectiveness of different GNN architectures. Our results highlight the dependency of model performance on the interplay between network structure and node features, providing insights for model selection in various scenarios. This study contributes to the field by offering a versatile tool for evaluating GNNs, thereby assisting in developing and selecting suitable models based on specific data characteristics.

cs.LG↗
arXiv2024

Feature-aware ultra-low dimensional reduction of real networks

In existing models and embedding methods of networked systems, node features describing their qualities are usually overlooked in favor of focusing solely on node connectivity. This study introduces $FiD$-Mercator, a model-based ultra-low dimensional reduction technique that integrates node features with network structure to create $D$-dimensional maps of complex networks in a hyperbolic space. This embedding method efficiently uses features as an initial condition, guiding the search of nodes' coordinates towards an optimal solution. The research reveals that downstream task performance improves with the correlation between network connectivity and features, emphasizing the importance of such correlation for enhancing the description and predictability of real networks. Simultaneously, hyperbolic embedding's ability to reproduce local network properties remains unaffected by the inclusion of features. The findings highlight the necessity for developing network embedding techniques capable of exploiting such correlations to optimize both network structure and feature association jointly in the future.

physics.soc-ph↗
arXiv2023

The D-Mercator method for the multidimensional hyperbolic embedding of real networks

One of the pillars of the geometric approach to networks has been the development of model-based mapping tools that embed real networks in its latent geometry. In particular, the tool Mercator embeds networks into the hyperbolic plane. However, some real networks are better described by the multidimensional formulation of the underlying geometric model. Here, we introduce $D$-Mercator, a model-based embedding method that produces multidimensional maps of real networks into the $(D+1)$-hyperbolic space, where the similarity subspace is represented as a $D$-sphere. We used $D$-Mercator to produce multidimensional hyperbolic maps of real networks and estimated their intrinsic dimensionality in terms of navigability and community structure. Multidimensional representations of real networks are instrumental in the identification of factors that determine connectivity and in elucidating fundamental issues that hinge on dimensionality, such as the presence of universality in critical behavior.

physics.soc-ph↗
arXiv2020

Determining crucial factors for the popularity of scientific articles

Using a set of over 70.000 records from PLOS One journal consisting of 37 lexical, sentiment and bibliographic variables we perform analysis backed with machine learning methods to predict the class of popularity of scientific papers defined by the number of times they have been viewed. Our study shows correlations among the features and recovers a threshold for the number of views that results in the best prediction results in terms of Matthew's correlation coefficient. Moreover, by creating a variable importance plot for random forest classifier, we are able to reduce the number of features while keeping similar predictability and determine crucial factors responsible for the popularity.

physics.soc-ph↗
arXiv2014

On the existence of bounded solutions for nonlinear second order neutral difference equations

\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} Δ\left(r_{n}\left(Δ\left(x_{n}+p_{n}x_{n-k}\right) \right) ^γ\right) +q_{n}x_{n}^α+a_{n}f(x_{n})=0. \end{equation*}% where $x:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $a,p,q:{\mathbb{N}}%_{0}\rightarrow {\mathbb{R}}$, $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}% \setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is a continuous function, and $k$ is a given positive integer, $γ\leq 1$ is ratio of odd positive integers, $α$ is a nonnegative constant. %$\sum a_{n}\left(t\right)$ converges uniformly on ${\mathbb{R}}$. %Here $\bN_0\colon =\left\{0,1,2, \dots \right\}$ and $\bN_k \colon = \left\{k, k+1, -k+2, \dots \right\}$ where $k$ is a given positive integer. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability and asymptotic stability are studied. Some earlier results are generalized. We note that the solution which we obtain does not directly correspond to a fixed point of a certain continuous operator since it is partially iterated. The method which we develop allows for considering through techniques connected with the measure of noncompactness also difference equations with memory. {\small \textbf{Keywords} Difference equation, measures of noncompactness, Darbo's fixed point theorem, boundedness, stability} {\small \textbf{AMS Subject classification} 39A10, 39A22, 39A30}

math.CA↗