Searcharxiv⌕ Search

arXiv subjects

Marián Boguñá

Publications and source records attributed to Marián Boguñá.

At least 19 recordsLinked to original sources
arXiv2026

All against the machine: the Solo score for rating skill in variable environments

We propose a distribution-free metric to rate individual skill in ``player-versus-environment'' settings, where participants face heterogeneous tasks without direct opponents. Such settings are common in digital platforms, games, education, finance, and the benchmark evaluation of AI agents. They combine high randomness, tasks of widely varying difficulty, and unknown heterogeneity across individuals. Our metric maps each task outcome to a bounded performance score with zero population mean and variance bounded by $1/3$, regardless of the outcome distribution, so that scores are directly comparable across tasks. Aggregating these scores over tasks yields an interpretable skill score for each individual (the ``Solo'' rating), and a reshuffling null model tests whether that score exceeds what chance alone would produce. We validate the approach on the progression of $2\times10^5$ players in the mobile games Candy Crush Saga and Bubble Witch 3 Saga. The metric identifies high- and low-skill players with high statistical confidence, their classification persists over hundreds of subsequent levels, and a windowed version of the score tracks changes in performance along progression.

physics.soc-ph↗
arXiv2026

Sampling biases in sprinkled causal sets

Poisson sprinkling provides a Lorentz-invariant way of generating causal sets from a continuum spacetime, but numerical work necessarily restricts the sprinkling to a finite region. For observables involving causal-set links, this finite-volume restriction can generate a bias that does not disappear by simply increasing the number of sprinkled points. The origin of the problem is geometric. At fixed proper time from an event in Minkowski spacetime, candidate links populate a non-compact hyperbolic space of rapidities. In dimensions $d+1$ with $d\geq 2$, this space is boundary dominated: the fraction of volume close to the boundary of a large region does not vanish in the infinite-volume limit. Different finite-volume regularizations can therefore produce different limiting normalized statistics. We relate this phenomenon to the failure of naive open-boundary thermodynamic limits on non-amenable spaces, derive explicitly the dependence of the link proper-time distribution on the finite cutoff, and distinguish a well-defined link-intensity profile from the ill-defined notion of a uniformly chosen link. We verify these results with independent Poisson-sprinkling simulations in $d+1$ dimensions with $d=1,2,3$. We then propose an event-centered product-window estimator with a controlled double-scaling limit, and discuss compact hyperbolic quotients as a possible boundary-free alternative.

gr-qc↗
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

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

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↗
arXiv2026

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↗
arXiv2026

Multiplexity amplifies geometry in networks

Many real-world network are multilayer, with nontrivial correlations across layers. Here we show that these correlations amplify geometry in networks. We focus on mutual clustering--a measure of the amount of triangles that are present in all layers among the same triplets of nodes--and find that this clustering is abnormally high in many real-world networks, even when clustering in each individual layer is weak. We explain this unexpected phenomenon using a simple multiplex network model with latent geometry: links that are most congruent with this geometry are the ones that persist across layers, amplifying the cross-layer triangle overlap. This result reveals a different dimension in which multilayer networks are radically distinct from their constituent layers.

physics.soc-ph↗
arXiv2026

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↗
arXiv2025

Extracting the geometric backbone of bipartite networks

Real bipartite networks combine degree-constrained random mixing with structured, locality-like rules. We introduce a statistical filter that benchmarks node-level bipartite clustering against degree-preserving randomizations to classify nodes as geometric (signal) or random-like (noise). In synthetic mixtures with known ground truth, the filter achieves high F-scores and sharpens inference of latent geometric parameters. Applied to four empirical systems -- metabolism, online group membership, plant-pollinator interactions, and languages -- it isolates recurrent neighborhoods while removing ubiquitous or weakly co-occurring entities. Filtering exposes a compact geometric backbone that disproportionately sustains connectivity under percolation and preserves downstream classifier accuracy in node-feature tasks, offering a simple, scalable way to disentangle structure from noise in bipartite networks.

physics.soc-ph↗
arXiv2025

Local d'Alembertian for causal sets

Causal set theory is an intrinsically nonlocal approach to quantum gravity, inheriting its nonlocality from Lorentzian nonlocality. This nonlocality causes problems in defining differential operators -- such as the d'Alembert operator, a cornerstone of any relativistic field theory -- within the causal set framework. It has been proposed that d'Alembertians in causal sets must themselves be nonlocal to respect causal set nonlocality. However, we show that such nonlocal d'Alembertians do not converge to the standard continuum d'Alembertian for some basic fields. To address this problem, we introduce a local d'Alembert operator for causal sets and demonstrate its convergence to its continuum counterpart for arbitrary fields in Minkowski spacetimes. Our construction leverages recent developments in measuring both timelike and spacelike distances in causal sets using only their intrinsic structure. This approach thus reconciles locality with Lorentz invariance, and paves a way toward defining converging discrete approximations to locality-based differential operators even in theories that are inherently nonlocal.

gr-qc↗
arXiv2025

Clustering Does Not Always Imply Latent Geometry

The latent space approach to complex networks has revealed fundamental principles and symmetries, enabling geometric methods. However, the conditions under which network topology implies geometricity remain unclear. We provide a mathematical proof and empirical evidence showing that the multiscale self-similarity of complex networks is a crucial factor in implying latent geometry. Using degree-thresholding renormalization, we prove that any random scale-free graph in a $d$-dimensional homogeneous and isotropic manifold is self-similar when interactions are pairwise. Hence, both clustering and self-similarity are required to imply geometricity. Our findings highlight that correlated links can lead to finite clustering without self-similarity, and therefore without inherent latent geometry. The implications are significant for network mapping and ensemble equivalence between graphs and continuous spaces.

physics.soc-ph↗
arXiv2024

Statistical mechanics of directed networks

Directed networks are essential for representing complex systems, capturing the asymmetry of interactions in fields such as neuroscience, transportation, and social networks. Directionality reveals how influence, information, or resources flow within a network, fundamentally shaping the behavior of dynamical processes and distinguishing directed networks from their undirected counterparts. Robust null models are crucial for identifying meaningful patterns in these representations, yet designing models that preserve key features remains a significant challenge. One such critical feature is reciprocity, which reflects the balance of bidirectional interactions in directed networks and provides insights into the underlying structural and dynamical principles that shape their connectivity. This paper introduces a statistical mechanics framework for directed networks, modeling them as ensembles of interacting fermions. By controlling reciprocity and other network properties, our formalism offers a principled approach to analyzing directed network structures and dynamics, introducing a new perspectives, and models and analytical tools for empirical studies.

cond-mat.dis-nn↗
arXiv2024

Renormalization of networks with weak geometric coupling

The Renormalization Group is crucial for understanding systems across scales, including complex networks. Renormalizing networks via network geometry, a framework in which their topology is based on the location of nodes in a hidden metric space, is one of the foundational approaches. However, the current methods assume that the geometric coupling is strong, neglecting weak coupling in many real networks. This paper extends renormalization to weak geometric coupling, showing that geometric information is essential to preserve self-similarity. Our results underline the importance of geometric effects on network topology even when the coupling to the underlying space is weak.

physics.soc-ph↗
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↗
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↗
arXiv2023

Feature-enriched hyperbolic network geometry

Graph-structured data provide a comprehensive description of complex systems, encompassing not only the interactions among nodes but also the intrinsic features that characterize these nodes. These features play a fundamental role in the formation of links within the network, making them valuable for extracting meaningful topological information. Notably, features are at the core of deep learning techniques such as Graph Convolutional Neural Networks (GCNs) and offer great utility in tasks like node classification, link prediction, and graph clustering. In this paper, we present a comprehensive framework that treats features as tangible entities and establishes a bipartite graph connecting nodes and features. By assuming that nodes sharing similarities should also share features, we introduce a hyperbolic geometric space where both nodes and features coexist, shaping the structure of both the node network and the bipartite network of nodes and features. Through this framework, we can identify correlations between nodes and features in real data and generate synthetic datasets that mimic the topological properties of their connectivity patterns. The approach provides insights into the inner workings of GCNs by revealing the intricate structure of the data.

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↗
arXiv2023

Geometric renormalization of weighted networks

The geometric renormalization technique for complex networks has successfully revealed the multiscale self-similarity of real network topologies and can be applied to generate replicas at different length scales. In this letter, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our findings demonstrate that weights in real networks exhibit multiscale self-similarity under a renormalization protocol that selects the connections with the maximum weight across increasingly longer length scales. We present a theory that elucidates this symmetry, and that sustains the selection of the maximum weight as a meaningful procedure. Based on our results, scaled-down replicas of weighted networks can be straightforwardly derived, facilitating the investigation of various size-dependent phenomena in downstream applications.

physics.soc-ph↗