arXiv subjects

M. Ángeles Serrano

Publications and source records attributed to M. Ángeles Serrano.

At least 19 recordsLinked to original sources

arXiv2026

Graphicality of power-law and double power-law degree sequences

The graphicality problem -- whether or not a sequence of integers can be used to create a simple graph -- is a key question in network theory and combinatorics, with many important practical applications. In this work, we study the graphicality of degree sequences distributed as a power-law with a size-dependent cutoff and as a double power-law with a size-dependent crossover. We combine the application of exact sufficient conditions for graphicality with heuristic conditions for nongraphicality which allow us to elucidate the physical reasons why some sequences are not graphical. For single power-laws we recover the known phase-diagram, we highlight the subtle interplay of distinct mechanisms violating graphicality and we explain why the infinite-size limit behavior is in some cases very far from being observed for finite sequences. For double power-laws we derive the graphicality of infinite sequences for all possible values of the degree exponents $γ_1$ and $γ_2$, uncovering a rich phase-diagram and pointing out the existence of six qualitatively distinct ways graphicality can be violated. The validity of theoretical arguments is supported by extensive numerical analysis.

cond-mat.dis-nn↗

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

Self-Similarity in Online Networks During Social Movements

Online platforms provide an infrastructure for social movements, leaving digital traces that can be modelled as networks to quantify how information, participation, and coordination emerge during episodes of collective action and evolve over time. In this work, we unveil the emergence of scale-invariant online interaction patterns in social movements through network analysis of three geographically and sociopolitically distinct massive mobilisation events. By constructing co-occurrence networks from Twitter (now X) hashtag data and applying a degree-thresholding renormalisation procedure, we demonstrate that these highly correlated social phenomena exhibit clear signatures of self-similarity at peak mobilisation times. These critical points are characterised by modular-to-nested transitions, both in the co-occurrence networks and the bi-partite ones, maxima in user participation, and clustering spectrum collapse across multiple network scales. Despite their geographical and sociopolitical diversity, all three movements display remarkably analogous self-similar properties. Furthermore, the results hint at the emergence of a latent metric structure that supports successful hyperbolic embedding, providing an estimate of effective social distance. Together, these findings suggest that self-similarity may constitute a universal organising principle of social movements during peak mobilisation phases, as it lays the groundwork for the rapid amplification of information across scales that is necessary for the successful coordination of collective action.

physics.soc-ph↗

arXiv2026

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↗

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↗

arXiv2026

Statistical Mechanics of Random Hyperbolic Graphs within the Fermionic Maximum-Entropy Framework

The intricate relations between elements in natural and human-made systems sustain the complex processes that shape our world, forming multiscale networks of interactions. These networks can be represented as graphs composed of nodes connected by links and, regardless of their domain, they share a set of fundamental structural properties. The family of network models in hyperbolic space constitutes one of the most advanced frameworks accounting for such properties, including sparsity, the small-world property, heterogeneity and hierarchical organization, high clustering, and scale invariance under network renormalization transformations. These geometric models also exhibit other intriguing phenomena, such as an anomalous, temperature-dependent phase transition between a geometric and a non-geometric phase. In simple graph representations, where network links are unweighted, the model can be derived within a statistical-mechanics framework by maximizing the Gibbs entropy of the graph ensemble subject to constraints imposed by observations, with links effectively behaving as fermionic particles. In this topical review, I revisit these derivations previously scattered across different sources and complement them, in order to properly contextualize and consolidate hyperbolic random graphs within the broad framework of the maximum-entropy principle in the statistical mechanics of complex networks. The approach presented here represents the least-biased prediction of the fundamental set of core network properties and establishes a principled framework for analyzing network structure, offering new perspectives and powerful analytical tools for both theoretical and empirical studies.

cond-mat.dis-nn↗

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

Quantum Key Distribution over Complex Networks

There exist several initiatives worldwide to deploy quantum key distribution (QKD) over existing fibre networks and achieve quantum-safe security at large scales. To understand the overall QKD network performance, it is required to transition from the analysis of individual links, as done so far, to the characterization of the network as a whole. In this work, we undertake this study by embedding QKD protocols on complex networks, which correctly model the existing fiber networks. We focus on networks with trusted nodes and on continuous-variable (CV) schemes, which have much higher key rates than their discrete-variable (DV) counterparts. In the effective CV network, however, many of the unique properties of complex networks, such as small-worldness and the presence of hubs, are lost due to the fast decay of the key rate with physical distance for CV systems. These properties can be restored when considering a hybrid network consisting of both CV and DV protocols, achieving at the same time high average rate and inter-connectivity. Our work opens the path to the study of QKD complex networks in existing infrastructures.

quant-ph↗

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

Network Renormalization

The renormalization group (RG) is a powerful theoretical framework developed to consistently transform the description of configurations of systems with many degrees of freedom, along with the associated model parameters and coupling constants, across different levels of resolution. It also provides a way to identify critical points of phase transitions and study the system's behaviour around them by distinguishing between relevant and irrelevant details, the latter being unnecessary to describe the emergent macroscopic properties. In traditional physical applications, the RG largely builds on the notions of homogeneity, symmetry, geometry and locality to define metric distances, scale transformations and self-similar coarse-graining schemes. More recently, various approaches have tried to extend RG concepts to the ubiquitous realm of complex networks where explicit geometric coordinates do not necessarily exist, nodes and subgraphs can have very different properties, and homogeneous lattice-like symmetries are absent. The strong heterogeneity of real-world networks significantly complicates the definition of consistent renormalization procedures. In this review, we discuss the main attempts, the most important advances, and the remaining open challenges on the road to network renormalization.

cond-mat.stat-mech↗

arXiv2024

The multiscale self-similarity of the weighted human brain connectome

Anatomical connectivity between different regions in the brain can be mapped to a network representation, the connectome, where the intensities of the links, the weights, influence its structural resilience and the functional processes it sustains. Yet, many features associated with the weights in the human brain connectome are not fully understood, particularly their multiscale organization. In this paper, we elucidate the architecture of weights, including weak ties, in multiscale hierarchical human brain connectomes reconstructed from empirical data. Our findings reveal multiscale self-similarity in the weighted statistical properties, including the ordering of weak ties, that remain consistent across the analyzed length scales of every individual and the group representatives. This phenomenon is effectively captured by a renormalization of the weighted structure applied to hyperbolic embeddings of the connectomes, based on a unique weighted geometric model that integrates links of all weights across all length scales. This eliminates the need for separate generative weighted connectivity rules for each scale or to replicate weak and strong ties at specific scales in brain connectomes. The observed symmetry represents a distinct signature of criticality in the weighted connectivity of human brain connectomes, aligning with the fractality observed in their topology, and raises important questions for future research, like the existence of a resolution threshold where the observed symmetry breaks, or whether it is preserved in cases of neurodegenerative disease or psychiatric disorder.

q-bio.NC↗

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↗