arXiv subjects

Renaud Lambiotte

Publications and source records attributed to Renaud Lambiotte.

At least 19 recordsLinked to original sources

arXiv2026

The impact of the path ensemble on path percolation

Traffic-induced failures, from packet loss in communication networks to congestion breakdown in transport systems, occur when flows progressively exhaust the edges they traverse. Path percolation models this process by removing edges along sampled origin-destination paths. Existing work assumes locally tree-like networks and deterministic shortest-path routing, leaving unclear how path degeneracy and routing stochasticity affect fragmentation in the clustered networks typical of real systems. We introduce a generalised path-percolation framework where paths are drawn from a temperature-controlled routing ensemble interpolating between geodesic and noisy transport. We argue based on box-covering renormalisation and our numerical experiments that, for any finite routing horizon $C$, the process coarse-grains to ordinary mean-field percolation. Routing details affect non-universal quantities, especially the percolation threshold $p_c$, through the entropy of the load distribution and the capacity of finite clusters to accommodate flow. Load entropy therefore acts as a robustness measure for networks under path-based failures. When the routing horizon is tuned to the mean-field correlation length, $C=N^{1/3}$, within a source-uniform ensemble, the system enters a crossover regime with scaling exponents distinct from shortest-path percolation with infinite budget. In this regime, path elongation becomes decoupled in time from structural fragmentation: the characteristic path length reaches a growing maximum, associated with routing temperature, asymptotically ahead of the collapse of the giant component. These results clarify how microscopic routing organisation shapes macroscopic resilience, and identify path elongation as a measurable precursor of failure in communication and transport infrastructure.

physics.soc-ph↗

arXiv2026

The nestedness of higher-order networks

In contrast to dyadic interactions, higher-order interactions may contain one another, with subgroups naturally embedded within larger groups. These containment patterns arise empirically in ecology, sociology, computer science and the science of science, and have been studied under the names nestedness, simpliciality, encapsulation, and inclusion. In this chapter, we review each of these measures and unify them through a mathematical object known as the encapsulation directed acyclic graph, formulating each measure as a function of its properties. We demonstrate that nested structure is prevalent in social systems across several domains, show that different measures capture complementary aspects of this structure, and find that the absence of nestedness can itself be a powerful indicator of the mesoscale organization of a system.

physics.soc-ph↗

arXiv2026

Coarse-graining nonequilibrium diffusions with Markov chains

We investigate nonequilibrium steady-state dynamics in both continuous- and discrete-state stochastic processes. Our analysis focuses on planar diffusion dynamics and their coarse-grained approximations by discrete-state Markov chains. Using finite-volume approximations, we derive an approximate master equation directly from the underlying diffusion and show that this discretisation preserves key features of the nonequilibrium steady-state. In particular, we show that the entropy production rate of the approximation converges as the number of discrete states goes to the limit. These results are illustrated with analytically solvable diffusions and numerical experiments on nonlinear processes, demonstrating how this approach can be used to explore the dependence of the entropy production rate on model parameters. Finally, we address the problem of inferring discrete-state Markov models from continuous stochastic trajectories. We show that discrete-state models significantly underestimate the true entropy production rate. However, we also show that they can provide tests to determine if a stationary planar diffusion is out of equilibrium. This property is illustrated with both simulated data and empirical trajectories from schooling fish.

cond-mat.stat-mech↗

arXiv2026

Structure-aware divergences for comparing probability distributions

Many natural and social science systems are described using probability distributions over elements that are related to each other: for instance, occupations with shared skills or species with similar traits. Standard information theory quantities such as entropies and $f$-divergences treat elements interchangeably and are blind to the similarity structure. We introduce a family of divergences that are sensitive to the geometry of the underlying domain. By virtue of being the Bregman divergences of structure-aware entropies, they provide a framework that retains several advantages of Kullback-Leibler divergence and Shannon entropy. Structure-aware divergences recover planted patterns in a synthetic clustering task that conventional divergences miss and are orders of magnitude faster than optimal transport distances. We demonstrate their applicability in economic geography and ecology, where structure plays an important role. Modelling different notions of occupation relatedness yields qualitatively different regionalisations of their geographic distribution. Our methods also reproduce established insights into functional $β$-diversity in ecology obtained with optimal transport methods.

cs.IT↗

arXiv2026

From First Principles to Multi-scale Decomposition:Mutual Information as a Segregation Index

Segregation is a multi-scale phenomenon that requires careful measurement. A segregation index implicitly defines how the demographic compositions of locations are compared. We identify two properties -- mean-minimisation and invariance -- that uniquely characterise the Kullback-Leibler divergence as a measure of demographic difference. Mean-minimiser makes the comparison consistent with population aggregation and invariance ensures that it behaves intuitively under demographic coarse-graining. The corresponding segregation index is mutual information, which can be decomposed across geographic and demographic scales to identify the contributions of regions and supergroups. We demonstrate how this reveals insights into ethnic residential segregation in England and Wales that would be inaccessible otherwise. By deriving mutual information from first principles, we identify situations in which it is the only suitable segregation index, and provide open source software to support multi-scale analysis.

physics.soc-ph↗

arXiv2026

Synchronization of higher-dimensional Kuramoto oscillators on networks: from scalar to matrix-weighted couplings

The Kuramoto model is the paradigmatic model to study synchronization in coupled oscillator systems. In its classical formulation, the oscillators move on the unit circle, each characterized by a scalar phase and a natural frequency, by interacting through a sinusoidal coupling. In this work, we propose a d-dimensional generalization in which oscillators are represented as unit vectors on the (d-1)-sphere and interact through a matrix-weighted network (MWN), a recently introduced framework where links are endowed with a matrix weight instead of a scalar one. We derive necessary conditions for global synchronization via a Master Stability Function approach: the existence of a synchronous solution requires identical frequency matrices across nodes and, in the MWN case, a coherence condition on the network structure. Through a suitable change of variables, the stability analysis reduces the full Nd-dimensional problem to a family of d-dimensional eigenvalue problems, each one parametrized by the eigenvalue of a suitable scalar weighted Laplacian, showing that the synchronous solution is locally stable for any positive coupling strength K on any connected network. Analytical results are complemented by numerical simulations.

cond-mat.stat-mech↗

arXiv2026

Drift-Diffusion Matching: Embedding dynamics in latent manifolds of asymmetric neural networks

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. Here we introduce a general framework, which we term drift-diffusion matching, for training continuous-time RNNs to represent arbitrary, nonlinear stochastic differential equations (SDEs), with given drift and diffusion coefficients, within a low-dimensional latent subspace. Allowing asymmetric connectivity, we show that RNNs can faithfully embed the drift and diffusion of a given SDE, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we construct RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.

cond-mat.dis-nn↗

arXiv2025

Needles in a haystack: using forensic network science to uncover insider trading

Although the automation and digitisation of anti-financial crime investigation has made significant progress in recent years, detecting insider trading remains a unique challenge, partly due to the limited availability of labelled data. To address this challenge, we propose using a data-driven networks approach that flags groups of corporate insiders who report coordinated transactions that are indicative of insider trading. Specifically, we leverage data on 2.9 million trades reported to the U.S. Securities and Exchange Commission (SEC) by company insiders (C-suite executives, board members and major shareholders) between 2014 and 2024. Our proposed algorithm constructs weighted edges between insiders based on the temporal similarity of their trades over the 10-year timeframe. Within this network we then uncover trends that indicate insider trading by focusing on central nodes and anomalous subgraphs. To highlight the validity of our approach we evaluate our findings with reference to two null models, generated by running our algorithm on synthetic empirically calibrated and shuffled datasets. The results indicate that our approach can be used to detect pairs or clusters of insiders whose behaviour suggests insider trading and/or market manipulation.

cs.SI↗

arXiv2025

Embedding networks with the random walk first return time distribution

We propose the first return time distribution (FRTD) of a random walk as an interpretable and mathematically grounded node embedding. The FRTD assigns a probability mass function to each node, allowing us to define a distance between any pair of nodes using standard metrics for discrete distributions. We present several arguments to motivate the FRTD embedding. First, we show that FRTDs are strictly more informative than eigenvalue spectra, yet insufficient for complete graph identification, thus placing FRTD equivalence between cospectrality and isomorphism. Second, we argue that FRTD equivalence between nodes captures structural similarity. Third, we empirically demonstrate that the FRTD embedding outperforms manually designed graph metrics in network alignment tasks. Finally, we show that random networks that approximately match the FRTD of a desired target also preserve other salient features. Together these results demonstrate the FRTD as a simple and mathematically principled embedding for complex networks.

cs.SI↗

arXiv2025

Global Synchronization in Matrix-Weighted Networks

Synchronization phenomena in complex systems are fundamental to understanding collective behavior across disciplines. While classical approaches model such systems by using scalar-weighted networks and simple diffusive couplings, many real-world interactions are inherently multidimensional and transformative. To address this limitation, Matrix-Weighted Networks (MWNs) have been introduced as a versatile framework where edges are associated with matrix weights that encode both interaction strength and directional transformation. In this work, we investigate the emergence and stability of global synchronization (GS) in MWNs by studying coupled Stuart-Landau (SL) oscillators, an archetypal model of nonlinear dynamics near a Hopf bifurcation. Besides the SL, we considered a generalization of regular oscillators to higher dimensions and also the Lorenz model as a prototype of chaotic oscillators. We derive a generalized Master Stability Function (MSF) tailored to MWNs and establish necessary and sufficient conditions for GS to occur. Central to our analysis is the concept of coherence, a structural property of MWNs ensuring path-independent transformations. Our results show that coherence is necessary to have global synchronization and provides a theoretical foundation for analyzing multidimensional dynamical processes in complex networked systems.

cond-mat.stat-mech↗

arXiv2025

Complex-Weighted Convolutional Networks: Provable Expressiveness via Complex Diffusion

Graph Neural Networks (GNNs) have achieved remarkable success across diverse applications, yet they remain limited by oversmoothing and poor performance on heterophilic graphs. To address these challenges, we introduce a novel framework that equips graphs with a complex-weighted structure, assigning each edge a complex number to drive a diffusion process that extends random walks into the complex domain. We prove that this diffusion is highly expressive: with appropriately chosen complex weights, any node-classification task can be solved in the steady state of a complex random walk. Building on this insight, we propose the Complex-Weighted Convolutional Network (CWCN), which learns suitable complex-weighted structures directly from data while enriching diffusion with learnable matrices and nonlinear activations. CWCN is simple to implement, requires no additional hyperparameters beyond those of standard GNNs, and achieves competitive performance on benchmark datasets. Our results demonstrate that complex-weighted diffusion provides a principled and general mechanism for enhancing GNN expressiveness, opening new avenues for models that are both theoretically grounded and practically effective.

cs.LG↗

arXiv2025

Manipulating Collective Opinion through Social Network Intervention

Social media platforms have transformed the dynamics of collective opinion formation, enabling rapid, large-scale interactions while simultaneously exposing online discourse to polarization and manipulation. Traditional models of opinion dynamics often predict convergence to a consensus, yet empirical evidence consistently highlights persistent polarization and radicalization, especially on contentious issues. This paper analytically investigates a mathematical model that captures the complex interplay of polarization, radicalization, and consensus within networked societies. By analyzing the emergence and stability of opinion clusters, we identify critical thresholds marking phase transitions in collective behavior, interpreted via a stability landscape. We further explore network-based interventions to manipulate the collective opinion, revealing that reducing inter-agent interactions can lead to unintended, irreversible shifts in opinion distributions. Our results underscore the dual-edged nature of intervention strategies, offering theoretical insight into the fragility and manipulability of public opinion in digital environments.

physics.soc-ph↗

arXiv2025

Nonequilibrium physics of brain dynamics

Information processing in the brain is coordinated by the dynamic activity of neurons and neural populations at a range of spatiotemporal scales. These dynamics, captured in the form of electrophysiological recordings and neuroimaging, show evidence of time-irreversibility and broken detailed balance suggesting that the brain operates in a nonequilibrium stationary state. Furthermore, the level of nonequilibrium, measured by entropy production or irreversibility appears to be a crucial signature of cognitive complexity and consciousness. The subsequent study of neural dynamics from the perspective of nonequilibrium statistical physics is an emergent field that challenges the assumptions of symmetry and maximum-entropy that are common in traditional models. In this review, we discuss the plethora of exciting results emerging at the interface of nonequilibrium dynamics and neuroscience. We begin with an introduction to the mathematical paradigms necessary to understand nonequilibrium dynamics in both continuous and discrete state-spaces. Next, we review both model-free and model-based approaches to analysing nonequilibrium dynamics in both continuous-state recordings and neural spike-trains, as well as the results of such analyses. We briefly consider the topic of nonequilibrium computation in neural systems, before concluding with a discussion and outlook on the field.

q-bio.NC↗

arXiv2025

Permutation-Invariant Spectral Learning via Dyson Diffusion

Diffusion models are central to generative modeling and have been adapted to graphs by diffusing adjacency matrix representations. The challenge of having up to $n!$ such representations for graphs with $n$ nodes is only partially mitigated by using permutation-equivariant learning architectures. Despite their computational efficiency, existing graph diffusion models struggle to distinguish certain graph families and their spectra, unless graph data are augmented with ad hoc features. This shortcoming stems from enforcing the inductive bias within the learning architecture. In this work, we leverage random matrix theory to analytically extract the spectral properties of the diffusion process, allowing us to push most of the inductive bias from the architecture into the dynamics. Building on this, we introduce the Dyson Diffusion Model, which employs Dyson's Brownian motion to capture the spectral dynamics of an Ornstein-Uhlenbeck process on the adjacency matrix. Furthermore, conditioned on the spectral dynamics, we formulate a Lie group diffusion, appropriately modeling the remaining degrees of freedom. Strikingly, the resulting learning problem becomes permutation invariant at the Lie algebra level. We demonstrate that the Dyson Diffusion Model learns graph spectra accurately and outperforms existing graph diffusion models.

stat.ML↗

arXiv2025

From reductionism to realism: Holistic mathematical modelling for complex biological systems

At its core, the physics paradigm adopts a reductionist approach, aiming to understand fundamental phenomena by decomposing them into simpler, elementary processes. While this strategy has been tremendously successful in physics, it has often fallen short in addressing fundamental questions in the biological sciences. This arises from the inherent complexity of biological systems, characterised by heterogeneity, polyfunctionality and interactions across spatiotemporal scales. Nevertheless, the traditional framework of complex systems modelling falls short, as its emphasis on broad theoretical principles has often failed to produce predictive, empirically-grounded insights. To advance towards actionable mathematical models in biology, we argue, using neuroscience as a case study, that it is necessary to move beyond reductionist approaches and instead embrace the complexity of biological systems - leveraging the growing availability of high-resolution data and advances in high-performance computing. We advocate for a holistic mathematical modelling paradigm that harnesses rich representational structures such as annotated and multilayer networks, employs agent-based models and simulation-based approaches, and focuses on the inverse problem of inferring system dynamics from observations. We emphasise that this approach is fully compatible with the search for fundamental biophysical principles, and highlight the potential it holds to drive progress in mathematical biology over the next two decades.

physics.bio-ph↗

arXiv2025

Gremban Expansion for Signed Networks: Algebraic and Combinatorial Foundations for Community-Faction Detection

This article deals with the characterization and detection of community and faction structures in signed networks. We approach the study of these mesoscale structures through the lens of the Gremban expansion. This graph operation lifts a signed graph to a larger unsigned graph, and allows the extension of standard techniques from unsigned to signed graphs. We develop the combinatorial and algebraic properties of the Gremban expansion, with a focus on its inherent involutive symmetry. The main technical result is a bijective correspondence between symmetry-respecting cut-sets in the Gremban expansion, and regular cut-sets and frustration sets in the signed graph (i.e., the combinatorial structures that underlie communities and factions respectively). This result forms the basis for our new approach to community-faction detection in signed networks, which makes use of spectral clustering techniques that naturally respect the required symmetries. We demonstrate how this approach distinguishes the two mesoscale structures, how to generalize the approach to multi-way clustering and discuss connections to network dynamical systems.

cs.DM↗

arXiv2025

Dynamic models of gentrification

The phenomenon of gentrification of an urban area is characterized by the displacement of lower-income residents due to rising living costs and an influx of wealthier individuals. This study presents an agent-based model that simulates urban gentrification through the relocation of three income groups -- low, middle, and high -- driven by living costs. The model incorporates economic and sociological theories to generate realistic neighborhood transition patterns. We introduce a temporal network-based measure to track the outflow of low-income residents and the inflow of middle- and high-income residents over time. Our experiments reveal that high-income residents trigger gentrification and that our network-based measure consistently detects gentrification patterns earlier than traditional count-based methods, potentially serving as an early detection tool in real-world scenarios. Moreover, the analysis also highlights how city density promotes gentrification. This framework offers valuable insights for understanding gentrification dynamics and informing urban planning and policy decisions.

physics.soc-ph↗

arXiv2025

Entropy-based models to randomize real-world hypergraphs

Network theory has often disregarded many-body relationships, solely focusing on pairwise interactions: neglecting them, however, can lead to misleading representations of complex systems. Hypergraphs represent a suitable framework for describing polyadic interactions. Here, we leverage the representation of hypergraphs based on the incidence matrix for extending the entropy-based approach to higher-order structures: in analogy with the Exponential Random Graphs, we introduce the Exponential Random Hypergraphs (ERHs). After exploring the asymptotic behaviour of thresholds generalising the percolation one, we apply ERHs to study real-world data. First, we generalise key network metrics to hypergraphs; then, we compute their expected value and compare it with the empirical one, in order to detect deviations from random behaviours. Our method is analytically tractable, scalable and capable of revealing structural patterns of real-world hypergraphs that differ significantly from those emerging as a consequence of simpler constraints.

cs.SI↗