Latent geometry emerging from network-driven processes

TL;DR

The paper addresses how latent geometry emerges from the interaction of network structure and dynamics by reviewing generative-model approaches that map diffusion-driven processes into latent spaces. It distinguishes fixed-time (single-scale) reconstructions from multi-scale (time-as-resolution) analyses and covers effective distance from discrete-time random walks, universal nonlinear-distance via perturbation response times, continuous-time diffusion distances, and Jacobian metrics from linearized dynamics. It shows these measures uncover mesoscale functional structure beyond topology and discusses applications to biological, social, and technological networks. The authors outline future directions, including information-geometry formulations and dynamical mean-field perspectives, to unify geometry with dynamical descriptions and enable robust predictions.

Abstract

Understanding network functionality requires integrating structure and dynamics, and emergent latent geometry induced by network-driven processes captures the low-dimensional spaces governing this interplay. Here, we focus on generative-model-based approaches, distinguishing two reconstruction classes: fixed-time methods, which infer geometry at specific temporal scales (e.g., equilibrium), and multi-scale methods, which integrate dynamics across near- and far-from-equilibrium scales. Over the past decade, these models have revealed functional organization in biological, social, and technological networks.

Figures (4)

• Figure 1: Functional organization from the interplay between structure and dynamics: latent geometry emerging from network-driven processes. Latent geometry based on structure (A) defines distances in terms of neighbors properties or structural properties (e.g., shortest paths), while neglecting dynamics. By taking dynamics into account (B), the global properties of signal propagation is uncovered and naturally taken into account for the analysis of network functionality with respect to specific generative models for information exchange among system's units. Two alternative approaches capture complementary information: single-scale distances (C), built upon asymptotic relaxation or effective rates, and multi-scale distances (D), which can be tuned, for instance, to detect mesoscopic network structures by using time as a multi-resolution parameter to probe network functionality.
• Figure 2: Effective distance for network dynamics. In brockmann2013hidden, epidemic spreading on the World Airtransportation Network is shown to hide remarkably regular patterns which are unveiled by the effective distance (A). The introduction of an effective velocity for global spreading $V_{\textrm{eff}}$, dependent on a set of structural and epidemic parameters, allows the approximation of the arrival time at node $i$ as $\tau_{i\leftarrow j}= d_{ij} / V_{\textrm{eff}}$, where $j$ is the outbreak node. In (B-C) the comparison between effective distances and arrival times for H1N1 (2009) and SARS (2003) outbreaks brockmann2013hidden. This approximation corresponds to a concentric front wave propagation with constant speed in the latent space (D). $\vert$ In hens2019spatiotemporalbontorin2023multi propagation times for signals are computed by taking into account fundamentally different network dynamics (E), again finding regular patterns within dynamical classes. Correspondingly, a notion of distance emerges, where nodes are close if their respective propagation time is small (F). Panels A-D have been adapted from brockmann2013hidden. Panel E has been adapted from bontorin2023multi.
• Figure 3: Multi-scale distances. (A) Network propagation states emerge from different source nodes. Pairs of nodes are assigned a time-dependent distance (B) according to the difference between their propagation states at a time $\tau$, thus unveiling different mesoscopic structures, with distances between nodes in different communities shrinking over longer time scales with respect to nodes in the same community; thus, an time-dependent embedding space emerges (C). In de2017diffusion the distance is computed according to the network states defined by the probability of being reached by a random walker starting at one of the source nodes $i$ or $j$. The set of pairwise diffusion distances results in a distance matrix across temporal scales (D) which can be then averaged (E) and compared with a configuration model to reveal a meaningful mesoscale structure in the diffusion process. (F) In barzon2024unraveling, empirical neural networks and functional brain mappings are analyzed through the Jacobian distance induced by a model of neural dynamics. (G) Adjusted mutual information (AMI) is optimized by the modular structure predicted by the Jacobian distance, which predicts more integrated modules than structural connectivity and network communicability (H). Panels D-E have been adapted from de2017diffusion. Panels F-H have been adapted from barzon2024unraveling.
• Figure 4: Applications of latent network geometry. In bertagnolli2021diffusion, diffusion metrics of multi-layer networks is explored through the lenses of classical, diffusive and physical with relaxation random walks, building phenomenological models for the emerging geometry of a broad class of complex phenomena. The supra-adjacency matrix (A-B) defines the multiplex structure of a network, whereas the supra-distance matrices (C), taken at different times, reveal network substructures at different scales and of different types as explored by different families of random walkers. $\vert$ (D) In klamser2023enhancing the latent geometry induced by the spreading a pathogen on the World Airtransportation Network from an outbreak airport is exploited to estimate the import risk klamser2024inferring of an infected individual. This effective distance on a human mobility network enters as a part of a pipeline to infer the pandemic potential of a Variant of Concern. Panels A-C have been adapted from bertagnolli2021diffusion. Panel D has been adapted from klamser2023enhancing.