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. In this Perspective, we review generative-model-based approaches, distinguishing two reconstruction classes: fixed-time methods, which infer geometry at specific temporal scales (e.g., equilibrium), and multi-resolution methods, which integrate dynamics across near- and far-from-equilibrium states. Over the past decade, these models have revealed functional organization in biological, social, and technological networks. Hence, we provide a unified overview of these methods, with particular attention to the underlying mathematical constructions. Further, we point to promising extensions of these frameworks, which combine the previously developed methods to other well-established analytical frameworks.

Complex systems are characterized by emergent patterns created by the nontrivial interplay between dynamical processes and the networks of interactions on which these processes unfold. Topological or dynamical descriptors alone are not enough to fully embrace this interplay in all its complexity, and many times one has to resort to dynamics-specific approaches that limit a comprehension of general principles. To address this challenge, we employ a metric—that we name Jacobian distance—which captures the spatiotemporal spreading of perturbations, enabling us to uncover the latent geometry inherent in network-driven processes. We compute the Jacobian distance for a broad set of nonlinear dynamical models on synthetic and real-world networks of high interest for applications from biological to ecological and social contexts. We show, analytically and computationally, that the process-driven latent geometry of a complex network is sensitive to both the specific features of the dynamics and the topological properties of the network. This translates into potential mismatches between the functional and the topological mesoscale organization, which we explain by means of the spectrum of the Jacobian matrix. Finally, we demonstrate that the Jacobian distance offers a clear advantage with respect to traditional methods when studying human brain networks. In particular, we show that it outperforms classical network communication models in explaining functional communities from structural data, therefore highlighting its potential in linking structure and function in the brain.

Network-based interactions allow one to model many technological and natural systems, where understanding information flow between nodes is important to predict their functioning. The complex interplay between network connectivity and dynamics can be captured by scaling laws overcoming the paradigm of information spread being solely dependent on network structure. Here, we capitalize on this paradigm to identify the relevant paths for perturbation propagation. We introduce a multi-pathways temporal distance between nodes that overcomes the limitation of focussing only on the shortest path. This metric predicts the latent geometry induced by the dynamics in which the signal propagation resembles the traveling wave solution of reaction-diffusion systems. We validate the framework on a set of synthetic dynamical models, showing that it outperforms existing approaches in predicting arrival times. On a set of empirical contact-based social systems, we show that it can be reliably used also for models of infectious diseases spread - such as the Susceptible-Infected-Susceptible - with remarkable accuracy in predicting the observed timing of infections. Our framework naturally encodes the concerted behavior of the ensemble of paths connecting two nodes in conveying perturbations, with applications ranging from regulatory dynamics within cells to epidemic spreading in social networks.

Microscopic structural damage, such as lesions in neural systems or disruptions in urban transportation networks, can impair the dynamics crucial for systems' functionality, such as electrochemical signals or human flows, or any other type of information exchange, respectively, at larger topological scales. Damage is usually modeled by progressive removal of components or connections and, consequently, systems' robustness is assessed in terms of how fast their structure fragments into disconnected subsystems. Yet, this approach fails to capture how damage hinders the propagation of information across scales, since system function can be degraded even in absence of fragmentation—e.g., pathological yet structurally integrated human brain. Here, we probe the response to damage of dynamical processes on the top of complex networks, to study how such an information flow is affected. We find that removal of nodes central for network connectivity might have insignificant effects, challenging the traditional assumption that structural metrics alone are sufficient to gain insights about how complex systems operate. Using a damaging protocol explicitly accounting for flow dynamics, we analyze synthetic and empirical systems, from biological to infrastructural ones, and show that it is possible to drive the system towards functional fragmentation before full structural disintegration.

Real systems are characterized by complex patterns of interactions between their units, by dynamical processes on them, and by the interplay of the two. It is well known that particular structures affect dynamical processes at different scales. Sometimes richly connected units are connected by costly, long-range links. In the brain, hubs form rich clubs for integrating information between different brain regions, and many biological and social networks show this same structural organization. It remains, however, unclear whether this structural organization alone enables a rapid communication between highly connected nodes or whether a functional rich club may emerge as a combination of direct links and longer paths between rich nodes. Here, we identify functional rich clubs through the diffusion geometry, providing a perspective on rich-club phenomena in complex networks. We show that weak structural rich clubs may be functionally stronger, thanks to bridge nodes, while diffusion inside strong structural rich clubs may be damped in modular networks.

Interconnected systems have to route information to function properly: At the lowest scale neural cells exchange electrochemical signals to communicate, while at larger scales animals and humans move between distinct spatial patches and machines exchange information via the Internet through communication protocols. Nontrivial patterns emerge from the analysis of information flows, which are not captured either by broadcasting, such as in random walks, or by geodesic routing, such as shortest paths. In fact, alternative models between those extreme protocols are still eluding us. Here we propose a class of stochastic processes, based on biased random walks, where agents are driven by a physical potential pervading the underlying network topology. By considering a generalized Coulomb dependence on the distance on destination(s), we show that it is possible to interpolate between random walk and geodesic routing in a simple and effective way. We demonstrate that it is not possible to find a one-size-fit-all solution to efficient navigation and that network heterogeneity or modularity has measurable effects. We illustrate how our framework can describe the movements of animals and humans, capturing with a stylized model some measurable features of the latter. From a methodological perspective, our potential-driven random walks open the doors to a broad spectrum of analytical tools, ranging from random-walk centralities to geometry induced by potential-driven network processes.

Information exchange in the human brain is crucial for vital tasks and to drive diseases. Neuroimaging techniques allow for the indirect measurement of information flows among brain areas and, consequently, for reconstructing connectomes analyzed through the lens of network science. However, standard analyses usually focus on a small set of network indicators and their joint probability distribution. Here, we propose an information-theoretic approach for the analysis of synthetic brain networks (based on generative models) and empirical brain networks, and to assess connectome's information capacity at different stages of dementia. Remarkably, our framework accounts for the whole network state, overcoming limitations due to limited sets of descriptors, and is used to probe human connectomes at different scales. We find that the spectral entropy of empirical data lies between two generative models, indicating an interpolation between modular and geometry-driven structural features. In fact, we show that the mesoscale is suitable for characterizing the differences between brain networks and their generative models. Finally, from the analysis of connectomes obtained from healthy and unhealthy subjects, we demonstrate that significant differences between healthy individuals and the ones affected by Alzheimer’s disease arise at the microscale (max. posterior probability smaller than 1%) and at the mesoscale (max. posterior probability smaller than 10%).

Complex networks are characterized by latent geometries induced by their topology or by the dynamics on the top of them. In the latter case, different network-driven processes induce distinct geometric features that can be captured by adequate metrics. Random walks, a proxy for a broad spectrum of processes, from simple contagion to metastable synchronization and consensus, have been recently used in [Phys. Rev. Lett. 118, 168301 (2017)] to define the class of diffusion geometry and pinpoint the functional mesoscale organization of complex networks from a genuine geometric perspective. Here, we firstly extend this class to families of distinct random walk dynamics -- including local and non-local information -- on the top of multilayer networks -- a paradigm for biological, neural, social, transportation, biological and financial systems -- overcoming limitations such as the presence of isolated nodes and disconnected components, typical of real-world networks. Secondly, we characterize the multilayer diffusion geometry of synthetic and empirical systems, highlighting the role played by different random search dynamics in shaping the geometric features of the corresponding diffusion manifolds.

Networks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, self-similarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity.

Collective phenomena emerge from the interaction of natural or artificial units with a complex organization. The interplay between structural patterns and dynamics might induce functional clusters that, in general, are different from topological ones. In biological systems, like the human brain, the overall functionality is often favored by the interplay between connectivity and synchronization dynamics, with functional clusters that do not coincide with anatomical modules in most cases. In social, sociotechnical, and engineering systems, the quest for consensus favors the emergence of clusters. Despite the unquestionable evidence for mesoscale organization of many complex systems and the heterogeneity of their interconnectivity, a way to predict and identify the emergence of functional modules in collective phenomena continues to elude us. Here, we propose an approach based on random walk dynamics to define the diffusion distance between any pair of units in a networked system. Such a metric allows us to exploit the underlying diffusion geometry to provide a unifying framework for the intimate relationship between metastable synchronization, consensus, and random search dynamics in complex networks, pinpointing the functional mesoscale organization of synthetic and biological systems.