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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. [ABSTRACT FROM AUTHOR]

When a large number of similar entities interact among each other and with their environment at a low scale, unexpected outcomes at higher spatio-temporal scales might spontaneously arise. This non-trivial phenomenon, known as emergence, characterizes a broad range of distinct complex systems—from physical to biological and social—and is often related to collective behaviour. It is ubiquitous, from non-living entities such as oscillators that under specific conditions synchronize, to living ones, such as birds flocking or fish schooling. Despite the ample phenomenological evidence of the existence of systems' emergent properties, central theoretical questions to the study of emergence remain unanswered, such as the lack of a widely accepted, rigorous definition of the phenomenon or the identification of the essential physical conditions that favour emergence. We offer here a general overview of the phenomenon of emergence and sketch current and future challenges on the topic. Our short review also serves as an introduction to the theme issue Emergent phenomena in complex physical and socio-technical systems: from cells to societies, where we provide a synthesis of the contents tackled in the issue and outline how they relate to these challenges, spanning from current advances in our understanding on the origin of life to the large-scale propagation of infectious diseases. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'. [ABSTRACT FROM AUTHOR]

Multilayer systems are coupled networks characterized by different contexts (layers) of interaction and have gained much attention recently due to their suitability to describe a broad spectrum of empirical complex systems. They are very fragile to percolation and first-neighbor failure propagation, but little is known about how they respond to non-local disruptions, as it occurs in failures induced by flow redistribution, for example. Acknowledging that many socio-technical and biological systems sustain a flow of some physical quantity, such as energy or information, across the their components, it becomes crucial to understand when the flow redistribution can cause global cascades of failures in order to design robust systems, to increase their resilience or to learn how to efficiently dismantle them. In this paper we study the impact that different multiplex topological features have on the robustness of the system when subjected to non-local cascade propagation. We first numerically demonstrate that this dynamics has a critical value at which a small initial perturbation effectively dismantles the entire network, and that the transition appears abruptly. Then we identify that the excess of flow caused by a failure is, in general, more homogeneously distributed the networks in which the average distance between nodes is small. Using this information we find that aggregated versions of multiplex networks tend to overestimate robustness, even though to make the system more robust can be achieved by increasing the number of layers. Our predictions are confirmed by simulated cascading failures in a real multilayer system. [ABSTRACT FROM AUTHOR]

We present an analytical framework to study the first-passage (FP) and first-return (FR) distributions for the broad family of models described by the one-dimensional Fokker-Planck equation in finite domains, identifying general properties of these distributions for different classes of models. When in the Fokker-Planck equation the diffusion coefficient is positive (nonzero) and the drift term is bounded, as in the case of a Brownian walker, both distributions may exhibit a power-law decay with exponent -3/2 for intermediate times. We discuss how the influence of an absorbing state changes this exponent. The absorbing state is characterized by a vanishing diffusion coefficient and/or a diverging drift term. Remarkably, the exponent of the Brownian walker class of models is still found, as long as the departure and arrival regions are far enough from the absorbing state, but the range of times where the power law is observed narrows. Close enough to the absorbing point, though, a new exponent may appear. The particular value of the exponent depends on the behavior of the diffusion and the drift terms of the Fokker-Planck equation. We focus on the case of a diffusion term vanishing linearly at the absorbing point. In this case, the FP and FR distributions are similar to those of the voter model, characterized by a power law with exponent -2. As an illustration of the general theory, we compare it with exact analytical solutions and extensive numerical simulations of a two-parameter voterlike family models. We study the behavior of the FP and FR distributions by tuning the importance of the absorbing points throughout changes of the parameters. Finally, the possibility of inferring relevant information about the steady-sate probability distribution of a model from the FP and FR distributions is addressed. [ABSTRACT FROM AUTHOR]

Aging is considered as the property of the elements of a system to be less prone to change states as they get older. We incorporate aging into the noisy voter model, a stochastic model in which the agents modify their binary state by means of noise and pairwise interactions. Interestingly, due to aging the system passes from a finite-size discontinuous transition between ordered (ferromagnetic) and disordered (paramagnetic) phases to a second order phase transition, well defined in the thermodynamic limit, belonging to the Ising universality class. We characterize it analytically by finding the stationary solution of an infinite set of mean field equations. The theoretical predictions are tested with extensive numerical simulations in low dimensional lattices and complex networks. We finally employ the aging properties to understand the symmetries broken in the phase transition. [ABSTRACT FROM AUTHOR]

Links in many real-world networks activate and deactivate in correspondence to the sporadic interactions between the elements of the system. The activation patterns may be irregular or bursty and play an important role on the dynamics of processes taking place in the network. Information or disease spreading in networks are paradigmatic examples of this situation. Besides burstiness, several correlations may appear in the process of link activation: memory effects imply temporal correlations, but also the existence of communities in the network may mediate the activation patterns of internal an external links. Here we study the competition of topological and temporal correlations in link activation and how they affect the dynamics of systems running on the network. Interestingly, both types of correlations by separate have opposite effects: one (topological) delays the dynamics of processes on the network, while the other (temporal) accelerates it. When they occur together, our results show that the direction and intensity of the final outcome depends on the competition in a non trivial way. [ABSTRACT FROM AUTHOR]

Localized perturbations in a real-world network have the potential to trigger cascade failures at the whole system level, hindering its operations and functions. Standard approaches analytically tackling this problem are mostly based either on static descriptions, such as percolation, or on models where the failure evolves through first-neighbor connections, crucially failing to capture the nonlocal behavior typical of real cascades. We introduce a dynamical model that maps the failure propagation across the network to a self-avoiding random walk that, at each step, has a probability to perform nonlocal jumps toward operational systems' units. Despite the inherent non-Markovian nature of the process, we are able to characterize the critical behavior of the system out of equilibrium, as well as the stopping time distribution of the cascades. Our numerical experiments on synthetic and empirical biological and transportation networks are in excellent agreement with theoretical expectation, demonstrating the ability of our framework to quantify the vulnerability to nonlocal cascade failures of complex systems with interconnected structure.

The voter model rules are simple, with agents copying the state of a random neighbor, but they lead to non-trivial dynamics. Besides opinion processes, the model has also applications for catalysis and species competition. Inspired by the temporal inhomogeneities found in human interactions, one can introduce ageing in the agents: the probability to update their state decreases with the time elapsed since the last change. This modified dynamics induces an approach to consensus via coarsening in single-layer complex networks. In this work, we investigate how a multilayer structure affects the dynamics of the ageing voter model. The system is studied as a function of the fraction of nodes sharing states across layers (multiplexity parameter q). We find that the dynamics of the system suffers a notable change at an intermediate value q*. Above it, the voter model always orders to an absorbing configuration. While below it a fraction of the realizations falls into dynamical traps associated to a spontaneous symmetry breaking. In this latter case, the majority opinion in the different layers takes opposite signs and the arrival at the absorbing state is indefinitely delayed due to ageing.