Resilience, a system's ability to adjust its activity to retain its basic functionality when errors, failures and environmental changes occur, is a defining property of many complex systems. Despite widespread consequences for human health, the economy and the environment, events leading to loss of resilience--from cascading failures in technological systems to mass extinctions in ecological networks--are rarely predictable and are often irreversible. These limitations are rooted in a theoretical gap: the current analytical framework of resilience is designed to treat low-dimensional models with a few interacting components, and is unsuitable for multi-dimensional systems consisting of a large number of components that interact through a complex network. Here we bridge this theoretical gap by developing a set of analytical tools with which to identify the natural control and state parameters of a multi-dimensional complex system, helping us derive effective one-dimensional dynamics that accurately predict the system's resilience. The proposed analytical framework allows us systematically to separate the roles of the system's dynamics and topology, collapsing the behaviour of different networks onto a single universal resilience function. The analytical results unveil the network characteristics that can enhance or diminish resilience, offering ways to prevent the collapse of ecological, biological or economic systems, and guiding the design of technological systems resilient to both internal failures and environmental changes.

Despite significant advances in characterizing the structural properties of complex networks, a mathematical framework that uncovers the universal properties of the interplay between the topology and the dynamics of complex systems continues to elude us. Here we develop a self-consistent theory of dynamical perturbations in complex systems, allowing us to systematically separate the contribution of the network topology and dynamics. The formalism covers a broad range of steady-state dynamical processes and offers testable predictions regarding the system's response to perturbations and the development of correlations. It predicts several distinct universality classes whose characteristics can be derived directly from the continuum equation governing the system's dynamics and which are validated on several canonical network-based dynamical systems, from biochemical dynamics to epidemic spreading. Finally, we collect experimental data pertaining to social and biological systems, demonstrating that we can accurately uncover their universality class even in the absence of an appropriate continuum theory that governs the system's dynamics.

Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. But the observational uncertainty induced by measurement noise and the energy required for control continue to be significant challenges in practical applications. Here we show that the variability of control energy and observational uncertainty for di erent directions of the state space depend strongly on the number of driver nodes. In particular, we find that if all nodes are directly driven, control is energetically feasible, as the maximum energy increases sublinearly with the system size. If, however, we aim to control a system through a single node, control in some directions is energetically prohibitive, increasing exponentially with the system size. For the cases in between, the maximum energy decays exponentially when the number of driver nodes increases. We validate our findings in several model and real networks, arriving at a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems.M any natural and man-made systems can be represented as networks 1-3 , where nodes are the system's components and links describe the interactions between them. Thanks to these interactions, perturbations of one node can alter the states of the other nodes 4-6 . This property has been exploited to control a network-that is, to move it from an initial state to a desired final state 7-9 -by manipulating the state variables of only a subset of its nodes 10,11 . Such control processes 10-26 play an important role in the regulation of protein expression 27 , the coordination of moving robots 28 , and the inhibition of undesirable social contagions 29 . At the same time the interdependence between nodes means that the states of a small number of sensor nodes contain sufficient information about the rest of the network, so that we can reconstruct the system's full internal state by accessing only a few outputs 30 . This can be utilized for biomarker design in cellular networks, or to monitor in real time the state and functionality of infrastructural 31 and socialecological 32 systems for early warning of failures or disasters 33 .Although recent advances in driver and sensor node identification constitute unavoidable steps towards controlling and observing real networks, in practice we continue to face significant challenges: the control of a large network may require a vast amount of energy 16-18 , and measurement noise 34 causes uncertainties in the observation process. To quantify these issues we formalize the dynamics of a controlled network with N nodes and N D external control inputs as 7-10ẋwhere the vector x(t) = [x 1 (t), x 2 (t), . . . , x N (t)] T describes the states of the N nodes at time t and x i (t) can represent the concentration of a metabolite in a metabolic network 35 , the geometric state of a chromosome in a chromosomal interaction network 14 , or the belief of an individual in opinion dynamics 29,36 . The vect...

Predicting physical and functional links between cellular components is a fundamental challenge of biology and network science. Yet, correlations, a ubiquitous input for biological link prediction, are affected by both direct and indirect effects, confounding our ability to identify true pairwise interactions. Here we exploit the fundamental properties of dynamical correlations in networks to develop a method to silence indirect effects. The method receives as input the observed correlations between node pairs and uses a matrix transformation to turn the correlation matrix into a highly discriminative silenced matrix, which enhances only the terms associated with direct causal links. Achieving perfect accuracy in model systems, we test the method against empirical data collected for the Escherichia coli regulatory interaction network, showing that it improves on the best preforming link prediction methods. Overall the silencing methodology helps translate the abundant correlation data into valuable local information, with applications ranging from link prediction to inferring the dynamical mechanisms governing biological networks.