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Observing and controlling complex networks are of paramount interest for understanding complex physical, biological and technological systems. Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. Yet, 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 different 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 to a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems.
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To understand the controllability of complex networks is a forefront problem relevant to different fields of science and engineering. Despite recent advances in network controllability theories, an outstanding issue is to understand the effect of network topology and nodal interactions on the controllability at the most fundamental level. Here we develop a universal framework based on local information only to unearth the most {em fundamental building blocks} that determine the controllability. In particular, we introduce a network dissection process to fully unveil the origin of the role of individual nodes and links in control, giving rise to a criterion for the much needed strong structural controllability. We theoretically uncover various phase-transition phenomena associated with the role of nodes and links and strong structural controllability. Applying our theory to a large number of empirical networks demonstrates that technological networks are more strongly structurally controllable (SSC) than many social and biological networks, and real world networks are generally much more SSC than their random counterparts with intrinsic resilience and adaptability as a result of human design and natural evolution.
Routing information through networks is a universal phenomenon in both natural and manmade complex systems. When each node has full knowledge of the global network connectivity, finding short communication paths is merely a matter of distributed computation. However, in many real networks nodes communicate efficiently even without such global intelligence. Here we show that the peculiar structural characteristics of many complex networks support efficient communication without global knowledge. We also describe a general mechanism that explains this connection between network structure and function. This mechanism relies on the presence of a metric space hidden behind an observable network. Our findings suggest that real networks in nature have underlying metric spaces that remain undiscovered. Their discovery would have practical applications ranging from routing in the Internet and searching social networks, to studying information flows in neural, gene regulatory networks, or signaling pathways.
Controlling complex networks is of paramount importance in science and engineering. Despite the recent development of structural-controllability theory, we continue to lack a framework to control undirected complex networks, especially given link weights. Here we introduce an exact-controllability paradigm based on the maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link-weight distributions. The framework reproduces the structural controllability of directed networks characterized by structural matrices. We explore the controllability of a large number of real and model networks, finding that dense networks with identical weights are difficult to be controlled. An efficient and accurate tool is offered to assess the controllability of large sparse and dense networks. The exact-controllability framework enables a comprehensive understanding of the impact of network properties on controllability, a fundamental problem towards our ultimate control of complex systems.
Nature, technology and society are full of complexity arising from the intricate web of the interactions among the units of the related systems (e.g., proteins, computers, people). Consequently, one of the most successful recent approaches to capturing the fundamental features of the structure and dynamics of complex systems has been the investigation of the networks associated with the above units (nodes) together with their relations (edges). Most complex systems have an inherently hierarchical organization and, correspondingly, the networks behind them also exhibit hierarchical features. Indeed, several papers have been devoted to describing this essential aspect of networks, however, without resulting in a widely accepted, converging concept concerning the quantitative characterization of the level of their hierarchy. Here we develop an approach and propose a quantity (measure) which is simple enough to be widely applicable, reveals a number of universal features of the organization of real-world networks and, as we demonstrate, is capable of capturing the essential features of the structure and the degree of hierarchy in a complex network. The measure we introduce is based on a generalization of the m-reach centrality, which we first extend to directed/partially directed graphs. Then, we define the global reaching centrality (GRC), which is the difference between the maximum and the average value of the generalized reach centralities over the network. We investigate the behavior of the GRC considering both a synthetic model with an adjustable level of hierarchy and real networks. Results for real networks show that our hierarchy measure is related to the controllability of the given system. We also propose a visualization procedure for large complex networks that can be used to obtain an overall qualitative picture about the nature of their hierarchical structure.
The generalized $H(n)$ Hirsch index of order $n$ has been recently introduced and shown to interpolate between the degree and the $K$-core centrality in networks. We provide a detailed analytical characterization of the properties of sets of nodes having the same $H(n)$, within the annealed network approximation. The connection between the Hirsch indices and the degree is highlighted. Numerical tests in synthetic uncorrelated networks and real-world correlated ones validate the findings. We also test the use of the Hirsch index for the identification of influential spreaders in networks, finding that it is in general outperformed by the recently introduced Non-Backtracking centrality.
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