Universality in network dynamics
TL;DR: A self-consistent theory of dynamical perturbations in complex systems is developed, allowing us to systematically separate the contribution of the network topology and dynamics.
Abstract: Models for the topology or dynamics of various networks abound, but until now, there has been no single universal framework for complex networks that can separate factors contributing to the topology and dynamics of networks across biological and social systems.
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TL;DR: 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.
Abstract: An analytical framework is proposed for a complex network to accurately predict its dynamic resilience and unveil the network characteristics that can enhance or diminish resilience. Failing nodes in a complex network, for example, stations in a power grid that are are switched off, can lead to a breakdown of the whole system. The ability of the network to adjust so that it still functions despite the errors is called its resilience. Although — at first glance — the points at which different networks lose their resilience seem to have little in common, Jainxi Gao and colleagues show here that, in fact, resilience has underlying universal features. They develop a universal resilience function that depends on a system's dynamics and topology, and show that this analytical framework readily describes ecological networks, power grids, and gene regulatory networks. Their framework may contribute to understanding the vulnerability of many additional natural and man-made systems. 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 systems1. Despite widespread consequences for human health2, the economy3 and the environment4, events leading to loss of resilience—from cascading failures in technological systems5 to mass extinctions in ecological networks6—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 components7, 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.
720 citations
Cites background from "Universality in network dynamics"
...Here resilience increases with βeff, as the larger is βeff, the deeper is the system into the active state, and farther from the critical transition at βeff c (equation (12))....
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509 citations
TL;DR: A method for assessing network structures from binary data based on Ising models, which combines logistic regression with model selection based on a Goodness-of-Fit measure to identify relevant relationships between variables that define connections in a network is presented.
Abstract: Network analysis is entering fields where network structures are unknown, such as psychology and the educational sciences. A crucial step in the application of network models lies in the assessment of network structure. Current methods either have serious drawbacks or are only suitable for Gaussian data. In the present paper, we present a method for assessing network structures from binary data. Although models for binary data are infamous for their computational intractability, we present a computationally efficient model for estimating network structures. The approach, which is based on Ising models as used in physics, combines logistic regression with model selection based on a Goodness-of-Fit measure to identify relevant relationships between variables that define connections in a network. A validation study shows that this method succeeds in revealing the most relevant features of a network for realistic sample sizes. We apply our proposed method to estimate the network of depression and anxiety symptoms from symptom scores of 1108 subjects. Possible extensions of the model are discussed.
430 citations
TL;DR: This corrects the article to show that the method used to derive the H2O2 “spatially aggregating force” is based on a two-step process, not a single step, like in the case of H1N1.
Abstract: Nature 530, 307–312 (2016); doi:10.1038/nature16948 In the last sentence of page 310 of this Letter, the parameter h should equal 2, rather than 1. In addition, after equation (4), the text should have stated ‘Aij > 0’ and ‘positive interactions’, to read “...the weighted connectivity matrix Aij > 0 captures the positive interactions between the nodes.
295 citations
Cites background from "Universality in network dynamics"
...Here resilience increases with βeff, as the larger is βeff, the deeper is the system into the active state, and farther from the critical transition at βeff c (equation (12))....
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TL;DR: It turns out that the extent to which one can exercise control via manipulation of a subset of nodes depends sensitively on the number of nodes perturbed.
Abstract: 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 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 at a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems. The complex interactions inherent in real-world networks grant us precise system control via manipulation of a subset of nodes. It turns out that the extent to which we can exercise this control depends sensitively on the number of nodes perturbed.
283 citations
References
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TL;DR: Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.
Abstract: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
39,297 citations
TL;DR: A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
Abstract: Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
33,771 citations
"Universality in network dynamics" refers background in this paper
...In such networks, the number of nodes at distance l from a node follows5 |K (l)| ∼ e (10) where...
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...For networks satisfying equation (10), for l < 〈l〉 we show that (Supplementary Section SIV) 0(l)= e (12) where β =m1−m0 up to a logarithmic correction, which depends on microscopic details of equation (1), for example, rate constants (Supplementary Section SIV....
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...In this case, the individual correlations Gij will decay with l , but this decay is driven entirely by the topological expansion of the network in equation (10), distributing the original perturbation over an exponentially increasing number of nodes....
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...According to equations (12) and (10), l(G)∼− lnG/(β+ 1)α, so P(G) follows (Supplementary Section SIV....
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TL;DR: This article proposes a method for detecting communities, built around the idea of using centrality indices to find community boundaries, and tests it on computer-generated and real-world graphs whose community structure is already known and finds that the method detects this known structure with high sensitivity and reliability.
Abstract: A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.
14,429 citations
Additional excerpts
...In such networks, the number of nodes at distance l from a node follows5 |K (l)| ∼ e (10) where e = 〈k2〉−〈k〉 〈k〉 (11) is the average nearest-neighbour degree....
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Book•
25 Mar 2010
TL;DR: This book brings together for the first time the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas.
Abstract: The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyze network data on a large scale, and the development of a variety of new theoretical tools has allowed us to extract new knowledge from many different kinds of networks.The study of networks is broadly interdisciplinary and important developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. This book brings together for the first time the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Subjects covered include the measurement and structure of networks in many branches of science, methods for analyzing network data, including methods developed in physics, statistics, and sociology, the fundamentals of graph theory, computer algorithms, and spectral methods, mathematical models of networks, including random graph models and generative models, and theories of dynamical processes taking place on networks.
10,567 citations
"Universality in network dynamics" refers background in this paper
...In such networks the number of nodes at distance l from a node follows [5] Barzel and Barabási Page 5...
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...Despite the profound diversity in the scale and purpose of networks observed in nature and technology, their topology shares several highly reproducible and often universal characteristics [1–8]: many real networks display the small world property [9], are scale-free [10], develop distinct community structure [11], and show degree correlations [12, 13]....
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TL;DR: This work aims to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture.
Abstract: The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
7,665 citations
"Universality in network dynamics" refers background in this paper
...Similarly, we define the stability of i as Si= 1 N ∑ j=1 AijGij (3)...
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...Denoting the leading terms of equations (6) and (7) by n0 and m0 respectively, and the leading non-vanishing terms by n1 and m1, we show that Si in equation (3) and Ii in equation (2) depend on the degree of node i as (Supplementary Section SIII....
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