1
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Gao Z, Luo W, Shen A. Asymptotically local synchronization in interdependent networks with unidirectional interlinks. PLoS One 2022; 17:e0267909. [PMID: 35511786 PMCID: PMC9070916 DOI: 10.1371/journal.pone.0267909] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2021] [Accepted: 04/18/2022] [Indexed: 11/19/2022] Open
Abstract
Synchronization in complex networks has been investigated for decades. Due to the particularity of the interlinks between networks, the synchronization in interdependent networks has received increasing interest. Since the interlinks are not always symmetric in interdependent networks, we focus on the synchronization in unidirectional interdependent networks to study the control scheme. The mathematical model is put forward and some factors are taken into consideration, such as different coupling functions and strengths. Firstly, the feasibility of the control scheme is proved theoretically by using Lyapunov stability theory and verified by simulations. Then, we find that the synchronization could be maintained in one sub-network by utilizing our control scheme while the nodes in the other sub-network are in chaos. The result indicates that the influence of interlinks can be decreased and the proposed scheme can guarantee the synchronization in one sub-network at least. Moreover, we also discuss the robust of our control scheme against the cascading failure. The scheme is verified by simulations to be effective while the disturbances occur.
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Affiliation(s)
- Zilin Gao
- School of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing, China
| | - Weimin Luo
- School of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing, China
| | - Aizhong Shen
- Faculty of Professional Finance and Accountancy, Shanghai Business School, Shanghai, China
- * E-mail:
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2
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Zhang H, Zhang W, Gao J. Synchronization of interconnected heterogeneous networks: The role of network sizes. Sci Rep 2019; 9:6154. [PMID: 30992507 PMCID: PMC6468008 DOI: 10.1038/s41598-019-42636-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2017] [Accepted: 04/04/2019] [Indexed: 11/09/2022] Open
Abstract
Increasing evidence shows that real networks interact with each other, forming a network of networks (NONs). Synchronization, a ubiquitous process in natural and engineering systems, has fascinatingly gained rising attentions in the context of NONs. Despite efforts to study the synchronization of NONs, it is still a challenge to understand how do the network sizes affect the synchronization and its phase diagram of NONs coupled with nonlinear dynamics. Here, we model such NONs as star-like motifs to analytically derive the critical values of both the internal and the external coupling strengths, at which a phase transition from synchronization to incoherence occurs. Our results show that the critical values strongly depend on the network sizes. Reducing the difference between network sizes will enhance the synchronization of the whole system, which indicates the irrationality of previous studies that assume the network sizes to be the same. The optimal connection strategy also changes as the network sizes change, a discovery contradicting to the previous conclusion that connecting the high-degree nodes of each network is always the most effective strategy to achieve synchronization unchangeably. This finding emphasizes the crucial role of network sizes which has been neglected in the previous studies and could contribute to the design of a global synchronized system.
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Affiliation(s)
- Huixin Zhang
- Shanghai Jiao Tong University, Automation, Shanghai, 200240, China
| | - Weidong Zhang
- Shanghai Jiao Tong University, Automation, Shanghai, 200240, China.
| | - Jianxi Gao
- Rensselaer Polytechnic Institute, Computer Science Department & Network Science and Technology Center, Troy, New York, 12180, USA.
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3
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Jalan S, Kumar A, Leyva I. Explosive synchronization in frequency displaced multiplex networks. CHAOS (WOODBURY, N.Y.) 2019; 29:041102. [PMID: 31042936 DOI: 10.1063/1.5092226] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2019] [Accepted: 03/21/2019] [Indexed: 06/09/2023]
Abstract
Motivated by the recent multiplex framework of complex networks, in this work, we investigate if explosive synchronization can be induced in the multiplex network of two layers. Using nonidentical Kuramoto oscillators, we show that a sufficient frequency mismatch between two layers of a multiplex network can lead to explosive inter- and intralayer synchronization due to mutual frustration in the completion of the synchronization processes of the layers, generating a hybrid transition without imposing any specific structure-dynamics correlation.
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Affiliation(s)
- Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Anil Kumar
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Inmaculada Leyva
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
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4
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Aguirre J, Catalán P, Cuesta JA, Manrubia S. On the networked architecture of genotype spaces and its critical effects on molecular evolution. Open Biol 2018; 8:180069. [PMID: 29973397 PMCID: PMC6070719 DOI: 10.1098/rsob.180069] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2018] [Accepted: 06/12/2018] [Indexed: 12/26/2022] Open
Abstract
Evolutionary dynamics is often viewed as a subtle process of change accumulation that causes a divergence among organisms and their genomes. However, this interpretation is an inheritance of a gradualistic view that has been challenged at the macroevolutionary, ecological and molecular level. Actually, when the complex architecture of genotype spaces is taken into account, the evolutionary dynamics of molecular populations becomes intrinsically non-uniform, sharing deep qualitative and quantitative similarities with slowly driven physical systems: nonlinear responses analogous to critical transitions, sudden state changes or hysteresis, among others. Furthermore, the phenotypic plasticity inherent to genotypes transforms classical fitness landscapes into multiscapes where adaptation in response to an environmental change may be very fast. The quantitative nature of adaptive molecular processes is deeply dependent on a network-of-networks multilayered structure of the map from genotype to function that we begin to unveil.
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Affiliation(s)
- Jacobo Aguirre
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain
- Programa de Biología de Sistemas, Centro Nacional de Biotecnología (CSIC), Madrid, Spain
| | - Pablo Catalán
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain
- Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Madrid, Spain
| | - José A Cuesta
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain
- Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, Zaragoza, Spain
- UC3M-BS Institute of Financial Big Data (IFiBiD), Universidad Carlos III de Madrid, Getafe, Madrid, Spain
| | - Susanna Manrubia
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain
- Programa de Biología de Sistemas, Centro Nacional de Biotecnología (CSIC), Madrid, Spain
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5
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Ansmann G. Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE. CHAOS (WOODBURY, N.Y.) 2018; 28:043116. [PMID: 31906633 DOI: 10.1063/1.5019320] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing the user to efficiently integrate differential equations from a higher-level interpreted language. The presented modules are particularly suited for large systems of differential equations such as those used to describe dynamics on complex networks. Through the selected method of input, the presented modules also allow almost complete automatization of the process of estimating regular as well as transversal Lyapunov exponents for ordinary and delay differential equations. We conceptually discuss the modules' design, analyze their performance, and demonstrate their capabilities by application to timely problems.
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Affiliation(s)
- Gerrit Ansmann
- Department of Epileptology, University of Bonn, Sigmund-Freud-Straße 25, 53105 Bonn, Germany
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6
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Yamamoto H, Kubota S, Shimizu FA, Hirano-Iwata A, Niwano M. Effective Subnetwork Topology for Synchronizing Interconnected Networks of Coupled Phase Oscillators. Front Comput Neurosci 2018; 12:17. [PMID: 29643771 PMCID: PMC5882810 DOI: 10.3389/fncom.2018.00017] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2017] [Accepted: 03/06/2018] [Indexed: 11/13/2022] Open
Abstract
A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks.
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Affiliation(s)
- Hideaki Yamamoto
- Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai, Japan
| | - Shigeru Kubota
- Graduate School of Science and Engineering, Yamagata University, Yamagata, Japan
| | - Fabio A Shimizu
- Research Institute of Electrical Communication, Tohoku University, Sendai, Japan
| | - Ayumi Hirano-Iwata
- Research Institute of Electrical Communication, Tohoku University, Sendai, Japan.,Advanced Institute for Materials Research, Tohoku University, Sendai, Japan
| | - Michio Niwano
- Research Institute of Electrical Communication, Tohoku University, Sendai, Japan
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7
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Makovkin S, Kumar A, Zaikin A, Jalan S, Ivanchenko M. Multiplexing topologies and time scales: The gains and losses of synchrony. Phys Rev E 2017; 96:052214. [PMID: 29347745 DOI: 10.1103/physreve.96.052214] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2017] [Indexed: 06/07/2023]
Abstract
Inspired by the recent interest in collective dynamics of biological neural networks immersed in the glial cell medium, we investigate the frequency and phase order, i.e., Kuramoto type of synchronization in a multiplex two-layer network of phase oscillators of different time scales and topologies. One of them has a long-range connectivity, exemplified by the Erdős-Rényi random network, and supports both kinds of synchrony. The other is a locally coupled two-dimensional lattice that can reach frequency synchronization but lacks phase order. Drastically different layer frequencies disentangle intra- and interlayer synchronization. We find that an indirect but sufficiently strong coupling through the regular layer can induce both phase order in the originally nonsynchronized random layer and global order, even when an isolated regular layer does not manifest it in principle. At the same time, the route to global synchronization is complex: an initial onset of (partial) synchrony in the regular layer, when its intra- and interlayer coupling is increased, provokes the loss of synchrony even in the originally synchronized random layer. Ultimately, a developed asynchronous dynamics in both layers is abruptly taken over by the global synchrony of both kinds.
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Affiliation(s)
- Sergey Makovkin
- Department of Applied Mathematics and Centre of Bioinformatics, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
| | - Anil Kumar
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Simrol, Indore, India
| | - Alexey Zaikin
- Department of Applied Mathematics and Centre of Bioinformatics, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
- Institute for Women's Health and Department of Mathematics, University College London, London, United Kingdom
| | - Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Simrol, Indore, India
- Centre for Bio-Science and Bio-Medical Engineering, Indian Institute of Technology Indore, Simrol, Indore, India
| | - Mikhail Ivanchenko
- Department of Applied Mathematics and Centre of Bioinformatics, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
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8
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Yubero P, Manrubia S, Aguirre J. The space of genotypes is a network of networks: implications for evolutionary and extinction dynamics. Sci Rep 2017; 7:13813. [PMID: 29062002 PMCID: PMC5653773 DOI: 10.1038/s41598-017-14048-x] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2017] [Accepted: 10/04/2017] [Indexed: 11/09/2022] Open
Abstract
The forcing that environmental variation exerts on populations causes continuous changes with only two possible evolutionary outcomes: adaptation or extinction. Here we address this topic by studying the transient dynamics of populations on complex fitness landscapes. There are three important features of realistic landscapes of relevance in the evolutionary process: fitness landscapes are rough but correlated, their fitness values depend on the current environment, and many (often most) genotypes do not yield viable phenotypes. We capture these properties by defining time-varying, holey, NK fitness landscapes. We show that the structure of the space of genotypes so generated is that of a network of networks: in a sufficiently holey landscape, populations are temporarily stuck in local networks of genotypes. Sudden jumps to neighbouring networks through narrow adaptive pathways (connector links) are possible, though strong enough local trapping may also cause decays in population growth and eventual extinction. A combination of analytical and numerical techniques to characterize complex networks and population dynamics on such networks permits to derive several quantitative relationships between the topology of the space of genotypes and the fate of evolving populations.
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Affiliation(s)
- Pablo Yubero
- Centro Nacional de Biotecnología, CSIC, c/Darwin 3, 28049, Madrid, Spain
| | - Susanna Manrubia
- Centro Nacional de Biotecnología, CSIC, c/Darwin 3, 28049, Madrid, Spain
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain
| | - Jacobo Aguirre
- Centro Nacional de Biotecnología, CSIC, c/Darwin 3, 28049, Madrid, Spain.
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain.
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9
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Wang B, Suzuki H, Aihara K. Enhancing synchronization stability in a multi-area power grid. Sci Rep 2016; 6:26596. [PMID: 27225708 PMCID: PMC4881022 DOI: 10.1038/srep26596] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2016] [Accepted: 04/29/2016] [Indexed: 11/12/2022] Open
Abstract
Maintaining a synchronous state of generators is of central importance to the normal operation of power grids, in which many networks are generally interconnected. In order to understand the condition under which the stability can be optimized, it is important to relate network stability with feedback control strategies as well as network structure. Here, we present a stability analysis on a multi-area power grid by relating it with several control strategies and topological design of network structure. We clarify the minimal feedback gain in the self-feedback control, and build the optimal communication network for the local and global control strategies. Finally, we consider relationship between the interconnection pattern and the synchronization stability; by optimizing the network interlinks, the obtained network shows better synchronization stability than the original network does, in particular, at a high power demand. Our analysis shows that interlinks between spatially distant nodes will improve the synchronization stability. The results seem unfeasible to be implemented in real systems but provide a potential guide for the design of stable power systems.
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Affiliation(s)
- Bing Wang
- School of Computer Engineering and Science, Shanghai University, No. 99 Shangda Road, Baoshan District, Shanghai 200444, P. R. China.,Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
| | - Hideyuki Suzuki
- Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - Kazuyuki Aihara
- Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
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10
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Abstract
Increasing evidence shows that real-world systems interact with one another via dependency connectivities. Failing connectivities are the mechanism behind the breakdown of interacting complex systems, e.g., blackouts caused by the interdependence of power grids and communication networks. Previous research analyzing the robustness of interdependent networks has been limited to undirected networks. However, most real-world networks are directed, their in-degrees and out-degrees may be correlated, and they are often coupled to one another as interdependent directed networks. To understand the breakdown and robustness of interdependent directed networks, we develop a theoretical framework based on generating functions and percolation theory. We find that for interdependent Erdős-Rényi networks the directionality within each network increases their vulnerability and exhibits hybrid phase transitions. We also find that the percolation behavior of interdependent directed scale-free networks with and without degree correlations is so complex that two criteria are needed to quantify and compare their robustness: the percolation threshold and the integrated size of the giant component during an entire attack process. Interestingly, we find that the in-degree and out-degree correlations in each network layer increase the robustness of interdependent degree heterogeneous networks that most real networks are, but decrease the robustness of interdependent networks with homogeneous degree distribution and with strong coupling strengths. Moreover, by applying our theoretical analysis to real interdependent international trade networks, we find that the robustness of these real-world systems increases with the in-degree and out-degree correlations, confirming our theoretical analysis.
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Affiliation(s)
- Xueming Liu
- Key Laboratory of Image Information Processing and Intelligent Control, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China; Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215
| | - H Eugene Stanley
- Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215;
| | - Jianxi Gao
- Center for Complex Network Research and Department of Physics, Northeastern University, Boston, MA 02115
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11
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Gong M, Ma L, Cai Q, Jiao L. Enhancing robustness of coupled networks under targeted recoveries. Sci Rep 2015; 5:8439. [PMID: 25675980 PMCID: PMC4327128 DOI: 10.1038/srep08439] [Citation(s) in RCA: 54] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2014] [Accepted: 01/19/2015] [Indexed: 11/12/2022] Open
Abstract
Coupled networks are extremely fragile because a node failure of a network would trigger a cascade of failures on the entire system. Existing studies mainly focused on the cascading failures and the robustness of coupled networks when the networks suffer from attacks. In reality, it is necessary to recover the damaged networks, and there are cascading failures in recovery processes. In this study, firstly, we analyze the cascading failures of coupled networks during recoveries. Then, a recovery robustness index is presented for evaluating the resilience of coupled networks to cascading failures in the recovery processes. Finally, we propose a technique aiming at protecting several influential nodes for enhancing robustness of coupled networks under the recoveries, and adopt six strategies based on the potential knowledge of network centrality to find the influential nodes. Experiments on three coupling networks demonstrate that with a small number of influential nodes protected, the robustness of coupled networks under the recoveries can be greatly enhanced.
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Affiliation(s)
- Maoguo Gong
- Key Lab of Intelligent Perception and Image Understanding of Ministry of Education, International Research Center for Intelligent Perception and Computation, Xidian University, Xi'an, Shaanxi Province 710071, China
| | - Lijia Ma
- Key Lab of Intelligent Perception and Image Understanding of Ministry of Education, International Research Center for Intelligent Perception and Computation, Xidian University, Xi'an, Shaanxi Province 710071, China
| | - Qing Cai
- Key Lab of Intelligent Perception and Image Understanding of Ministry of Education, International Research Center for Intelligent Perception and Computation, Xidian University, Xi'an, Shaanxi Province 710071, China
| | - Licheng Jiao
- Key Lab of Intelligent Perception and Image Understanding of Ministry of Education, International Research Center for Intelligent Perception and Computation, Xidian University, Xi'an, Shaanxi Province 710071, China
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12
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Papo D, Zanin M, Pineda-Pardo JA, Boccaletti S, Buldú JM. Functional brain networks: great expectations, hard times and the big leap forward. Philos Trans R Soc Lond B Biol Sci 2014; 369:20130525. [PMID: 25180303 PMCID: PMC4150300 DOI: 10.1098/rstb.2013.0525] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
Abstract
Many physical and biological systems can be studied using complex network theory, a new statistical physics understanding of graph theory. The recent application of complex network theory to the study of functional brain networks has generated great enthusiasm as it allows addressing hitherto non-standard issues in the field, such as efficiency of brain functioning or vulnerability to damage. However, in spite of its high degree of generality, the theory was originally designed to describe systems profoundly different from the brain. We discuss some important caveats in the wholesale application of existing tools and concepts to a field they were not originally designed to describe. At the same time, we argue that complex network theory has not yet been taken full advantage of, as many of its important aspects are yet to make their appearance in the neuroscience literature. Finally, we propose that, rather than simply borrowing from an existing theory, functional neural networks can inspire a fundamental reformulation of complex network theory, to account for its exquisitely complex functioning mode.
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Affiliation(s)
- David Papo
- Center for Biomedical Technology, Universidad Politécnica de Madrid, Madrid, Spain
| | - Massimiliano Zanin
- Faculdade de Cîencias e Tecnologia, Departamento de Engenharia, Electrotécnica, Universidade Nova de Lisboa, Lisboa, Portugal Innaxis Foundation and Research Institute, Madrid, Spain
| | | | | | - Javier M Buldú
- Center for Biomedical Technology, Universidad Politécnica de Madrid, Madrid, Spain Complex Systems Group, Universidad Rey Juan Carlos, Móstoles, Spain
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13
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Abstract
Abstract
Network science has attracted much attention in recent years due to its interdisciplinary applications. We witnessed the revolution of network science in 1998 and 1999 started with small-world and scale-free networks having now thousands of high-profile publications, and it seems that since 2010 studies of ‘network of networks’ (NON), sometimes called multilayer networks or multiplex, have attracted more and more attention. The analytic framework for NON yields a novel percolation law for n interdependent networks that shows that percolation theory of single networks studied extensively in physics and mathematics in the last 50 years is a specific limit of the rich and very different general case of n coupled networks. Since then, properties and dynamics of interdependent and interconnected networks have been studied extensively, and scientists are finding many interesting results and discovering many surprising phenomena. Because most natural and engineered systems are composed of multiple subsystems and layers of connectivity, it is important to consider these features in order to improve our understanding of such complex systems. Now the study of NON has become one of the important directions in network science. In this paper, we review recent studies on the new emerging area—NON. Due to the fast growth of this field, there are many definitions of different types of NON, such as interdependent networks, interconnected networks, multilayered networks, multiplex networks and many others. There exist many datasets that can be represented as NON, such as network of different transportation networks including flight networks, railway networks and road networks, network of ecological networks including species interacting networks and food webs, network of biological networks including gene regulation network, metabolic network and protein–protein interacting network, network of social networks and so on. Among them, many interdependent networks including critical infrastructures are embedded in space, introducing spatial constraints. Thus, we also review the progress on study of spatially embedded networks. As a result of spatial constraints, such interdependent networks exhibit extreme vulnerabilities compared with their non-embedded counterparts. Such studies help us to understand, realize and hopefully mitigate the increasing risk in NON.
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Affiliation(s)
- Jianxi Gao
- Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
- Center for Complex Network Research and Department of Physics, Northeastern University, Boston, MA 02115, USA
| | - Daqing Li
- School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China
- Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China
| | - Shlomo Havlin
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
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14
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Nishikawa T. Introduction to Focus Issue: synchronization and cascading processes in complex networks. CHAOS (WOODBURY, N.Y.) 2011; 21:025101. [PMID: 21721779 DOI: 10.1063/1.3605467] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
The study of collective dynamics in complex networks has emerged as a next frontier in the science of networks. This Focus Issue presents the latest developments on this exciting front, focusing in particular on synchronous and cascading dynamics, which are ubiquitous forms of network dynamics found in a wide range of physical, biological, social, and technological systems.
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Affiliation(s)
- Takashi Nishikawa
- Department of Mathematics, Clarkson University, Potsdam, New York 13699, USA.
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