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Mizutaka S, Mori K, Hasegawa T. Synergistic epidemic spreading in correlated networks. Phys Rev E 2022; 106:034305. [PMID: 36266882 DOI: 10.1103/physreve.106.034305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2021] [Accepted: 08/04/2022] [Indexed: 06/16/2023]
Abstract
We investigate the effect of degree correlation on a susceptible-infected-susceptible (SIS) model with a nonlinear cooperative effect (synergy) in infectious transmissions. In a mean-field treatment of the synergistic SIS model on a bimodal network with tunable degree correlation, we identify a discontinuous transition that is independent of the degree correlation strength unless the synergy is absent or extremely weak. Regardless of synergy (absent or present), a positive and negative degree correlation in the model reduces and raises the epidemic threshold, respectively. For networks with a strongly positive degree correlation, the mean-field treatment predicts the emergence of two discontinuous jumps in the steady-state infected density. To test the mean-field treatment, we provide approximate master equations of the present model. We quantitatively confirm that the approximate master equations agree with not only all qualitative predictions of the mean-field treatment but also corresponding Monte Carlo simulations.
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Affiliation(s)
- Shogo Mizutaka
- Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi 924-1292, Japan
| | - Kizashi Mori
- Graduate School of Science and Engineering, Ibaraki University, 2-1-1 Bunkyo, Mito 310-8512, Japan
| | - Takehisa Hasegawa
- Graduate School of Science and Engineering, Ibaraki University, 2-1-1 Bunkyo, Mito 310-8512, Japan
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2
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Bolfe M, Metz FL, Guzmán-González E, Castillo IP. Analytic solution of the two-star model with correlated degrees. Phys Rev E 2021; 104:014147. [PMID: 34412227 DOI: 10.1103/physreve.104.014147] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2021] [Accepted: 07/07/2021] [Indexed: 11/06/2022]
Abstract
Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime. The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors). We compute exactly the network free energy and show that this model undergoes a first-order transition to a condensed phase. For non-negative degree correlations between next nearest neighbors, the degree distribution inside the condensed phase has a single peak at the largest degree, while for negative degree correlations between next nearest neighbors the condensed phase is characterized by a bimodal degree distribution. We calculate the degree assortativities and show they are nonmonotonic functions of the model parameters, with a discontinuous behavior at the first-order transition. The first-order critical line terminates at a second-order critical point, whose location in the phase diagram can be accurately determined. Our results can help to develop more detailed models of complex networks with correlated degrees.
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Affiliation(s)
- Maíra Bolfe
- Physics Department, Federal University of Santa Maria, 97105-900 Santa Maria, Brazil
| | - Fernando L Metz
- Physics Institute, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil and London Mathematical Laboratory, 8 Margravine Gardens, London W6 8RH, United Kingdom
| | - Edgar Guzmán-González
- Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340, México
| | - Isaac Pérez Castillo
- Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340, México
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Ganaie MA, Ghosh S, Mendola N, Tanveer M, Jalan S. Identification of chimera using machine learning. CHAOS (WOODBURY, N.Y.) 2020; 30:063128. [PMID: 32611090 DOI: 10.1063/1.5143285] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2019] [Accepted: 05/07/2020] [Indexed: 06/11/2023]
Abstract
Chimera state refers to the coexistence of coherent and non-coherent phases in identically coupled dynamical units found in various complex dynamical systems. Identification of chimera, on one hand, is essential due to its applicability in various areas including neuroscience and, on the other hand, is challenging due to its widely varied appearance in different systems and the peculiar nature of its profile. Therefore, a simple yet universal method for its identification remains an open problem. Here, we present a very distinctive approach using machine learning techniques to characterize different dynamical phases and identify the chimera state from given spatial profiles generated using various different models. The experimental results show that the performance of the classification algorithms varies for different dynamical models. The machine learning algorithms, namely, random forest, oblique random forest based on Tikhonov, axis-parallel split, and null space regularization achieved more than 96% accuracy for the Kuramoto model. For the logistic maps, random forest and Tikhonov regularization based oblique random forest showed more than 90% accuracy, and for the Hénon map model, random forest, null space, and axis-parallel split regularization based oblique random forest achieved more than 80% accuracy. The oblique random forest with null space regularization achieved consistent performance (more than 83% accuracy) across different dynamical models while the auto-encoder based random vector functional link neural network showed relatively lower performance. This work provides a direction for employing machine learning techniques to identify dynamical patterns arising in coupled non-linear units on large-scale and for characterizing complex spatiotemporal patterns in real-world systems for various applications.
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Affiliation(s)
- M A Ganaie
- Discipline of Mathematics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Saptarshi Ghosh
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Naveen Mendola
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - M Tanveer
- Discipline of Mathematics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
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Sarkar C, Jalan S. Spectral properties of complex networks. CHAOS (WOODBURY, N.Y.) 2018; 28:102101. [PMID: 30384632 DOI: 10.1063/1.5040897] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.
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Affiliation(s)
- Camellia Sarkar
- Centre for Biosciences and Biomedical Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Sarika Jalan
- Centre for Biosciences and Biomedical Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
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Fujiki Y, Takaguchi T, Yakubo K. General formulation of long-range degree correlations in complex networks. Phys Rev E 2018; 97:062308. [PMID: 30011590 DOI: 10.1103/physreve.97.062308] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2017] [Indexed: 11/07/2022]
Abstract
We provide a general framework for analyzing degree correlations between nodes separated by more than one step (i.e., beyond nearest neighbors) in complex networks. One joint and four conditional probability distributions are introduced to fully describe long-range degree correlations with respect to degrees k and k^{'} of two nodes and shortest path length l between them. We present general relations among these probability distributions and clarify the relevance to nearest-neighbor degree correlations. Unlike nearest-neighbor correlations, some of these probability distributions are meaningful only in finite-size networks. Furthermore, as a baseline to determine the existence of intrinsic long-range degree correlations in a network other than inevitable correlations caused by the finite-size effect, the functional forms of these probability distributions for random networks are analytically evaluated within a mean-field approximation. The utility of our argument is demonstrated by applying it to real-world networks.
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Affiliation(s)
- Yuka Fujiki
- Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
| | - Taro Takaguchi
- National Institute of Information and Communications Technology, Tokyo 184-8795, Japan
| | - Kousuke Yakubo
- Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
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Ghosh S, Jalan S. Engineering chimera patterns in networks using heterogeneous delays. CHAOS (WOODBURY, N.Y.) 2018; 28:071103. [PMID: 30070528 DOI: 10.1063/1.5042133] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2018] [Accepted: 06/26/2018] [Indexed: 06/08/2023]
Abstract
Symmetry breaking spatial patterns, referred to as chimera states, have recently been catapulted into the limelight due to their coexisting coherent and incoherent hybrid dynamics. Here, we present a method to engineer a chimera state by using an appropriate distribution of heterogeneous time delays on the edges of a network. The time delays in interactions, intrinsic to natural or artificial complex systems, are known to induce various modifications in spatiotemporal behaviors of the coupled dynamics on networks. Using a coupled chaotic map with the identical coupling environment, we demonstrate that control over the spatial location of the incoherent region of a chimera state in a network can be achieved by appropriately introducing time delays. This method allows for the engineering of tailor-made one cluster or multi-cluster chimera patterns. Furthermore, borrowing a measure of eigenvector localization from the spectral graph theory, we introduce a spatial inverse participation ratio, which provides a robust way for the identification of the chimera state. This report highlights the necessity to consider the heterogeneous time delays to develop applications for the chimera states in particular and understand coupled dynamical systems in general.
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Affiliation(s)
- Saptarshi Ghosh
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
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Kumar A, Baptista MS, Zaikin A, Jalan S. Mirror node correlations tuning synchronization in multiplex networks. Phys Rev E 2017; 96:062301. [PMID: 29347340 DOI: 10.1103/physreve.96.062301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2017] [Indexed: 06/07/2023]
Abstract
We show that the degree-degree correlations have a major impact on global synchronizability (GS) of multiplex networks, enabling the specification of synchronizability by only changing the degree-degree correlations of the mirror nodes while maintaining the connection architecture of the individual layer unaltered. If individual layers have nodes that are mildly correlated, the multiplex network is best synchronizable when the mirror degrees are strongly negatively correlated. If individual layers have nodes with strong degree-degree correlations, mild correlations among the degrees of mirror nodes are the best strategy for the optimization of GS. Global synchronization also depend on the density of connections, a phenomenon not observed in a single layer network. The results are crucial to understand, predict, and specify behavior of systems having multiple types of connections among the interacting units.
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Affiliation(s)
- Anil Kumar
- Complex Systems Laboratory, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Murilo S Baptista
- Institute of Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Alexey Zaikin
- Department of Mathematics, Institute for Women's Health, University College London, London WC1H 0AY, United Kingdom
- Department of Applied Mathematics, Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod 603950, Russia
| | - Sarika Jalan
- Complex Systems Laboratory, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
- Centre for Bio-Science and Bio-Medical Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
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Pradhan P, Yadav A, Dwivedi SK, Jalan S. Optimized evolution of networks for principal eigenvector localization. Phys Rev E 2017; 96:022312. [PMID: 28950611 DOI: 10.1103/physreve.96.022312] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2017] [Indexed: 05/11/2023]
Abstract
Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.
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Affiliation(s)
- Priodyuti Pradhan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Alok Yadav
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Sanjiv K Dwivedi
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
- Centre for Biosciences and Biomedical Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
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