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Papo D, Buldú JM. Does the brain behave like a (complex) network? I. Dynamics. Phys Life Rev 2024; 48:47-98. [PMID: 38145591 DOI: 10.1016/j.plrev.2023.12.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2023] [Accepted: 12/10/2023] [Indexed: 12/27/2023]
Abstract
Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.
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Affiliation(s)
- D Papo
- Department of Neuroscience and Rehabilitation, Section of Physiology, University of Ferrara, Ferrara, Italy; Center for Translational Neurophysiology, Fondazione Istituto Italiano di Tecnologia, Ferrara, Italy.
| | - J M Buldú
- Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, Madrid, Spain
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2
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The dynamic nature of percolation on networks with triadic interactions. Nat Commun 2023; 14:1308. [PMID: 36894591 PMCID: PMC9998640 DOI: 10.1038/s41467-023-37019-5] [Citation(s) in RCA: 10] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2022] [Accepted: 02/24/2023] [Indexed: 03/11/2023] Open
Abstract
Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, we show that percolation can be turned into a fully fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, in which a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time and that the order parameter undergoes a period doubling and a route to chaos. We provide a general theory for triadic percolation which accurately predicts the full phase diagram on random graphs as confirmed by extensive numerical simulations. We find that triadic percolation on real network topologies reveals a similar phenomenology. These results radically change our understanding of percolation and may be used to study complex systems in which the functional connectivity is changing in time dynamically and in a non-trivial way, such as in neural and climate networks.
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3
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Punetha N, Wetzel L. How clock heterogeneity affects synchronization and can enhance stability. Phys Rev E 2022; 106:054216. [PMID: 36559456 DOI: 10.1103/physreve.106.054216] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2022] [Accepted: 10/05/2022] [Indexed: 06/17/2023]
Abstract
The production process of integrated electronic circuitry inherently leads to large heterogeneities on the component level. For electronic clock networks this implies detuned intrinsic frequencies and differences in coupling strength and the characteristic time delays associated with signal transmission, processing, and feedback. Using a phase-model description, we study the effects of such component heterogeneity on the dynamical properties of synchronization in networks of mutually delay-coupled Kuramoto oscillators. We test the theory against experimental results and circuit-level simulations in a prototype system of mutually delay-coupled electronic clocks, so-called phase-locked loops. Interestingly, our results show that heterogeneity in the system can actually enhance the stability of synchronized states. That means that beyond the optimizations that can be achieved by tuning homogeneous coupling strengths, time delays, and loop-filter cut-off frequencies, heterogeneities in these system parameters enable much better optimization of perturbation decay rates, stabilization of synchronous states, and tuning of phase differences between the clocks. Our theory enables the design of custom-fit synchronization layers according to the specific requirements and properties of electronic systems, such as operational frequencies, phase relations, and, e.g., transmission delays. These results are not restricted to electronic systems, because signal transmission, processing, and feedback delays are common to networks of spatially distributed and coupled autonomous oscillators.
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Affiliation(s)
- Nirmal Punetha
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Amity University Haryana, Gurgaon, Haryana 122413, India
| | - Lucas Wetzel
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Center for Advancing Electronics (CFAED), Würzburger Straße 46, 01187, Dresden, Germany
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4
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Punetha N, Wetzel L. Heterogeneity-induced synchronization in delay-coupled electronic oscillators. Phys Rev E 2022; 106:L052201. [PMID: 36559447 DOI: 10.1103/physreve.106.l052201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2022] [Accepted: 10/04/2022] [Indexed: 06/17/2023]
Abstract
We study synchronization in networks of delay-coupled electronic oscillators, so-called phase-locked loops (PLLs). Using a phase-model description, we study the collective dynamics of mutually coupled PLLs and report the phenomenon of heterogeneity-induced synchronization. This phenomenon refers to the observation that heterogeneity in the system's parameters can induce synchronization by stabilizing the states which are unstable without such heterogeneity. In systems where component heterogeneity can be tuned and controlled, we show how the complex collective self-organized dynamics can be guided towards synchronized states with specific operational frequencies and phase relations. This is of importance for the technical applicability of self-organized dynamics. In electrical engineering, for example, where components can be strongly heterogeneous, our theoretical framework can inform the design process for networks of spatially distributed PLLs. The results presented here are also useful in understanding the collective dynamics in ensembles of phase oscillators with time-delayed interactions, inertia, and heterogeneity.
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Affiliation(s)
- Nirmal Punetha
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Amity University Haryana, Gurugram (Manesar), 122413 Haryana, India
| | - Lucas Wetzel
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Center for Advancing Electronics (CFAED), Würzburger Straße 46, 01187 Dresden, Germany
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5
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Schäfer B, Pesch T, Manik D, Gollenstede J, Lin G, Beck HP, Witthaut D, Timme M. Understanding Braess’ Paradox in power grids. Nat Commun 2022; 13:5396. [PMID: 36104335 PMCID: PMC9474455 DOI: 10.1038/s41467-022-32917-6] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2021] [Accepted: 08/23/2022] [Indexed: 11/09/2022] Open
Abstract
AbstractThe ongoing energy transition requires power grid extensions to connect renewable generators to consumers and to transfer power among distant areas. The process of grid extension requires a large investment of resources and is supposed to make grid operation more robust. Yet, counter-intuitively, increasing the capacity of existing lines or adding new lines may also reduce the overall system performance and even promote blackouts due to Braess’ paradox. Braess’ paradox was theoretically modeled but not yet proven in realistically scaled power grids. Here, we present an experimental setup demonstrating Braess’ paradox in an AC power grid and show how it constrains ongoing large-scale grid extension projects. We present a topological theory that reveals the key mechanism and predicts Braessian grid extensions from the network structure. These results offer a theoretical method to understand and practical guidelines in support of preventing unsuitable infrastructures and the systemic planning of grid extensions.
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Nikfard T, Tabatabaei YH, Shahbazi F. Contrariety and inhibition enhance synchronization in a small-world network of phase oscillators. Phys Rev E 2021; 104:054213. [PMID: 34942811 DOI: 10.1103/physreve.104.054213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2020] [Accepted: 11/11/2021] [Indexed: 11/07/2022]
Abstract
We numerically study Kuramoto model synchronization consisting of the two groups of conformist-contrarian and excitatory-inhibitory phase oscillators with equal intrinsic frequency. We consider random and small-world (SW) topologies for the connectivity network of the oscillators. In random networks, regardless of the contrarian to conformist connection strength ratio, we found a crossover from the π-state to the blurred π-state and then a continuous transition to the incoherent state by increasing the fraction of contrarians. However, for the excitatory-inhibitory model in a random network, we found that for all the values of the fraction of inhibitors, the two groups remain in phase and the transition point of fully synchronized to an incoherent state reduced by strengthening the ratio of inhibitory to excitatory links. In the SW networks we found that the order parameters for both models do not show monotonic behavior in terms of the fraction of contrarians and inhibitors. Up to the optimal fraction of contrarians and inhibitors, the synchronization rises by introducing the number of contrarians and inhibitors and then falls. We discuss that the nonmonotonic behavior in synchronization is due to the weakening of the defects already formed in the pure conformist and excitatory agent model in SW networks. We found that in SW networks, the optimal fraction of contrarians and inhibitors remain unchanged for the rewiring probabilities up to ∼0.15, above which synchronization falls monotonically, like the random network. We also showed that in the conformist-contrarian model, the optimal fraction of contrarians is independent of the strength of contrarian links. However, in the excitatory-inhibitory model, the optimal fraction of inhibitors is approximately proportional to the inverse of inhibition strength.
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Affiliation(s)
- Tayebe Nikfard
- Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
| | | | - Farhad Shahbazi
- Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
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7
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Kohler J, Wunderling N, Donges JF, Vollmer J. Complex networks of interacting stochastic tipping elements: Cooperativity of phase separation in the large-system limit. Phys Rev E 2021; 104:044301. [PMID: 34781496 DOI: 10.1103/physreve.104.044301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2021] [Accepted: 08/26/2021] [Indexed: 11/07/2022]
Abstract
Tipping elements in the Earth system have received increased scientific attention over recent years due to their nonlinear behavior and the risks of abrupt state changes. While being stable over a large range of parameters, a tipping element undergoes a drastic shift in its state upon an additional small parameter change when close to its tipping point. Recently, the focus of research broadened towards emergent behavior in networks of tipping elements, like global tipping cascades triggered by local perturbations. Here, we analyze the response to the perturbation of a single node in a system that initially resides in an unstable equilibrium. The evolution is described in terms of coupled nonlinear equations for the cumulants of the distribution of the elements. We show that drift terms acting on individual elements and offsets in the coupling strength are subdominant in the limit of large networks, and we derive an analytical prediction for the evolution of the expectation (i.e., the first cumulant). It behaves like a single aggregated tipping element characterized by a dimensionless parameter that accounts for the network size, its overall connectivity, and the average coupling strength. The resulting predictions are in excellent agreement with numerical data for Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks of different size and with different coupling parameters.
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Affiliation(s)
- Jan Kohler
- Institute for Theoretical Physics, University of Leipzig, 04103 Leipzig, Germany, EU.,Earth System Analysis, Potsdam-Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany, EU
| | - Nico Wunderling
- Earth System Analysis, Potsdam-Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany, EU.,Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany, EU.,Department of Physics, Humboldt University of Berlin, 12489 Berlin, Germany, EU
| | - Jonathan F Donges
- Earth System Analysis, Potsdam-Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany, EU.,Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden, EU
| | - Jürgen Vollmer
- Institute for Theoretical Physics, University of Leipzig, 04103 Leipzig, Germany, EU
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Rozum JC, Gómez Tejeda Zañudo J, Gan X, Deritei D, Albert R. Parity and time reversal elucidate both decision-making in empirical models and attractor scaling in critical Boolean networks. SCIENCE ADVANCES 2021; 7:eabf8124. [PMID: 34272246 PMCID: PMC8284893 DOI: 10.1126/sciadv.abf8124] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2020] [Accepted: 06/03/2021] [Indexed: 05/14/2023]
Abstract
We present new applications of parity inversion and time reversal to the emergence of complex behavior from simple dynamical rules in stochastic discrete models. Our parity-based encoding of causal relationships and time-reversal construction efficiently reveal discrete analogs of stable and unstable manifolds. We demonstrate their predictive power by studying decision-making in systems biology and statistical physics models. These applications underpin a novel attractor identification algorithm implemented for Boolean networks under stochastic dynamics. Its speed enables resolving a long-standing open question of how attractor count in critical random Boolean networks scales with network size and whether the scaling matches biological observations. Via 80-fold improvement in probed network size (N = 16,384), we find the unexpectedly low scaling exponent of 0.12 ± 0.05, approximately one-tenth the analytical upper bound. We demonstrate a general principle: A system's relationship to its time reversal and state-space inversion constrains its repertoire of emergent behaviors.
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Affiliation(s)
- Jordan C Rozum
- Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA.
| | - Jorge Gómez Tejeda Zañudo
- Eli and Edythe L. Broad Institute of MIT and Harvard, Cambridge, MA 02142, USA
- Department of Medical Oncology, Dana-Farber Cancer Institute, Harvard Medical School, Boston, MA 02115, USA
| | - Xiao Gan
- Network Science Institute and Department of Physics, Northeastern University, Boston, MA 02115, USA
- Channing Division of Network Medicine, Department of Medicine, Brigham and Women's Hospital, Harvard Medical School, Boston, MA 02115, USA
| | - Dávid Deritei
- Department of Molecular Biology, Semmelweis University, Budapest, Hungary
| | - Réka Albert
- Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA.
- Department of Biology, The Pennsylvania State University, University Park, PA 16802, USA
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9
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Arola-Fernández L, Skardal PS, Arenas A. Geometric unfolding of synchronization dynamics on networks. CHAOS (WOODBURY, N.Y.) 2021; 31:061105. [PMID: 34241326 DOI: 10.1063/5.0053837] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2021] [Accepted: 05/17/2021] [Indexed: 06/13/2023]
Abstract
We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Specifically, each additional term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncated series grows geometrically with the second largest eigenvalue of the normalized adjacency matrix, analogously to the rate of convergence to the stationary distribution of a random walk. Last, we derive a local approximation for the synchronized state by truncating the spatial series, at the first neighborhood term, to illustrate the practical advantages of our approach.
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Affiliation(s)
- Lluís Arola-Fernández
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | | | - Alex Arenas
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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10
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Porfiri M. Validity and Limitations of the Detection Matrix to Determine Hidden Units and Network Size from Perceptible Dynamics. PHYSICAL REVIEW LETTERS 2020; 124:168301. [PMID: 32383942 DOI: 10.1103/physrevlett.124.168301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2019] [Revised: 03/04/2020] [Accepted: 03/20/2020] [Indexed: 06/11/2023]
Abstract
Determining the size of a network dynamical system from the time series of some accessible units is a critical problem in network science. Recent work by Haehne et al. [Phys. Rev. Lett. 122, 158301 (2019).PRLTAO0031-900710.1103/PhysRevLett.122.158301] has presented a model-free approach to address this problem, by studying the rank of a detection matrix that collates sampled time series of perceptible nodes from independent experiments. Here, we unveil a profound connection between the rank of the detection matrix and the control-theoretic notion of observability, upon which we conclude when and how it is feasible to exactly infer the size of a network dynamical system.
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Affiliation(s)
- Maurizio Porfiri
- Department of Mechanical and Aerospace Engineering and Department of Biomedical Engineering, New York University, Tandon School of Engineering, Brooklyn, New York 11201, USA
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11
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Braess’s paradox and programmable behaviour in microfluidic networks. Nature 2019; 574:647-652. [DOI: 10.1038/s41586-019-1701-6] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2018] [Accepted: 08/01/2019] [Indexed: 01/01/2023]
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Haehne H, Casadiego J, Peinke J, Timme M. Detecting Hidden Units and Network Size from Perceptible Dynamics. PHYSICAL REVIEW LETTERS 2019; 122:158301. [PMID: 31050518 DOI: 10.1103/physrevlett.122.158301] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2018] [Revised: 02/12/2019] [Indexed: 06/09/2023]
Abstract
The number of units of a network dynamical system, its size, arguably constitutes its most fundamental property. Many units of a network, however, are typically experimentally inaccessible such that the network size is often unknown. Here we introduce a detection matrix that suitably arranges multiple transient time series from the subset of accessible units to detect network size via matching rank constraints. The proposed method is model-free, applicable across system types and interaction topologies, and applies to nonstationary dynamics near fixed points, as well as periodic and chaotic collective motion. Even if only a small minority of units is perceptible and for systems simultaneously exhibiting nonlinearities, heterogeneities, and noise, exact size detection is feasible. We illustrate applicability for a paradigmatic class of biochemical reaction networks.
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Affiliation(s)
- Hauke Haehne
- Institute of Physics and ForWind, University of Oldenburg, 26111 Oldenburg, Germany
| | - Jose Casadiego
- Chair for Network Dynamics, Institute for Theoretical Physics and Center for Advancing Electronics Dresden (cfaed), Technical University of Dresden, 01062 Dresden, Germany
| | - Joachim Peinke
- Institute of Physics and ForWind, University of Oldenburg, 26111 Oldenburg, Germany
| | - Marc Timme
- Chair for Network Dynamics, Institute for Theoretical Physics and Center for Advancing Electronics Dresden (cfaed), Technical University of Dresden, 01062 Dresden, Germany
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