1
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Kim M, Radicchi F. Shortest-Path Percolation on Random Networks. PHYSICAL REVIEW LETTERS 2024; 133:047402. [PMID: 39121422 DOI: 10.1103/physrevlett.133.047402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2024] [Revised: 05/28/2024] [Accepted: 06/20/2024] [Indexed: 08/11/2024]
Abstract
We propose a bond-percolation model intended to describe the consumption, and eventual exhaustion, of resources in transport networks. Edges forming minimum-length paths connecting demanded origin-destination nodes are removed if below a certain budget. As pairs of nodes are demanded and edges are removed, the macroscopic connected component of the graph disappears, i.e., the graph undergoes a percolation transition. Here, we study such a shortest-path-percolation transition in homogeneous random graphs where pairs of demanded origin-destination nodes are randomly generated, and fully characterize it by means of finite-size scaling analysis. If budget is finite, the transition is identical to the one of ordinary percolation, where a single giant cluster shrinks as edges are removed from the graph; for infinite budget, the transition becomes more abrupt than the one of ordinary percolation, being characterized by the sudden fragmentation of the giant connected component into a multitude of clusters of similar size.
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Affiliation(s)
- Minsuk Kim
- Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, Indiana 47408, USA
| | - Filippo Radicchi
- Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, Indiana 47408, USA
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2
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Luo Z, Chen W, Nagler J. Universality of explosive percolation under product and sum rule. Phys Rev E 2023; 108:034108. [PMID: 37849098 DOI: 10.1103/physreve.108.034108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2023] [Accepted: 08/11/2023] [Indexed: 10/19/2023]
Abstract
We study explosive percolation processes on random graphs for the so-called product rule (PR) and sum rule (SR), in which M candidate edges are randomly selected from all possible ones at each time step, and the edge with the smallest product or sum of the sizes of the two components that would be joined by the edge is added to the graph, while all other M-1 candidate edges are being discarded. These two rules are prototypical "explosive" percolation rules, which exhibit an extremely abrupt yet continuous phase transition in the thermodynamic limit. Recently, it has been demonstrated that PR and SR belong to the same universality class for two competing edges, i.e., M=2. Here we investigate whether the claimed PR-SR universality is valid for higher-order models with M larger than 2. Based on traditional finite-size scaling theory and largest-gap scaling, we obtain the percolation threshold and the critical exponents of the order parameter, susceptibility, and the derivative of entropy for PR and SR for M from 2 to 9. Our results strongly suggest PR-SR universality, for any fixed M.
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Affiliation(s)
- Ziting Luo
- LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China
| | - Wei Chen
- Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
- Zhongguancun Laboratory, Beijing 100094, People's Republic of China
- Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China
| | - Jan Nagler
- Deep Dynamics, Centre for Human and Machine Intelligence, Frankfurt School of Finance and Management, Frankfurt am Main 60322, Germany
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3
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Nguyen CPT, Schoenherr P, Salje EKH, Seidel J. Crackling noise microscopy. Nat Commun 2023; 14:4963. [PMID: 37587105 PMCID: PMC10432464 DOI: 10.1038/s41467-023-40665-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2022] [Accepted: 08/02/2023] [Indexed: 08/18/2023] Open
Abstract
Crackling noise is a scale-invariant phenomenon found in various driven nonlinear dynamical material systems as a response to external stimuli such as force or external fields. Jerky material movements in the form of avalanches can span many orders of magnitude in size and follow universal scaling rules described by power laws. The concept was originally studied as Barkhausen noise in magnetic materials and now is used in diverse fields from earthquake research and building materials monitoring to fundamental research involving phase transitions and neural networks. Here, we demonstrate a method for nanoscale crackling noise measurements based on AFM nanoindentation, where the AFM probe can be used to study the crackling of individual nanoscale features, a technique we call crackling noise microscopy. The method is successfully applied to investigate the crackling of individual topological defects, i.e. ferroelectric domain walls. We show that critical exponents for avalanches are altered at these nanoscale features, leading to a suppression of mixed-criticality, which is otherwise present in domains. The presented concept opens the possibility of investigating the crackling of individual nanoscale features in a wide range of material systems.
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Affiliation(s)
- Cam-Phu Thi Nguyen
- School of Materials Science and Engineering, UNSW Sydney, Sydney, NSW, 2052, Australia
| | - Peggy Schoenherr
- School of Materials Science and Engineering, UNSW Sydney, Sydney, NSW, 2052, Australia
- ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET), UNSW Sydney, Sydney, NSW, 2052, Australia
| | - Ekhard K H Salje
- Department of Earth Sciences, Cambridge University, Cambridge, UK.
| | - Jan Seidel
- School of Materials Science and Engineering, UNSW Sydney, Sydney, NSW, 2052, Australia.
- ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET), UNSW Sydney, Sydney, NSW, 2052, Australia.
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4
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Park J, Yi S, Kahng B. Hysteresis and criticality in hybrid percolation transitions. CHAOS (WOODBURY, N.Y.) 2020; 30:051102. [PMID: 32491902 DOI: 10.1063/5.0008189] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2020] [Accepted: 04/10/2020] [Indexed: 06/11/2023]
Abstract
Phase transitions (PTs) are generally classified into second-order and first-order transitions, each exhibiting different intrinsic properties. For instance, a first-order transition exhibits latent heat and hysteresis when a control parameter is increased and then decreased across a transition point, whereas a second-order transition does not. Recently, hybrid percolation transitions (HPTs) are issued in diverse complex systems, in which the features of first-order and second-order PTs occur at the same transition point. Thus, the question whether hysteresis appears in an HPT arises. Herein, we investigate this fundamental question with a so-called restricted Erdős-Rényi random network model, in which a cluster fragmentation process is additionally proposed. A hysteresis curve of the order parameter was obtained. Depending on when the reverse process is initiated, the shapes of hysteresis curves change, and the critical behavior of the HPT is conserved throughout the forward and reverse processes.
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Affiliation(s)
- Jinha Park
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
| | - Sudo Yi
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, South Korea
| | - B Kahng
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
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5
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Park J, Yi S, Choi K, Lee D, Kahng B. Interevent time distribution, burst, and hybrid percolation transition. CHAOS (WOODBURY, N.Y.) 2019; 29:091102. [PMID: 31575135 DOI: 10.1063/1.5121775] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2019] [Accepted: 08/28/2019] [Indexed: 06/10/2023]
Abstract
Understanding of a hybrid percolation transitions (HPTs) induced by cluster coalescence, exhibiting a jump in the giant cluster size and a critical behavior of finite clusters, is fundamental and intriguing. Here, we uncover the underlying mechanism using the so-called restricted-random network model, in which clusters are ranked by size and partitioned into small- and large-cluster sets. As clusters are merged and their rankings are updated, they may move back and forth across the set boundary. The intervals of these crossings exhibit a self-organized critical (SOC) behavior with two power-law exponents. During this process, a bump is formed and eliminated in the cluster size distribution, characterizing the criticality of the HPT. This SOC behavior is in contrast to the critical branching process, which governs the avalanche dynamics of the HPT in the pruning process. Finally, we find that a burst of such crossing events occurs and signals the upcoming abrupt transition.
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Affiliation(s)
- Jinha Park
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
| | - Sudo Yi
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
| | - K Choi
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
| | - Deokjae Lee
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
| | - B Kahng
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
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6
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Molkenthin N, Schröder M, Timme M. Adhesion-Induced Discontinuous Transitions and Classifying Social Networks. PHYSICAL REVIEW LETTERS 2018; 121:138301. [PMID: 30312048 DOI: 10.1103/physrevlett.121.138301] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2017] [Revised: 06/08/2018] [Indexed: 06/08/2023]
Abstract
Transition points mark qualitative changes in the macroscopic properties of large complex systems. Explosive transitions, exhibiting properties of both continuous and discontinuous phase transitions, have recently been uncovered in network growth processes. Real networks not only grow but often also restructure; yet common network restructuring processes, such as small world rewiring, do not exhibit phase transitions. Here, we uncover a class of intrinsically discontinuous transitions emerging in network restructuring processes controlled by adhesion-the preference of a chosen link to remain connected to its end node. Deriving a master equation for the temporal network evolution and working out an analytic solution, we identify genuinely discontinuous transitions in nongrowing networks, separating qualitatively distinct phases with monotonic and with peaked degree distributions. Intriguingly, our analysis of empirical data indicates a separation between the same two forms of degree distributions distinguishing abstract from face-to-face social networks.
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Affiliation(s)
- Nora Molkenthin
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
| | - Malte Schröder
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed) and Institute for Theoretical Physics, Technical University of Dresden, 01069 Dresden, Germany
| | - Marc Timme
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed) and Institute for Theoretical Physics, Technical University of Dresden, 01069 Dresden, Germany
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7
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Zimmer P, Kruse K, Nagler J. Anomalous percolation features in molecular evolution. Phys Rev E 2018; 98:022408. [PMID: 30253595 DOI: 10.1103/physreve.98.022408] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2018] [Indexed: 06/08/2023]
Abstract
Self-replication underlies every species of living beings and simple physical intuition dictates that some sort of autocatalysis invariably constitutes a necessary ingredient for the emergence of molecular life. This led Worst et al. [E. G. Worst, P. Zimmer, E. Wollrab, K. Kruse, and A. Ott, New J. Phys. 18, 103003 (2016)NJOPFM1367-263010.1088/1367-2630/18/10/103003] to study a model of molecular evolution of self-replicating molecules where spontaneous ligation and simple autocatalysis are in competition for their building blocks. We revisit this model, where irreversible aggregation leads to a transition from a regime of small molecules to macromolecules, and find an array of anomalous percolation features, some of them predicted for very specific percolation processes [R. M. D'Souza and J. Nagler, Nat. Phys. 11, 531 (2015)1745-247310.1038/nphys3378].
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Affiliation(s)
- P Zimmer
- Theoretical Physics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany
| | - K Kruse
- NCCR Chemical Biology, Departments of Biochemistry and Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland and Theoretical Physics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany
| | - J Nagler
- Deep Dynamics Group & Centre for Human and Machine Intelligence, Frankfurt School of Finance & Management, Frankfurt, Germany and Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
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8
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Schröder M, Nagler J, Timme M, Witthaut D. Hysteretic Percolation from Locally Optimal Individual Decisions. PHYSICAL REVIEW LETTERS 2018; 120:248302. [PMID: 29957012 DOI: 10.1103/physrevlett.120.248302] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2017] [Revised: 04/10/2018] [Indexed: 06/08/2023]
Abstract
The emergence of large-scale connectivity underlies the proper functioning of many networked systems, ranging from social networks and technological infrastructure to global trade networks. Percolation theory characterizes network formation following stochastic local rules, while optimization models of network formation assume a single controlling authority or one global objective function. In socioeconomic networks, however, network formation is often driven by individual, locally optimal decisions. How such decisions impact connectivity is only poorly understood to date. Here, we study how large-scale connectivity emerges from decisions made by rational agents that individually minimize costs for satisfying their demand. We establish that the solution of the resulting nonlinear optimization model is exactly given by the final state of a local percolation process. This allows us to systematically analyze how locally optimal decisions on the microlevel define the structure of networks on the macroscopic scale.
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Affiliation(s)
- Malte Schröder
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed) and Institute for Theoretical Physics, Technical University of Dresden, 01069 Dresden, Germany
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
| | - Jan Nagler
- ETH Zürich, Wolfgang-Pauli-Strasse 27, CH-8093 Zürich, Switzerland
- Computational Social Science, Department of Humanities, Social and Political Sciences, ETH Zurich, Clausiusstrasse 50, CH-8092 Zurich, Switzerland
| | - Marc Timme
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed) and Institute for Theoretical Physics, Technical University of Dresden, 01069 Dresden, Germany
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
| | - Dirk Witthaut
- Forschungszentrum Jülich, Institute for Energy and Climate Research-Systems Analysis and Technology Evaluation (IEK-STE), 52428 Jülich, Germany
- Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany
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9
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Trevelyan AJ, Tsekenis G, Corwin EI. Degree product rule tempers explosive percolation in the absence of global information. Phys Rev E 2018; 97:020301. [PMID: 29548192 DOI: 10.1103/physreve.97.020301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2017] [Indexed: 06/08/2023]
Abstract
We introduce a guided network growth model, which we call the degree product rule process, that uses solely local information when adding new edges. For small numbers of candidate edges our process gives rise to a second-order phase transition, but becomes first order in the limit of global choice. We provide the set of critical exponents required to characterize the nature of this percolation transition. Such a process permits interventions which can delay the onset of percolation while tempering the explosiveness caused by cluster product rule processes.
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Affiliation(s)
- Alexander J Trevelyan
- Materials Science Institute and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
| | - Georgios Tsekenis
- Materials Science Institute and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
| | - Eric I Corwin
- Materials Science Institute and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
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10
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Schröder M, Araújo NAM, Sornette D, Nagler J. Controlling percolation with limited resources. Phys Rev E 2017; 96:062302. [PMID: 29347337 DOI: 10.1103/physreve.96.062302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2017] [Indexed: 06/07/2023]
Abstract
Connectivity, or the lack thereof, is crucial for the function of many man-made systems, from financial and economic networks over epidemic spreading in social networks to technical infrastructure. Often, connections are deliberately established or removed to induce, maintain, or destroy global connectivity. Thus, there has been a great interest in understanding how to control percolation, the transition to large-scale connectivity. Previous work, however, studied control strategies assuming unlimited resources. Here, we depart from this unrealistic assumption and consider the effect of limited resources on the effectiveness of control. We show that, even for scarce resources, percolation can be controlled with an efficient intervention strategy. We derive such an efficient strategy and study its implications, revealing a discontinuous transition as an unintended side effect of optimal control.
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Affiliation(s)
- Malte Schröder
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
| | - Nuno A M Araújo
- Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, P-1749-016 Lisboa, Portugal
- Centro de Física Teórica e Computacional, Universidade de Lisboa, 1749-016 Lisboa, Portugal
| | - Didier Sornette
- Department of Management, Technology and Economics (D-MTEC), ETH Zürich, Scheuchzerstrasse 7, CH-8092 Zurich, Switzerland
| | - Jan Nagler
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zürich, Wolfgang-Pauli-Strasse 27, HIT, CH-8093 Zurich, Switzerland
- Computational Social Science, Department of Humanities, Social and Political Sciences, ETH Zürich, Clausiusstrasse 50, CH-8092 Zurich, Switzerland
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11
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Böttcher L, Nagler J, Herrmann HJ. Critical Behaviors in Contagion Dynamics. PHYSICAL REVIEW LETTERS 2017; 118:088301. [PMID: 28282207 PMCID: PMC7219437 DOI: 10.1103/physrevlett.118.088301] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2016] [Indexed: 05/03/2023]
Abstract
We study the critical behavior of a general contagion model where nodes are either active (e.g., with opinion A, or functioning) or inactive (e.g., with opinion B, or damaged). The transitions between these two states are determined by (i) spontaneous transitions independent of the neighborhood, (ii) transitions induced by neighboring nodes, and (iii) spontaneous reverse transitions. The resulting dynamics is extremely rich including limit cycles and random phase switching. We derive a unifying mean-field theory. Specifically, we analytically show that the critical behavior of systems whose dynamics is governed by processes (i)-(iii) can only exhibit three distinct regimes: (a) uncorrelated spontaneous transition dynamics, (b) contact process dynamics, and (c) cusp catastrophes. This ends a long-standing debate on the universality classes of complex contagion dynamics in mean field and substantially deepens its mathematical understanding.
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Affiliation(s)
- L Böttcher
- ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - J Nagler
- ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - H J Herrmann
- ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
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12
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Böttcher L, Luković M, Nagler J, Havlin S, Herrmann HJ. Failure and recovery in dynamical networks. Sci Rep 2017; 7:41729. [PMID: 28155876 PMCID: PMC5290536 DOI: 10.1038/srep41729] [Citation(s) in RCA: 42] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2016] [Accepted: 12/28/2016] [Indexed: 12/17/2022] Open
Abstract
Failure, damage spread and recovery crucially underlie many spatially embedded networked systems ranging from transportation structures to the human body. Here we study the interplay between spontaneous damage, induced failure and recovery in both embedded and non-embedded networks. In our model the network's components follow three realistic processes that capture these features: (i) spontaneous failure of a component independent of the neighborhood (internal failure), (ii) failure induced by failed neighboring nodes (external failure) and (iii) spontaneous recovery of a component. We identify a metastable domain in the global network phase diagram spanned by the model's control parameters where dramatic hysteresis effects and random switching between two coexisting states are observed. This dynamics depends on the characteristic link length of the embedded system. For the Euclidean lattice in particular, hysteresis and switching only occur in an extremely narrow region of the parameter space compared to random networks. We develop a unifying theory which links the dynamics of our model to contact processes. Our unifying framework may help to better understand controllability in spatially embedded and random networks where spontaneous recovery of components can mitigate spontaneous failure and damage spread in dynamical networks.
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Affiliation(s)
- L. Böttcher
- ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093
Zurich, Switzerland
| | - M. Luković
- ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093
Zurich, Switzerland
| | - J. Nagler
- ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093
Zurich, Switzerland
| | - S. Havlin
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts
02215, USA
- Department of Physics, Bar-Ilan University, 52900
Ramat-Gan, Israel
| | - H. J. Herrmann
- ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093
Zurich, Switzerland
- Departamento de Física, Universidade Federal do Ceará, 60451-970
Fortaleza, Ceará, Brazil
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13
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Cho YS, Mazza MG, Kahng B, Nagler J. Genuine non-self-averaging and ultraslow convergence in gelation. Phys Rev E 2016; 94:022602. [PMID: 27627355 DOI: 10.1103/physreve.94.022602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2015] [Indexed: 06/06/2023]
Abstract
In irreversible aggregation processes droplets or polymers of microscopic size successively coalesce until a large cluster of macroscopic scale forms. This gelation transition is widely believed to be self-averaging, meaning that the order parameter (the relative size of the largest connected cluster) attains well-defined values upon ensemble averaging with no sample-to-sample fluctuations in the thermodynamic limit. Here, we report on anomalous gelation transition types. Depending on the growth rate of the largest clusters, the gelation transition can show very diverse patterns as a function of the control parameter, which includes multiple stochastic discontinuous transitions, genuine non-self-averaging and ultraslow convergence of the transition point. Our framework may be helpful in understanding and controlling gelation.
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Affiliation(s)
- Y S Cho
- Center for Complex Systems Studies and CTP, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
| | - M G Mazza
- Max Planck Institute for Dynamics and Self-Organization (MPI DS) Göttingen, Am Faßberg 17, 37077 Germany
| | - B Kahng
- Center for Complex Systems Studies and CTP, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
| | - J Nagler
- Center for Complex Systems Studies and CTP, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
- Max Planck Institute for Dynamics and Self-Organization (MPI DS) Göttingen, Am Faßberg 17, 37077 Germany
- Computational Physics, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
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14
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Saberi AA, Ebrahimnazhad Rahbari SH, Dashti-Naserabadi H, Abbasi A, Cho YS, Nagler J. Universality in boundary domain growth by sudden bridging. Sci Rep 2016; 6:21110. [PMID: 26899304 PMCID: PMC4761969 DOI: 10.1038/srep21110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2015] [Accepted: 01/18/2016] [Indexed: 11/09/2022] Open
Abstract
We report on universality in boundary domain growth in cluster aggregation in the limit of maximum concentration. Maximal concentration means that the diffusivity of the clusters is effectively zero and, instead, clusters merge successively in a percolation process, which leads to a sudden growth of the boundary domains. For two-dimensional square lattices of linear dimension L, independent of the models studied here, we find that the maximum of the boundary interface width, the susceptibility χ, exhibits the scaling χ ~ Lγ with the universal exponent γ = 1. The rapid growth of the boundary domain at the percolation threshold, which is guaranteed to occur for almost any cluster percolation process, underlies the the universal scaling of χ.
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Affiliation(s)
- A A Saberi
- Department of Physics, University of Tehran, P.O. Box 14395-547,Tehran, Iran.,School of Physics and Accelerators, Institute for research in Fundamental Science (IPM) P.O. 19395-5531, Tehran, Iran
| | - S H Ebrahimnazhad Rahbari
- School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea.,Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran
| | - H Dashti-Naserabadi
- Department of Physics, Plasma and Condensed Matter Computational Laboratory, Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran
| | - A Abbasi
- Physics and Accelerators Research School, NSRTI 11365-3486, Tehran, Iran
| | - Y S Cho
- Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
| | - J Nagler
- Computational Physics, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
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15
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Elçi EM, Weigel M, Fytas NG. Fragmentation of fractal random structures. PHYSICAL REVIEW LETTERS 2015; 114:115701. [PMID: 25839290 DOI: 10.1103/physrevlett.114.115701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2014] [Indexed: 06/04/2023]
Abstract
We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model allows us to discuss a wide range of systems with fractal properties including trees as well as dense clusters. We present exact results for the densities of fragmenting edges and the distribution of fragment sizes for critical clusters in two dimensions. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes. The resulting power laws are shown to encode characteristic fingerprints of the fragmented objects.
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Affiliation(s)
- Eren Metin Elçi
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, England
| | - Martin Weigel
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, England
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 7, D-55099 Mainz, Germany
| | - Nikolaos G Fytas
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, England
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16
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Sheinman M, Sharma A, Alvarado J, Koenderink GH, MacKintosh FC. Anomalous discontinuity at the percolation critical point of active gels. PHYSICAL REVIEW LETTERS 2015; 114:098104. [PMID: 25793855 DOI: 10.1103/physrevlett.114.098104] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2015] [Indexed: 06/04/2023]
Abstract
We develop a percolation model motivated by recent experimental studies of gels with active network remodeling by molecular motors. This remodeling was found to lead to a critical state reminiscent of random percolation (RP), but with a cluster distribution inconsistent with RP. Our model not only can account for these experiments, but also exhibits an unusual type of mixed phase transition: We find that the transition is characterized by signatures of criticality, but with a discontinuity in the order parameter.
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Affiliation(s)
- M Sheinman
- Department of Physics and Astronomy, VU University, 1081 HV Amsterdam, Netherlands
- Max Planck Institute for Molecular Genetics, 14195 Berlin, Germany
| | - A Sharma
- Department of Physics and Astronomy, VU University, 1081 HV Amsterdam, Netherlands
| | - J Alvarado
- FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
- Department of Mechanical Engineering, Hatsopoulos Microfluids Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - G H Koenderink
- FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - F C MacKintosh
- Department of Physics and Astronomy, VU University, 1081 HV Amsterdam, Netherlands
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17
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Bastas N, Kosmidis K, Giazitzidis P, Maragakis M. Method for estimating critical exponents in percolation processes with low sampling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062101. [PMID: 25615038 DOI: 10.1103/physreve.90.062101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2014] [Indexed: 06/04/2023]
Abstract
In phase-transition phenomena, the estimation of the critical point is crucial for the calculation of the various critical exponents and the determination of the universality class they belong to. However, this is not an easy task, since a large amount of realizations is needed to eliminate the noise in the data. In this paper, we introduce a novel method for the simultaneous estimation of the critical point p(c) and the critical exponent β/ν, applied for the case of "explosive" bond percolation on two-dimensional square lattices and Erdös-Rényi networks. The results show that with only a few hundred realizations, it is possible to acquire accurate values for these quantities. Guidelines are given at the end for the applicability of the method to other cases as well.
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Affiliation(s)
- N Bastas
- Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece
| | - K Kosmidis
- Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece and School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany
| | - P Giazitzidis
- Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece
| | - M Maragakis
- Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece and Department of Economics, University of Macedonia, 54006 Thessaloniki, Greece
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18
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Krejčí P, Lamba H, Melnik S, Rachinskii D. Analytical solution for a class of network dynamics with mechanical and financial applications. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032822. [PMID: 25314496 DOI: 10.1103/physreve.90.032822] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2013] [Indexed: 06/04/2023]
Abstract
We show that for a certain class of dynamics at the nodes the response of a network of any topology to arbitrary inputs is defined in a simple way by its response to a monotone input. The nodes may have either a discrete or continuous set of states and there is no limit on the complexity of the network. The results provide both an efficient numerical method and the potential for accurate analytic approximation of the dynamics on such networks. As illustrative applications, we introduce a quasistatic mechanical model with objects interacting via frictional forces and a financial market model with avalanches and critical behavior that are generated by momentum trading strategies.
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Affiliation(s)
- P Krejčí
- Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
| | - H Lamba
- Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030, USA
| | - S Melnik
- MACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland
| | - D Rachinskii
- Department of Applied Mathematics, University College Cork, Western Road, Cork, Ireland and Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
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19
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Stollmeier F, Geisel T, Nagler J. Possible origin of stagnation and variability of earth's biodiversity. PHYSICAL REVIEW LETTERS 2014; 112:228101. [PMID: 24949790 DOI: 10.1103/physrevlett.112.228101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/06/2014] [Indexed: 06/03/2023]
Abstract
The magnitude and variability of Earth's biodiversity have puzzled scientists ever since paleontologic fossil databases became available. We identify and study a model of interdependent species where both endogenous and exogenous impacts determine the nonstationary extinction dynamics. The framework provides an explanation for the qualitative difference of marine and continental biodiversity growth. In particular, the stagnation of marine biodiversity may result from a global transition from an imbalanced to a balanced state of the species dependency network. The predictions of our framework are in agreement with paleontologic databases.
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Affiliation(s)
- Frank Stollmeier
- Network Dynamics Group, Max Planck Institute for Dynamics and Self-Organization (MPI DS) Göttingen, Am Fassberg 17, 37077 Göttingen, Germany and Max Planck Institute for Dynamics and Self-Organization (MPI DS) Göttingen, and Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Am Fassberg 17, 37077 Göttingen, Germany
| | - Theo Geisel
- Max Planck Institute for Dynamics and Self-Organization (MPI DS) Göttingen, and Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Am Fassberg 17, 37077 Göttingen, Germany
| | - Jan Nagler
- Max Planck Institute for Dynamics and Self-Organization (MPI DS) Göttingen, and Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Am Fassberg 17, 37077 Göttingen, Germany and Computational Physics, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
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20
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Chen W, Schröder M, D'Souza RM, Sornette D, Nagler J. Microtransition cascades to percolation. PHYSICAL REVIEW LETTERS 2014; 112:155701. [PMID: 24785054 DOI: 10.1103/physrevlett.112.155701] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2013] [Indexed: 06/03/2023]
Abstract
We report the discovery of a discrete hierarchy of microtransitions occurring in models of continuous and discontinuous percolation. The precursory microtransitions allow us to target almost deterministically the location of the transition point to global connectivity. This extends to the class of intrinsically stochastic processes the possibility to use warning signals anticipating phase transitions in complex systems.
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Affiliation(s)
- Wei Chen
- Institute of Computing Technology, Chinese Academy of Sciences, 6 Kexueyuan South Road, Haidian, Beijing 100190, China
| | - Malte Schröder
- Max Planck Institute for Dynamics and Self-Organization (MPI DS), Am Fassberg 17, 37077 Göttingen, Germany and Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Am Fassberg 17, 37077 Göttingen, Germany
| | - Raissa M D'Souza
- University of California, Davis, California 95616, USA and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
| | - Didier Sornette
- Department of Management, Technology and Economics (D-MTEC), ETH Zurich, Scheuchzerstrasse 7, CH-8092 Zurich, Switzerland
| | - Jan Nagler
- Max Planck Institute for Dynamics and Self-Organization (MPI DS), Am Fassberg 17, 37077 Göttingen, Germany and Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Am Fassberg 17, 37077 Göttingen, Germany and Department of Management, Technology and Economics (D-MTEC), ETH Zurich, Scheuchzerstrasse 7, CH-8092 Zurich, Switzerland
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21
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Chen W, Cheng X, Zheng Z, Chung NN, D'Souza RM, Nagler J. Unstable supercritical discontinuous percolation transitions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042152. [PMID: 24229160 DOI: 10.1103/physreve.88.042152] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2013] [Indexed: 06/02/2023]
Abstract
The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can have additional exotic behaviors, such as displaying a "Devil's staircase" of discrete jumps in the supercritical regime. Here we investigate whether the location of the largest jump coincides with the percolation threshold for a range of processes, such as Erdős-Rényipercolation, percolation via edge competition and via growth by overtaking. We find that the largest jump asymptotically occurs at the percolation transition for Erdős-Rényiand other processes exhibiting global continuity, including models exhibiting an "explosive" transition. However, for percolation processes exhibiting genuine discontinuities, the behavior is substantially richer. In percolation models where the order parameter exhibits a staircase, the largest discontinuity generically does not coincide with the percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on the model parameter. Distinct parameter regimes well in the supercritical regime feature unstable discontinuous transitions-a novel and unexpected phenomenon in percolation. We thus demonstrate that seemingly and genuinely discontinuous percolation transitions can involve a rich behavior in supercriticality, a regime that has been largely ignored in percolation.
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Affiliation(s)
- Wei Chen
- Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China and School of Mathematical Sciences, Peking University, Beijing, China and University of California, Davis, California 95616, USA
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