1
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Xue L, Gao S, Gallos LK, Levy O, Gross B, Di Z, Havlin S. Nucleation phenomena and extreme vulnerability of spatial k-core systems. Nat Commun 2024; 15:5850. [PMID: 38992015 PMCID: PMC11239893 DOI: 10.1038/s41467-024-50273-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2023] [Accepted: 07/05/2024] [Indexed: 07/13/2024] Open
Abstract
K-core percolation is a fundamental dynamical process in complex networks with applications that span numerous real-world systems. Earlier studies focus primarily on random networks without spatial constraints and reveal intriguing mixed-order transitions. However, real-world systems, ranging from transportation and communication networks to complex brain networks, are not random but are spatially embedded. Here, we study k-core percolation on two-dimensional spatially embedded networks and show that, in contrast to regular percolation, the length of connections can control the transition type, leading to four different types of phase transitions associated with interesting phenomena and a rich phase diagram. A key finding is the existence of a metastable phase where microscopic localized damage, independent of system size, can cause a macroscopic phase transition, a result which cannot be achieved in traditional percolation. In this case, local failures spontaneously propagate the damage radially until the system collapses, a phenomenon analogous to the nucleation process.
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Affiliation(s)
- Leyang Xue
- International Academic Center of Complex Systems, Beijing Normal University, Zhuhai, 519087, China
- School of Systems Science, Beijing Normal University, Beijing, 100875, China
- Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel
| | - Shengling Gao
- Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel
- School of Mathematical Sciences, Beihang University, 100191, Beijing, China
| | | | - Orr Levy
- Department of Immunobiology, Yale University School of Medicine, New Haven, CT, USA
- Howard Hughes Medical Institute, Chevy Chase, MD, USA
| | - Bnaya Gross
- Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel
| | - Zengru Di
- International Academic Center of Complex Systems, Beijing Normal University, Zhuhai, 519087, China.
- School of Systems Science, Beijing Normal University, Beijing, 100875, China.
| | - Shlomo Havlin
- Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel.
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2
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Aguilar-Janita M, Khalil N, Leyva I, Sendiña-Nadal I. Cooperation transitions in social games induced by aspiration-driven players. Phys Rev E 2024; 109:024107. [PMID: 38491644 DOI: 10.1103/physreve.109.024107] [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: 08/03/2023] [Accepted: 01/16/2024] [Indexed: 03/18/2024]
Abstract
Cooperation and defection are social traits whose evolutionary origin is still unresolved. Recent behavioral experiments with humans suggested that strategy changes are driven mainly by the individuals' expectations and not by imitation. This work theoretically analyzes and numerically explores an aspiration-driven strategy updating in a well-mixed population playing games. The payoffs of the game matrix and the aspiration are condensed into just two parameters that allow a comprehensive description of the dynamics. We find continuous and abrupt transitions in the cooperation density with excellent agreement between theory and the Gillespie simulations. Under strong selection, the system can display several levels of steady cooperation or get trapped into absorbing states. These states are still relevant for experiments even when irrational choices are made due to their prolonged relaxation times. Finally, we show that for the particular case of the prisoner dilemma, where defection is the dominant strategy under imitation mechanisms, the self-evaluation update instead favors cooperation nonlinearly with the level of aspiration. Thus, our work provides insights into the distinct role between imitation and self-evaluation with no learning dynamics.
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Affiliation(s)
- M Aguilar-Janita
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
| | - N Khalil
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
| | - I Leyva
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, Pozuelo de Alarcón, 28223 Madrid, Spain
| | - I Sendiña-Nadal
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, Pozuelo de Alarcón, 28223 Madrid, Spain
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3
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Jhun B, Choi H, Lee Y, Lee J, Kim CH, Kahng B. Prediction and mitigation of nonlocal cascading failures using graph neural networks. CHAOS (WOODBURY, N.Y.) 2023; 33:013115. [PMID: 36725647 DOI: 10.1063/5.0107420] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2022] [Accepted: 12/13/2022] [Indexed: 06/18/2023]
Abstract
Cascading failures in electrical power grids, comprising nodes and links, propagate nonlocally. After a local disturbance, successive resultant can be distant from the source. Since avalanche failures can propagate unexpectedly, care must be taken when formulating a mitigation strategy. Herein, we propose a strategy for mitigating such cascading failures. First, to characterize the impact of each node on the avalanche dynamics, we propose a novel measure, that of Avalanche Centrality (AC). Then, based on the ACs, nodes potentially needing reinforcement are identified and selected for mitigation. Compared with heuristic measures, AC has proven to be efficient at reducing avalanche size; however, due to nonlocal propagation, calculating ACs can be computationally burdensome. To resolve this problem, we use a graph neural network (GNN). We begin by training a GNN using a large number of small networks; then, once trained, the GNN can predict ACs efficiently in large networks and real-world topological power grids in manageable computational time. Thus, under our strategy, mitigation in large networks is achieved by reinforcing nodes with large ACs. The framework developed in this study can be implemented in other complex processes that require longer computational time to simulate large networks.
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Affiliation(s)
- Bukyoung Jhun
- CCSS and CTP, Seoul National University, Seoul 08826, South Korea
| | - Hoyun Choi
- CCSS and CTP, Seoul National University, Seoul 08826, South Korea
| | - Yongsun Lee
- CCSS and CTP, Seoul National University, Seoul 08826, South Korea
| | - Jongshin Lee
- CCSS and CTP, Seoul National University, Seoul 08826, South Korea
| | - Cook Hyun Kim
- CCSS and CTP, Seoul National University, Seoul 08826, South Korea
| | - B Kahng
- Center for Complex Systems and KI for Grid Modernization, Korea Institute of Energy Technology, Naju, Jeonnam 58217, South Korea
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4
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Gross B, Bonamassa I, Havlin S. Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks. PHYSICAL REVIEW LETTERS 2022; 129:268301. [PMID: 36608183 DOI: 10.1103/physrevlett.129.268301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2022] [Revised: 10/24/2022] [Accepted: 11/29/2022] [Indexed: 06/17/2023]
Abstract
We study the critical features of the order parameter's fluctuations near the threshold of mixed-order phase transitions in randomly interdependent spatial networks. Remarkably, we find that although the structure of the order parameter is not scale invariant, its fluctuations are fractal up to a well-defined correlation length ξ^{'} that diverges when approaching the mixed-order transition threshold. We characterize the self-similar nature of these critical fluctuations through their effective fractal dimension d_{f}^{'}=3d/4, and correlation length exponent ν^{'}=2/d, where d is the dimension of the system. By analyzing percolation and magnetization, we demonstrate that d_{f}^{'} and ν^{'} are the same for both, i.e., independent of the symmetry of the process for any d of the underlying networks.
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Affiliation(s)
- Bnaya Gross
- Department of Physics, Bar Ilan University, Ramat Gan, Israel
| | - Ivan Bonamassa
- Department of Physics, Bar Ilan University, Ramat Gan, Israel
- Department of Network and Data Science, CEU, Quellenstrasse 51, A-1100 Vienna, Austria
| | - Shlomo Havlin
- Department of Physics, Bar Ilan University, Ramat Gan, Israel
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5
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Jhun B, Choi H. Abrupt transition of the efficient vaccination strategy in a population with heterogeneous fatality rates. CHAOS (WOODBURY, N.Y.) 2022; 32:093140. [PMID: 36182386 DOI: 10.1063/5.0087627] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2022] [Accepted: 09/01/2022] [Indexed: 06/16/2023]
Abstract
An insufficient supply of an effective SARS-CoV-2 vaccine in most countries demands an effective vaccination strategy to minimize the damage caused by the disease. Currently, many countries vaccinate their population in descending order of age (i.e., descending order of fatality rate) to minimize the deaths caused by the disease; however, the effectiveness of this strategy needs to be quantitatively assessed. We employ the susceptible-infected-recovered-dead model to investigate various vaccination strategies. We constructed a metapopulation model with heterogeneous contact and fatality rates and investigated the effectiveness of vaccination strategies to reduce epidemic mortality. We found that the fatality-based strategy, which is currently employed in many countries, is more effective when the contagion rate is high and vaccine supply is low, but the contact-based method outperforms the fatality-based strategy when there is a sufficiently high supply of the vaccine. We identified a discontinuous transition of the optimal vaccination strategy and path-dependency analogous to hysteresis. This transition and path-dependency imply that combining the fatality-based and contact-based strategies is ineffective in reducing the number of deaths. Furthermore, we demonstrate that such phenomena occur in real-world epidemic diseases, such as tuberculosis and COVID-19. We also show that the conclusions of this research are valid even when the complex epidemic stages, efficacy of the vaccine, and reinfection are considered.
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Affiliation(s)
- Bukyoung Jhun
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
| | - Hoyun Choi
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
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6
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Diaz-Diaz F, San Miguel M, Meloni S. Echo chambers and information transmission biases in homophilic and heterophilic networks. Sci Rep 2022; 12:9350. [PMID: 35672432 PMCID: PMC9174247 DOI: 10.1038/s41598-022-13343-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2022] [Accepted: 05/23/2022] [Indexed: 12/04/2022] Open
Abstract
We study how information transmission biases arise by the interplay between the structural properties of the network and the dynamics of the information in synthetic scale-free homophilic/heterophilic networks. We provide simple mathematical tools to quantify these biases. Both Simple and Complex Contagion models are insufficient to predict significant biases. In contrast, a Hybrid Contagion model—in which both Simple and Complex Contagion occur—gives rise to three different homophily-dependent biases: emissivity and receptivity biases, and echo chambers. Simulations in an empirical network with high homophily confirm our findings. Our results shed light on the mechanisms that cause inequalities in the visibility of information sources, reduced access to information, and lack of communication among distinct groups.
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Affiliation(s)
- Fernando Diaz-Diaz
- IFISC (UIB-CSIC), Institute for Cross-Disciplinary Physics and Complex Systems, Campus Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain
| | - Maxi San Miguel
- IFISC (UIB-CSIC), Institute for Cross-Disciplinary Physics and Complex Systems, Campus Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain
| | - Sandro Meloni
- IFISC (UIB-CSIC), Institute for Cross-Disciplinary Physics and Complex Systems, Campus Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain.
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7
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Kim JH, Goh KI. K-selective percolation: A simple model leading to a rich repertoire of phase transitions. CHAOS (WOODBURY, N.Y.) 2022; 32:023115. [PMID: 35232055 DOI: 10.1063/5.0081253] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2021] [Accepted: 01/19/2022] [Indexed: 06/14/2023]
Abstract
We propose a K-selective percolation process as a model for iterative removals of nodes with a specific intermediate degree in complex networks. In the model, a random node with degree K is deactivated one by one until no more nodes with degree K remain. The non-monotonic response of the giant component size on various synthetic and real-world networks implies a conclusion that a network can be more robust against such a selective attack by removing further edges. From a theoretical perspective, the K-selective percolation process exhibits a rich repertoire of phase transitions, including double transitions of hybrid and continuous, as well as reentrant transitions. Notably, we observe a tricritical-like point on Erdős-Rényi networks. We also examine a discontinuous transition with unusual order parameter fluctuation and distribution on simple cubic lattices, which does not appear in other percolation models with cascade processes. Finally, we perform finite-size scaling analysis to obtain critical exponents on various transition points, including those exotic ones.
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Affiliation(s)
- Jung-Ho Kim
- Department of Physics, Korea University, Seoul 02841, South Korea
| | - K-I Goh
- Department of Physics, Korea University, Seoul 02841, South Korea
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8
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Kureh YH, Porter MA. Fitting in and breaking up: A nonlinear version of coevolving voter models. Phys Rev E 2020; 101:062303. [PMID: 32688568 DOI: 10.1103/physreve.101.062303] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2019] [Accepted: 02/07/2020] [Indexed: 11/07/2022]
Abstract
We investigate a nonlinear version of coevolving voter models, in which node states and network structure update as a coupled stochastic process. Most prior work on coevolving voter models has focused on linear update rules with fixed and homogeneous rewiring and adopting probabilities. By contrast, in our nonlinear version, the probability that a node rewires or adopts is a function of how well it "fits in" with the nodes in its neighborhood. To explore this idea, we incorporate a local-survey parameter σ_{i} that encodes the fraction of neighbors of an updating node i that share its opinion state. In an update, with probability σ_{i}^{q} (for some nonlinearity parameter q), the updating node rewires; with complementary probability 1-σ_{i}^{q}, the updating node adopts a new opinion state. We study this mechanism using three rewiring schemes: after an updating node deletes one of its discordant edges, it then either (1) "rewires-to-random" by choosing a new neighbor in a random process; (2) "rewires-to-same" by choosing a new neighbor in a random process from nodes that share its state; or (3) "rewires-to-none" by not rewiring at all (akin to "unfriending" on social media). We compare our nonlinear coevolving voter model to several existing linear coevolving voter models on various network architectures. Relative to those models, we find in our model that initial network topology plays a larger role in the dynamics and that the choice of rewiring mechanism plays a smaller role. A particularly interesting feature of our model is that, under certain conditions, the opinion state that is held initially by a minority of the nodes can effectively spread to almost every node in a network if the minority nodes view themselves as the majority. In light of this observation, we relate our results to recent work on the majority illusion in social networks.
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Affiliation(s)
- Yacoub H Kureh
- Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095, USA
| | - Mason A Porter
- Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095, USA
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9
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Gross B, Sanhedrai H, Shekhtman L, Havlin S. Interconnections between networks acting like an external field in a first-order percolation transition. Phys Rev E 2020; 101:022316. [PMID: 32168699 DOI: 10.1103/physreve.101.022316] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2019] [Accepted: 02/05/2020] [Indexed: 11/07/2022]
Abstract
Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we show how the effects of such interconnections can be described as an external field for interdependent networks experiencing a first-order percolation transition. We find that the critical exponents γ and δ, related to the external field, can also be defined for first-order transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents (for first and second order) can even be found within a single model of interdependent networks, depending on the dependency coupling strength. Nevertheless, in both cases both sets satisfy Widom's identity, δ-1=γ/β, which further supports the validity of their definitions. Furthermore, we find that both Erdős-Rényi and scale-free networks have the same values of the exponents in the first-order regime, implying that these models are in the same universality class. In addition, we find that in k-core percolation the values of the critical exponents related to the field are the same as for interdependent networks, suggesting that these systems also belong to the same universality class.
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Affiliation(s)
- Bnaya Gross
- Department of Physics, Bar Ilan University, Ramat Gan, Israel
| | | | - Louis Shekhtman
- Network Science Institute, Northeastern University, Boston 02115, USA
| | - Shlomo Havlin
- Department of Physics, Bar Ilan University, Ramat Gan, Israel.,Institute of Innovative Research, Tokyo Institute of Technology, Midori-ku, Yokohama 226-8503, Japan
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10
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Kryven I, Bianconi G. Enhancing the robustness of a multiplex network leads to multiple discontinuous percolation transitions. Phys Rev E 2019; 100:020301. [PMID: 31574739 DOI: 10.1103/physreve.100.020301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2019] [Indexed: 06/10/2023]
Abstract
Determining design principles that boost the robustness of interdependent networks is a fundamental question of engineering, economics, and biology. It is known that maximizing the degree correlation between replicas of the same node leads to optimal robustness. Here we show that increased robustness might also come at the expense of introducing multiple phase transitions. These results reveal yet another possible source of fragility of multiplex networks that has to be taken into the account during network optimization and design.
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Affiliation(s)
- Ivan Kryven
- Mathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
| | - Ginestra Bianconi
- School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, United Kingdom The Alan Turing Institute, the British Library, London NW1 2DB, United Kingdom
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11
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Choi K, Lee D, Cho YS, Thiele JC, Herrmann HJ, Kahng B. Critical phenomena of a hybrid phase transition in cluster merging dynamics. Phys Rev E 2017; 96:042148. [PMID: 29347575 DOI: 10.1103/physreve.96.042148] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/19/2017] [Indexed: 11/07/2022]
Abstract
Recently, a hybrid percolation transition (HPT) that exhibits both a discontinuous transition and critical behavior at the same transition point has been observed in diverse complex systems. While the HPT induced by avalanche dynamics has been studied extensively, the HPT induced by cluster merging dynamics (HPT-CMD) has received little attention. Here, we aim to develop a theoretical framework for the HPT-CMD. We find that two correlation-length exponents are necessary for characterizing the giant cluster and finite clusters separately. The conventional formula of the fractal dimension in terms of the critical exponents is not valid. Neither the giant nor finite clusters are fractals, but they have fractal boundaries. A finite-size scaling method for the HPT-CMD is also introduced.
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Affiliation(s)
- K Choi
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - Deokjae Lee
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - Y S Cho
- Department of Physics, Chonbuk National University, Jeonju 54896, Korea
| | - J C Thiele
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zürich, 8093 Zürich, Switzerland
| | - H J Herrmann
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zürich, 8093 Zürich, Switzerland
| | - B Kahng
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
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12
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Abstract
Hybrid percolation transitions (HPTs) induced by cascading processes have been observed in diverse complex systems such as k-core percolation, breakdown on interdependent networks and cooperative epidemic spreading models. Here we present the microscopic universal mechanism underlying those HPTs. We show that the discontinuity in the order parameter results from two steps: a durable critical branching (CB) and an explosive, supercritical (SC) process, the latter resulting from large loops inevitably present in finite size samples. In a random network of N nodes at the transition the CB process persists for O(N 1/3) time and the remaining nodes become vulnerable, which are then activated in the short SC process. This crossover mechanism and scaling behavior are universal for different HPT systems. Our result implies that the crossover time O(N 1/3) is a golden time, during which one needs to take actions to control and prevent the formation of a macroscopic cascade, e.g., a pandemic outbreak.
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13
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Choi W, Lee D, Kahng B. Critical behavior of a two-step contagion model with multiple seeds. Phys Rev E 2017; 95:062115. [PMID: 28709296 PMCID: PMC7217524 DOI: 10.1103/physreve.95.062115] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2017] [Indexed: 11/07/2022]
Abstract
A two-step contagion model with a single seed serves as a cornerstone for understanding the critical behaviors and underlying mechanism of discontinuous percolation transitions induced by cascade dynamics. When the contagion spreads from a single seed, a cluster of infected and recovered nodes grows without any cluster merging process. However, when the contagion starts from multiple seeds of O(N) where N is the system size, a node weakened by a seed can be infected more easily when it is in contact with another node infected by a different pathogen seed. This contagion process can be viewed as a cluster merging process in a percolation model. Here we show analytically and numerically that when the density of infectious seeds is relatively small but O(1), the epidemic transition is hybrid, exhibiting both continuous and discontinuous behavior, whereas when it is sufficiently large and reaches a critical point, the transition becomes continuous. We determine the full set of critical exponents describing the hybrid and the continuous transitions. Their critical behaviors differ from those in the single-seed case.
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Affiliation(s)
- Wonjun Choi
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea
| | - Deokjae Lee
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea
| | - B Kahng
- CCSS, CTP, and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea
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14
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Kong LW, Li M, Liu RR, Wang BH. Percolation on networks with weak and heterogeneous dependency. Phys Rev E 2017; 95:032301. [PMID: 28415263 DOI: 10.1103/physreve.95.032301] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2016] [Indexed: 11/07/2022]
Abstract
In real networks, the dependency between nodes is ubiquitous; however, the dependency is not always complete and homogeneous. In this paper, we propose a percolation model with weak and heterogeneous dependency; i.e., dependency strengths could be different between different nodes. We find that the heterogeneous dependency strength will make the system more robust, and for various distributions of dependency strengths both continuous and discontinuous percolation transitions can be found. For Erdős-Rényi networks, we prove that the crossing point of the continuous and discontinuous percolation transitions is dependent on the first five moments of the dependency strength distribution. This indicates that the discontinuous percolation transition on networks with dependency is determined not only by the dependency strength but also by its distribution. Furthermore, in the area of the continuous percolation transition, we also find that the critical point depends on the first and second moments of the dependency strength distribution. To validate the theoretical analysis, cases with two different dependency strengths and Gaussian distribution of dependency strengths are presented as examples.
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Affiliation(s)
- Ling-Wei Kong
- School of the Gifted Young, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Ming Li
- School of Engineering Science, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Run-Ran Liu
- Alibaba Research Center for Complexity Sciences, Hangzhou Normal University, Hangzhou 311121, People's Republic of China
| | - Bing-Hong Wang
- Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
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15
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Choi W, Lee D, Kahng B. Mixed-order phase transition in a two-step contagion model with a single infectious seed. Phys Rev E 2017; 95:022304. [PMID: 28297964 DOI: 10.1103/physreve.95.022304] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2016] [Indexed: 01/21/2023]
Abstract
Percolation is known as one of the most robust continuous transitions, because its occupation rule is intrinsically local. As one of the ways to break the robustness, occupation is allowed to more than one species of particles and they occupy cooperatively. This generalized percolation model undergoes a discontinuous transition. Here we investigate an epidemic model with two contagion steps and characterize its phase transition analytically and numerically. We find that even though the order parameter jumps at a transition point r_{c}, then increases continuously, it does not exhibit any critical behavior: the fluctuations of the order parameter do not diverge at r_{c}. However, critical behavior appears in mean outbreak size, which diverges at the transition point in a manner that the ordinary percolation shows. Such a type of phase transition is regarded as a mixed-order phase transition. We also obtain scaling relations of cascade outbreak statistics when the order parameter jumps at r_{c}.
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Affiliation(s)
- Wonjun Choi
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - Deokjae Lee
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - B Kahng
- CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
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16
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Lee D, Jo M, Kahng B. Critical behavior of k-core percolation: Numerical studies. Phys Rev E 2016; 94:062307. [PMID: 28085402 DOI: 10.1103/physreve.94.062307] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2016] [Indexed: 06/06/2023]
Abstract
k-core percolation has served as a paradigmatic model of discontinuous percolation for a long time. Recently it was revealed that the order parameter of k-core percolation of random networks additionally exhibits critical behavior. Thus k-core percolation exhibits a hybrid phase transition. Unlike the critical behaviors of ordinary percolation that are well understood, those of hybrid percolation transitions have not been thoroughly understood yet. Here, we investigate the critical behavior of k-core percolation of Erdős-Rényi networks. We find numerically that the fluctuations of the order parameter and the mean avalanche size diverge in different ways. Thus, we classify the critical exponents into two types: those associated with the order parameter and those with finite avalanches. The conventional scaling relations hold within each set, however, these two critical exponents are coupled. Finally we discuss some universal features of the critical behaviors of k-core percolation and the cascade failure model on multiplex networks.
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Affiliation(s)
- Deokjae Lee
- Department of Physics and Astronomy, CCSS, CTP, Seoul National University, Seoul 08826, Korea
| | - Minjae Jo
- Department of Physics and Astronomy, CCSS, CTP, Seoul National University, Seoul 08826, Korea
| | - B Kahng
- Department of Physics and Astronomy, CCSS, CTP, Seoul National University, Seoul 08826, Korea
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