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Fazli D, Khanjanianpak M, Azimi-Tafreshi N. Control of cascading failures using protective measures. Sci Rep 2024; 14:14444. [PMID: 38910163 PMCID: PMC11194283 DOI: 10.1038/s41598-024-65379-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2023] [Accepted: 06/19/2024] [Indexed: 06/25/2024] Open
Abstract
Cascading failures, triggered by a local perturbation, can be catastrophic and cause irreparable damages in a wide area. Hence, blocking the devastating cascades is an important issue in real world networks. One of the ways to control the cascade is to use protective measures, so that the agents decide to be protected against failure. Here, we consider a coevolution of the linear threshold model for the spread of cascading failures and a decision-making game based on the perceived risk of failure. Protected agents are less vulnerable to failure and in return the size of the cascade affects the agent's decision to get insured. We find at what range of protection efficiency and cost of failure, the global cascades stop. Also we observe that in some range of protection efficiency, a bistable region emerges for the size of cascade and the prevalence of protected agents. Moreover, we show how savings or the ability of agents to repair can prevent cascades from occurring.
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Affiliation(s)
- Davood Fazli
- Physics Department, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45137-66736, Iran
| | - Mozhgan Khanjanianpak
- Pasargad Institute for Advanced Innovative Solutions (PIAIS), Tehran, 1991633357, Iran
| | - Nahid Azimi-Tafreshi
- Physics Department, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45137-66736, Iran.
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2
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Tong B, Roth DS, Buldyrev SV. Cascading failures in bipartite networks with directional support links. Phys Rev E 2024; 109:064307. [PMID: 39020941 DOI: 10.1103/physreve.109.064307] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2023] [Accepted: 04/30/2024] [Indexed: 07/20/2024]
Abstract
We study the cascading failures in a system of two interdependent networks whose internetwork supply links are directional. We will show that, by utilizing generating function formalism, the cascading process can be modeled by a set of recursive relations. Most importantly, the functions involved in these relations are solely dependent upon the choice of the degree distribution of ingoing links. Simulation results in the limit of very large networks based on different choices of degree distributions for outgoing links, e.g., Kronecker delta, Poisson and Pareto, are indeed identical and are in excellent agreement with the theory. However, for Pareto distribution with the shape parameter 1<α<2, the convergence is slow. In general, directional networks can be more vulnerable or less vulnerable than their bidirectional counterparts. For three special settings of interdependent networks, we analytically compare their vulnerability. For practical applications it is important to predict if a system responds to the size of the initial attack continuously or if there is catastrophic collapse of the system if the attack exceeds a specific transition size. We analytically show that systems with lower average degrees are more resilient against this abrupt transition. We also establish an equivalence of this transition with the liquid-gas transition in statistical mechanics. In the last section, we derive the set of recursive relation to describe the cascading process where the initial attack is not restricted to a single network.
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Jiang J, Yu X, Wang B, Ma L, Guan Y. DECAF: An interpretable deep cascading framework for ICU mortality prediction. Artif Intell Med 2022; 138:102437. [PMID: 36990582 DOI: 10.1016/j.artmed.2022.102437] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2021] [Revised: 07/28/2022] [Accepted: 10/28/2022] [Indexed: 11/09/2022]
Abstract
Medical risk detection is an important topic and a challenging task to improve the performance of clinical practices in Intensive Care Units (ICU). Although many bio-statistical learning and deep learning approaches have provided patient-specific mortality predictions, these existing methods lack interpretability that is crucial to gain adequate insight on why such predictions would work. In this paper, we introduce cascading theory to model the physiological domino effect and provide a novel approach to dynamically simulate the deterioration of patients' conditions. We propose a general DEep CAscading Framework (DECAF) to predict the potential risks of all physiological functions at each clinical stage. Compared with other feature-based and/or score-based models, our approach has a range of desirable properties, such as being interpretable, applicable with multi prediction tasks, and learnable from medical common sense and/or clinical experience knowledge. Experiments on a medical dataset (MIMIC-III) of 21,828 ICU patients show that DECAF reaches up to 89.30 % on AUROC, which surpasses the best competing methods for mortality prediction.
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Hidden transition in multiplex networks. Sci Rep 2022; 12:3973. [PMID: 35273259 PMCID: PMC8913666 DOI: 10.1038/s41598-022-07913-x] [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: 11/24/2021] [Accepted: 02/14/2022] [Indexed: 11/23/2022] Open
Abstract
Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, \documentclass[12pt]{minimal}
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\begin{document}$$P(0)\,=\,0$$\end{document}P(0)=0. Above a critical value of a control parameter, the removal of a tiny fraction \documentclass[12pt]{minimal}
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\begin{document}$$\Delta$$\end{document}Δ of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, \documentclass[12pt]{minimal}
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\begin{document}$$\Pi \,=\,P(1)$$\end{document}Π=P(1). We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point \documentclass[12pt]{minimal}
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\begin{document}$$\Pi _c$$\end{document}Πc, respectively. In the limit \documentclass[12pt]{minimal}
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\begin{document}$$\Pi \,>\,\Pi _\text {c}$$\end{document}Π>Πc takes a time \documentclass[12pt]{minimal}
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\begin{document}$$T \propto 1/(\Pi -\Pi _\text {c})$$\end{document}T∝1/(Π-Πc), while there is an exponential relaxation below \documentclass[12pt]{minimal}
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\begin{document}$$\Pi _\text {c}$$\end{document}Πc with a relaxation time \documentclass[12pt]{minimal}
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\begin{document}$$\tau \propto 1/[\Pi _\text {c}-\Pi ]$$\end{document}τ∝1/[Πc-Π].
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Three Decades in Econophysics—From Microscopic Modelling to Macroscopic Complexity and Back. ENTROPY 2022; 24:e24020271. [PMID: 35205566 PMCID: PMC8870777 DOI: 10.3390/e24020271] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/06/2022] [Revised: 02/10/2022] [Accepted: 02/11/2022] [Indexed: 01/27/2023]
Abstract
We explore recent contributions to research in Econophysics, switching between Macroscopic complexity and microscopic modelling, showing how each leads to the other and detailing the everyday applicability of both approaches and the tools they help develop. Over the past decades, the world underwent several major crises, leading to significant increase in interdependence and, thus, complexity. We show here that from the perspective of network science, these processes become more understandable and, to some extent, also controllable.
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Abstract
Research on the robustness of networks, and in particular the Internet, has gained critical importance in recent decades because more and more individuals, societies and firms rely on this global network infrastructure for communication, knowledge transfer, business processes and e-commerce. In particular, modeling the structure of the Internet has inspired several novel graph metrics for assessing important topological robustness features of large complex networks. This survey provides a comparative overview of these metrics, presents their strengths and limitations for analyzing the robustness of the Internet topology, and outlines a conceptual tool set in order to facilitate their future adoption by Internet research and practice but also other areas of network science.
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Kornbluth Y, Cwilich G, Buldyrev SV, Soltan S, Zussman G. Distribution of blackouts in the power grid and the Motter and Lai model. Phys Rev E 2021; 103:032309. [PMID: 33862809 DOI: 10.1103/physreve.103.032309] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2020] [Accepted: 02/09/2021] [Indexed: 11/07/2022]
Abstract
Carreras, Dobson, and colleagues have studied empirical data on the sizes of the blackouts in real grids and modeled them with computer simulations using the direct current approximation. They have found that the resulting blackout sizes are distributed as a power law and suggested that this is because the grids are driven to the self-organized critical state. In contrast, more recent studies found that the distribution of cascades is bimodal resulting in either a very small blackout or a very large blackout, engulfing a finite fraction of the system. Here we reconcile the two approaches and investigate how the distribution of the blackouts changes with model parameters, including the tolerance criteria and the dynamic rules of failure of the overloaded lines during the cascade. In addition, we study the same problem for the Motter and Lai model and find similar results, suggesting that the physical laws of flow on the network are not as important as network topology, overload conditions, and dynamic rules of failure.
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Affiliation(s)
- Yosef Kornbluth
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
| | - Gabriel Cwilich
- Department of Physics, Yeshiva University, New York, New York, USA
| | | | - Saleh Soltan
- Princeton University, Princeton, New Jersey, USA
| | - Gil Zussman
- Department of Electrical Engineering, Columbia University, New York, New York, USA
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Breen L, Gaule PB, Canonici A, Walsh N, Collins DM, Cremona M, Hennessy BT, Duffy MJ, Crown J, Donovan NO, Eustace AJ. Targeting c-Met in triple negative breast cancer: preclinical studies using the c-Met inhibitor, Cpd A. Invest New Drugs 2020; 38:1365-1372. [PMID: 32318883 DOI: 10.1007/s10637-020-00937-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2020] [Accepted: 04/07/2020] [Indexed: 11/26/2022]
Abstract
Introduction Triple negative breast cancer (TNBC) represents a heterogeneous subtype of breast cancer that carries a poorer prognosis. There remains a need to identify novel drivers of TNBC, which may represent targets to treat the disease. c-Met overexpression is linked with decreased survival and is associated with the basal subtype of breast cancer. Cpd A, a kinase inhibitor selective/specific for Met kinase has demonstrated preclinical anti-cancer efficacy in TNBC. We aimed to assess the anti-cancer efficacy of Cpd A when combined with Src kinase, ErbB-family or hepatocyte growth factor (HGF) inhibitors in TNBC cell lines. Methods We determined the anti-proliferative effects of Cpd A, rilotumumab, neratinib and saracatinib tested alone and in combination in a panel of TNBC cells by acid phosphatase assays. We performed reverse phase protein array analysis of c-Met and IGF1Rβ expression and phosphorylation of c-Met (Y1234/1235) in TNBC cells and correlated their expression/phosphorylation with Cpd A sensitivity. We examined the impact of Cpd A, neratinib and saracatinib tested alone and in combination on invasive potential and colony formation.Results TNBC cells are not inherently sensitive to Cpd A, and neither c-Met expression nor phosphorylation are biomarkers of sensitivity to Cpd A. Cpd A enhanced the anti-proliferative effects of neratinib in vitro; however, this effect was limited to cell lines with innate sensitivity to Cpd A. Cpd A had limited anti-invasive effects but it reduced colony formation in the TNBC cell line panel.Conclusions Despite Cpd A having a potential role in reducing cancer cell metastasis, identification of strong predictive biomarkers of c-Met sensitivity would be essential to the development of a c-Met targeted treatment for an appropriately selected cohort of TNBC patients.
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Affiliation(s)
- Laura Breen
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland
| | - Patricia B Gaule
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland
| | - Alexandra Canonici
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland
| | - Naomi Walsh
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland
| | - Denis M Collins
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland
| | - Mattia Cremona
- Medical Oncology Group, Department of Molecular Medicine, Beaumont Hospital, Royal College of Surgeons in Ireland, Dublin, Ireland
| | - Bryan T Hennessy
- Medical Oncology Group, Department of Molecular Medicine, Beaumont Hospital, Royal College of Surgeons in Ireland, Dublin, Ireland
| | - Michael J Duffy
- UCD Clinical Research Centre, St. Vincent's University Hospital, Dublin, Ireland
- UCD School of Medicine, Conway Institute of Biomolecular and Biomedical Research, University College Dublin, Dublin, Ireland
| | - John Crown
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland
- Department of Medical Oncology, St Vincent's University Hospital, Dublin, Ireland
| | - Norma O' Donovan
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland
| | - Alex J Eustace
- Molecular Therapeutics for Cancer in Ireland, National Institute for Cellular Biotechnology, Dublin City University, Dublin, Ireland.
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9
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Di Muro MA, Buldyrev SV, Braunstein LA. Reversible bootstrap percolation: Fake news and fact checking. Phys Rev E 2020; 101:042307. [PMID: 32422807 DOI: 10.1103/physreve.101.042307] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2019] [Accepted: 03/17/2020] [Indexed: 06/11/2023]
Abstract
Bootstrap percolation has been used to describe opinion formation in society and other social and natural phenomena. The formal equation of the bootstrap percolation may have more than one solution, corresponding to several stable fixed points of the corresponding iteration process. We construct a reversible bootstrap percolation process, which converges to these extra solutions displaying a hysteresis typical of discontinuous phase transitions. This process provides a reasonable model for fake news spreading and the effectiveness of fact checking. We show that sometimes it is not sufficient to discard all the sources of fake news in order to reverse the belief of a population that formed under the influence of these sources.
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Affiliation(s)
- Matías A Di Muro
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, (7600) Mar del Plata, Argentina
| | - Sergey V Buldyrev
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA and Politecnico di Milano, Department of Management, Economics and Industrial Engineering, Via Lambruschini 4, BLD 26, 20156 Milano, Italy
| | - Lidia A Braunstein
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, (7600) Mar del Plata, Argentina and Physics Department, Boston University, Boston, Massachusetts 02215, USA
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10
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Zhang H, Zhou J, Zou Y, Tang M, Xiao G, Stanley HE. Asymmetric interdependent networks with multiple-dependence relation. Phys Rev E 2020; 101:022314. [PMID: 32168681 DOI: 10.1103/physreve.101.022314] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2019] [Accepted: 01/31/2020] [Indexed: 11/07/2022]
Abstract
In this paper, we study the robustness of interdependent networks with multiple-dependence (MD) relation which is defined that a node is interdependent on several nodes on another layer, and this node will fail if any of these dependent nodes are failed. We propose a two-layered asymmetric interdependent network (AIN) model to address this problem, where the asymmetric feature is that nodes in one layer may be dependent on more than one node in the other layer with MD relation, while nodes in the other layer are dependent on exactly one node in this layer. We show that in this model the layer where nodes are allowed to have MD relation exhibits different types of phase transitions (discontinuous and hybrid), while the other layer only presents discontinuous phase transition. A heuristic theory based on message-passing approach is developed to understand the structural feature of interdependent networks and an intuitive picture for the emergence of a tricritical point is provided. Moreover, we study the correlation between the intralayer degree and interlayer degree of the nodes and find that this correlation has prominent impact to the continuous phase transition but has feeble effect on the discontinuous phase transition. Furthermore, we extend the two-layered AIN model to general multilayered AIN, and the percolation behaviors and properties of relevant phase transitions are elaborated.
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Affiliation(s)
- Hang Zhang
- School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
| | - Jie Zhou
- School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China.,Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts, 02215, USA
| | - Yong Zou
- School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
| | - Ming Tang
- School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
| | - Gaoxi Xiao
- School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
| | - H Eugene Stanley
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts, 02215, USA
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Di Muro MA, Valdez LD, Stanley HE, Buldyrev SV, Braunstein LA. Insights into bootstrap percolation: Its equivalence with k-core percolation and the giant component. Phys Rev E 2019; 99:022311. [PMID: 30934313 DOI: 10.1103/physreve.99.022311] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2018] [Indexed: 06/09/2023]
Abstract
K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been suggested in recent years that they could be complementary. In this manuscript we provide a rigorous analysis that shows that for any degree and threshold distributions heterogeneous bootstrap percolation can be mapped into heterogeneous k-core percolation and vice versa, if the functionality thresholds in both processes satisfy a complementary relation. Another interesting problem in bootstrap and k-core percolation is the fraction of nodes belonging to their giant connected components P_{∞b} and P_{∞c}, respectively. We solve this problem analytically for arbitrary randomly connected graphs and arbitrary threshold distributions, and we show that P_{∞b} and P_{∞c} are not complementary. Our theoretical results coincide with computer simulations in the limit of very large graphs. In bootstrap percolation, we show that when using the branching theory to compute the size of the giant component, we must consider two different types of links, which are related to distinct spanning branches of active nodes.
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Affiliation(s)
- Matías A Di Muro
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, (7600) Mar del Plata, Argentina
| | - Lucas D Valdez
- Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA
| | - H Eugene Stanley
- Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA
| | - Sergey V Buldyrev
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA and Politecnico di Milano, Department of Management, Economics and Industrial Engineering, Via Lambruschini 4, BLD 26, 20156 Milano, Italy
| | - Lidia A Braunstein
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, (7600) Mar del Plata, Argentina and Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA
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Wandelt S, Sun X, Feng D, Zanin M, Havlin S. A comparative analysis of approaches to network-dismantling. Sci Rep 2018; 8:13513. [PMID: 30202039 PMCID: PMC6131543 DOI: 10.1038/s41598-018-31902-8] [Citation(s) in RCA: 61] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2018] [Accepted: 08/29/2018] [Indexed: 11/24/2022] Open
Abstract
Estimating, understanding, and improving the robustness of networks has many application areas such as bioinformatics, transportation, or computational linguistics. Accordingly, with the rise of network science for modeling complex systems, many methods for robustness estimation and network dismantling have been developed and applied to real-world problems. The state-of-the-art in this field is quite fuzzy, as results are published in various domain-specific venues and using different datasets. In this study, we report, to the best of our knowledge, on the analysis of the largest benchmark regarding network dismantling. We reimplemented and compared 13 competitors on 12 types of random networks, including ER, BA, and WS, with different network generation parameters. We find that network metrics, proposed more than 20 years ago, are often non-dominating competitors, while many recently proposed techniques perform well only on specific network types. Besides the solution quality, we also investigate the execution time. Moreover, we analyze the similarity of competitors, as induced by their node rankings. We compare and validate our results on real-world networks. Our study is aimed to be a reference for selecting a network dismantling method for a given network, considering accuracy requirements and run time constraints.
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Affiliation(s)
- Sebastian Wandelt
- National Key Laboratory of CNS/ATM, School of Electronic and Information Engineering, Beihang University, 100191, Beijing, China
- National Engineering Laboratory of Multi-Modal Transportation Big Data, 100191, Beijing, China
- Beijing Advanced Innovation Center for Big Data-based Precision Medicine, Beihang University, 100083, Beijing, China
| | - Xiaoqian Sun
- National Key Laboratory of CNS/ATM, School of Electronic and Information Engineering, Beihang University, 100191, Beijing, China.
- National Engineering Laboratory of Multi-Modal Transportation Big Data, 100191, Beijing, China.
| | - Daozhong Feng
- National Key Laboratory of CNS/ATM, School of Electronic and Information Engineering, Beihang University, 100191, Beijing, China
| | - Massimiliano Zanin
- Centro de Tecnologica Biomedica, Universidad Politecnica de Madrid, 28223, Madrid, Spain
- Faculdade de Ciecias e Tecnologia, Universidade Nova de Lisboa, 2829-516, Caparica, Portugal
| | - Shlomo Havlin
- Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel
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Kornbluth Y, Barach G, Tuchman Y, Kadish B, Cwilich G, Buldyrev SV. Network overload due to massive attacks. Phys Rev E 2018; 97:052309. [PMID: 29906843 DOI: 10.1103/physreve.97.052309] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Indexed: 06/08/2023]
Abstract
We study the cascading failure of networks due to overload, using the betweenness centrality of a node as the measure of its load following the Motter and Lai model. We study the fraction of survived nodes at the end of the cascade p_{f} as a function of the strength of the initial attack, measured by the fraction of nodes p that survive the initial attack for different values of tolerance α in random regular and Erdös-Renyi graphs. We find the existence of a first-order phase-transition line p_{t}(α) on a p-α plane, such that if p<p_{t}, the cascade of failures leads to a very small fraction of survived nodes p_{f} and the giant component of the network disappears, while for p>p_{t}, p_{f} is large and the giant component of the network is still present. Exactly at p_{t}, the function p_{f}(p) undergoes a first-order discontinuity. We find that the line p_{t}(α) ends at a critical point (p_{c},α_{c}), in which the cascading failures are replaced by a second-order percolation transition. We find analytically the average betweenness of nodes with different degrees before and after the initial attack, we investigate their roles in the cascading failures, and we find a lower bound for p_{t}(α). We also study the difference between localized and random attacks.
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Affiliation(s)
- Yosef Kornbluth
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Gilad Barach
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA
| | - Yaakov Tuchman
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA
| | - Benjamin Kadish
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA
| | - Gabriel Cwilich
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA
- Donostia International Physics Center (DIPC), Paseo Manuel Lardizabal 4, 20018 Donostia-San Sebastian, Spain
| | - Sergey V Buldyrev
- Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA
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