Resilience of networks with community structure behaves as if under an external field

Gaogao Dong; Jingfang Fan; Louis M. Shekhtman; Saray Shai; Ruijin Du; Lixin Tian; Xiaosong Chen; H. Eugene Stanley; Shlomo Havlin

doi:10.1073/pnas.1801588115

Resilience of networks with community structure behaves as if under an external field

Gaogao Dong
, Jingfang Fan
, Louis M. Shekhtman
, Saray Shai
, Ruijin Du
, Lixin Tian
, Xiaosong Chen
, H. Eugene Stanley
, Shlomo Havlin

Department of Physics - at Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

123 Scopus citations

Abstract

Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic– paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.

Original language	English
Pages (from-to)	6911-6915
Number of pages	5
Journal	Proceedings of the National Academy of Sciences of the United States of America
Volume	115
Issue number	27
DOIs	https://doi.org/10.1073/pnas.1801588115
State	Published - 3 Jul 2018

Bibliographical note

Publisher Copyright:
© 2018 National Academy of Sciences. All Rights Reserved.

Funding

We thank Dr. Lucas Daniel Valdez for useful discussions. The Boston University Center for Polymer Studies is supported by NSF Grants PHY-1505000, CMMI-1125290, and CHE-1213217; and by Defense Threat Reduction Agency Grant HDTRA1-14-1-0017. We acknowledge the Israel–Italian collaborative project Network Aware Cyber Security (NECST); the Israel Science Foundation; Major Program of National Natural Science Foundation of China Grants 71690242 and 91546118; the Office of Naval Research; the Japan Science Foundation; and US–Israel Binational Science Foundation–NSF. This work was partially supported by National Natural Science Foundation of China Grants 61403171, 71403105, 2015M581738, and 1501100B; and Key Research Program of Frontier Sciences, Chinese Academy of Sciences Grant QYZDJ-SSW-SYS019. J.F. thanks the fellowship program funded by the Planning and Budgeting Committee of the Council for Higher Education of Israel. Israel–Italian collaborative project Network Aware Cyber Security (NECST); the Israel Science Foundation; Major Program of National Natural Science Foundation of China Grants 71690242 and 91546118; the Office of Naval Research; the Japan Science Foundation; and US–Israel Binational Science Foundation–NSF. This work was partially supported by National Natural Science Foundation of China Grants 61403171, 71403105, 2015M581738, and 1501100B; and Key Research Program of Frontier Sciences, Chinese Academy of Sciences Grant QYZDJ-SSW-SYS019. J.F. thanks the fellowship program funded by the Planning and Budgeting Committee of the Council for Higher Education of Israel. ACKNOWLEDGMENTS. We thank Dr. Lucas Daniel Valdez for useful discussions. The Boston University Center for Polymer Studies is supported by NSF Grants PHY-1505000, CMMI-1125290, and CHE-1213217; and by Defense Threat Reduction Agency Grant HDTRA1-14-1-0017. We acknowledge the Fig. 4. ∂r∂S(r,p) as a function of pc − p with r = 0.0001 for λ = 4.5, r = 0.0001 for λ = 3.35, and r = 0.0005 for λ = 2.8, Left and Right show numerical and simulation results. The slopes of red dashed lines are equal to −γ. The simulation results were averaged over 1,000 realizations with kmin = 2, kmax = 106, N1 = N2 = 108, and Minter = N1.

Funders	Funder number
Japan Science Foundation
Key Research Program of Frontier Sciences
National Natural
Science Foundation of China
US-Israel Binational Science Foundation
National Science Foundation	CMMI-1125290, PHY-1505000, CHE-1213217
Office of Naval Research
Defense Threat Reduction Agency	HDTRA1-14-1-0017
Japan Health Sciences Foundation
FP7 Security
National Stroke Foundation
United States-Israel Binational Science Foundation
National Natural Science Foundation of China	91546118, 1501100B, 71403105, 61403171, 71690242, 2015M581738
Chinese Academy of Sciences	QYZDJ-SSW-SYS019
Israel Science Foundation
Council for Higher Education

Keywords

Community structure
External field
Percolation
Resilience
Universality

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10.1073/pnas.1801588115

Cite this

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title = "Resilience of networks with community structure behaves as if under an external field",

abstract = "Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic– paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.",

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AU - Dong, Gaogao

AU - Fan, Jingfang

AU - Shekhtman, Louis M.

AU - Shai, Saray

AU - Du, Ruijin

AU - Tian, Lixin

AU - Chen, Xiaosong

AU - Eugene Stanley, H.

AU - Havlin, Shlomo

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N2 - Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic– paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.

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Resilience of networks with community structure behaves as if under an external field

Abstract

Bibliographical note

Funding

Keywords

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