Interconnections between networks acting like an external field in a first-order percolation transition

Bnaya Gross, Hillel Sanhedrai, Louis Shekhtman, Shlomo Havlin

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15 Scopus citations


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 Erdos-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.

Original languageEnglish
Article number022316
JournalPhysical Review E
Issue number2
StatePublished - Feb 2020

Bibliographical note

Publisher Copyright:
© 2020 American Physical Society.


We thank Ivan Bonamassa for very useful discussions related to this project. We also thank the Italian Ministry of Foreign Affairs and International Cooperation jointly with the Israeli Ministry of Science, Technology, and Space (MOST); the Israel Science Foundation; ONR; the Japan Science Foundation with MOST; BSF-NSF; ARO; the BIU Center for Research in Applied Cryptography and Cyber Security; and DTRA (Grants No. HDTRA-1-14-1-0017 and No. HDTRA-1-19-1-0016) for financial support.

FundersFunder number
Israeli Ministry of Science, Technology, and Space
Japan Science Foundation
Office of Naval Research
Army Research OfficeHDTRA-1-14-1-0017, HDTRA-1-19-1-0016
Ministry of Science and Technology
Israel Science Foundation
Ministry for Foreign Affairs


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