Optimal resilience of modular interacting networks

Gaogao Dong, Fan Wang, Louis M. Shekhtman, Michael M. Danziger, Jingfang Fan, Ruijin Du, Jianguo Liu, Lixin Tian, H. Eugene Stanley, Shlomo Havlin

Research output: Contribution to journalArticlepeer-review

30 Scopus citations


Coupling between networks is widely prevalent in real systems and has dramatic effects on their resilience and functional properties. However, current theoretical models tend to assume homogeneous coupling where all the various subcomponents interact with one another, whereas real-world systems tend to have various different coupling patterns. We develop two frameworks to explore the resilience of such modular networks, including specific deterministic coupling patterns and coupling patterns where specific subnetworks are connected randomly. We find both analytically and numerically that the location of the percolation phase transition varies nonmonotonically with the fraction of interconnected nodes when the total number of interconnecting links remains fixed. Furthermore, there exists an optimal fraction rof interconnected nodes where the system becomes optimally resilient and is able to withstand more damage. Our results suggest that, although the exact location of the optimal rvaries based on the coupling patterns, for all coupling patterns, there exists such an optimal point. Our findings provide a deeper understanding of network resilience and show how networks can be optimized based on their specific coupling patterns.

Original languageEnglish
Article numbere1922831118
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number22
StatePublished - 1 Jun 2021
Externally publishedYes


  • Interacting network
  • Optimal phenomenon
  • Percolation
  • Resilience

ASJC Scopus subject areas

  • General


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