title: Critical tipping point distinguishing two types of transitions in modular network structures.
abstract: Modularity is a key organizing principle in real-world large-scale complex networks. The relatively sparse interactions between modules are critical to the functionality of the system and are often the first to fail. We model such failures as site percolation targeting interconnected nodes, those connecting between modules. We find, using percolation theory and simulations, that they lead to a "tipping point" between two distinct regimes. In one regime, removal of interconnected nodes fragments the modules internally and causes the system to collapse. In contrast, in the other regime, while only attacking a small fraction of nodes, the modules remain but become disconnected, breaking the entire system. We show that networks with broader degree distribution might be highly vulnerable to such attacks since only few nodes are needed to interconnect the modules, consequently putting the entire system at high risk. Our model has the potential to shed light on many real-world phenomena, and we briefly consider its implications on recent advances in the understanding of several neurocognitive processes and diseases.
authors.affiliation: School of Computer Science, University of St Andrews, St Andrews, Fife KY16 9SX, Scotland, United Kingdom.|Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA.
authors.forename: Saray
authors.initials: S
authors.lastname: Shai
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
authors.forename: Dror Y
authors.initials: DY
authors.lastname: Kenett
authors.affiliation: Gonda Brain Research Center, Bar-Ilan University, 52900 Ramat-Gan, Israel.
authors.forename: Yoed N
authors.initials: YN
authors.lastname: Kenett
authors.affiliation: Gonda Brain Research Center, Bar-Ilan University, 52900 Ramat-Gan, Israel.|Department of Psychology, Bar-Ilan University, 52900 Ramat-Gan, Israel.
authors.forename: Miriam
authors.initials: M
authors.lastname: Faust
authors.affiliation: School of Computer Science, University of St Andrews, St Andrews, Fife KY16 9SX, Scotland, United Kingdom.
authors.forename: Simon
authors.initials: S
authors.lastname: Dobson
authors.affiliation: Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel.
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin