Fragility and anomalous susceptibility of weakly interacting networks

Abstract

Percolation is a fundamental concept that brought new understanding on the robustness properties of complex systems. Here we consider percolation on weakly interacting networks, that is, network layers coupled together by much less interlinks than the connections within each layer. For these kinds of structures, both continuous and abrupt phase transition are observed in the size of the giant component. The continuous (second-order) transition corresponds to the formation of a giant cluster inside one layer, and has a well defined percolation threshold. The abrupt transition instead corresponds to the merger of coexisting giant clusters among different layers, and is characterised by a remarkable uncertainty in the percolation threshold, which in turns causes an anomalous trend in the observed susceptibility. We develop a simple mathematical model able to describe this phenomenon and to estimate the critical threshold for which the abrupt transition is more likely to occur. Remarkably, finite-size scaling analysis in the abrupt region supports the hypothesis of a genuine first-order phase transition.

Enhancing the robustness of a multiplex network leads to multiple discontinuous percolation transitions

“…wherẽ P (B) = k>0 P (k, B), andP (k) = B>0 P (k, B). (15) In this case, the interdependent percolation can be studied by investigating the critical properties of Eqs. (7) and (8) supplied with P (k, B) = P null (k, B).…”

“…Other disciplines, as material science and chemical synthesis, are on the way to adopt the network science toolbox, less for analysis purpose, but mainly for its potential for optimisation and rational design [4][5][6][7][8]. Therefore, predicting the robustness of multilayer networks [9][10][11], assessing the risk of large avalanches of failures [9][10][11][12][13][14][15], and designing robust multilayer architectures [16][17][18] are the key theoretical questions entailing implications for engineering, economics, and biology. Recent works on percolation in multilayer networks revealed that the correlation between intra-layer degrees of replica nodes [16,17] and link overlap [19][20][21] have profound consequences in determining the response of a multiplex network to random damage.…”

Percolation in networks with local homeostatic plasticity

“…The size of LCC indicates what fraction of the whole network stays connected after damage, and in contexts such as transportation or communication—connected means functional. Studying percolation helps to understand the contribution of the network structure to its resilience, and both first- and second-order phase transitions in the size of LCC have been observed across a wealth of studies 1 , 4 – 8 . The most notable example refers to explaining the spreading of a disease driven by the susceptible–infected–recovered contact process.…”

Exact Rank Reduction of Network Models

“…Through networked models, researchers have improved the understanding of complex systems in terms of easily interpretable analytical relations. A paramount example is the large literature on critical phenomena on complex networks: synchronization [13], spreading processes [14][15][16][17][18][19][20], or percolation [21][22][23], to cite a few. These models usually encode network structure into a small set of parameters and generative rules.…”