Jan 30, 2018 | 12:00—13:30
Many real world networks have groups of similar nodes which are vulnerable to the same failure or adversary. Nodes can be colored in such a way that colors encode the shared vulnerabilities. Using multiple paths to avoid these vulnerabilities can greatly improve network robustness. Color-avoiding percolation provides a theoretical framework for analyzing this scenario, focusing on the maximal set of nodes which can be connected via multiple color-avoiding paths. We explicitly account for the fact that the same particular link can be part of different paths avoiding different colors. This fact was previously accounted for with a heuristic approximation. We compare this approximation with a new, more exact theory and show that the new theory is substantially more accurate for many avoided colors. Further, we formulate our new theory with differentiated node functions, as senders/receivers or as transmitters. In both functions, nodes can be explicitly trusted or avoided. With only one avoided color we obtain standard percolation. With one by one avoiding additional colors, we can understand the critical behavior of color avoiding percolation. For heterogeneous color frequencies, we find that the colors with the largest frequencies control the critical threshold and exponent. Colors of small frequencies have only a minor influence on color avoiding connectivity, thus allowing for approximations. We show that modularity of network provides an interesting analogue in the study of networks to the Ising model in the presence of a magnetic field. Lastly, we find from the perspective of statistical physics, that found exponents define new universality classes characterizing higher-order transitions with corresponding higher-order derivative analogues of \gamma. To the best of our knowledge the present study represents the first time that higher-order transitions are observed in percolation type systems with their higher-order scaling exponents defined and measured.