János Kertész (CEU, BME, Aalto)

Tuesday 2017-07-25 14.15 – 15.00

Lecture room AS3, TUAS-building

## The hybrid percolation transition

Percolation is a paradigmatic example of second order phase transitions where the order parameter changes continuously (in contrast to first order transitions). However, there are dynamic versions of the percolation model, where the order parameter has a jump at the transition. Such models include the Watts cascade model, the interdependent network model, k-core percolation and the SWIR epidemic model. Interestingly, all these models show features of both first order and second order transitions at the same time: There is a discontinuity in the order parameter and there are scaling phehomena too. We show that there are two sets of exponents describing the transition in these cases: One characterizes the behavior of the order parameter, the other one that of the avalanches (or cascades). There is a scaling law connecting these exponents and one of them is given exactly for the case of infinite range interdependent networks. We show on the Erdős-Rényi graph that there is a universal mechanism behind the hybrid percolation transition: The process starts with the evolution of a cascade as a critical branching tree (hence the scaling properties) during which latent nodes accumulate. Due to the finitness of the samples loops occure which ignite some latent nodes leading rapidly to a global cascade and a jump in the order parameter. These phenomena separate in time, enabling intervention during the first period and giving an interpretation of the “golden time” in epidemiology. We show that the dependence of the length of the golden time on the size N of the system is ~N^1/3 if the process starts from an O(1) part of the system and it is ~N^1/4 if from an O(N) part.