## Solution of the explosive percolation quest

By: Sergey Dorogovtsev

From: Universidade de Aveiro

At: Instituto de Investigação Interdisciplinar, B1-01

[2012-03-20]

Percolation refers to the emergence of a giant connected (percolating) cluster in a disordered system when the number of connections between nodes exceeds some critical value. The percolation phase transitions were believed to be continuous until recently when in a new so-called Ã¢â‚¬Å“explosive percolationÃ¢â‚¬Â problem for a Metropolis-like algorithm driven evolution, a discontinuous phase transition was reported based on computer experiments. The direct analysis of evolution equations for this process showed however that this transition is actually continuous though with quite unusual values of critical exponents. One should stress that the most exciting problem is not the continuity of the Ã¢â‚¬Å“explosive percolationÃ¢â‚¬Â transition but rather its unique features, sharply distinct from standard percolation. The question is: what is the nature of this strange transition? Here we present a complete solution of this intriguing quest. For a wide class of representative models, we develop a strict scaling theory which provides the full set of scaling functions and critical exponents for each of the models with any desired precision. This theory indicates the relevant order parameter and susceptibility for the problem, and explains the continuous nature of this transition and its unusual properties.

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[3] S. N. Dorogovtsev, Lectures on Complex Networks (Oxford University Press, 2010);