Dynamics on and of Random Graphs

Objetivo

In the past 25 years, networks, such as social, economic and communication networks, have
become society’s backbone and arteries. Remarkable insight into network structure has been
derived using random graphs, and into network functionality using random processes living on
them. This double randomness gives rise to deep and fascinating mathematical problems at the
core of modern probability theory.
My focus in this proposal is on solving challenges around tipping points, or phase transitions,
of random processes on random graphs. These challenges have so far resisted efforts, yet their
solution is now feasible due to recent breakthroughs, including my own. My proposal embarks on
the ambitious and timely endeavour to develop a novel transformative theory to determine the
behaviour close to the phase transitions of paradigmatic processes on random graphs, with a focus
on percolation and spin systems. The holy grail is to give a tangible and effective criterion when
the random graph disorder is relevant or irrelevant for the process on it, i.e. when the behaviour is
mean field. Throughout my proposal, percolative properties, describing how connectivity depends
on model parameters, are a leitmotif.

My innovative approach is to use dynamics of and on random graphs. Dynamics arise in three
intertwined ways. (1) Key graph properties can be distilled by exploring the graph. (2) Phase
transitions and their universality classes are key to understanding random processes on graphs.
(3) Dynamic graphs change over time in a way that may depend on the process on them, a setting
so-far hardly studied in mathematics. Despite the fact that these three forms of dynamics each
have attracted ample attention, my approach leverages their effective combination, a key example
being percolation on preferential attachment models.

I am a leader worldwide in the mathematics of networks, and uniquely positioned to tackle the
above challenges, whose solution will provide a major breakthrough.

Ámbito científico (EuroSciVoc)

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