Community Research and Development Information Service - CORDIS



Project ID: 339096
Funded under: FP7-IDEAS-ERC
Country: Israel

Mid-Term Report Summary - HI-DIM COMBINATORICS (High-dimensional combinatorics)

Networks (aka graphs) appear everywhere: It can be a social network that depicts connections between humans, or a biological pathway that reflects the way proteins interact in some organism, or the web of business operations and how they trade, the examples are plenty. The theory of random graphs is crucial in the study of large-scale networks. This theory was initiated by Erdos and Renyi over 50 years ago and has been a constant source of inspiration in such studies, both theoretical and applied. One the great discoveries of Erdos and Renyi was the so-called phase transition in evolution of random graphs. Namely, they discovered that as the random graph evolves, at some well-defined moment, and very quickly. a giant component emerges.
A major weakness in this classical theory is that it deals only with systems in which the interactions are in pairs. Needless to say, there are many real-life situations where the basic interactions involve several parties at once. For example, most of the interesting biological processes involve several proteins in mutual interaction. One of the main goals of the present project was to develop the mathematical machinery and theory that is necessary for the elucidation of such processes. We have now reached that goal and we know what the phase transition looks like in systems where interactions are multi-way. One remarkable discovery that we have made is that in this "high-dimensional" scenario, the phase transition is much more abrupt than in the case of graphs.

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