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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - CombLimit (Limit Theory in Combinatorics)

Teaser

Summary

Large networks are present everywhere is our world. We have communication networks, traffic networks, social networks, we use our network of brain cells, etc.
Graphs are the mathematical abstraction of networks. Properties of large graphs are analyzed by their limit when the size tends to infinity.
This is analogous to how a physical medium is defined as a continuous object describing the case when the number of particles tends to infinity.
This is a very recent area of mathematics and the goal is to contribute to its general mathematical theory.
This will lead to a better understanding of our world.

Work performed

We extended our understanding of random graphs to locally transitive graphs via inequalities of IID-processes.
The problem of finding an almost largest independent set in random graphs is a benchmark problem of the topic, and we made a significant contribution about the problem.
For example, while we know that almost maximum size independent sets cannot be constructed locally in random regular graphs of large degree, now we get closer to prove that it is not true for small degree.
This leads to the understanding of an unexpected phase transition, a change in the structure of random graphs.

Final results

The project initiated a new research direction. Statistical physicists have non-rigorous (and sometimes wrong) conjectures about the structures of random graphs.
In the recent years, some of these intuitions are transformed to mathematical proofs by extremely long mathematical papers.
But we realized that some of these ideas are similar to ours, and it would be very beneficial to connect these areas.
A new research group about these problems has already formed at RÃ©nyi Institute. Our first big but very plausible goal is to understand, simplify and generalize the 78 pages long recent Acta Mathematica paper about the independence ratio of random regular graphs. The ultimate goal is to connect graph limit theory with statistical physics, hereby discovering and deeply understanding unknown phase transitions, and hereby getting a better understanding the structures of large graphs. This may give new explanation of real-life phenomena such as ``regime changes\'\' or aging.