Periodic Reporting for period 1 - ExtSpGraphLim (Extremal Sparse Graphs and Graph Limits)
Okres sprawozdawczy: 2017-09-01 do 2019-08-31
The importance of understanding large networks cannot be exaggerated as they surround us everywhere, let it be the internet, the network of people (facebook) or the neural network of a brain. Our focus was to develop tools to investiagte the above mentioned problems from a mathematical point of view, and not to study particular networks. Shortly, our goal was to improve on the already existent ideas, let it be the zero-free region of graph polynomials, the understanding of a statistical physical heuristics called Bethe approximation or advancing tools like the use of graph covers. These questions also lead to pure mathematical questions like Sidorenko's conjecture.
1. (with B. Szegedy) On Sidorenko's conjecture for determinants and Gaussian Markov random fields
2. (with F. Bencs) Note on the zero-free region of the hard-core model
3. (with M. Borbényi) On degree-contsrained subgraphs and orientations
4. (with A. Imolay) Covers, factors and orientations
The first paper resolves a determinantal version of the Sidorenko's conjecture and it also builds a bridge between homomorpism numbers (important statistics of networks) and Gaussian Markov random fields, a very important concept in statistics and probability theory.
In the second paper we extended the zero-free region of the hard-core model on bounded degree graphs. Hard-core model is one of the most fundamental models in statistical physics. It has also the speciality that many other models like the monomer-dimer model or the Widom-Rowlinson model can be reduced to the understanding of this model. Giving a zero-free region of such a model directly means that we can efficiently approximate many important quantities in certain parameter regimes of the model.
In the third paper we study a tool called gauge transformation. This is a fantastic tool that enables one to transform a computational problem to another one. It also provides a surprising path to the statistical physical heuristics called Bethe approximation. In this paper we used this tool to study subgraph counting problems together with counting orientations. We gained new proofs and insights to old mathematical results.
Is some sense the fourth paper was motivated by the third paper. We wanted to give simpler proofs for certain facts of the third paper that does not use the deep machinery of gauge transformation. This lead to an interesting elementary method that seems very powerful. This method is practically a double counting method providing unusual recursion formula for certain natural graph parameters. This method also turned out to be related to the method of graph covers, a method that were earlier developed to study Bethe approximation. I found it extremely satisfactory how all these ideas fit together.