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Section: New Results

Random Graph and Matrix Models

Participants : Konstantin Avrachenkov, Andrei Bobu.

Random geometric graphs

Random geometric graphs are good examples of random graphs with a tendency to demonstrate community structure. Vertices of such a graph are represented by points in Euclid space $R^d$, and edge appearance depends on the distance between the points. Random geometric graphs were extensively explored and many of their basic properties are revealed. However, in the case of growing dimension $d\to \infty$ practically nothing is known; this regime corresponds to the case of data with many features, a case commonly appearing in practice. In [30], K. Avrachenkov and A. Bobu focus on the cliques of these graphs in the situation when average vertex degree grows significantly slower than the number of vertices $n$ with $n\to \infty$ and $d\to \infty$. They show that under these conditions random geometric graphs do not contain cliques of size 4 a.s. As for the size 3, they present new bounds on the expected number of triangles in the case ${log}^2\left(n\right)\ll d\ll {log}^3\left(n\right)$ that improve previously known results.

Network geometries are typically characterized by having a finite spectral dimension (SD), that characterizes the return time distribution of a random walk on a graph. The main purpose of this work is to determine the SD of random geometric graphs (RGGs) in the thermodynamic regime, in which the average vertex degree is constant. The spectral dimension depends on the eigenvalue density (ED) of the RGG normalized Laplacian in the neighborhood of the minimum eigenvalues. In fact, the behavior of the ED in such a neighborhood characterizes the random walk. Therefore, in [33] K. Avrachenkov together with L. Cottatellucci (FAU, Germany and Eurecom) and M. Hamidouche (Eurecom) first provide an analytical approximation for the eigenvalues of the regularized normalized Laplacian matrix of RGGs in the thermodynamic regime. Then, we show that the smallest non zero eigenvalue converges to zero in the large graph limit. Based on the analytical expression of the eigenvalues, they show that the eigenvalue distribution in a neighborhood of the minimum value follows a power-law tail. Using this result, they find that the SD of RGGs is approximated by the space dimension $d$ in the thermodynamic regime.

In [42] K. Avrachenkov together with L. Cottatellucci (FAU, Germany and Eurecom) and M. Hamidouche (Eurecom) have analyzed the limiting eigenvalue distribution (LED) of random geometric graphs. In particular, they study the LED of the adjacency matrix of RGGs in the connectivity regime, in which the average vertex degree scales as $log\left(n\right)$ or faster. In the connectivity regime and under some conditions on the radius $r$, they show that the LED of the adjacency matrix of RGGs converges to the LED of the adjacency matrix of a deterministic geometric graph (DGG) with nodes in a grid as the size of the graph $n$ goes to infinity. Then, for $n$ finite, they use the structure of the DGG to approximate the eigenvalues of the adjacency matrix of the RGG and provide an upper bound for the approximation error.