Topology of random -dimensional cubical complexes

We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

Topological Data Analysis reveals robust alterations in the whole-brain and frontal lobe functional connectomes in Attention-Deficit/Hyperactivity Disorder

The functional organization of the brain network (connectome) has been widely studied as a graph; however, methodological issues may affect the results, such as the brain parcellation scheme or the selection of a proper threshold value. Instead of exploring the brain in terms of a static connectivity threshold, this work explores its algebraic topology as a function of the filtration value (i.e., the connectivity threshold), a process termed the Rips filtration in Topological Data Analysis. Specifically, we characterized the transition from all nodes being isolated to being connected into a single component as a function of the filtration value, in a public dataset of children with attention-deficit/hyperactivity disorder (ADHD) and typically developing children. Results were highly congruent when using four different brain segmentations (atlases), and exhibited significant differences for the brain topology of children with ADHD, both at the whole brain network and at the functional sub-network levels, particularly involving the frontal lobe and the default mode network. Therefore, this approach may contribute to identify the neurophysio-pathology of ADHD, reducing the bias of connectomics-related methods.HighlightsTopological Data Analysis was implemented in functional connectomes.Betti curves were assessed based on the area under the curve, slope and kurtosis.The explored variables were robust along four different brain atlases.ADHD showed lower areas, suggesting decreased functional segregation.Frontal and default mode networks showed the greatest differences between groups.Graphical Abstract

Large deviation principles for connectable receivers in wireless networks

We study large deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large deviation principle for the rescaled process of these receivers as the connection threshold tends to 0. Finally, we show how these results can be used to develop importance sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

On the relation between graph distance and Euclidean distance in random geometric graphs

Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).