Simplicial Homology of Random Configurations

Abstract

Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

“…Using this estimate in (28) and combining with (27) and the triangle inequality for d 3 , we obtain the result.…”

“…Our results deal with the situation of increasing intensity in the case where the functional of interest has the form of a Poisson U -statistic. As well as in the context described above, our theory can thus be applied directly to numbers of k-simplices of random simplical complexes (see [3]) and to subgraph counting in random geometric graphs (see [8] and [15]) with a fixed distance threshold. For problems that were previously considered in the literature for fixed intensity and increasing observation windows, such as the numbers of k-clusters [1], statistics of rather general random geometric graphs [9], or proximity functionals of nonintersecting k-flat processes [18], our results provide complementary central limit theorems for fixed windows and increasing intensity.…”

Disaster recovery in wireless networks: A homology-based algorithm

“…They both are of complexity O(N a ): computations of N a positions. For the uniform method we have to add the complexity of computing the coverage via the Betti numbers, which is of the order of the number of triangles times the number of edges that is O ((N a + N i ) 5 ( r a ) 6 ) for a square of side a according to [5].…”

Homology based algorithm for disaster recovery in wireless networks

“…The determinantal method is the more complex because it takes into account the position of existing vertices. To these complexities we have to add the complexity of computing the coverage via the Betti numbers which is of the order of the number of triangles times the number of edges that is O ((N a + N i ) 5 ( a ) 6 ) according to [5].…”