Homological percolation and the Euler characteristic

Abstract

In this paper we study the connection between the phenomenon of homological percolation (the formation of "giant" cycles in persistent homology), and the zeros of the expected Euler characteristic curve. We perform an experimental study that covers four different models: site-percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields.All the models are generated on the flat torus T d , for d = 2, 3, 4. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homological-percolation. Our results also provide some insight about the approximation error.Further study of this connection could have powerful implications both in the study of percolation theory, and in the field of Topological Data Analysis.

Geometric functionals of fractal percolation

“…So far, this is in accordance with the observations in classical percolation models. Since p c > 1 2 , one would by analogy expect p 0 to be an upper bound for p c . This is also what the naive heuristics suggests for our model: below p c the limit set F is totally disconnected and therefore, if V 0 (F) is naively interpreted as the 'Euler characteristic' of F (i.e., as #components -#holes), then it should be positive for all p < p c .…”

“…Morphometric methods to estimate thresholds in percolation models have been proposed in [20] and intensively studied in the physics literature [12,19,21,22]; see also the recent study in homological percolation [1] using topological data analysis. These methods are based on additive functionals from integral geometry, in particular the Euler characteristic, and rely on the observation that in many percolation models the expected Euler characteristic per site (as a function of the model parameter p)-which can easily be computed analytically in many models-has a zero close to the percolation threshold of the model.…”

“…on the thresholds that capture their dependence on system parameters such as the degree of anisotropy [12]. More precisely, across many models, the zero of the Euler characteristic has been observed to be a lower bound for the percolation threshold p c if p c < 1 2 , and to be an upper bound for p c if p c > 1 2 . For self-matching lattices with p c = 1 2 (such as the triangular lattice), the zero of the Euler characteristic is also found to be 1 2 .…”

“…More precisely, across many models, the zero of the Euler characteristic has been observed to be a lower bound for the percolation threshold p c if p c < 1 2 , and to be an upper bound for p c if p c > 1 2 . For self-matching lattices with p c = 1 2 (such as the triangular lattice), the zero of the Euler characteristic is also found to be 1 2 . This case is the only one up to now in which the relation between Euler characteristic and percolation thresholds has been proven rigorously using symmetry arguments; see e.g.…”

The Euler Characteristic and Topological Phase Transitions in Complex Systems

“…Those transitions can be seen as a generalization of percolation transitions to higher dimensional objects and were observed for the random clique complex of the Erdos-Reyni graph and for functional brain networks [21]. Later, the zeros of the Euler characteristic were also investigated independently for the models of site-percolation on lattices, the Poisson-Boolean model, and Gaussian random fields [22]. In addition to the discovery that the zeros of the Euler characteristic are good estimators for the emergence of giant shades, and that it depicts signal changes in the curvature of simplicial complexes [21], it was also observed later that the zeros of the Euler characteristic are also associated with the emergence of giant k-cycles [22].…”