Notes on Percolation Analysis of Sampled Scalar Fields

Abstract

Percolation analysis is used to explore the connectivity of randomly connected infinite graphs. In the finite case, a closely related percolation function captures the relative volume of the largest connected component in a scalar field's superlevel set. While prior work has shown that random scalar fields with little spatial correlation yield a sharp transition in this function, little is known about its behavior on real data. In this work, we explore how different characteristics of a scalar field -such as its histogram or degree of structure -influence the shape of the percolation function. We estimate the critical value and transition width of the percolation function, and propose a corresponding normalization scheme that relates these values to known results on infinite graphs. In our experiments, we find that percolation analysis can be used to analyze the degree of structure in Gaussian random fields. On a simulated turbulent duct flow data set we observe that the critical values are stable and consistent across time. Our normalization scheme indeed aids comparison between data sets and relation to infinite graphs.

Towards replacing physical testing of granular materials with a Topology-based Model

“…A related topological approach, Percolation theory, studies the connectivity of an infinite network in terms of the size and extent of the largest connected component, as vertices or edges are included/excluded from the network [2,8]. A discontinuity in the resulting percolation function provides a threshold that describes an intrinsic porosity property of the material [27]. It has been adapted to finite domains and large material images [14].…”

Distributed Percolation Analysis for Turbulent Flows

“…The impact of different kinds of normalization has further been explored by Köpp and Friederici et al [22], who also study the general application of percolation analysis on sampled data.…”