Characterizing spatial point processes by percolation transitions

Abstract

A set of discrete individual points located in an embedding continuum space can be seen as percolating or non-percolating, depending on the radius of the discs/spheres associated with each of them. This problem is relevant in theoretical ecology to analyze, e.g., the spatial percolation of a tree species in a tropical forest or a savanna. Here, we revisit the problem of aggregating random points in continuum systems (from 2 to 6-dimensional Euclidean spaces) to analyze the nature of the corresponding percolation transition in spatial point processes. This problem finds a natural description in terms of the canonical ensemble but not in the usual grand-canonical one, customarily employed to describe percolation transitions. This leads us to analyze the question of ensemble equivalence and study whether the resulting canonical continuum percolation transition shares its universal properties with standard percolation transitions, analyzing diverse homogeneous and heterogeneous spatial point processes. We, therefore, provide a powerful tool to characterize and classify a vast class of natural point patterns, revealing their fundamental properties based on percolation phase transitions.

Evidence of scale-free clusters of vegetation in tropical rainforests

“…Through continuum clustering techniques [12], we present evidence of scale-free clusters of vegetation in empirical data of BCI, shedding much light on their spatial aggregation properties and correlation scales. We compare actual data with different simulated emergent spatial point processes, showing the wide variety of scales that play a crucial role in natural rainforests, and uncovering emergent critical dynamics hitherto unknown.…”

“…For example, neuronal avalanches, i.e, cascades of activations clustered in time, have been crucial to scrutinize the emergent dynamical behavior in neural populations [11]. However, unlike von Neumann neighborhood in discrete systems, the clustering statistics in continuous embeddings [12], either temporal or spatial, rely on nearest-neighbors distance assumptions with an intrinsic degree of freedom: there is not a unique way to define clusters in the system. The time-bin issue in determining neuronal avalanches is an example of this [1,13].…”

“…The most common way to characterize the aggregation properties of a fixed number of points, N , distributed in a continuous embedding space relies in usual continuum percolation techniques [12,24,32]. Hence, based on some predefined distance r, two individual points will belong to the same cluster if their Euclidean distance is less than or equal to r [33,34].…”

“…A percolating cluster of points exists if a path connects all the points satisfying the previous condition. Let us remark that, the mean nearest-neighbor distance (MNN), defined as the nearest neighbor mean center-to-center distance between points, makes us possible to study a point process independently of the area where it takes place, and study it only depending on the system size [12,24]. Thus, we normalize the control parameter by the inter-particle distance of the point process, i.e., defining r = r/MNN, and producing a non-dimensional version of the distance parameter, which allows us to interpret cluster analysis in the language of statistical mechanics and percolation phase transitions.…”

“…Sketch of the clustering process of a fixed number of points for different interaction distances. Upper insets illustrate the clusters at criticality for different homogeneous and inhomogeneous Poisson point processes (see [12] for further examples).…”