Statistical spatial graphs to study the topology and geometry of large karst networks

Abstract:

Because of the lack of data and the complexity of the patterns observed by speleologists, karst network modeling is a challenge. In this work, we study topological and geometrical properties of large karst networks trough spatial graph models. A statistical mechanics approach is used to explore the set of directed and (possibly) weighted spatial graphs on a given space to extract subset of graphs of given properties. Within this framework, it's possible to generate sets of karst networks having realistic properties (in terms of spatial extension, degree distribution of nodes, number of cycles, total available volume for water transport, ...) and passing trough a given set of fixed points (for example some known inlets and outlets).
Spatial graph analogues of the traditional statistical ensembles are defined (the microcanonical, canonical and grand canonical ensembles).
We show how choice of (pseudo) Hamiltonians and ensemble parameters influence the structure of typical generated graphs (tree graphs, many loops graphs, hight clustering coefficient graphs, small total length graphs, ...). Two kind of Hamiltonians are studied: - the structural ones, $i.e.$ Hamiltonians which are functions of the topology and the geometrical properties of the graph and - Hamiltonians related to the dynamical process occurring on the graph. An important dynamics related Hamiltonian is the measure of the total dissipated energy by groundwater flow trough the karst network.
An ensemble of special interest is the grand canonical ensemble in which the number of nodes (excepted the given set outlets and inlets) and the number of edges are not imposed. This ensemble is useful to investigate how a network can emerge on a prescribed space. We give an interpretation to the parameters defining this ensemble, show some realisations, and discuss the interest of this ensemble for karst generation.