Average topological properties of successive neighbours of cells in random networks

Abstract

Abstract Random planar (trivalent) networks are studied with the purpose of finding correlations between the number i of sides of a cell (i cell) and average properties of its successive neighbours. The properties investigated are the average number n i k of neighbours of order k of i cells and the average number m i k of sides of these neighbours. The neighbours of a given order k are classified into various types according to their ‘proximity’ to k+1 neighbours. Approximate relations are anticipated, such as a linear variation in n i k with both i and k, and a generalization of the Aboav-Weaire relation to more distant neighbours, that is a linear relation between n i k m i k and i. These relations were ‘experimentally’ assessed by analysing two types of random trivalent network (Voronoi and Poisson) using image analysis methods.