AVERAGE TOPOLOGICAL PROPERTIES OF SUCCESSIVE NEIGHBORS OF CELLS IN RANDOM NETWORKS

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(k)i] of neighbours of order k of i cells and the average number [m(k)i] 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(k)i] with both i and k, and a generalization of the Aboav-Weaire relation to more distant neighbours, that is a linear relation between [n(k)i][m(k)i] and i. These relations were 'experimentally' assessed by analysing two types of random trivalent network (Voronoi and Poisson) using image analysis methods.