Scaling of Average Shortest Distance of Two Colored Substitution Networks

Abstract

In this paper, two colored substitution networks by substitution rules are introduced. In order to calculate the sum of all shortest distances, we discuss the following both cases: two nodes are in adjacent branches or non-adjacent ones. The most difficult problem is to compute the sum of all shortest distances whose nodes are in two symmetrical branches. We were very surprised to find one efficient method. The key to the method is to use the geodesic distance between two initial nodes in previous generation. Consequently, we first derive two formulas of the geodesic distances between two initial nodes. Then, we obtain the analytic expressions of the sum of the shortest distances between the initial node A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (or B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ), and other nodes in the substitution networks. Finally, using the obtained results of geodesic distance we study the total shortest distances of the colored substitution networks. And then, the leading term of the average shortest distance is obtained. The obtained results show that the colored substitution networks grow unbounded, while the leading scaling of the average shortest distance exhibits a sublinear dependence on the network order.