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 $A_{i}$ (or $B_{i}$), 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.

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两种有色替代网络的平均最短距离的标度

在本文中，通过替换规则介绍了两种彩色替换网络。为了计算所有最短距离的总和，我们讨论以下两种情况：两个节点在相邻的分支或不相邻的分支中。最困难的问题是计算节点在两个对称分支中的所有最短距离的总和。我们非常惊讶地发现了一种有效的方法。该方法的关键是利用上一代中两个初始节点之间的测地线距离。因此，我们首先推导出两个初始节点之间的测地线距离的两个公式。然后，我们得到初始节点之间的最短距离之和的解析表达式$A_{i}$ （或者 $B_{i}$)，以及替代网络中的其他节点。最后，使用测地距离的获得结果，我们研究了彩色替代网络的总最短距离。然后，得到平均最短距离的前导项。获得的结果表明，彩色替代网络无限增长，而平均最短距离的领先缩放表现出对网络顺序的次线性依赖性。