In this paper, in order to calculate the average path length for unweighted and weighted hierarchical networks, we define the sum of the distances from each node to the vertex and the bottom nodes. For the unweighted network, we show that the average path length grows with the size of [Formula: see text] as [Formula: see text]. For the weighted network, we prove its weighted path length approaches a constant related to the weighting factor [Formula: see text] and parameter [Formula: see text]. In particular, for [Formula: see text], the average weighted path length of the network tends to a specific value [Formula: see text].

中文翻译：

---

一类分层网络的平均加权路径长度

在本文中，为了计算未加权和加权分层网络的平均路径长度，我们定义了每个节点到顶点和底部节点的距离之和。对于未加权的网络，我们表明平均路径长度随着 [公式：参见文本] 的大小而增长，如 [公式：参见文本]。对于加权网络，我们证明其加权路径长度接近与权重因子[公式：见文本]和参数[公式：见文本]相关的常数。特别是对于【公式：见正文】，网络的平均加权路径长度趋于特定值【公式：见正文】。