带约束树模型测地线距离的精确解
Exact solutions for geodesic distance on treelike models with some constraints
摘要：
测地线距离，通常称为最短路径长度，已被证明在各种学科中非常有用。它在搜索引擎中扮演着重要的角色，因此在过去的几十年里受到了广泛的关注，特别是几乎所有适合于搜索信息的数据结构和相应的算法都是基于树状模型生成的。因此，本文详细研究了由一阶细分、（$1,m$）-星分形运算和$m$ -顶点运算生成的树模型的测地距离。与图的测地距离计算的常用方法，如枚举法和矩阵乘法相比，本文利用了图论和映射语言中的顶点覆盖概念，提出了一种新的计算图的测地距离的方法。对于每一种树模型，我们当然得到一个精确解的测地距离使用我们的方法。借助于计算机模拟，我们证实了分析结果与模拟完全一致。此外，我们还报道了其中两类树模型的一些有趣的结构性质。一个服从指数度分布，在许多复杂的网络，相比之下，另一个拥有所有的，但叶顶点的度相同，并显示出更均匀的拓扑结构比前者。此外，两者都具有某种意义上的自相似性，但后者表现出分形性质。
英文摘要：
Geodesic distance, commonly called shortest path length, has proved useful in a great variety of disciplines. It has been playing a significant role in search engine at present and so attracted considerable attention at the last few decades, particularly, almost all data structures and corresponding algorithms suitable to searching information generated based on treelike models. Hence, we, in this paper, study in detail geodesic distance on some treelike models which can be generated by three different types of operations, including first-order subdivision, ($1,m$)-star-fractal operation and $m$-vertex-operation. Compared to the most best used approaches for calculating geodesic distance on graphs, for instance, enumeration method and matrix multiplication, we take useful advantage of a novel method consisting in spirit of the concept of vertex cover in the language of graph theory and mapping. For each kind of treelike model addressed here, we certainly obtain an exact solution for its geodesic distance using our method. With the help of computer simulations, we confirm that the analytical results are in perfect agreement with simulations. In addition, we also report some intriguing structure properties on treelike models of two types among them. The one obeys exponential degree distribution seen in many complex networks, by contrast, the other possesses all but leaf vertices with identical degree and shows more homogeneous topological structure than the former. Besides that, the both have, in some sense, self-similar feature but instead the latter exhibits fractal property.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
