In various industrial products including parallel computing systems, it is expected that the performance of the whole system will be improved by applying a network topology with smaller diameter and average shortest path length (ASPL). Such network topologies can be defined as the order/degree problem in graph theory by modeling a network topology as an undirected graph. Although previous research indicates that the random graph has both small diameter and ASPL due to the small-world effect, there is still room for improvement. In this paper, we propose an algorithm for the order/degree problem that optimizes random graphs. The feature of the proposed algorithm is that by giving symmetry to the graph, the solution search performance is improved and the calculation time for obtaining the diameter and APSL is greatly reduced. The proposed algorithm is evaluated using various graphs, including a huge graph with one million vertices presented by Graph Golf, an international competition for the order/degree problem. The results show that the proposed algorithm can generate a network topology with sufficiently small diameter and APSL. In addition, we also simulate the latency, performance of the parallel benchmarks, and bisection on the generated network topologies. The results show that the generated network topology performs better than the random topology and a conventional $k$-ary $n$-cube topology.

在包括并行计算系统在内的各种工业产品中，预计通过应用具有更小直径和平均最短路径长度（ASPL）的网络拓扑结构来提高整个系统的性能。通过将网络拓扑建模为无向图，可以将此类网络拓扑定义为图论中的阶/度问题。尽管之前的研究表明随机图由于小世界效应而具有小直径和ASPL，但仍有改进的空间。在本文中，我们提出了一种优化随机图的阶/度问题的算法。该算法的特点是通过赋予图形对称性，提高了解的搜索性能，大大减少了获得直径和APSL的计算时间。所提出的算法使用各种图进行评估，包括由 Graph Golf 呈现的具有一百万个顶点的巨大图，这是一个针对阶/度问题的国际竞赛。结果表明，所提出的算法可以生成具有足够小直径和APSL的网络拓扑。此外，我们还模拟了并行基准测试的延迟、性能以及生成的网络拓扑结构的二分。结果表明，生成的网络拓扑比随机拓扑和传统的网络拓扑表现更好。我们还模拟了并行基准测试的延迟、性能以及生成的网络拓扑结构的二分。结果表明，生成的网络拓扑比随机拓扑和传统的网络拓扑表现更好。我们还模拟了并行基准测试的延迟、性能以及生成的网络拓扑结构的二分。结果表明，生成的网络拓扑比随机拓扑和传统的网络拓扑表现更好。$克$-ary $n$-立方体拓扑。