[论文解读] Discovering the structure of complex networks by minimizing cyclic bandwidth sum

Ronan Hamon, Pierre Borgnat+2 more|arXiv (Cornell University)|Oct 22, 2014

Complex Network Analysis Techniques参考文献 29被引用 7

一句话总结

本文提出一种新颖的两步启发式方法，以最小化复杂网络中的循环带宽和（CBS），改进能反映底层图拓扑结构的顶点标记。通过使用基于Jaccard的路径遍历和贪心路径合并，该方法在合成网络和真实网络中均实现了接近最优的CBS值，显著提升了信号平滑度和可视化质量。

ABSTRACT

Getting a labeling of vertices close to the structure of the graph has been proved to be of interest in many applications e.g., to follow smooth signals indexed by the vertices of the network. This question can be related to a graph labeling problem known as the cyclic bandwidth sum problem. It consists in finding a labeling of the vertices of an undirected and unweighted graph with distinct integers such that the sum of (cyclic) difference of labels of adjacent vertices is minimized. Although theoretical results exist that give optimal value of cyclic bandwidth sum for standard graphs, there are neither results in the general case, nor explicit methods to reach this optimal result. In addition to this lack of theoretical knowledge, only a few methods have been proposed to approximately solve this problem. In this paper, we introduce a new heuristic to find an approximate solution for the cyclic bandwidth sum problem, by following the structure of the graph. The heuristic is a two-step algorithm: the first step consists of traversing the graph to find a set of paths which follow the structure of the graph, using a similarity criterion based on the Jaccard index to jump from one vertex to the next one. The second step is the merging of all obtained paths, based on a greedy approach that extends a partial solution by inserting a new path at the position that minimizes the cyclic bandwidth sum. The effectiveness of the proposed heuristic, both in terms of performance and time execution, is shown through experiments on graphs whose optimal value of CBS is known as well as on real-world networks, where the consistence between labeling and topology is highlighted. An extension to weighted graphs is also proposed.

研究动机与目标

• 为解决一般图中近似最优循环带宽和（CBS）缺乏有效方法的问题。
• 开发一种启发式方法，生成与复杂网络结构拓扑一致的顶点标记。
• 通过将顶点排序与图连通性模式对齐，提升信号平滑度和可视化质量。
• 将该方法扩展至加权图，扩大其在真实世界网络数据中的适用性。
• 在已知最优图和真实世界网络上评估性能，证明标记与拓扑之间的一致性。

提出的方法

• 第一步：使用基于Jaccard指数的相似性准则遍历图，识别遵循网络结构组织的路径。
• 第二步：通过贪心算法合并已识别的路径，将每条新路径插入使总循环带宽和最小的位置。
• 该算法使用循环标记方案，将CBS计算为相邻顶点标签绝对差值之和，模顶点数。
• 通过将边权重纳入Jaccard相似性和CBS计算，将该方法扩展至加权图。
• 通过多次运行的CBS值的中位数和中位数绝对偏差（MAD）评估启发式的鲁棒性。
• 该方法在基准图（如环形图、社区图）、真实世界网络以及Harwell-Boeing数据集上进行了测试。

实验结果

研究问题

• RQ1 能否开发一种启发式方法，以近似一般无向、无权图的最优循环带宽和？
• RQ2 所提出的标记在顶点排序方面在多大程度上保留了复杂网络的结构拓扑？
• RQ3 与随机或任意标记相比，该方法在信号平滑度和可视化质量方面的表现如何？
• RQ4 该方法能否有效扩展至加权图，同时保持低CBS值？
• RQ5 该启发式方法在具有已知最优解的图上是否能达到接近最优的CBS值？

主要发现

• 所提出的启发式方法在环形图上实现了最优CBS值（例如，50个顶点的环形图为11236），证实了在已知基准上的正确性。
• 对于含四个社区的图，该方法在k=10时的CBS中位数为26338，与已知最优值26292非常接近。
• 在Harwell-Boeing数据集中，该方法在多次运行中始终实现已知最优CBS，中位数CBS等于最小值（例如，第一张图为11236）。
• 对于无标度（SF）和小世界（SW）随机图，CBS中位数接近理论最小值，标准差较低（例如，k=10时SF图为373 ± 7）。
• 该方法显著提升了信号平滑度：在环形图上，合理标记可产生平滑的谐波信号，而随机标记则导致高频噪声。
• 在可视化任务中，所提出的标记减少了边交叉，并增强了圆形布局的清晰度，如图1和图2所示。