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Getting a labeling of vertices close to the structure of the graph has beenproved to be of interest in many applications e.g., to follow smooth signalsindexed by the vertices of the network. This question can be related to a graphlabeling problem known as the cyclic bandwidth sum problem. It consists infinding a labeling of the vertices of an undirected and unweighted graph withdistinct integers such that the sum of (cyclic) difference of labels ofadjacent vertices is minimized. Although theoretical results exist that giveoptimal value of cyclic bandwidth sum for standard graphs, there are neitherresults 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 beenproposed to approximately solve this problem. In this paper, we introduce a newheuristic 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 whichfollow the structure of the graph, using a similarity criterion based on theJaccard index to jump from one vertex to the next one. The second step is themerging of all obtained paths, based on a greedy approach that extends apartial solution by inserting a new path at the position that minimizes thecyclic bandwidth sum. The effectiveness of the proposed heuristic, both interms of performance and time execution, is shown through experiments on graphswhose optimal value of CBS is known as well as on real-world networks, wherethe consistence between labeling and topology is highlighted. An extension toweighted graphs is also proposed.

Discovering the structure of complex networks by minimizing cyclic bandwidth sum