Ronan Hamon

Networks as signals, with an application to a bike sharing system

Dynamic graphs are commonly used for describing networks with a time evolution. A method has been proposed to transform these graphs into a collection of signals indexed by vertices. This approach is here further explored in a number of different directions. First, the importance of a good indexing of a graph is stressed, and a solution is proposed using a node labeling algorithm which follows the structure of the graph. Second, a spectral analysis of identified signals is performed to compute features linked to graph properties such as regularity or structure in communities. Finally, these features can be tracked over time to evidence the structure evolution of the graph. As a case study, the approach is applied to a dynamic graph based on a dataset of trips made using the bike sharing system Vlov in use in Lyon, France. This is shown to offer specific insights on behaviors of bike users over time in two districts of the city.

Nonnegative matrix factorization to find features in temporal networks

Temporal networks describe a large variety of systems having a temporal evolution. Characterization and visualization of their evolution are often an issue especially when the amount of data becomes huge. We propose here an approach based on the duality between graphs and signals. Temporal networks are represented at each time instant by a collection of signals, whose spectral analysis reveals connection between frequency features and structure of the network. We use nonnegative matrix factorization (NMF) to find these frequency features and track them along time. Transforming back these features into subgraphs reveals the underlying structures which form a decomposition of the temporal network.

Extraction of Temporal Network Structures From Graph-Based Signals

A new framework to track the structure of temporal networks with a signal processing approach is introduced. The method is based on the duality between static networks and signals, obtained using a multidimensional scaling technique, that makes possible the study of the network structure from frequency patterns of the corresponding signals. In this paper, we propose an approach to identify structures in temporal networks by extracting the most significant frequency patterns and their activation coefficients over time, using non-negative matrix factorization of the temporal spectra. The framework, inspired by audio decomposition, allows transforming back these frequency patterns into networks, to highlight the evolution of the underlying structure of the network over time. The effectiveness of the method is first evidenced on a synthetic example, prior being used to study a temporal network of face-to-face contacts. The extracted subnetworks highlight significant structures decomposed on time intervals that validates the relevance of the approach on real-world data.

Relabelling vertices according to the network structure by minimizing the cyclic bandwidth sum

Getting a labelling of vertices close to the structure of the graph has been proved to be of interest in many applications, e.g. to follow signals indexed by the vertices of the network. This question can be related to a graph labelling problem known as the cyclic bandwidth sum problem (CBSP). It consists of finding a labelling 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. In this paper, we introduce a new heuristic to follow the structure of the graph, by finding an approximate solution for the CBSP. Although theoretical results exist that give optimal value of cyclic bandwidth sum (CBS) for standard graphs, there are neither results for real-world complex networks, nor explicit methods to reach this optimal result. Furthermore, only a few methods have been proposed to approximately solve this problem. The heuristic we propose 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 CBS. The effectiveness of the proposed heuristic is shown through experiments on graphs whose optimal value of CBS is known, as well as on complex networks, where the consistency between labelling and topology is highlighted.