Development of algorithms for the conception and the analysis of graphs and complex networks

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A graph is a mathematical object consisting of vertices connected by edges if necessary. Graphs are useful for modeling some situations or to facilitate the visualization of complex data. ******My research project is related to the graphs, namely the search graph having desired characteristics, the study of particular graphs and the representation of complex information using graphs. ******1. Searching for extreme graphs ******Since my Ph.D, I work on finding extreme graphs, graphs that have special properties. Besides allowing to identify mathematical properties of graphs or finding results in mathematical chemistry (molecules may be represented by graphs), this approach recently allowed one of my students during his Ph.D., to find a family of graphs that correspond to a robust telecommunication networks. We plan to continue in this direction and use optimization for the design of telecommunications networks. There are still relevant applications in that field, however I am now considering extending my research area to some other, related problems. ******2. The study of complex networks ******Complex networks, such as social networks are typically graphs of large size. The complex network analysis is primarily based on two aspects: the identification of important vertices and communities detection. Identification of important vertices is based upon the centrality and allows the identification of individuals who have a key role. An application may be the viral marketing problem where the goal is to identify influent individuals. Communities detection in a network is an application of clustering, an area where I work for years. ******3. The representation of complex information ******The last point, the representation of complex information, is an area in which I entered during a recent work on the representation of the writing process. This process includes a temporal component and a spatial one. Some programs allow the recording of the activity of a person while she writes with a word processor or by hand with a graphical tablet. The records yield a large amount of data that is difficult to analyse by a human. We found that such data may be modelled by a graph. This is a new representation technique that is perceived by linguists, psychologists and educationalists as an advance toward a better understanding of the phenomenon. There are reasons to believe that such a representation could be extended to other processes involving space and time, with possible applications in industrial engineering.