Periodic Reporting for period 1 - PROXNET (Modelling Complex Networks Through Graph Editing Problems)
Reporting period: 2018-02-01 to 2020-01-31
The goal of the PROXNET project is to open a new way for analysing, modelling and managing complex networks, through graph editing problems. The reason why these networks are said to be complex is that they are loosely structured, due to the part of uncertainty and randomness they contain. On the other hand, the real-world context where they come from strongly constrains their organisation and gives them some specific structure. The difficulty in retrieving this structure is that it is altered by the noise resulting from the uncertainty and randomness that these networks contain. In the PROXNET project, we retrieve the hidden structures of complex networks thanks to graph editing problems, which consist in changing some adjacencies of the graph in order to obtain a desired property. We develop the algorithms necessary to solve graph editing problems on huge instances of graphs, we apply them to real-world datasets and use the results obtained in order to design new models of complex networks.
We designed an algorithm for computing a minimal editing of an aribtrary graph into a cograph that runs in time linear with respect to the size of the input graph. This complexity, which is optimal, could not be achieved for any other non-trivial editing problem before. We implemented our algorithm and tested it on real-world complex networks. We thereby obtained two main results. Firstly, our algorithm is able to treat real-world networks of very large size, up to a dozen of millions of edges, and secondly, there are some complex networks that are close to being cographs, i.e. only a limited proportion of their adjacencies need to be changed in order to turn them into a cograph. These observations proves the soundnes and the interest of the leading idea of the PROXNET project: representing complex networks as almost structured graphs.
The results we obtain open the way to a new line of research in the field of complex networks. Thanks to the algorithm we developped, it is now feasible, in practice, to compare the structure of real-world networks to the structure of some mathematically defined class of graphs, even for real-world networks of very large size. This paves the way for similar works for other target classes of graphs. Moreover, the first tests we made show that some complex networks are surprisingly close to having such a perfect mathematical structure. This observation opens many exciting possibilities for analysing, modelling, understanding and exploiting these networks, in the parctical contexts where they come from.