## Section: Scientific Foundations

### Theory and Structural Dynamic Properties of dynamic Networks

Participants : Guillaume Chelius, Eric Fleury, Qinna Wang.

- Characterization of the dynamics of complex networks.
We need to focus on intrinsic properties of evolving/dynamic complex networks. New notions (as opposed to classical static graph properties) have to be introduced: rate of vertices or links appearances or disappearances, the duration of link presences or absences. Moreover, more specific properties related to the dynamics have to be defined and are somehow related to the way to model a dynamical graph.

To go further in the Classical graph notions like the definition of path, connected components and k -core have to be revisited in this context. Advanced properties need also to be defined in order to apprehend the intrinsic dynamic structural issues of such dynamic graphs. The notion of communities (dense group of nodes) is important in any social / interaction network context and may play an important role within an epidemic process. To transpose the static graph community concept into the dynamical graph framework is a challenging task and appears necessary in order to better understand how the structure of graphs evolves in time. In these contexte we define the following objectives:

**Toward a dynamic graph model and theory.**We want to design new notions, methods and models for the analysis of dynamic graphs. For the static case, graph theory has defined a vast and consistent set of notions and methods such as degrees, paths, centrality measures, cliques. These notions and methods are completely lacking for the study of dynamic graphs; not only do we not have basic notions allowing the description of a graph and its dynamics, but we also lack the basic definitions and algorithms for manipulating dynamic graphs. Our goal is therefore to fill this lack, by providing the basic notions for manipulating dynamic graphs as well as the notions and indicators to describe its dynamic meaningfully (as complex networks theory does for static complex networks).

**Dynamic communities.**The detection of dynamic communities is particularly appealing to describe dynamic networks. In order to extend the static case, one may apply existing community detection methods to successive snapshots of dynamic networks. This is however not totally satisfying for two main reasons: first, this would take a large amount of time (directly proportional to the data span); moreover, having a temporal succession of independent communities is not sufficient and we loose valuable information and dependencies. We also need to investigate the temporal links, study the time granularity and look for time periods that could be compressed within a single snapshot.

**Tools for dynamic graph visualization.**Designing generic and pure graph visualization tools is clearly out of the scope of the D-NET project. Efficient graph drawing tools or network analysis toolkit/software are now available (e.g., GUESS, TULIP, Sonivis, Network Workbench). However, the drawback of most softwares is that the dynamics is not taken into account. Since we will study the hierarchy of dynamics through the definition of communities we plan to extend graph drawing methods by using the communities' structures. We also plan to handle the time evolution in the network analysis toolkit. A tool like TULIP is well designed and could be improved by allowing operations (selection, grouping, sub graph computation...) to take place on the time dimension as well.