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## Section: New Results

### Model for Time-Varying Graphs.

Participant : Éric Fleury.

We propose a novel model for representing finite discrete Time-Varying Graphs (TVGs). The major application of such a model is for the modelling and representation of dynamic networks. In our proposed model, an edge is able to connect a node $u$ at a given time instant ${t}_{a}$ to any other node $v$ ($u$ possibly equal to $v$) at any other time instant ${t}_{b}$ (${t}_{a}$ possibly equal to ${t}_{b}$), leading to the concept that such an edge can be represented by an ordered quadruple of the form $\left(u,{t}_{a},v,{t}_{b}\right)$. Building upon this basic concept, our proposed model defines a TVG as an object $H=\left(V,E,T\right)$, where $V$ is the set of nodes, $E\subseteq V×T×V×T$ is the set of edges, and $T$ is the finite set of time instants on which the TVG is defined. We show how key concepts, such as degree, path, and connectivity, are handled in our model. We also analyse the data structures used for the representation of dynamic networks built following our proposed model and demonstrate that, for most practical cases, the asymptotic memory complexity of our TVG representation model is determined by the cardinality of the set of edges. (See [20] )