A Unifying Model for Representing Time-Varying Graphs

Graph-based models form a fundamental aspect of data representation in Data Sciences and play a key role in modeling complex networked systems. In particular, recently there is an ever-increasing interest in modeling dynamic complex networks, i.e. networks in which the topological structure (nodes and edges) may vary over time. In this context, we propose a novel model for representing finite discrete Time-Varying Graphs (TVGs), which are typically used to model dynamic complex networked systems. We analyze the data structures built from our proposed model and demonstrate that, for most practical cases, the asymptotic memory complexity of our model is in the order of the cardinality of the set of edges. Further, we show that our proposal is an unifying model that can represent several previous (classes of) models for dynamic networks found in the recent literature, which in general are unable to represent each other. In contrast to previous models, our proposal is also able to intrinsically model cyclic (i.e. periodic) behavior in dynamic networks. These representation capabilities attest the expressive power of our proposed unifying model for TVGs. We thus believe our unifying model for TVGs is a step forward in the theoretical foundations for data analysis of complex networked systems.Comment: Also appears in the Proc. of the IEEE International Conference on Data Science and Advanced Analytics (IEEE DSAA'2015

A Unifying Model for Representing Time-Varying Graphs

We propose a novel model for representing finite discrete Time-Varying Graphs~(TVGs). We show how key concepts, such as degree, path, and connectivity, are handled in our model. We also analyze the data structures built following our proposed model and demonstrate that, for most practical cases, the asymptotic memory complexity of our model is restricted to the cardinality of the set of edges. Moreover, we prove that if the TVG nodes can be considered as independent entities at each time instant, the analyzed TVG is isomorphic to a directed static graph. This is an important theoretical result since this allows the use of the isomorphic directed graph as a tool to analyze both the properties of a TVG and the behavior of dynamic processes over a TVG. We also show that our unifying model can represent several previous (classes of) models for dynamic networks found in the recent literature, which in general are unable to represent each other. In contrast to previous models, our proposal is also able to intrinsically model cyclic~(i.e. periodic) behavior in dynamic networks. These representation capabilities attest the expressive power of our proposed unifying model for TVGs.Nous proposons un modèle (TVG pour \emph{Time-Varying Graphs}) pour représenter les graphes dynamiques (\emph{i.e.}, des graphes susceptibles d'évoluer au cours du temps). Nous montrons qu elles définitionsclefs comme le degré, la notion de chemin, de connectivité sont prise en compte par ce modèle. Une analyse de la complexité des structures de données nécessaire à la représentation de ce modèle montre que la complexité asymptotique est en $O(m)$ (cardinalité du nombre d'arêtes du graphe dynamique). Si les sommets d'un TVG peuvent être considérés comme des entités indépendantes à chaque instant, alors on démontre que le graphe TVG est isomorphe à un graphe orienté static. Notre modèle permet de représenter et de prendre en compte les différentes propositions existantes qui n'étaient pas en mesure de se représenter les unes les autres

Time-Varying Graphs and Dynamic Networks

The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems -- delay-tolerant networks, opportunistic-mobility networks, social networks -- obtaining closely related insights. Indeed, the concepts discovered in these investigations can be viewed as parts of the same conceptual universe; and the formal models proposed so far to express some specific concepts are components of a larger formal description of this universe. The main contribution of this paper is to integrate the vast collection of concepts, formalisms, and results found in the literature into a unified framework, which we call TVG (for time-varying graphs). Using this framework, it is possible to express directly in the same formalism not only the concepts common to all those different areas, but also those specific to each. Based on this definitional work, employing both existing results and original observations, we present a hierarchical classification of TVGs; each class corresponds to a significant property examined in the distributed computing literature. We then examine how TVGs can be used to study the evolution of network properties, and propose different techniques, depending on whether the indicators for these properties are a-temporal (as in the majority of existing studies) or temporal. Finally, we briefly discuss the introduction of randomness in TVGs.Comment: A short version appeared in ADHOC-NOW'11. This version is to be published in Internation Journal of Parallel, Emergent and Distributed System

Discovering Patterns of Interest in IP Traffic Using Cliques in Bipartite Link Streams

Studying IP traffic is crucial for many applications. We focus here on the detection of (structurally and temporally) dense sequences of interactions, that may indicate botnets or coordinated network scans. More precisely, we model a MAWI capture of IP traffic as a link streams, i.e. a sequence of interactions $(t_1 , t_2 , u, v)$ meaning that devices $u$ and $v$ exchanged packets from time $t_1$ to time $t_2$ . This traffic is captured on a single router and so has a bipartite structure: links occur only between nodes in two disjoint sets. We design a method for finding interesting bipartite cliques in such link streams, i.e. two sets of nodes and a time interval such that all nodes in the first set are linked to all nodes in the second set throughout the time interval. We then explore the bipartite cliques present in the considered trace. Comparison with the MAWILab classification of anomalous IP addresses shows that the found cliques succeed in detecting anomalous network activity