Clémence Magnien

Computing maximal cliques in link streams

We introduce delta-cliques, that generalize graph cliques to link streams/time-varying graphs.We provide a greedy algorithm to compute all delta-cliques of a link stream.Implementation available on http://www.github.com/JordanV/delta-cliques. A link stream is a collection of triplets ( t , u , v ) indicating that an interaction occurred between u and v at time t. We generalize the classical notion of cliques in graphs to such link streams: for a given Δ, a Δ-clique is a set of nodes and a time interval such that all pairs of nodes in this set interact at least once during each sub-interval of duration Δ. We propose an algorithm to enumerate all maximal (in terms of nodes or time interval) cliques of a link stream, and illustrate its practical relevance to a real-world contact trace.

Outskewer: Using Skewness to Spot Outliers in Samples and Time Series

Finding outliers in datasets is a classical problem of high interest for (dynamic) social network analysis. However, most methods rely on assumptions which are rarely met in practice, such as prior knowledge of some outliers or about normal behavior. We propose here Out skewer, a new approach based on the notion of skewness (a measure of the symmetry of a distribution) and its evolution when extremal values are removed one by one. Our method is easy to set up, it requires no prior knowledge on the system, and it may be used on-line. We illustrate its performance on two data sets representative of many use-cases: evolution of ego-centered views of the internet topology, and logs of queries entered into a search engine.

Stream Graphs and Link Streams for the Modeling of Interactions over Time

Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the intrinsically temporal and structural nature of interactions, which calls for a dedicated formalism. In this paper, we generalize graph concepts to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher level objects such as quotient graphs, line graphs, k-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex cases.

Impact of power-law topology on IP-level routing dynamics: Simulation results

This paper focuses on the Internet IP-level routing topology and proposes relevant explanations to its apparent dynamics.We first represent this topology as a power-law random graph. Then, we incorporate to the graph two well known factors responsible for the observed dynamics, which are load balancing and route evolution. Finally, we simulate on the graph traceroute-like measurements. Repeating the process many times, we obtain several graph instances that we use to model the dynamics. Our results show that we are able to capture on power-law graphs the dynamic behaviors observed on the Internet. We find that the results on power-law graphs, while qualitatively similar to the one of Erdös-Rényi random graphs, highly differ quantitatively; for instance, the rate of discovery of new nodes in power-law graphs is extremely low compared to the rate in Erdös-Rényi graphs.

Classes of lattices induced by chip firing (and sandpile) dynamics

In this paper we study three classes of models widely used in physics, computer science and social science: the chip firing game (CFG), the Abelian sandpile model (ASM) and the CFG on a mutating graph (or mutating CFG (MCFG)). These three models are variations of a game on a graph, in the vertices of which are stored chips, and where vertices can be fired, sending one chip along each outgoing edge. We study the set of configurations reachable from a given initial configuration, called the configuration space of a model. It is known that the order induced over the configurations by the evolution rule is a lattice, a special kind of partially ordered set. Our aim is to compare for inclusion the classes of lattices induced by these models. We show that the MCFG and the CFG induce exactly the same class of lattices, and we give new results towards the characterization of the class of lattices induced by the ASM.

Finding remarkably dense sequences of contacts in link streams

A link stream is a set of quadruplets (b, e, u, v) meaning that a link exists between u and v from time b to time e. Link streams model many real-world situations like contacts between individuals, connections between devices, and others. Much work is currently devoted to the generalization of classical graph and network concepts to link streams. We argue that the density is a valuable notion for understanding and characterizing links streams. We propose a method to capture specific groups of links that are structurally and temporally densely connected and show that they are meaningful for the description of link streams. To find such groups, we use classical graph community detection algorithms, and we assess obtained groups. We apply our method to several real-world contact traces (captured by sensors) and demonstrate the relevance of the obtained structures.

Internal links and pairs as a new tool for the analysis of bipartite complex networks

Many real-world complex networks are best modeled as bipartite (or 2-mode) graphs, where nodes are divided into two sets with links connecting one side to the other. However, there is currently a lack of methods to analyze properly such graphs as most existing measures and methods are suited to classical graphs. A usual but limited approach consists in deriving 1-mode graphs (called projections) from the underlying bipartite structure, though it causes important loss of information and data storage issues. We introduce here internal links and pairs as a new notion useful for a bipartite analysis, which gives insights into the information lost by projecting the bipartite graph. We illustrate the relevance of these concepts in several real-world instances, illustrating how it enables to discriminate behaviors among various cases when we compare them to a benchmark of random graphs. Then, we show that we can draw benefit from this concept for both modeling complex networks and storing them in a compact format.

Coding distributive lattices with edge firing games

In this note, we show that any distributive lattice is isomorphic to the set of reachable configurations of an Edge Firing Game. Together with the result of James Propp, saying that the set of reachable configurations of any Edge Firing Game is always a distributive lattice, this shows that the two concepts are equivalent.

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

Enumerating maximal cliques in link streams with durations

(t_1 , t_2 , u, v)