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

Exploring Graphs with Time Constraints by Unreliable Collections of Mobile Robots

A graph environment must be explored by a collection of mobile robots. Some of the robots, a priori unknown, may turn out to be unreliable. The graph is weighted and each node is assigned a deadline. The exploration is successful if each node of the graph is visited before its deadline by a reliable robot. The edge weight corresponds to the time needed by a robot to traverse the edge. Given the number of robots which may crash, is it possible to design an algorithm, which will always guarantee the exploration, independently of the choice of the subset of unreliable robots by the adversary? We find the optimal time, during which the graph may be explored. Our approach permits to find the maximal number of robots, which may turn out to be unreliable, and the graph is still guaranteed to be explored. We concentrate on line graphs and rings, for which we give positive results. We start with the case of the collections involving only reliable robots. We give algorithms finding optimal times needed for exploration when the robots are assigned to fixed initial positions as well as when such starting positions may be determined by the algorithm. We extend our consideration to the case when some number of robots may be unreliable. Our most surprising result is that solving the line exploration problem with robots at given positions, which may involve crash-faulty ones, is NP-hard. The same problem has polynomial solutions for a ring and for the case when the initial robots' positions on the line are arbitrary. The exploration problem is shown to be NP-hard for star graphs, even when the team consists of only two reliable robots

The Next 700 Impossibility Results in Time-Varying Graphs

We address highly dynamic distributed systems modeled by time-varying graphs (TVGs). We interest in proof of impossibility results that often use informal arguments about convergence. First, we provide a distance among TVGs to define correctly the convergence of TVG sequences. Next, we provide a general framework that formally proves the convergence of the sequence of executions of any deterministic algorithm over TVGs of any convergent sequence of TVGs. Finally, we illustrate the relevance of the above result by proving that no deterministic algorithm exists to compute the underlying graph of any connected-over-time TVG, i.e., any TVG of the weakest class of long-lived TVGs

Computational Controversy

Climate change, vaccination, abortion, Trump: Many topics are surrounded by fierce controversies. The nature of such heated debates and their elements have been studied extensively in the social science literature. More recently, various computational approaches to controversy analysis have appeared, using new data sources such as Wikipedia, which help us now better understand these phenomena. However, compared to what social sciences have discovered about such debates, the existing computational approaches mostly focus on just a few of the many important aspects around the concept of controversies. In order to link the two strands, we provide and evaluate here a controversy model that is both, rooted in the findings of the social science literature and at the same time strongly linked to computational methods. We show how this model can lead to computational controversy analytics that have full coverage over all the crucial aspects that make up a controversy.Comment: In Proceedings of the 9th International Conference on Social Informatics (SocInfo) 201

Storage and Search in Dynamic Peer-to-Peer Networks

We study robust and efficient distributed algorithms for searching, storing, and maintaining data in dynamic Peer-to-Peer (P2P) networks. P2P networks are highly dynamic networks that experience heavy node churn (i.e., nodes join and leave the network continuously over time). Our goal is to guarantee, despite high node churn rate, that a large number of nodes in the network can store, retrieve, and maintain a large number of data items. Our main contributions are fast randomized distributed algorithms that guarantee the above with high probability (whp) even under high adversarial churn: 1. A randomized distributed search algorithm that (whp) guarantees that searches from as many as $n - o(n)$ nodes ($n$ is the stable network size) succeed in ${O}(\log n)$-rounds despite ${O}(n/\log^{1+\delta} n)$ churn, for any small constant $\delta > 0$, per round. We assume that the churn is controlled by an oblivious adversary (that has complete knowledge and control of what nodes join and leave and at what time, but is oblivious to the random choices made by the algorithm). 2. A storage and maintenance algorithm that guarantees (whp) data items can be efficiently stored (with only $\Theta(\log{n})$ copies of each data item) and maintained in a dynamic P2P network with churn rate up to ${O}(n/\log^{1+\delta} n)$ per round. Our search algorithm together with our storage and maintenance algorithm guarantees that as many as $n - o(n)$ nodes can efficiently store, maintain, and search even under ${O}(n/\log^{1+\delta} n)$ churn per round. Our algorithms require only polylogarithmic in $n$ bits to be processed and sent (per round) by each node. To the best of our knowledge, our algorithms are the first-known, fully-distributed storage and search algorithms that provably work under highly dynamic settings (i.e., high churn rates per step).Comment: to appear at SPAA 201

Computing maximal cliques in link streams

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 $\Delta$, a $\Delta$-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 $\Delta$. 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 on a real-world contact trace

Dynamic Monopolies in Colored Tori

The {\em information diffusion} has been modeled as the spread of an information within a group through a process of social influence, where the diffusion is driven by the so called {\em influential network}. Such a process, which has been intensively studied under the name of {\em viral marketing}, has the goal to select an initial good set of individuals that will promote a new idea (or message) by spreading the "rumor" within the entire social network through the word-of-mouth. Several studies used the {\em linear threshold model} where the group is represented by a graph, nodes have two possible states (active, non-active), and the threshold triggering the adoption (activation) of a new idea to a node is given by the number of the active neighbors. The problem of detecting in a graph the presence of the minimal number of nodes that will be able to activate the entire network is called {\em target set selection} (TSS). In this paper we extend TSS by allowing nodes to have more than two colors. The multicolored version of the TSS can be described as follows: let $G$ be a torus where every node is assigned a color from a finite set of colors. At each local time step, each node can recolor itself, depending on the local configurations, with the color held by the majority of its neighbors. We study the initial distributions of colors leading the system to a monochromatic configuration of color $k$, focusing on the minimum number of initial $k$-colored nodes. We conclude the paper by providing the time complexity to achieve the monochromatic configuration

Reliable Communication in a Dynamic Network in the Presence of Byzantine Faults

We consider the following problem: two nodes want to reliably communicate in a dynamic multihop network where some nodes have been compromised, and may have a totally arbitrary and unpredictable behavior. These nodes are called Byzantine. We consider the two cases where cryptography is available and not available. We prove the necessary and sufficient condition (that is, the weakest possible condition) to ensure reliable communication in this context. Our proof is constructive, as we provide Byzantine-resilient algorithms for reliable communication that are optimal with respect to our impossibility results. In a second part, we investigate the impact of our conditions in three case studies: participants interacting in a conference, robots moving on a grid and agents in the subway. Our simulations indicate a clear benefit of using our algorithms for reliable communication in those contexts