Temporal Reachability Graphs

While a natural fit for modeling and understanding mobile networks, time-varying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in time-varying ones. In this paper, we introduce the notion of temporal reachability graphs. A (tau,delta)-reachability graph} is a time-varying directed graph derived from an existing connectivity graph. An edge exists from one node to another in the reachability graph at time t if there exists a journey (i.e., a spatiotemporal path) in the connectivity graph from the first node to the second, leaving after t, with a positive edge traversal time tau, and arriving within a maximum delay delta. We make three contributions. First, we develop the theoretical framework around temporal reachability graphs. Second, we harness our theoretical findings to propose an algorithm for their efficient computation. Finally, we demonstrate the analytic power of the temporal reachability graph concept by applying it to synthetic and real-life datasets. On top of defining clear upper bounds on communication capabilities, reachability graphs highlight asymmetric communication opportunities and offloading potential.Comment: In proceedings ACM Mobicom 201

Graph Metrics for Temporal Networks

Temporal networks, i.e., networks in which the interactions among a set of elementary units change over time, can be modelled in terms of time-varying graphs, which are time-ordered sequences of graphs over a set of nodes. In such graphs, the concepts of node adjacency and reachability crucially depend on the exact temporal ordering of the links. Consequently, all the concepts and metrics proposed and used for the characterisation of static complex networks have to be redefined or appropriately extended to time-varying graphs, in order to take into account the effects of time ordering on causality. In this chapter we discuss how to represent temporal networks and we review the definitions of walks, paths, connectedness and connected components valid for graphs in which the links fluctuate over time. We then focus on temporal node-node distance, and we discuss how to characterise link persistence and the temporal small-world behaviour in this class of networks. Finally, we discuss the extension of classic centrality measures, including closeness, betweenness and spectral centrality, to the case of time-varying graphs, and we review the work on temporal motifs analysis and the definition of modularity for temporal graphs.Comment: 26 pages, 5 figures, Chapter in Temporal Networks (Petter Holme and Jari Saram\"aki editors). Springer. Berlin, Heidelberg 201

From RT-LOTOS to Time Petri Nets new foundations for a verification platform

The formal description technique RT-LOTOS has been selected as intermediate language to add formality to a real-time UML profile named TURTLE. For this sake, an RT-LOTOS verification platform has been developed for early detection of design errors in real-time system models. The paper discusses an extension of the platform by inclusion of verification tools developed for Time Petri Nets. The starting point is the definition of RT-LOTOS to TPN translation patterns. In particular, we introduce the concept of components embedding Time Petri Nets. The translation patterns are implemented in a prototype tool which takes as input an RT-LOTOS specification and outputs a TPN in the format admitted by the TINA tool. The efficiency of the proposed solution has been demonstrated on various case studies

Efficiently answering reachability and path queries on temporal bipartite graphs

Bipartite graphs are naturally used to model relationships between two different types of entities, such as people-location, authorpaper, and customer-product. When modeling real-world applications like disease outbreaks, edges are often enriched with temporal information, leading to temporal bipartite graphs. While reachability has been extensively studied on (temporal) unipartite graphs, it remains largely unexplored on temporal bipartite graphs. To fill this research gap, in this paper, we study the reachability problem on temporal bipartite graphs. Specifically, a vertex u reaches a vertex w in a temporal bipartite graph G if u and w are connected through a series of consecutive wedges with time constraints. Towards efficiently answering if a vertex can reach the other vertex, we propose an index-based method by adapting the idea of 2-hop labeling. Effective optimization strategies and parallelization techniques are devised to accelerate the index construction process. To better support real-life scenarios, we further show how the index is leveraged to efficiently answer other types of queries, e.g., singlesource reachability query and earliest-arrival path query. Extensive experiments on 16 real-world graphs demonstrate the effectiveness and efficiency of our proposed techniques

Simple, strict, proper, happy: A study of reachability in temporal graphs

Dynamic networks are a complex subject. Not only do they inherit the complexity of static networks (as a particular case); they are also sensitive to definitional subtleties that are a frequent source of confusion and incomparability of results in the literature. In this paper, we take a step back and examine three such aspects in more details, exploring their impact in a systematic way; namely, whether the temporal paths are required to be \emph{strict} (i.e., the times along a path must increasing, not just be non-decreasing), whether the time labeling is \emph{proper} (two adjacent edges cannot be present at the same time) and whether the time labeling is \emph{simple} (an edge can have only one presence time). In particular, we investigate how different combinations of these features impact the expressivity of the graph in terms of reachability. Our results imply a hierarchy of expressivity for the resulting settings, shedding light on the loss of generality that one is making when considering either combination. Some settings are more general than expected; in particular, proper temporal graphs turn out to be as expressive as general temporal graphs where non-strict paths are allowed. Also, we show that the simplest setting, that of \emph{happy} temporal graphs (i.e., both proper and simple) remains expressive enough to emulate the reachability of general temporal graphs in a certain (restricted but useful) sense. Furthermore, this setting is advocated as a target of choice for proving negative results. We illustrates this by strengthening two known results to happy graphs (namely, the inexistence of sparse spanners, and the hardness of computing temporal components). Overall, we hope that this article can be seen as a guide for choosing between different settings of temporal graphs, while being aware of the way these choices affect generality.Comment: 20 pages, 6 figure