Besides the complexity in time or in number of messages, a common approach for analyzing distributed algorithms is to look at the assumptions they make on the underlying network. We investigate this question from the perspective of network dynamics. In particular, we ask how a given property on the evolution of the network can be rigorously proven as necessary or sufficient for a given algorithm. The main contribution of this paper is to propose the combination of two existing tools in this direction: local computations by means of graph relabelings, and evolving graphs. Such a combination makes it possible to express fine-grained properties on the network dynamics, then examine what impact those properties have on the execution at a precise, intertwined, level. We illustrate the use of this framework through the analysis of three simple algorithms, then discuss general implications of this work, which include (i) the possibility to compare distributed algorithms on the basis of their topological requirements, (ii) a formal hierarchy of dynamic networks based on these requirements, and (iii) the potential for mechanization induced by our framework, which we believe opens a door towards automated analysis and decision support in dynamic networks.
In a different context, @cite_2 were proposed as a combinatorial model for dynamic networks. The initial purpose of this model was to provide a suitable representation of (FSDNs), in order to compute optimal communication routes such as shortest, fastest and foremost journeys @cite_9 . In such a context, the evolution of the network was known beforehand. In the present work, we use evolving graphs in a very different purpose, which is to express properties on the network dynamics. It is important to keep in mind that the analyzed algorithms are never supposed to know the evolution of the network beforehand.
@cite_9 @cite_2
1984196269 1982459859
New technologies and the deployment of mobile and nomadic services are driving the emergence of complex communications networks, that have a highly dynamic behavior. This naturally engenders new route-discovery problems under changing conditions over these networks. In this paper, we use and extend a recently proposed graph theoretic model, which helps capture the evolving characteristic of such networks, in order to propose and formally analyze least cost journeys (the analog of paths in usual graphs) in a class of dynamic networks, where the changes in the topology can be predicted in advance. Cost measures investigated here are hop count (shortest journeys), arrival date (foremost journeys), and time span (fastest journeys). New technologies and the deployment of mobile and nomadic services are driving the emergence of complex communications networks, that have a highly dynamic behavior. This naturally engenders new route-discovery problems under changing conditions over these networks. Unfortunately, the temporal variations in the network topology are hard to be effectively captured in a classical graph model. In this paper, we use and extend a recently proposed graph theoretic model, which helps capture the evolving characteristic of such networks, in order to propose and formally analyze least cost journey (the analog of paths in usual graphs) in a class of dynamic networks, where the changes in the topology can be predicted in advance. Cost measures investigated here are hop count (shortest journeys), arrival date (foremost journeys), and time span (fastest journeys). Most highly dynamic infrastructure-less networks have in common that the assumption of connectivity does not necessarily hold at a given instant. Still, communication routes can be available between any pair of nodes over time and space. These networks (variously called delay-tolerant, disruptive-tolerant, challenged) are naturally modeled as time-varying graphs (or evolving graphs), where the existence of an edge is a function of time. In this paper we study deterministic computations under unstructured mobility, that is when the edges of the graph appear infinitely often but without any (known) pattern. In particular, we focus on the problem of broadcasting with termination detection. We explore the problem with respect to three possible metrics: the date of message arrival (foremost), the time spent doing the broadcast (fastest), and the number of hops used by the broadcast (shortest). We prove that the solvability and complexity of this problem vary with the metric considered, as well as with the type of knowledge a priori available to the entities. These results draw a complete computability map for this problem when mobility is unstructured. The number of telecommunication networks deployed in a dynamic environment is quickly growing. This trend exists both in everyday life (e.g., smartphones, vehicles, and commercial satellites) and in a military context (e.g., dismounted soldiers or swarms of UAVs). Unfortunately, few theoretical tools to date have enabled the study of dynamic networks in a formal and rigorous way. As a result, it is hard and sometimes impossible to guarantee, mathematically, that a given algorithm will reach its objectives once deployed in real conditions. In this report, we identify a collection of recent theoretical tools whose purpose is to model, describe, and leverage dynamic networks in a formal way. These tools include a dynamic graph formalism, various computational models, and communication models for distributed networks. We extend many graph theoretical concepts towards a dynamic variant and show how these new variants impact the solution of classical distributed problems. The report also presents a hierarchy of dynamic networks based on dynamic graph properties, thereby offering a combinatorial alternative to the well-known mobility models typically used in simulations.