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
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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 topology of dynamic networks are hard to be effectively captured in a classical graph model. In this paper, we use evolving graphs, which helps capture the dynamic characteristics of such networks, in order to compute multicast trees with minimum overall transmission time for a class of wireless mobile dynamic networks. We first show that computing different types of strongly connected components in evolving digraphs is NP-Complete, and then propose an algorithm to build all rooted directed minimum spanning trees in strongly connected dynamic networks. 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). The assessment of routing protocols for mobile wireless networks is a difficult task, because of the networks' dynamic behavior and the absence of benchmarks. However, some of these networks, such as intermittent wireless sensors networks, periodic or cyclic networks, and some delay tolerant networks (DTNs), have more predictable dynamics, as the temporal variations in the network topology can be considered as deterministic, which may make them easier to study. Recently, a graph theoretic model--the evolving graphs--was proposed to help capture the dynamic behavior of such networks, in view of the construction of least cost routing and other algorithms. The algorithms and insights obtained through this model are theoretically very efficient and intriguing. However, there is no study about the use of such theoretical results into practical situations. Therefore, the objective of our work is to analyze the applicability of the evolving graph theory in the construction of efficient routing protocols in realistic scenarios. In this paper, we use the NS2 network simulator to first implement an evolving graph based routing protocol, and then to use it as a benchmark when comparing the four major ad hoc routing protocols (AODV, DSR, OLSR and DSDV). Interestingly, our experiments show that evolving graphs have the potential to be an effective and powerful tool in the development and analysis of algorithms for dynamic networks, with predictable dynamics at least. In order to make this model widely applicable, however, some practical issues still have to be addressed and incorporated into the model, like adaptive algorithms. We also discuss such issues in this paper, as a result of our experience. This paper presents a new approach for learning in structured domains (SDs) using a constructive neural network for graphs (NN4G). The new model allows the extension of the input domain for supervised neural networks to a general class of graphs including both acyclic cyclic, directed undirected labeled graphs. In particular, the model can realize adaptive contextual transductions, learning the mapping from graphs for both classification and regression tasks. In contrast to previous neural networks for structures that had a recursive dynamics, NN4G is based on a constructive feedforward architecture with state variables that uses neurons with no feedback connections. The neurons are applied to the input graphs by a general traversal process that relaxes the constraints of previous approaches derived by the causality assumption over hierarchical input data. Moreover, the incremental approach eliminates the need to introduce cyclic dependencies in the definition of the system state variables. In the traversal process, the NN4G units exploit (local) contextual information of the graphs vertices. In spite of the simplicity of the approach, we show that, through the compositionality of the contextual information developed by the learning, the model can deal with contextual information that is incrementally extended according to the graphs topology. The effectiveness and the generality of the new approach are investigated by analyzing its theoretical properties and providing experimental results.