Many modern networks are reconfigurable, in the sense that the topology of the network can be changed by the nodes in the network. For example, peer-to-peer, wireless and ad-hoc networks are reconfigurable. More generally, many social networks, such as a company’s organizational chart; infrastructure networks, such as an airline’s transportation network; and biological networks, such as the human brain, are also reconfigurable. Modern reconfigurable networks have a complexity unprecedented in the history of engineering, resembling more a dynamic and evolving living animal rather than a structure of steel designed from a blueprint. Unfortunately, our mathematical and algorithmic tools have not yet developed enough to handle this complexity and fully exploit the flexibility of these networks. We believe that it is no longer possible to build networks that are scalable and never have node failures. Instead, these networks should be able to admit small, and maybe, periodic failures and still recover like skin heals from a cut. This process, where the network can recover itself by maintaining key invariants in response to attack by a powerful adversary is what we call self-healing. Here, we present several fast and provably good distributed algorithms for self-healing in reconfigurable dynamic networks. Each of these algorithms have different properties, a different set of gaurantees and limitations. We also discuss future directions and theoretical questions we would like to answer.
A responsive approach was followed by the authors in @cite_1 @cite_38 , which proposed a simple line algorithm for self-healing to maintain network connectivity. This algorithm has obvious drawbacks with regard to properties such as diameter maintenance but has served as a useful starting point for our research.
@cite_38 @cite_1
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Healing algorithms play a crucial part in distributed P2P networks where failures occur continuously and frequently. Several self-healing algorithms have been suggested recently [IPDPS'08, PODC'08, PODC'09, PODC'11] in a line of work that has yielded gradual improvements in the properties ensured on the graph. This work motivates a strong general phenomenon of edge-preserving healing that aims at obtaining self-healing algorithms with the constraint that all original edges in the graph (not deleted by the adversary), be retained in every intermediate graph. The previous algorithms, in their nascent form, are not explicitly edge preserving. In this paper, we show they can be suitably modified (We introduce Xheal+, an edge-preserving version of Xheal[PODC'11]). Towards this end, we present a general self-healing model that unifies the previous models. The main contribution of this paper is not in the technical complexity, rather in the simplicity with which the edge-preserving property can be ensured and the message that this is a crucial property with several benefits. In particular, we highlight this by showing that, almost as an immediate corollary, subgraph densities are preserved or increased. Maintaining density is a notion motivated by the fact that in certain distributed networks, certain nodes may require and initially have a larger number of inter-connections. It is vital that a healing algorithm, even amidst failures, respect these requirements. Our suggested modifications yield such subgraph density preservation as a by product. In addition, edge preservation helps maintain any subgraph induced property that is monotonic. Also, algorithms that are edge-preserving require minimal alteration of edges which can be an expensive cost in healing - something that has not been modeled in any of the past work. We present an algorithm to self-heal reconfigurable networks. This algorithm reconfigures the network during an attack to protect two critical invariants. First, it insures that the network remains connected. Second, it insures that no node increases its degree by more than O(log n). We prove that our algorithm can successfully maintain these invariants even for large networks under massive attack by a computationally unbounded adversary. We consider the problem of self-healing in reconfigurable networks (e.g. peer-to-peer and wireless mesh networks) that are under repeated attack by an omniscient adversary and propose a fully distributed algorithm, Xheal, that maintains good expansion and spectral properties of the network, also keeping the network connected. Moreover, Xheal does this while allowing only low stretch and degree increase per node. The algorithm heals global properties like expansion and stretch while only doing local changes and using only local information. We use a model similar to that used in recent work on self-healing. In our model, over a sequence of rounds, an adversary either inserts a node with arbitrary connections or deletes an arbitrary node from the network. The network responds by quick "repairs," which consist of adding or deleting edges in an efficient localized manner. These repairs preserve the edge expansion, spectral gap, and network stretch, after adversarial deletions, without increasing node degrees by too much, in the following sense. At any point in the algorithm, the expansion of the graph will be either 'better' than the expansion of the graph formed by considering only the adversarial insertions (not the adversarial deletions) or the expansion will be, at least, a constant. Also, the stretch i.e. the distance between any pair of nodes in the healed graph is no more than a O(log n) factor. Similarly, at any point, a node v whose degree would have been d in the graph with adversarial insertions only, will have degree at most O(κ d) in the actual graph, for a small parameter κ. We also provide bounds on the second smallest eigenvalue of the Laplacian which captures key properties such as mixing time, conductance, congestion in routing etc. Our distributed data structure has low amortized latency and bandwidth requirements. Our work improves over the self-healing algorithms Forgiving tree [PODC 2008] and Forgiving graph [PODC 2009] in that we are able to give guarantees on degree and stretch, while at the same time preserving the expansion and spectral properties of the network. In self-healing networks, end-to-end restoration schemes have been considered more advantageous than line restoration schemes because of a possible cost reduction of the total capacity to construct a fully restorable network. This paper clarifies the benefit of end-to-end restoration schemes quantitatively through a comparative analysis of the minimum link capacity installation cost. A jointly optimal capacity and flow assignment algorithm is developed for the self-healing ATM networks based on end-to-end and line restoration. Several networks with diverse topological characteristics as well as multiple projected traffic demand patterns are employed in the experiments to see the effect of various network parameters. The results indicate that the network topology has a significant impact on the required resource installation cost for each restoration scheme. Contrary to a wide belief in the economic advantage of the end-to-end restoration scheme, this study reveals that the attainable gain could be marginal for a well-connected and or unbalanced network.