Xheal: a localized self-healing algorithm using expanders

Abstract

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(κd) 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. * A shorter version of this paper published at PODC 2011,

Error Correction Coding Meets Cyber-Physical Systems: Message-Passing Analysis of Self-Healing Interdependent Networks

“…A number of works [25]- [30] have been devoted to selfhealing single-layer networks. Several authors [26], [27] studied the concept of self-healing networks through distributed communication protocols that set up new links to recover the system connection.…”

Enabling Robust and Efficient Distributed Computation in Dynamic Peer-to-Peer Networks

“…On the other hand, an important aspect of our protocol is that it works even when the network is continually changing in an highly dynamic and adversarial manner. Most prior algorithms (e.g., [24], [35], [36], [23], [37], [38]) will only work under the assumption that the network will eventually stabilize and stop changing or there is a "repair" time for maintenance when there are no further changes (till the repair/maintenance is finished); these algorithms do not work under high continuous adversarial churn. Law and Siu [24] provide a distributed algorithm for maintaining an expander in the presence of limited number of insertions/deletions; their algorithm does not work for high continuous adversarial churn.…”

“…Law and Siu [24] provide a distributed algorithm for maintaining an expander in the presence of limited number of insertions/deletions; their algorithm does not work for high continuous adversarial churn. In [36] it is shown how to maintain the expansion property of a network in the self-healing model (there are no changes during repair) where the adversary can delete/insert one node in every step. In the same model, [23] present a protocol that maintains constant node degrees and constant expansion (both with probability 1) against an adaptive adversary, while requiring only logarithmic (in the network size) messages, time, and topology changes per deletion/insertion.…”

Local-on-Average Distributed Tasks

“…In this setting, the distributed local algorithm is allowed to add edges to the structure following adversarial removal of vertices or edges, in order to preserve certain graph properties (e.g., diameter, maximum degree, expansion) of the original network (i.e., before the attack), cf. [26,50,51]. Note that this approach requires the network to be reconfigurable, in the sense that it should be possible to change the network topology.…”