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

The aggregation problem assumes that every process starts an execution with a unique token (an abstraction for data). The goal is to collect these tokens at a minimum number of processes by the end of the execution. This problem is particularly relevant to mobile networks where peer-to-peer communication is cheap (e.g., using 802.11 or Bluetooth), but uploading data to a central server can be costly (e.g., using 3G/4G). With this in mind, we study this problem in a dynamic network model, in which the communication graph can change arbitrarily from round to round. We start by exploring global bounds. First we prove a negative result that shows that in general dynamic graphs no algorithm can achieve any measure of competitiveness against the optimal offline algorithm. Guided by this impossibility result, we focus our attention to dynamic graphs where every node interacts, at some point in the execution, with at least a p-fraction of the total number of nodes in the graph. We call these graphs p-clusters. We describe a distributed algorithm that in p-clusters aggregates the tokens to O(log n) processes with high probability. We then turn our attention to local bounds. Specifically we ask whether its possible to aggregate to O(log n) processes in parts of the graph that locally form a p-cluster. Here we prove a negative result: this is only possible if the local p-clusters are sufficiently isolated from the rest of the graph. We then match this result with an algorithm that achieves the desired aggregation given (close to) the minimal required p-cluster isolation. Together, these results imply a “paradox of connectivity”: in some graphs, increasing connectivity can lead to inherently worse aggregation performance. We conclude by considering what it seems to be a promising performance metric to circumvent our lower bounds for local aggregation algorithms. However, perhaps surprisingly, we show that no aggregation algorithm can perform well with respect to this metric, even in very well connected and very well isolated clusters. 1