Picking up the Pieces: Self-Healing in reconfigurable networks

We consider the problem of self-healing in networks that are reconfigurable in the sense that they can change their topology during an attack. Our goal is to maintain connectivity in these networks, even in the presence of repeated adversarial node deletion, by carefully adding edges after each attack. We present a new algorithm, DASH, that provably ensures that: 1) the network stays connected even if an adversary deletes up to all nodes in the network; and 2) no node ever increases its degree by more than 2 log n, where n is the number of nodes initially in the network. DASH is fully distributed; adds new edges only among neighbors of deleted nodes; and has average latency and bandwidth costs that are at most logarithmic in n. DASH has these properties irrespective of the topology of the initial network, and is thus orthogonal and complementary to traditional topology- based approaches to defending against attack. We also prove lower-bounds showing that DASH is asymptotically optimal in terms of minimizing maximum degree increase over multiple attacks. Finally, we present empirical results on power-law graphs that show that DASH performs well in practice, and that it significantly outperforms naive algorithms in reducing maximum degree increase.

Edge-preserving self-healing: Keeping network backbones densely connected

Healing algorithms play a crucial part in distributed peer-to-peer networks where failures occur continuously and frequently. Whereas there are approaches for robustness that rely largely on built-in redundancy, we adopt a responsive approach that is more akin to that of biological networks e.g. the brain. The general goal of self-healing distributed graphs is to maintain certain network properties while recovering from failure quickly and making bounded alterations locally. Several self-healing algorithms have been suggested in the recent literature [IPDPS'08, PODC'08, PODC'09, PODC'11]; they heal various network properties while fulfilling competing requirements such as having low degree increase while maintaining connectivity, expansion and low stretch of the network. In this work, we augment the previous algorithms by adding the notion of edge-preserving self-healing which requires the healing algorithm to not delete any edges originally present or adversarialy inserted. This reflects the cost of adding additional edges but more importantly it immediately follows that edge preservation helps maintain any subgraph induced property that is monotonic, in particular important properties such as graph and subgraph densities. Density is an important network property and in certain distributed networks, maintaining it preserves high connectivity among certain subgraphs and backbones. We introduce a general model of self-healing, and introduce xheal+, an edge-preserving version of xheal[PODC'11]. © 2012 IEEE.

Mitigation of Fatigue Damage in Self-Healing Vascular Materials

The fatigue response of an epoxy matrix containing vasculature for the delivery of liquid healing agents is investigated. The release of a rapidly curing, two-part epoxy healing chemistry into the wake of a propagating crack reduces the rate of crack extension by shielding the crack tip from the full range of applied stress intensity factor. Crack propagation is studied for a variety of loading conditions, with the maximum applied stress intensity factor ranging from 62 to 84% of the quasi-static fracture toughness of the material. At the highest level of applied load, the rate of mechanical damage is so fast that the healing agents do not fully mix and polymerize, and the effect of healing is minimal. The self-healing response is most effective at impeding the slower propagating cracks, with complete crack arrest occurring at the lowest level of applied load, and reductions of 79–84% in the rate of crack extension at intermediate loads.

Xheal: A localized self-healing algorithm using expanders

We consider the problem of self-healing in reconfigurable networks e.g., peer-to-peer and wireless mesh networks. For such networks under repeated attack by an omniscient adversary, we propose a fully distributed algorithm, Xheal, that maintains good expansion and spectral properties of the network, while 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 also provide bounds on the second smallest eigenvalue of the Laplacian which captures key properties such as mixing time, conductance, congestion in routing etc. Xheal has low amortized latency and bandwidth requirements. Our work improves over the self-healing algorithms Forgiving tree [PODC 2008] andForgiving 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.

The forgiving tree: a self-healing distributed data structure

We consider the problem of self-healing in peer-to-peer networks that are under repeated attack by an omniscient adversary. We assume that the following process continues for up to n rounds where n is the total number of nodes initially in the network: the adversary deletesan arbitrary node from the network, then the network responds by quickly adding a small number of new edges.

We present a distributed data structure that ensures two key properties. First, the diameter of the network is never more than O(log Delta) times its original diameter, where Delta is the maximum degree of the network initially. We note that for many peer-to-peer systems, Delta is polylogarithmic, so the diameter increase would be a O(loglog n) multiplicative factor. Second, the degree of any node never increases by more than 3 over its original degree. Our data structure is fully distributed, has O(1) latency per round and requires each node to send and receive O(1) messages per round. The data structure requires an initial setup phase that has latency equal to the diameter of the original network, and requires, with high probability, each node v to send O(log n) messages along every edge incident to v. Our approach is orthogonal and complementary to traditional topology-based approaches to defending against attack.

Algorithms for Self-Healing Networks

Many modern networks are \emph{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 \emph{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. %in the final dissertation that this document is proposed to lead to.

Self-healing systems and virtual structures

Modern networks are large, highly complex and dynamic. Add to that the mobility of the agents comprising many of these networks. It is difficult or even impossible for such systems to be managed centrally in an efficient manner. It is imperative for such systems to attain a degree of self-management. Self-healing i.e. the capability of a system in a good state to recover to another good state in face of an attack, is desirable for such systems. In this paper, we discuss the self-healing model for dynamic reconfigurable systems. In this model, an omniscient adversary inserts or deletes nodes from a network and the algorithm responds by adding a limited number of edges in order to maintain invariants of the network. We look at some of the results in this model and argue for their applicability and further extensions of the results and the model. We also look at some of the techniques we have used in our earlier work, in particular, we look at the idea of maintaining virtual graphs mapped over the existing network and assert that this may be a useful technique to use in many problem domains.

DEX: Self-Healing Expanders

We present a fully-distributed self-healing algorithm DEX, that maintains a constant degree expander network in a dynamic setting. To the best of our knowledge, our algorithm provides the first efficient distributed construction of expanders - whose expansion properties hold deterministically - that works even under an all-powerful adaptive adversary that controls the dynamic changes to the network (the adversary has unlimited computational power and knowledge of the entire network state, can decide which nodes join and leave and at what time, and knows the past random choices made by the algorithm). Previous distributed expander constructions typically provide only probabilistic guarantees on the network expansion which rapidly degrade in a dynamic setting, in particular, the expansion properties can degrade even more rapidly under adversarial insertions and deletions. Our algorithm provides efficient maintenance and incurs a low overhead per insertion/deletion by an adaptive adversary: only O(log n) rounds and O(log n) messages are needed with high probability (n is the number of nodes currently in the network). The algorithm requires only a constant number of topology changes. Moreover, our algorithm allows for an efficient implementation and maintenance of a distributed hash table (DHT) on top of DEX, with only a constant additional overhead. Our results are a step towards implementing efficient self-healing networks that have guaranteed properties (constant bounded degree and expansion) despite dynamic changes.

DEX: self-healing expanders

Compact Routing Messages in Self-Healing Trees

Existing compact routing schemes, e.g., Thorup and Zwick [SPAA 2001] and Chechik [PODC 2013], often have no means to tolerate failures, once the system has been setup and started. This paper presents, to our knowledge, the first self-healing compact routing scheme. Besides, our schemes are developed for low memory nodes, i.e., nodes need only O(log2 n) memory, and are thus, compact schemes.
We introduce two algorithms of independent interest: The first is CompactFT, a novel compact version (using only O(log n) local memory) of the self-healing algorithm Forgiving Tree of Hayes et al. [PODC 2008]. The second algorithm (CompactFTZ) combines CompactFT with Thorup-Zwick’s treebased compact routing scheme [SPAA 2001] to produce a fully compact self-healing routing scheme. In the self-healing model, the adversary deletes nodes one at a time with the affected nodes self-healing locally by adding few edges. CompactFT recovers from each attack in only O(1) time and ∆ messages, with only +3 degree increase and O(log∆) graph diameter increase, over any sequence of deletions (∆ is the initial maximum degree).
Additionally, CompactFTZ guarantees delivery of a packet sent from sender s as long as the receiver has not been deleted, with only an additional O(y log ∆) latency, where y is the number of nodes that have been deleted on the path between s and t. If t has been deleted, s gets informed and the packet removed from the network.

Self-healing of cracks during ductile regime machining of silicon: Insights from molecular dynamics simulation

During nanoindentation and ductile-regime machining of silicon, a phenomenon known as “self-healing” takes place in that the microcracks, microfractures, and small spallings generated during the machining are filled by the plastically flowing ductile phase of silicon. However, this phenomenon has not been observed in simulation studies. In this work, using a long-range potential function, molecular dynamics simulation was used to provide an improved explanation of this mechanism. A unique phenomenon of brittle cracking was discovered, typically inclined at an angle of 45° to 55° to the cut surface, leading to the formation of periodic arrays of nanogrooves being filled by plastically flowing silicon during cutting. This observation is supported by the direct imaging. The simulated X-ray diffraction analysis proves that in contrast to experiments, Si-I to Si-II (beta tin) transformation during ductile-regime cutting is highly unlikely and solid-state amorphisation of silicon caused solely by the machining stress rather than the cutting temperature is the key to its brittle-ductile transition observed during the MD simulations