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.

Motivated by the need for designing efficient and robust fully-distributed computation in highly dynamic networks such as Peer-to-Peer (P2P) networks, we study distributed protocols for constructing and maintaining dynamic network topologies with good expansion properties. Our goal is to maintain a sparse (bounded degree) expander topology despite heavy churn (i.e., nodes joining and leaving the network continuously over time). We assume that the churn is controlled by an adversary that has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power, but is oblivious to the random choices made by the algorithm. Our main contribution is a randomized distributed protocol that guarantees with high probability the maintenance of a constant degree graph with high expansion even under continuous high adversarial churn. Our protocol can tolerate a churn rate of up to $O(n/\text{polylog}(n))$ per round (where $n$ is the stable network size). Our protocol is efficient, lightweight, and scalable, and it incurs only $O(\text{polylog}(n))$ overhead for topology maintenance: only polylogarithmic (in $n$) bits needs to be processed and sent by each node per round and any node's computation cost per round is also polylogarithmic. The given protocol is a fundamental ingredient that is needed for the design of efficient fully-distributed algorithms for solving fundamental distributed computing problems such as agreement, leader election, search, and storage in highly dynamic P2P networks and enables fast and scalable algorithms for these problems that can tolerate a large amount of churn.

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.