# Distributed Computation in Dynamic Networks

In this report we investigate distributed computation in dynamic networks in which the network topology changes from round to round. We consider a worst-case model in which the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors for the current round are before they broadcast their messages. The model is intended to capture mobile networks and wireless networks, in which mobility and interference render communication unpredictable. The model allows the study of the fundamental computation power of dynamic networks. In particular, it captures mobile networks and wireless networks, in which mobility and interference render communication unpredictable. In contrast to much of the existing work on dynamic networks, we do not assume that the network eventually stops changing; we require correctness and termination even in networks that change continually. We introduce a stability property called T-interval connectivity (for T >= 1), which stipulates that for every T consecutive rounds there exists a stable connected spanning subgraph. For T = 1 this means that the graph is connected in every round, but changes arbitrarily between rounds. Algorithms for the dynamic graph model must cope with these unceasing changes. We show that in 1-interval connected graphs it is possible for nodes to determine the size of the network and compute any computable function of their initial inputs in O(n^2) rounds using messages of size O(log n + d), where d is the size of the input to a single node. Further, if the graph is T-interval connected for T > 1, the computation can be sped up by a factor of T, and any function can be computed in O(n + n^2 / T) rounds using messages of size O(log n + d). We also give two lower bounds on the gossip problem, which requires the nodes to disseminate k pieces of information to all the nodes in the network. We show an Omega(n log k) bound on gossip in 1-interval connected graphs against centralized algorithms, and an Omega(n + nk / T) bound on exchanging k pieces of information in T-interval connected graphs for a restricted class of randomized distributed algorithms. The T-interval connected dynamic graph model is a novel model, which we believe opens new avenues for research in the theory of distributed computing in wireless, mobile and dynamic networks.