Adaptive Self-Organization in Anonymous Dynamic Networks

We introduce the problem of adaptive self-organization in which the nodes of an anonymous, synchronous dynamic network must distributively change the collective distribution of their responses (or "colors") as a function of time-varying environmental signals, even when these signals are only perceived locally and the network topology changes adversarially. Specifically, a signal adversary may change the type of signal and which node(s) witness that signal arbitrarily between rounds. If a signal (or lack thereof) $s$ persists in the system for sufficiently long, the dynamic network must stabilize such that nodes' colors reach and remain in a distribution closely approximating $r(s)$, a goal distribution defined by the problem instance. We first prove that if nodes are deterministic, the only solvable instances of adaptive-self organization are those with homogeneous goal distributions, i.e., those where all nodes must stabilize with the same color. We then present a linear-time, logarithmic-memory, deterministic algorithm for this subclass of instances that works even when the multiplicity and location of signal witnesses change arbitrarily. When nodes know $n$, the number of nodes in the network, a small adaptation of this algorithm achieves a stronger convergence property in which adversarial edge and signal dynamics are entirely unable to disturb stabilized configurations. Finally, we present a randomized extension of these algorithms that solves arbitrary (i.e., not necessarily homogeneous) instances of adaptive self-organization with high probability when nodes know the goal distributions.