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.

翻译：我们提出自适应自组织问题：在匿名、同步的动态网络中，节点需根据随时间变化的环境信号，分布式地改变其响应（或“颜色”）的集体分布。即使这些信号仅被局部感知且网络拓扑结构受到对抗性改变，节点仍需完成这一任务。具体而言，信号对手可在各轮次之间任意改变信号类型以及感知该信号的节点集合。若某个信号（或信号缺失）$s$在系统中持续足够长时间，动态网络必须稳定下来，使节点的颜色达到并保持在一个近似于$r(s)$（由问题实例定义的目标分布）的分布状态。我们首先证明：若节点是确定性的，则唯一可解的自适应自组织实例是那些具有均匀目标分布的实例，即所有节点必须以相同颜色稳定下来。随后，我们提出一种线性时间、对数空间的确定性算法，适用于这类子问题，即便信号见证者的数量和位置发生任意变化，该算法仍有效。当节点已知网络节点数$n$时，对该算法进行小幅调整可实现更强的收敛性质，使得对抗性的边与信号动态完全无法干扰已稳定的配置。最后，我们提出这些算法的随机化扩展版本，在节点已知目标分布的情况下，能够以高概率求解任意（即不一定均匀的）自适应自组织实例。