Besides the complexity in time or in number of messages, a common approach for analyzing distributed algorithms is to look at the assumptions they make on the underlying network. We investigate this question from the perspective of network dynamics. In particular, we ask how a given property on the evolution of the network can be rigorously proven as necessary or sufficient for a given algorithm. The main contribution of this paper is to propose the combination of two existing tools in this direction: local computations by means of graph relabelings, and evolving graphs. Such a combination makes it possible to express fine-grained properties on the network dynamics, then examine what impact those properties have on the execution at a precise, intertwined, level. We illustrate the use of this framework through the analysis of three simple algorithms, then discuss general implications of this work, which include (i) the possibility to compare distributed algorithms on the basis of their topological requirements, (ii) a formal hierarchy of dynamic networks based on these requirements, and (iii) the potential for mechanization induced by our framework, which we believe opens a door towards automated analysis and decision support in dynamic networks.
Distributed algorithms can be expressed using a variety of communication models ( message passing, mailboxes, shared memory). Although a vast majority of algorithms is designed in one of these models -- predominantly the message passing model --, the very fact that one of them is chosen implies that the obtained results ( positive or negative characterizations and associated proofs) are limited to the scope of this model. This problem of diversity among formalisms and results, already pointed out twenty years ago in @cite_17 , led researchers to consider higher abstractions when studying fundamental properties of distributed systems.
@cite_17
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Distributed computing models typically assume reliable communication between processors. While such assumptions often hold for engineered networks, e.g., due to underlying error correction protocols, their relevance to biological systems, wherein messages are often distorted before reaching their destination, is quite limited. In this study we aim at bridging this gap by rigorously analyzing a model of communication in large anonymous populations composed of simple agents which interact through short and highly unreliable messages. We focus on the rumor-spreading problem and the majority-consensus problem, two fundamental tasks in distributed computing, and initiate their study under communication noise. Our model for communication is extremely weak and follows the push gossip communication paradigm: In each synchronous round each agent that wishes to send information delivers a message to a random anonymous agent. This communication is further restricted to contain only one bit (essentially representing an opinion). Lastly, the system is assumed to be so noisy that the bit in each message sent is flipped independently with probability 1 2-e, for some small Ae >0. Even in this severely restricted, stochastic and noisy setting we give natural protocols that solve the noisy rumor-spreading and the noisy majority-consensus problems efficiently. Our protocols run in O(log n e2) rounds and use O(n log n e2) messages bits in total, where n is the number of agents. These bounds are asymptotically optimal and, in fact, are as fast and message efficient as if each agent would have been simultaneously informed directly by the source. Our efficient, robust, and simple algorithms suggest balancing between silence and transmission, synchronization, and majority-based decisions as important ingredients towards understanding collective communication schemes in anonymous and noisy populations.