Fully Decentralized Estimation of Some Global Properties of a Network

It is often beneficial to architect networks and overlays as fully decentralized systems, in the sense that any computation (e. g., routing or search) would only use local information, and no single node would have a complete view or control over the whole network. Yet sometimes it also important to compute global properties of the network. In this paper we propose a fully decentralized algorithm to compute some global properties that can be derived from the spectrum of the network. More specifically, we compute the most significant eigenvalues of a descriptive matrix closely related to the adjacency matrix of the network graph. Such spectral properties can then lead to, for example, the "mixing time" of a network, which can be used to parametrize random walks and related search algorithms typical of peer-to-peer networks. Our key insight is to view the network as a linear dynamic system whose impulse response can be computed efficiently and locally by each node. We then use this impulse response to identify the spectral properties of the network. This algorithm is completely decentralized and requires only minimal local state and local communication. We show experimentally that the algorithm works well on different kinds of networks and in the presence of network instability.