  The study of dynamical systems on networks, describing complex interactive
processes, provides insight into how network structure affects global
behaviour. Yet many methods for network dynamics fail to cope with large or
partially-known networks, a ubiquitous situation in real-world applications.
Here we propose a localised method, applicable to a broad class of dynamical
models on networks, whereby individual nodes monitor and store the evolution of
their own state and use these values to approximate, via a simple computation,
their own steady state solution. Hence the nodes predict their own final state
without actually reaching it. Furthermore, the localised formulation enables
nodes to compute global network metrics without knowledge of the full network
structure. The method can be used to compute global rankings in the network
from local information; to detect community detection from fast, local
transient dynamics; and to identify key nodes that compute global network
metrics ahead of others. We illustrate some of the applications of the
algorithm by efficiently performing web-page ranking for a large internet
network and identifying the dynamic roles of inter-neurons in the C. Elegans
neural network. The mathematical formulation is simple, widely applicable and
easily scalable to real-world datasets suggesting how local computation can
provide an approach to the study of large-scale network dynamics.
