  For spatial and network data, we consider models formed from a Markov random
field (MRF) structure and the specification of a conditional distribution for
each observation. Fast simulation from such MRF models is often an important
consideration, particularly when repeated generation of large numbers of data
sets is required. However, a standard Gibbs strategy for simulating from MRF
models involves single-site updates, performed with the conditional univariate
distribution of each observation in a sequential manner, whereby a complete
Gibbs iteration may become computationally involved even for moderate samples.
As an alternative, we describe a general way to simulate from MRF models using
Gibbs sampling with "concliques" (i.e., groups of non-neighboring
observations). Compared to standard Gibbs sampling, this simulation scheme can
be much faster by reducing Gibbs steps and independently updating all
observations per conclique at once. The speed improvement depends on the number
of concliques relative to the sample size for simulation, and
order-of-magnitude speed increases are possible with many MRF models (e.g.,
having appropriately bounded neighborhoods). We detail the simulation method,
establish its validity, and assess its computational performance through
numerical studies, where speed advantages are shown for several spatial and
network examples.
