The problem of optimal arrangement of nodes of a random weighted graph is
studied in this thesis. The nodes of graphs under study are fixed, but their edges
are random and established according to the so called edge-probability function.
This function is assumed to depend on the weights attributed to the pairs of graph
nodes (or distances between them) and a statistical parameter. It is the purpose
of experimentation to make inference on the statistical parameter and thus to
extract as much information about it as possible. We also distinguish between two
different experimentation scenarios: progressive and instructive designs.
We adopt a utility-based Bayesian framework to tackle the optimal design
problem for random graphs of this kind. Simulation based optimisation methods,
mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain
the solution. We study optimal design problem for the inference based on partial
observations of random graphs by employing data augmentation technique.
We prove that the infinitely growing or diminishing node configurations asymptotically
represent the worst node arrangements. We also obtain the exact solution
to the optimal design problem for proximity graphs (geometric graphs) and numerical
solution for graphs with threshold edge-probability functions.
We consider inference and optimal design problems for finite clusters from bond
percolation on the integer lattice Zd and derive a range of both numerical and
analytical results for these graphs. We introduce inner-outer plots by deleting
some of the lattice nodes and show that the ‘mostly populated’ designs are not
necessarily optimal in the case of incomplete observations under both progressive
and instructive design scenarios.
Finally, we formulate a problem of approximating finite point sets with lattice
nodes and describe a solution to this problem
