A formal definition of random graphs is introduced which is applicable to graphical pattern recognition problems. The definition is used to formulate rigorously the structural-contextual dichotomy of random graphs. The probability of outcome graphs is expressed as the product of two terms, one due to the statistical variability of structure among the outcome graphs and the other due to their contextual variability. Expressions are obtained to estimate the various probability, typicality, and entropy measures. The members in an ensemble of signed digraphs are interpreted as outcome graphs of a random graph. The synthesized random graph is used to quantify the structural, contextual, and overall typicality of the outcome graphs with respect to the random graph.

The characterization of graph morphisms in terms of the subgraphs of the Cartesian graph product is extended and used to develop algorithms for an optimal graph monomorphism problem. The objective functional considered is defined as the sum of the weights associated with vertex and arc mappings. A reduction algorithm is proposed to obtain sharp lower bounds on the value of the solution. The lower bounds are used in a branch-and-bound algorithm for the optimal graph monomorphism problem.

A theoretical planning model that consists of a composite set of modal networks for serving the population in an urban area or region is presented. By varying and controling various parameters in the model, an equilibrium of modal choices can be obtained by seeking the condition that no user can alter his path without experiencing an increase in cost. The equivalence between equilibrium conditions and a nonlinear programming problem will be established, following a fundamental theorem. Thus, under small perturbations in the composite network, either through changes in the frequency of certain trips or changes in the structure of the network, it is possible to linearize about the observed equilibrium to determine the effects of perturbations.