A mathematical programming approach to fitting general graphs

We present an algorithm for fitting general graphs to proximity data. The algorithm utilizes a mathematical programming procedure based on a penalty function approach to impose additivity constraints upon parameters. For a user-specified number of links, the algorithm seeks to provide the connected network that gives the least-squares approximation to the proximity data with the specified number of links, allowing for linear transformations of the data. The network distance is the minimum-path-length metric for connected graphs. As a limiting case, the algorithm provides a tree where each node corresponds to an object, if the number of links is set equal to the number of objects minus one. A Monte Carlo investigation indicates that the resulting networks tend to fall within one percentage point of the least-squares solution in terms of the variance accounted for, but do not always attain this global optimum. The network model is discussed in relation to ordinal network representations (Klauer 1989) and NETSCAL (Hutchinson 1989), and applied to several well-known data sets.

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