Mapping Networks and Trees with Multidimensional Scaling of Proximities

Network methodology typically has two separate stages: (1) constructing a graph from relational data, and (2) drawing the graph on a map to comprehend its structure. Multidimensional scaling (MDS) is discussed as a distance-driven graph drawing method. It is shown that popular drawing methods in computer science that minimize the potential energy of a spring model are equivalent to a simple form of MDS without optimal transformation of the graphical distance. They share a weighted least squares loss function (Kruskal’s stress). The best way to minimize Stress (Guttman’s algorithm) is shown to be a particular force-directed updating scheme in terms of a spring model. With several analyses of two examples (a simple graph and an additive tree), it is shown that using shortest path distances in the graph as input to MDS gives better drawing results than just using its adjacency matrix. Inclusion of data weights in Stress to emphasize good fit of small distances turns out not to be essential. It may even be detrimental to correct representation of line length in a weighted graph.

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