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Abstract:
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There are different ways to calculate transitivity in weighted graphs. Most real weighted graphs are semimetric (i.e., symmetric, reflexive, but not transitive). Modelers typically embed these graphs in a metric space using the Dijkstra algorithm, which is in effect a metric closure that causes some distortion of the topology of graphs. We relate metric properties of distance graphs with the transitive properties of proximity graphs. The isomorphism between these graphs is formulated via t-norm/t-conorm functions. We show (1) the most intuitive t-norm pair (max, min) leads to an ultra-metric closure with the loss of scale-free structure. (2) The metric closure leads to a t-norm pair with poor axiomatics. A sounder transitive closure for complex networks is given by Hamacher's t-norm pair, which is shown to better preserve the scale-free structure of real networks.
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