Collaboration: People, Papers, Average Graphs, Durfee Squares and Metric Dimension

Utilizing several methods, this note shows that, in any collaboration network analysis, paper exclusion not only creates a loss of information, but can lead to incorrect interpretation of network structure because interpretation of vertex degree in the authors only graph is not well defined. Because the bipartite authors with papers graph is the actual social network, metric dimension is used to show that the relative distance structure of the bipartite graph is often defined by the structure of the papers, not that of the authors. Due to the NP-hard nature of metric dimension, methods that increase computational efficiency for the bipartite authors and papers graph are explored. With a departmental collaboration focus, public data for 245 professors from mathematics, physics and biology departments of three U.S. public universities is analyzed with network structure compared using metric dimension. By discipline, an average graph is defined with average graphs constructed from the collected data for the authors only structure, for the bipartite authors with papers structure and for papers only. Social analysis of the collected data shows that a 27\% change in the total number of hubs, along with identifying different professors as hubs, when the authors only graphs are compared to bipartite graphs reiterating the need for paper inclusion in any collaboration study.