The Isomorphic Version of Brualdi's and Sanderson's Nestedness

The discrepancy BR for an m × n 0, 1-matrix from Brualdi and Sanderson in 1998 is defined as the minimum number of 1 s that need to be shifted in each row to the left to achieve its Ferrers matrix, i.e., each row consists of consecutive 1 s followed by consecutive 0 s. For ecological bipartite networks, BR describes a nested set of relationships. Since two different labelled networks can be isomorphic, but possess different discrepancies due to different adjacency matrices, we define a metric determining the minimum discrepancy in an isomorphic class. We give a reduction to k ≤ n minimum weighted perfect matching problems. We show on 289 ecological matrices (given as a benchmark by Atmar and Patterson in 1995) that classical discrepancy can underestimate the nestedness by up to 30%.