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The isomorphic version of Brualdi’s and Sanderson’s nestedness. (English) Zbl 1462.05212
Summary: The discrepancy BR for an \(m \times n\) \(0, 1\)-matrix from R. A. Brualdi and J. G. Sanderson [“Nested species subsets, gaps, and discrepancy”, Oecologia 119, 256–264 (1998; doi:10.1007/s004420050784)] 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 \(k \leq n\) minimum weighted perfect matching problems. We show on 289 ecological matrices (given as a benchmark by W. Atmar and B. D. Patterson [“The nestedness temperature calculator: a visual basic program, Including 294 presence-absence matrices”, Chicago, IL: AICS Research, Inc. (1995)]) that classical discrepancy can underestimate the nestedness by up to 30%.
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05A05 Permutations, words, matrices
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05E10 Combinatorial aspects of representation theory
15B34 Boolean and Hadamard matrices
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