which matrices show perfect nestedness or the absence of nestedness? an analytical study on the performance of nodf and wnodf

Nestedness is a concept employed to describe a particular pattern of organization in species interaction networks and in site-by-species incidence matrices. Currently the most widely used nestedness index is the NODF (Nestedness metric based on Overlap and Decreasing Fill), initially presented for binary data and later extended to quantitative data, WNODF. In this manuscript we present a rigorous formulation of this index for both cases, NODF and WNODF. In addition, we characterize the matrices corresponding to the two extreme cases, (W)NODF=1 and (W)NODF=0, representing a perfectly nested pattern andthe absence of nestedness respectively. After permutations of rows and columns if necessary, the perfectly nested pattern is a full triangular matrix, which must of course besquare, with additional inequalities between the elements for WNODF. On the other hand there are many patterns characterized by the total absence of nestedness. Indeed, any binary matrix (whether square or rectangular) with uniform row and column sums (or marginals) satisfies this condition: the chessboard and a pattern reflecting an underlying annular ecological gradient, which we shall call gradient-like, are symmetrical or nearly symmetrical examples from this class.