Nestedness and τ-temperature in ecological networks

The τ-temperature is a measure of disorder of bipartite networks that is based on the total Manhattan distance of the adjacency matrix. Two properties of this measure are that it does not depend on permutations of lines or columns that have the same connectivity and it is completely determined by connectivities of lines and columns. The normalisation of τ is done by an uniform random matrix whose elements were previously sorted. τ shows no bias against uniform random matrices of several occupations, ρ, sizes, L, and shapes. The scaling of the total Manhattan distance of a random matrix is Drand ∝ L3ρ while the same scaling for a full nested matrix is Dnest ∝ L3ρ3/2. We test τ for a large set of empirical matrices to verify these scalings. The index τ correlates better with the temperature of Atmar than with the NODF index of nestedness. We conclude this work by discussing differences between nestedness indices and order/disorder indices.