Coupling Geometry on Binary Bipartite Networks: Hypotheses Testing on Pattern Geometry and Nestedness

Abstract

Upon a matrix representation of a binary bipartite network, via the permutation invariance, a coupling geometry is computed to approximate the minimum energy macrostate of a network's system. Such a macrostate is supposed to constitute the intrinsic structures of the system, so that the coupling geometry can be taken as the information content of the network or even the nonparametric minimum sufficient statistics of the network data. Based on this, pertinent null and alternative hypotheses, such as nestedness, are to be formulated according to the macrostate. That is, any efficient testing statistic needs to be a function of this coupling geometry. These conceptual architectures and mechanisms are, by and large, still missing in community ecology literature and have rendered misconceptions to be prevalent in this research area. Here, the algorithmically computed coupling geometry is shown to consist of deterministic multiscale block patterns, which are framed by two marginal ultrametric trees on row and column axes, and stochastic uniform randomness within each block found on the finest scale. Functionally, a series of increasingly larger ensembles of matrix mimicries is derived by conforming to the multiscale block configurations. Here, matrix mimicking is meant to be subject to constraints of row and column sums sequences. Based on such a series of ensembles, a profile of distributions becomes a natural device for checking the validity of testing statistics or structural indexes. An energy-based index is used for testing whether network data indeed contain structural geometry. A new version of block-based nestedness index is also proposed. Its validity is checked and compared with the existing ones. A computing paradigm, called Data Mechanics, and its application in one real data network are illustrated throughout the developments and discussions in this paper.