An analytic approach to the measurement of nestedness in bipartite networks

We present an index that measures the nestedness pattern of bipartite networks, a problem that arises in theoretical ecology. Our measure is derived using the sum of distances of the occupied elements in the incidence matrix of the network. This index quantifies directly the deviation of a given matrix from the nested pattern. In the simplest case the distance of the matrix element ai,jai,j is di,j=i+jdi,j=i+j, the Manhattan distance. A generic distance is obtained as di,j=(iχ+jχ)1/χdi,j=(iχ+jχ)1/χ. The nestedness index is defined by ν=1−τν=1−τ, where ττ is the “temperature” of the matrix. We construct the temperature index using two benchmarks: the distance of the complete nested matrix that corresponds to zero temperature and the distance of the average random matrix where the temperature is defined as one. We discuss an important feature of the problem: matrix occupancy ρρ. We address this question using a metric index χχ that adjusts for matrix occupancy.