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Identifying the rank of species in a social or ecological network is adifficult task, since the rank of each species is invariably determined bycomplex interactions stipulated with other species. Simply put, the rank of aspecies is a function of the ranks of all other species through the adjacencymatrix of the network. A common system of ranking is to order species in such away that their neighbours form maximally nested sets, a problem called nestedmaximization problem (NMP). Here we show that the NMP can be formulated as aninstance of the Quadratic Assignment Problem, one of the most importantcombinatorial optimization problem widely studied in computer science,economics, and operations research. We tackle the problem by StatisticalPhysics techniques: we derive a set of self-consistent nonlinear equationswhose fixed point represents the optimal rankings of species in an arbitrarybipartite mutualistic network, which generalize the Fitness-Complexityequations widely used in the field of economic complexity. Furthermore, wepresent an efficient algorithm to solve the NMP that outperformsstate-of-the-art network-based metrics and genetic algorithms. Eventually, ourtheoretical framework may be easily generalized to study the relationshipbetween ranking and network structure beyond pairwise interactions, e.g. inhigher-order networks.

Ranking species in complex ecosystems through nestedness maximization