Identifying the rank of species in a social or ecological network is a
difficult task, since the rank of each species is invariably determined by
complex interactions stipulated with other species. Simply put, the rank of a
species is a function of the ranks of all other species through the adjacency
matrix of the network. A common system of ranking is to order species in such a
way that their neighbours form maximally nested sets, a problem called nested
maximization problem (NMP). Here we show that the NMP can be formulated as an
instance of the Quadratic Assignment Problem, one of the most important
combinatorial optimization problem widely studied in computer science,
economics, and operations research. We tackle the problem by Statistical
Physics techniques: we derive a set of self-consistent nonlinear equations
whose fixed point represents the optimal rankings of species in an arbitrary
bipartite mutualistic network, which generalize the Fitness-Complexity
equations widely used in the field of economic complexity. Furthermore, we
present an efficient algorithm to solve the NMP that outperforms
state-of-the-art network-based metrics and genetic algorithms. Eventually, our
theoretical framework may be easily generalized to study the relationship
between ranking and network structure beyond pairwise interactions, e.g. in
higher-order networks.Comment: 28 pages; 2 figure