Statistical properties of mutualistic-competitive random networks

•We investigate the spectral properties of matrices constructed from the union of bipartite random networks with their respective two one-mode projections.•We apply a statistical approach, based on random matrix theory (RMT) techniques, to mutualistic random networks with projected edges that emulate intra-group competitive interactions.•The measures we employ (the ratio of consecutive eigenvalue spacings, the Shannon entropy related to the eigenvectors of the adjacency matrix, and the Randic index) exhibit a universal behavior as a function of the average degree.•Unexpectedly, we show that real-world ecological networks follow with a reasonably good agreement the universal behavior reported for the mutualistic random networks. Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m−n. Our RME depends on three parameters: the network size n, the size of the smaller set m, and the connectivity between the two sets α, where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r, the Shannon entropy of the eigenvectors S, and the Randić index R. First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ≡ξ(n,m,α) that scales the average normalized measures (with X representing r, S and R). Specifically, we show that (i) ξ∝αn with a weak dependence on m, and (ii) for ξ10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1/10