Statistical properties of mutualistic-competitive 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 $\alpha$, where $\alpha$ is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the 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 Randi\'c index $R$. First, within a random matrix theory approach (i.e., a statistical approach), we identify a parameter $\xi\equiv\xi(n,m,\alpha)$ that scales the average normalized measures $\left< \overline{X} \right>$ (with $X$ representing $r$, $S$ and $R$). Specifically, we show that (i) $\xi\propto \alpha n$ with a weak dependence on $m$, and (ii) for $\xi<1/10$ most vertices in the mutualistic network are isolated, while for $\xi>10$ 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 $1/10<\xi<10$. Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves $\left< \overline{X} \right>$ vs. $\xi$ show good correspondence with the properties of real-world networks.