Abstract
Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of n nodes that is decomposed into two disjoint subsets, having m and n−m vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter α ∈ [0, 1] that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter ξ ≡ ξ(n, m, α) that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when ξ < 1/10 (ξ > 10) the eigenvectors are localized (extended), whereas the localization–to–delocalization transition occurs in the interval 1/10 < ξ < 10. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed ξ, the spectral properties of our graph model are also universal.
Highlights
The latest developments in network science have largely contributed to a better understanding of the structure and dynamics of many real-wold complex systems [1,2,3]
It is worth stressing that once we have found that ξ exists and that this parameter scales the eigenvector properties of the model of random bipartite graphs here studied, it is natural to expect that other properties of the graph model would scale with the same parameter
To characterize the spectral properties of the random bipartite graph model, we use the ratios of consecutive energy-level spacings r, which are defined as follows
Summary
The latest developments in network science have largely contributed to a better understanding of the structure and dynamics of many real-wold complex systems [1,2,3]. As a matter of fact, research done during the last 20 years have allowed to take key steps in our comprehension of seemingly diverse phenomena such as the large-scale spreading of diseases [4,5], information dissemination [2], cascading failures [6], diffusion dynamics [7,8,9] and more recently, on how multilayer systems work [10,11,12]. These advances are at a theoretical level.
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