Spectral analysis of complex network with tunable degree distribution and clustering

Network models provide a natural way to describe real data set in diverse fields as for example biology, sociology, ecology, internet, global economy and many others. Real networks display topological features, such as heavy tail degree distribution, high clustering coefficient and assortativity or disassortativity, that can not be modeled by a totally regular or random graphs. Many random network models has been proposed with the aim to capture features regularly found in empirical networks. Two of most studied classes of such models are the Scale-free network model (SF) and the Small-world network model (SW). The original SF model display power law degree distribution and vanishing clustering coefficient in the limit of large networks.
The SW model interpolates between regular and random network by means a parameter $p$, that characterizes the disorder in the system. For small values of $p$ the model exhibits high clustering coefficient. Since each graph/network can be represented by a matrix (adjacency, laplacian or normalized laplacian) its spectra (eigenvalues and eigenvectors) encodes information about its topology. The fluctuations of networks spectra can be analysed through the framework of Random Matrices Theory (RMT). Results for SW model have revealed that as the disorder level increases (decreasing clustering coefficient) the spectral fluctuations follows the description of a transition between two different ensemble of RMT (Poisson-GOE transition).
The main focus of this work is to clarify the effects of clustering on the spectral properties of complex networks. The clustering quantifies the likelihood that two neighbors of a vertex are neighbors themselves and it is related to number of triangles in the network (the vast majority of real networks display a high density of triangles). The spectra of totally random networks (no correlation between vertices) is well described by GOE ensemble of RMT. According to RMT it is expected that network models with high clustering (correlation) deviates from GOE prediction.
Here we analyse under the RMT framework the spectral properties of different classes of random networks where both the degree-dependent clustering coefficient and the degree distribution are tunable with special attention to the Scale-free networks.