Analysis of Complex Networks

4.7 Conclusion

The discovery of X-rays more than a century ago has increased our knowledge in many fields, such as the structure of matter, cosmology, security in technology, and X-ray diagnostics, among others. The existence of a tool, like X-rays and other spectroscopic techniques, permits us to understand the internal structure of the systems under study from molecules and materials to the human body. In a similar way, spectral graph theory is the X-ray machine for studying complex networks. As we have shown here, the use of graph spectral techniques permits us to analyze the local and global structure of complex networks.

Using graph spectral theory it is possible to “see” how central a node is based on its weighted participation in all substructures present in the graph. The same techniques permit us to analyze whether a network is homogeneous or modular. In the last case it permits us to classify their structures according to certain universal structural classes, regardless of whether it is representing a cell or a society. In addition, the spectral techniques explained in this chapter permit us to identify the communities existing in a complex network, as well as the bipartivity structure of certain substructures present in such systems. Many other characteristics of complex networks could be investigated using the spectra of graphs. Some of them have already been described by the scientists working in this field, others are still waiting for the development of the appropriate ...