Origin and implications of zero degeneracy in networks spectra.

@article{Yadav2015OriginAI,
  title={Origin and implications of zero degeneracy in networks spectra.},
  author={Alok Yadav and Sarika Jalan},
  journal={Chaos},
  year={2015},
  volume={25 4},
  pages={
          043110
        }
}
The spectra of many real world networks exhibit properties which are different from those of random networks generated using various models. One such property is the existence of a very high degeneracy at the zero eigenvalue. In this work, we provide all the possible reasons behind the occurrence of the zero degeneracy in the network spectra, namely, the complete and partial duplications, as well as their implications. The power-law degree sequence and the preferential attachment are the… 

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References

SHOWING 1-10 OF 63 REFERENCES
Spectra of complex networks.
TLDR
It is shown that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real-world networks, e.g., of the Internet.
Eigenvalue spectra of modular networks.
TLDR
This work obtains in a unified fashion the spectrum of a large family of operators, including the adjacency, Laplacian, and normalized LaPLacian matrices, for networks with generic modular structure, in the limit of large degrees.
Spectral analysis and the dynamic response of complex networks.
TLDR
It is shown that the spectral density of hierarchical networks follows a very different pattern, which can be used as a fingerprint of modularity, related to the homeostatic response of the network.
Statistical mechanics of complex networks
TLDR
A simple model based on these two principles was able to reproduce the power-law degree distribution of real networks, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network.
Graph spectra and the detectability of community structure in networks
TLDR
Using methods from random matrix theory, the spectra of networks that display community structure are calculated, and it is shown that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.
Emergence of scaling in random networks
TLDR
A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
Measures of degeneracy and redundancy in biological networks.
TLDR
Functional measures of the degeneracy and redundancy of a system with respect to a set of outputs are developed and promise to be useful in characterizing and understanding the functional robustness and adaptability of biological networks.
Random graphs with arbitrary degree distributions and their applications.
TLDR
It is demonstrated that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.
Graph Spectra for Complex Networks
TLDR
This self-contained book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks, and the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks.
...
...