Network Density of States

Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.Comment: 10 pages, 7 figure

Exploring self-similarity of complex cellular networks: The edge-covering method with simulated annealing and log-periodic sampling

Song, Havlin and Makse (2005) have recently used a version of the box-counting method, called the node-covering method, to quantify the self-similar properties of 43 cellular networks: the minimal number $N_V$ of boxes of size $\ell$ needed to cover all the nodes of a cellular network was found to scale as the power law $N_V \sim (\ell+1)^{-D_V}$ with a fractal dimension $D_V=3.53\pm0.26$. We propose a new box-counting method based on edge-covering, which outperforms the node-covering approach when applied to strictly self-similar model networks, such as the Sierpinski network. The minimal number $N_E$ of boxes of size $\ell$ in the edge-covering method is obtained with the simulated annealing algorithm. We take into account the possible discrete scale symmetry of networks (artifactual and/or real), which is visualized in terms of log-periodic oscillations in the dependence of the logarithm of $N_E$ as a function of the logarithm of $\ell$. In this way, we are able to remove the bias of the estimator of the fractal dimension, existing for finite networks. With this new methodology, we find that $N_E$ scales with respect to $\ell$ as a power law $N_E \sim \ell^{-D_E}$ with $D_E=2.67\pm0.15$ for the 43 cellular networks previously analyzed by Song, Havlin and Makse (2005). Bootstrap tests suggest that the analyzed cellular networks may have a significant log-periodicity qualifying a discrete hierarchy with a scaling ratio close to 2. In sum, we propose that our method of edge-covering with simulated annealing and log-periodic sampling minimizes the significant bias in the determination of fractal dimensions in log-log regressions.Comment: 19 elsart pages including 9 eps figure

Probability of local bifurcation type from a fixed point: A random matrix perspective

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance.Comment: 21 pages, 19 figure