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Register a new userDegeneracies in the length spectra of metric graphs
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The spectral theory of quantum graphs is related via an exact trace formula with the spectrum of the lengths of periodic orbits (cycles) on the graphs. The latter is a degenerate spectrum, and understanding its structure (i.e.,finding out how many different lengths exist for periodic orbits with a given period and the average number of periodic orbits with the same length) is necessary for the systematic study of spectral fluctuations using the trace formula. This is a combinatorial problem which we solve exactly for complete (fully connected) graphs with arbitrary number of vertices.
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An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graphs non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graphs first Betti number $beta$. We study the distribution of the nodal surplus values in the countably infinite set of the graphs eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing $beta$. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.
We use the Fourier based Gabetta-Toscani-Wennberg (GTW) metric $d_2$ to study the rate of convergence to equilibrium for the Kac model in $1$ dimension. We take the initial velocity distribution of the particles to be a Borel probability measure $mu$ on $mathbb{R}^n$ that is symmetric in all its variables, has mean $vec{0}$ and finite second moment. Let $mu_t(dv)$ denote the Kac-evolved distribution at time $t$, and let $R_mu$ be the angular average of $mu$. We give an upper bound to $d_2(mu_t, R_mu)$ of the form $min{ B e^{-frac{4 lambda_1}{n+3}t}, d_2(mu,R_mu)}$, where $lambda_1 = frac{n+2}{2(n-1)}$ is the gap of the Kac model in $L^2$ and $B$ depends only on the second moment of $mu$. We also construct a family of Schwartz probability densities ${f_0^{(n)}: mathbb{R}^nrightarrow mathbb{R}}$ with finite second moments that shows practically no decrease in $d_2(f_0(t), R_{f_0})$ for time at least $frac{1}{2lambda}$ with $lambda$ the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in [14].
We investigate the exact relation existing between the stability equation for the solutions of a mechanical system and the geodesic deviation equation of the associated geodesic problem in the Jacobi metric constructed via the Maupertuis-Jacobi Principle. We conclude that the dynamical and geometrical approaches to the stability/instability problem are not equivalent.
Many real-world networks exhibit a high degeneracy at few eigenvalues. We show that a simple transformation of the networks adjacency matrix provides an understanding of the origins of occurrence of high multiplicities in the networks spectra. We find that the eigenvectors associated with the degenerate eigenvalues shed light on the structures contributing to the degeneracy. Since these degeneracies are rarely observed in model graphs, we present results for various cancer networks. This approach gives an opportunity to search for structures contributing to degeneracy which might have an important role in a network.
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