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Register a new userRandom walks and Laplacians on hypergraphs: When do they match?
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We develop a general theory of random walks on hypergraphs which includes, as special cases, the different models that are found in literature. In particular, we introduce and analyze general random walk Laplacians for hypergraphs, and we compare them to hypergraph normalized Laplacians that are not necessarily related to random walks, but which are motivated by biological and chemical networks. We show that, although these two classes of Laplacians coincide in the case of graphs, they appear to have important conceptual differences in the general case. We study the spectral properties of both classes, as well as their applications to Coupled Hypergraph Maps: discrete-time dynamical systems that generalize the well-known Coupled Map Lattices on graphs. Our results also show why for some hypergraph Laplacian variants one expects more classical results from (weighted) graphs to generalize directly, while these results must fail for other hypergraph Laplacians.
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Classical turning surfaces of Kohn-Sham potentials, separating classically-allowed regions (CARs) from classically-forbidden regions (CFRs), provide a useful and rigorous approach to understanding many chemical properties of molecules. Here we calculate such surfaces for several paradigmatic solids. Our study of perfect crystals at equilibrium geometries suggests that CFRs are absent in metals, rare in covalent semiconductors, but common in ionic and molecular crystals. A CFR can appear at a monovacancy in a metal. In all materials, CFRs appear or grow as the internuclear distances are uniformly expanded. Calculations with several approximate density functionals and codes confirm these behaviors. A classical picture of conduction suggests that CARs should be connected in metals, and disconnected in wide-gap insulators. This classical picture is confirmed in the limits of extreme uniform compression of the internuclear distances, where all materials become metals without CFRs, and extreme expansion, where all materials become insulators with disconnected and widely-separated CARs around the atoms.
We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph.
We apply a simple, one-equation, galaxy formation model on top of the halos and subhalos of a high-resolution dark matter cosmological simulation to study how dwarf galaxies acquire their mass and, for better mass resolution, on over 10^5 halo merger trees, to predict when they form their stars. With the first approach, we show that the large majority of galaxies within group- and cluster-mass halos have acquired the bulk of their stellar mass through gas accretion and not via galaxy mergers. We deduce that most dwarf ellipticals are not built up by galaxy mergers. With the second approach, we constrain the star formation histories of dwarfs by requiring that star formation must occur within halos of a minimum circular velocity set by the evolution of the temperature of the IGM, starting before the epoch of reionization. We qualitatively reproduce the downsizing trend of greater ages at greater masses and predict an upsizing trend of greater ages as one proceeds to masses lower than m_crit. We find that the fraction of galaxies with very young stellar populations (more than half the mass formed within the last 1.5 Gyr) is a function of present-day mass in stars and cold gas, which peaks at 0.5% at m_crit=10^6-8 M_Sun, corresponding to blue compact dwarfs such as I Zw 18. We predict that the baryonic mass function of galaxies should not show a maximum at masses above 10^5.5, M_Sun, and we speculate on the nature of the lowest mass galaxies.
We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (mathcal{L}u)(i) = -sum_{j sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i in V$, we define the diffusion distance to a set $B subset V$ as the smallest number of steps $d_{B}(i) in mathbb{N}$ required for half of all random walks started in $i$ and moving randomly with respect to the weights $p_{ij}$ to visit $B$ within $d_{B}(i)$ steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if $u$ satisfies $mathcal{L}u = lambda u$ on $V$ and $$ B = left{ i in V: - varepsilon leq u(i) leq varepsilon right} eq emptyset,$$ then, for all $i in V$, $$ d_{B}(i) log{left( frac{1}{|1-lambda|} right) } geq log{left( frac{ |u(i)| }{|u|_{L^{infty}}} right)} - log{left(frac{1}{2} + varepsilonright)}.$$ $d_B(i)$ is a remarkably good approximation of $|u|$ in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.
Let $Omega$ be a curvilinear polygon and $Q^gamma_{Omega}$ be the Laplacian in $L^2(Omega)$, $Q^gamma_{Omega}psi=-Delta psi$, with the Robin boundary condition $partial_ u psi=gamma psi$, where $partial_ u$ is the outer normal derivative and $gamma>0$. We are interested in the behavior of the eigenvalues of $Q^gamma_Omega$ as $gamma$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^gamma_Omega$ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with $partial Omega$. In the particular case of a polygon with straight edges the first eigenpairs are exponentially close to those of the model operators. Finally, we prove a Weyl asymptotics for the eigenvalue counting function of $Q^gamma_Omega$ for a threshold depending on $gamma$, and show that the leading term is the same as for smooth domains.
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