Exact Expectations of Minimal Spanning Trees for Graphs With Random Edge Weights

Two methods are used to compute the expected value of the length of the minimal spanning tree (MST) of a graph whose edges are assigned lengths which are independent and uniformly distributed. The first method yields an exact formula in terms of the Tutte polynomial. As an illustration, the expected length of the MST of the Petersen graph is found to be 34877/12012 = 2.9035 .... A second, more elementary, method for computing the expected length of the MST is then derived by conditioning on the length of the shortest edge. Both methods in principle apply to any finite graph. To illustrate the method we compute the expected lengths of the MSTs for complete graphs

Phase Transition in the Aldous-Shields Model of Growing Trees

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to c^{-l} where c is a positive parameter and l is the distance of the perimeter site from the root. For c=1, this model corresponds to random binary search trees and for c=2 it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value c=sqrt{2}. While for c>sqrt{2} the variance is proportional to the mean and the distribution is normal, for c<sqrt{2} the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at c=sqrt{m}.Comment: Latex 17 pages, 6 figure

Invariance Properties of Schoenberg's Tone Row System

1 online resource (PDF, 24 pages

Updating models for restoration and management of fiery ecosystems

© 2015 Elsevier B.V. Scientific models that guide restoration/management protocols should be reviewed periodically as new data become available. We examine ecological concepts used to guide restoration of pine savannas and woodlands, historically prominent but now rare habitats in the southern North American Coastal Plain. For many decades, pine savanna management has been guided predominantly by a biome-centric succession model. Pine savannas have been considered early-successional communities that, in the absence of fire, transition rapidly toward closed-canopy hardwood forests. Recurrent fires have been viewed as exogenous disturbances that maintain savanna ecosystems as a sub-climax, blocking succession to an equilibrium steady state (closed-canopy forests). Over recent decades, a vegetation-fire feedback model has emerged in which pine savannas are conceptualized as persistent, non-equilibrium communities maintained by endogenous, co-evolutionary vegetation-fire feedbacks. Endemic plant species are resistant to fires and specialized for post-fire conditions generated by frequent lightning fires, primarily within a distinct fire season. These species produce pyrogenic fine fuels that are easily ignited. The resulting fire regimes, entrained by these vegetation-fire feedbacks, are predicted to result in persistent pine savannas. Local variation over space and time in evolutionary feedback mechanisms between pyrogenic vegetation and fire regimes produces heterogeneous landscapes. Disturbances of these feedbacks, such as human fire suppression, are postulated to result in rapid transition to communities lacking feedback elements, such as closed-canopy forest and those without pyrogenic species. Succession-based management focuses on reversing the transition to forest, primarily by removing hardwoods and reintroducing fire as a disturbance. However, we advocate restoration and management approaches that target reinstitution of functional vegetation-fire feedbacks. Such approaches should favor native pyrogenic plant species and reinstitute fire regimes that mimic historical, evolutionarily derived fire regimes. Vegetation-fire feedback concepts should be useful in addressing resistance and resilience of fiery ecosystems worldwide to inherent changes in feedback mechanisms, constituting a framework useful in addressing global management challenges

Almost uniform sampling via quantum walks

Many classical randomized algorithms (e.g., approximation algorithms for #P-complete problems) utilize the following random walk algorithm for {\em almost uniform sampling} from a state space $S$ of cardinality $N$: run a symmetric ergodic Markov chain $P$ on $S$ for long enough to obtain a random state from within $\epsilon$ total variation distance of the uniform distribution over $S$. The running time of this algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} (\log N + \log \epsilon^{-1}))$, where $\delta$ is the spectral gap of $P$. We present a natural quantum version of this algorithm based on repeated measurements of the {\em quantum walk} $U_t = e^{-iPt}$. We show that it samples almost uniformly from $S$ with logarithmic dependence on $\epsilon^{-1}$ just as the classical walk $P$ does; previously, no such quantum walk algorithm was known. We then outline a framework for analyzing its running time and formulate two plausible conjectures which together would imply that it runs in time $O(\delta^{-1/2} \log N \log \epsilon^{-1})$ when $P$ is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac

Boundary non-crossings of Brownian pillow

Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]^2\to R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability \psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall s,t\in [0,1]}. Further we investigate the asymptotic behaviour of $\psi(u;\gamma h)$ with $\gamma$ tending to infinity, and solve a related minimisation problem.Comment: 14 page

First Passage Properties of the Erdos-Renyi Random Graph

We study the mean time for a random walk to traverse between two arbitrary sites of the Erdos-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-monotonically with p near the percolation transition. Much of this behavior can be understood by simple heuristic arguments.Comment: 10 pages, 9 figures, 2-column revtex4 forma

Political branding: sense of identity or identity crisis? An investigation of the transfer potential of the brand identity prism to the UK Conservative Party

Brands are strategic assets and key to achieving a competitive advantage. Brands can be seen as a heuristic device, encapsulating a series of values that enable the consumer to make quick and efficient choices. More recently, the notion of a political brand and the rhetoric of branding have been widely adopted by many political parties as they seek to differentiate themselves, and this has led to an emerging interest in the idea of the political brand. Therefore, this paper examines the UK Conservative Party brand under David Cameron’s leadership and examines the applicability of Kapferer’s brand identity prism to political branding. This paper extends and operationalises the brand identity prism into a ‘political brand identity network’ which identifies the inter-relatedness of the components of the corporate political brand and the candidate political brand. Crucial for practitioners, this model can demonstrate how the brand is presented and communicated to the electorate and serves as a useful mechanism to identify consistency within the corporate and candidate political brands

Exact eigenvalue spectrum of a class of fractal scale-free networks

The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.Comment: Definitive version accepted for publication in EPL (Europhysics Letters

Exact field ionization rates in the barrier suppression-regime from numerical TDSE calculations

Numerically determined ionization rates for the field ionization of atomic hydrogen in strong and short laser pulses are presented. The laser pulse intensity reaches the so-called "barrier suppression ionization" regime where field ionization occurs within a few half laser cycles. Comparison of our numerical results with analytical theories frequently used shows poor agreement. An empirical formula for the "barrier suppression ionization"-rate is presented. This rate reproduces very well the course of the numerically determined ground state populations for laser pulses with different length, shape, amplitude, and frequency. Number(s): 32.80.RmComment: Enlarged and newly revised version, 22 pages (REVTeX) + 8 figures in ps-format, submitted for publication to Physical Review A, WWW: http://www.physik.tu-darmstadt.de/tqe