Entropy and Hausdorff Dimension in Random Growing Trees

We investigate the limiting behavior of random tree growth in preferential attachment models. The tree stems from a root, and we add vertices to the system one-by-one at random, according to a rule which depends on the degree distribution of the already existing tree. The so-called weight function, in terms of which the rule of attachment is formulated, is such that each vertex in the tree can have at most K children. We define the concept of a certain random measure mu on the leaves of the limiting tree, which captures a global property of the tree growth in a natural way. We prove that the Hausdorff and the packing dimension of this limiting measure is equal and constant with probability one. Moreover, the local dimension of mu equals the Hausdorff dimension at mu-almost every point. We give an explicit formula for the dimension, given the rule of attachment

Mutation, selection, and ancestry in branching models: a variational approach

We consider the evolution of populations under the joint action of mutation and differential reproduction, or selection. The population is modelled as a finite-type Markov branching process in continuous time, and the associated genealogical tree is viewed both in the forward and the backward direction of time. The stationary type distribution of the reversed process, the so-called ancestral distribution, turns out as a key for the study of mutation-selection balance. This balance can be expressed in the form of a variational principle that quantifies the respective roles of reproduction and mutation for any possible type distribution. It shows that the mean growth rate of the population results from a competition for a maximal long-term growth rate, as given by the difference between the current mean reproduction rate, and an asymptotic decay rate related to the mutation process; this tradeoff is won by the ancestral distribution. Our main application is the quasispecies model of sequence evolution with mutation coupled to reproduction but independent across sites, and a fitness function that is invariant under permutation of sites. Here, the variational principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure

A Parsec Scale Accelerating Radio Jet in the Giant Radio Galaxy NGC315

Observations of the core of the giant radio galaxy NGC315 made with VLBI interferometers are discussed in the context of a relativistic jet. The sidedness asymmetry suggests Doppler favoritism from a relativistic jet. The presence of moving features in the jet as well as jet counter--jet brightness ratios hint at an accelerating, relativistic jet. An increasing jet velocity is also supported by a comparison of the jet's observed properties with the predictions of an adiabatic expansion model. On the parsec scale, the jet is unpolarized at a wavelength of 6 cm to a very high degree in clear distinction to the high polarization seen on the kiloparsec scale.Comment: 24 pages with 8 figures. ApJ in pres

3C 129 at 90cm: Evidence for a Radio Relic?

We present a new wide-field map of the radio galaxy 3C 129 and its companion galaxy 3C 129.1 at lambda=90 cm. We see a distinct steep-spectrum feature near the head of 3C 129, extending in a direction perpendicular to the radio tails. We propose that this Crosspiece might consist of fossil radio plasma, which has been re-energized by the compression of the bow shock of the supersonically moving galaxy 3C 129. One possible origin of the fossil radio plasma could be the tail of a nearby head-tail radio galaxy. We discuss the implications of, and give testable predictions for this scenario.Comment: 8 pages, 2 figures, accepted for publication in A

Characterizing the Initial Phase of Epidemic Growth on some Empirical Networks

A key parameter in models for the spread of infectious diseases is the basic reproduction number $R_0$, which is the expected number of secondary cases a typical infected primary case infects during its infectious period in a large mostly susceptible population. In order for this quantity to be meaningful, the initial expected growth of the number of infectious individuals in the large-population limit should be exponential. We investigate to what extent this assumption is valid by performing repeated simulations of epidemics on selected empirical networks, viewing each epidemic as a random process in discrete time. The initial phase of each epidemic is analyzed by fitting the number of infected people at each time step to a generalised growth model, allowing for estimating the shape of the growth. For reference, similar investigations are done on some elementary graphs such as integer lattices in different dimensions and configuration model graphs, for which the early epidemic behaviour is known. We find that for the empirical networks tested in this paper, exponential growth characterizes the early stages of the epidemic, except when the network is restricted by a strong low-dimensional spacial constraint, such as is the case for the two-dimensional square lattice. However, on finite integer lattices of sufficiently high dimension, the early development of epidemics shows exponential growth.Comment: To be included in the conference proceedings for SPAS 2017 (International Conference on Stochastic Processes and Algebraic Structures), October 4-6, 201

Multifrequency VLA observations of the FR I radio galaxy 3C 31: morphology, spectrum and magnetic field

We present high-quality VLA images of the FR I radio galaxy 3C 31 in the frequency range 1365 to 8440 MHz with angular resolutions from 0.25 to 40 arcsec. Our new images reveal complex, well resolved filamentary substructure in the radio jets and tails. We also use these images to explore the spectral structure of 3C 31 on large and small scales. We infer the apparent magnetic field structure by correcting for Faraday rotation. Some of the intensity substructure in the jets is clearly related to structure in their apparent magnetic field: there are arcs of emission where the degree of linear polarization increases, with the apparent magnetic field parallel to the ridges of the arcs. The spectral indices are significantly steeper (0.62) within 7 arcsec of the nucleus than between 7 and 50 arcsec (0.52 - 0.57). The spectra of the jet edges are also slightly flatter than the average for their surroundings. At larger distances, the jets are clearly delimited from surrounding larger-scale emission both by their flatter radio spectra and by sharp brightness gradients. The spectral index of 0.62 in the first 7 arcsec of 3C 31's jets is very close to that found in other FR I galaxies where their jets first brighten in the radio and where X-ray synchrotron emission is most prominent. Farther from the nucleus, where the spectra flatten, X-ray emission is fainter relative to the radio. The brightest X-ray emission from FR I jets is therefore not associated with the flattest radio spectra, but with a particle-acceleration process whose characteristic energy index is 2.24. The spectral flattening with distance from the nucleus occurs where our relativistic jet models require deceleration, and the flatter-spectra at the jet edges may be associated with transverse velocity shear. (Slightly abridged)Comment: 17 pages, 13 figures, accepted for publication in MNRA

A weighted configuration model and inhomogeneous epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight. It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that $R_0$ can be substantially over- or underestimated if the correlation between degree and weight is not taken into account

The 74MHz System on the Very Large Array

The Naval Research Laboratory and the National Radio Astronomy Observatory completed implementation of a low frequency capability on the VLA at 73.8 MHz in 1998. This frequency band offers unprecedented sensitivity (~25 mJy/beam) and resolution (~25 arcsec) for low-frequency observations. We review the hardware, the calibration and imaging strategies, comparing them to those at higher frequencies, including aspects of interference excision and wide-field imaging. Ionospheric phase fluctuations pose the major difficulty in calibrating the array. Over restricted fields of view or at times of extremely quiescent ionospheric ``weather'', an angle-invariant calibration strategy can be used. In this approach a single phase correction is devised for each antenna, typically via self-calibration. Over larger fields of view or at times of more normal ionospheric ``weather'' when the ionospheric isoplanatic patch size is smaller than the field of view, we adopt a field-based strategy in which the phase correction depends upon location within the field of view. This second calibration strategy was implemented by modeling the ionosphere above the array using Zernike polynomials. Images of 3C sources of moderate strength are provided as examples of routine, angle-invariant calibration and imaging. Flux density measurements indicate that the 74 MHz flux scale at the VLA is stable to a few percent, and tied to the Baars et al. value of Cygnus A at the 5 percent level. We also present an example of a wide-field image, devoid of bright objects and containing hundreds of weaker sources, constructed from the field-based calibration. We close with a summary of lessons the 74 MHz system offers as a model for new and developing low-frequency telescopes. (Abridged)Comment: 73 pages, 46 jpeg figures, to appear in ApJ