Jonathan Jordan

A RANDOM HIERARCHICAL LATTICE: THE SERIES-PARALLEL GRAPH AND ITS PROPERTIES

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

Degree sequences of geometric preferential attachment graphs

We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter α is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in Bollobás et al. (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results.

Phase transitions for random geometric preferential attachment graphs

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

Preferential attachment with choice

We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses

Reversible Complex Hénon Maps

r

The spectra of the laplacians of fractal graphs not satisfying spectral decimation

vertices according to a preferential rule and connects to the vertex in the selection with the

COMB GRAPHS AND SPECTRAL DECIMATION

s

Spectrum of the Laplacian of an asymmetric fractal graph

th highest degree. For meek choice, where

Non-convergence of proportions of types in a preferential attachment graph with three co-existing types

s>1

The degree sequences and spectra of scale-free random graphs

, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where