Exchanges in complex networks: income and wealth distributions

We investigate the wealth evolution in a system of agents that exchange wealth through a disordered network in presence of an additive stochastic Gaussian noise. We show that the resulting wealth distribution is shaped by the degree distribution of the underlying network and in particular we verify that scale free networks generate distributions with power-law tails in the high-income region. Numerical simulations of wealth exchanges performed on two different kind of networks show the inner relation between the wealth distribution and the network properties and confirm the agreement with a self-consistent solution. We show that empirical data for the income distribution in Australia are qualitatively well described by our theoretical predictions.Comment: 8 pages, 11 figure

Some factors affecting journey-times on urban roads.

This thesis discusses the results of two types of study, (a) of the relationships between journey-time and traffic flow during peak periods on two sections of urban road, each about 0.4 miles in length, and (b) of relationships between journey-times and a number of land-use variables over about 50 miles of suburban main roads in off-peak periods. The results of study (a) show that the most significant relationships occurred when the data was analysed using non-linear or two-regime models. In the latter case the two regimes were separated by a critical flow. Above the critical flow changes in flow were correlated with changes in journey-time and below the critical flow the changes appeared to be random. Further evidence is provided to show that the use of different sampling intervals can give rise to different empirical relationships with the same data. A queuing model postulated by Davidson (1968) has been shown to give a satisfactory agreement with the results of study (a). A model is developed in this thesis which accounts for consistent changes in saturation flow arising from the use of different sampling intervals in Davidson's model. Additionally, an empirical index has been derived which reflects the changes in activity during the peak hour, and journey-time is highly correlated with this index. Study (b) shows that after the effects of major intersections have been accounted for, journey-time was significantly correlated with a number of factors of land-use along the suburban main roads which were studied. Furthermore, a number of mathematical techniques have been used which permit the calculation of indices of the variation of the surrounding conditions and these techniques were shown to produce robust and repeatable results. Journey-time was significantly correlated with an Activity Index which was associated with a number of land-use parameters

The Hausdorff dimension of graphs of prevalent continuous functions

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain results on the `horizon problem' for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function

A note on the 1-prevalence of continuous images with full Hausdorff dimension

We consider the Banach space consisting of real-valued continuous functions on an arbitrary compact metric space. It is known that for a prevalent (in the sense of Hunt, Sauer and Yorke) set of functions the Hausdorff dimension of the image is as large as possible, namely 1. We extend this result by showing that `prevalent' can be replaced by `1-prevalent', i.e. it is possible to \emph{witness} this prevalence using a measure supported on a one dimensional subspace. Such one dimensional measures are called \emph{probes} and their existence indicates that the structure and nature of the prevalence is simpler than if a more complicated `infinite dimensional' witnessing measure has to be used.Comment: 8 page

Effects of the Charge-Dipole Interaction on the Coagulation of Fractal Aggregates

A numerical model with broad applications to complex (dusty) plasmas is presented. The self-consistent N-body code allows simulation of the coagulation of fractal aggregates, including the charge-dipole interaction of the clusters due to the spatial arrangement of charge on the aggregate. It is shown that not only does a population of oppositely charged particles increase the coagulation rate, the inclusion of the charge-dipole interaction of the aggregates as well as the electric dipole potential of the dust ensemble decreases the gelation time by a factor of up to twenty. It is further shown that these interactions can also stimulate the onset of gelation, or "runaway growth," even in a population of particles charged to a monopotential where previously it was believed that like-charged grains would inhibit coagulation. Gelation is observed to occur due to the formation of high-mass aggregates with fractal dimensions greater than two which act as seeds for runaway growth.Comment: 9 page

Structural Phases of Bounded Three-Dimensional Screened Coulomb Clusters (Finite Yukawa System)

The formation of three-dimensional (3D) dust clusters within a complex plasma modeled as a spatially confined Yukawa system is simulated using the box_tree code. Similar to unscreened Coulomb clusters, the occurrence of concentric shells with characteristic occupation numbers was observed. Both the occupation numbers and radii were found to depend on the Debye length. Ground and low energy meta-stable states of the shielded 3D Coulomb clusters were determined for 4<N<20. The structure and energy of the clusters in different states was analyzed for various Debye lengths. Structural phase transitions, including inter-shell structural phase transitions and intra-shell structural phase transitions, were observed for varying Debye length and the critical value for transitions calculated