Metastability for the contact process on the preferential attachment graph

We consider the contact process on the preferential attachment graph. The work of Berger, Borgs, Chayes and Saberi [BBCS1] confirmed physicists predictions that the contact process starting from a typical vertex becomes endemic for an arbitrarily small infection rate $\lambda$ with positive probability. More precisely, they showed that with probability $\lambda^{\Theta (1)}$, it survives for a time exponential in the largest degree. Here we obtain sharp bounds for the density of infected sites at a time close to exponential in the number of vertices (up to some logarithmic factor).Comment: 45 pages; accepted for publication in Internet Mathematic

Metastability for the contact process on the configuration model with infinite mean degree

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two. We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an exponent larger than $2$ obtained in \cite{CD,MMVY,MVY}.Comment: Proposition 6.2 replaced by a weaker version (after a gap in its proof was mentioned to us by Daniel Valesin). Does not affect the two main theorems of the pape

Super-exponential extinction time of the contact process on random geometric graphs

In this paper, we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connecting radius tending to infinity. We obtain that for any infection rate $\lambda >0$, the contact process on these graphs survives a time super-exponential in the number of vertices.Comment: Accepted for publication in Combinatorics, Probability and Computin

Extinction time for the contact process on general graphs

23p.International audienceWe consider the contact process on finite and connected graphs and study the behavior of the extinction time, that is, the amount of time that it takes for the infection to disappear in the process started from full occupancy. We prove, without any restriction on the graph G, that if the infection rate λ is larger than the critical rate of the one-dimensional process, then the extinction time grows faster than exp{|G|/(log |G|) κ } for any constant κ > 1, where |G| denotes the number of vertices of G. Also for general graphs, we show that the extinction time divided by its expectation converges in distribution, as the number of vertices tends to infinity, to the exponential distribution with parameter 1. These results complement earlier work of Mountford, Mourrat, Valesin and Yao, in which only graphs of bounded degrees were considered, and the extinction time was shown to grow exponentially in n; here we also provide a simpler proof of this fact

The occurence and effects of short paths in scale-free geometric random graphs

We investigate a large class of geometric random graphs defined on a Poisson point process in Euclidean space, where each vertex carries an independent random mark. On this vertex set edges are established at random, such that the class is only determined by upper and lower bounds on the connection probabilities between finitely many pairs of vertices, which depend crucially on the marks and the spatial distance of each pair of vertices. This class includes different geometric random graphs emerging from real-world network models, such as a version of spatial preferential attachment (where marks can be understood as birth times), and continuum percolation, such as the soft Boolean model, as well as a whole range of further graph models with scale-free degree distribution and edges between distant vertices. For this class of geometric random graphs we study the occurence of short paths leading to ultrasmallness of the graphs, i.e. that the graph distance of a pair of distant vertices grows at most of doubly logarithmic order in the spatial distance of the pair. We give a sharp criterion for the absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two very distant vertices. Unlike in non-spatial scale-free network models and spatially embedded random graphs such as scale-free percolation the boundary of the ultrasmall regime and the limit theorem depend not only on the power-law exponent of the degree distribution but also on the rate of decay of the probability of an edge connecting two vertices with typical marks in terms of their Euclidean distance. Furthermore, we study the effect of the short paths in the ultrasmall regime on the survival of the contact process on geometric random graphs in this class. We show that the non-extinction probability is positive for any positive choice of the infection rate and give precise asymptotics for it when the infection rate decays to zero. On finite spatial restrictions of the graphs we show that the extinction time is of exponential order of the size of the graphs

Percolation by cumulative merging and phase transition for the contact process on random graphs

Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging smaller clusters and cumulating their weights. For several classical random weighted graphs, we show that there exists a phase transition regarding the existence of an infinite cluster. The motivation for introducing this partition arises from a connection with the contact process as it roughly describes the geometry of the sets where the process survives for a long time. We give a sufficient condition on a graph to ensure that the contact process has a non trivial phase transition in terms of the existence of an infinite cluster. As an application, we prove that the contact process admits a sub-critical phase on d-dimensional random geometric graphs and on random Delaunay triangulations. To the best of our knowledge, these are the first examples of graphs with unbounded degrees where the critical parameter is shown to be strictly positive.Comment: 50 pages, many figure

Selective recovery of salts from a ternary eutectic system in EFC using seeding

Industrial and mining saline streams are often multi-component in nature. Much research within Eutectic Freeze Crystallization (EFC) has focused on the crystallization of ice and single salts from aqueous solutions. However, as a single salt and ice are crystallized, the concentration of the non-crystallizing salt species increase until the system is saturated with more than two species. In such a situation, the sequence and rate of crystallization of each species depends on both the kinetics of crystallization of each salt and the interaction between the different species. Seeding could be employed to control kinetics and thereby achieve selective recovery from multi-supersaturated systems. The aim of this study was therefore to determine the effect of seeding on the yield and purity of the salt product in a system supersaturated with two salts and ice. A eutectic MgSO4-Na2SO4-H2O system was chosen for this study as these salts are prevalent in saline waste streams in South Africa. A continuous 2 ℓ jacketed, scraped and stirred glass crystallizer was seeded with Na2SO4∙10H2O, MgSO4∙11H2O and ice. The initial salt seed loading and initial supersaturation were varied. The operating conditions used were 30 minutes residence time, a coolant temperature of -11°C, and operating temperature of approximately - 5.0 to -5.1°C. An increase in seeding mass was found to increase the yield and proportion of the seeded salt in the salt product due to an increase in salt growth rate. However, in all experiments it was found that MgSO4∙11H2O crystallized out at fractions higher than the eutectic thermodynamic ratio, indicating a higher selectivity towards this salt. Furthermore, the introduction of 30 g of MgSO4∙11H2O seeds produced a pure salt product (above 99.4 wt.% purity) and the highest salt yield. A similar mass of either seeding material resulted in a similar total mass of salt product. This was attributed to MgSO4∙11H2O crystallizing as the majority salt, and therefore its kinetics played a major role in the total salt yield. Initial supersaturation was found to have no significant effect on steady state salt purity and yield. This study showed that multiple steady states exist within this system at the same operating conditions but different initial seeding conditions. Seeding was found to have the potential to engineer the salt purity of the overflow and underflow split fractions by changing the individual salt average particle sizes. Therefore, this study showed that selectivity recovery of one salt is possible in a multi supersaturated system through seed engineering

Topics in complex systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Fundamental laws of physics, although successful in explaining many phenomena observed in nature and society, cannot account for the behaviour of complex, non-Hamiltonian systems. Much effort has been devoted to better understanding the topological properties of these systems. Neither ordered nor disordered, these systems of high variability are found in many areas of science. Studies on sandpiles, earthquakes and lattice gases have all yielded evidence of complexity in the form of power law distributions. This scalefree characteristic is believed to be the hall-mark of complexity known as self-organised criticality. Systems in the self-organised critical state regulate themselves and are resistant to error and attacks. The aim of this thesis is to further current knowledge of complex systems by proposing and analysing three models of real systems. Statistical mechanics and numerical simulations are used to analyse these models. The first model mimics herd behaviour in social groups and encompasses growth and addition. It has been found that when the growth rate is fast enough, the group size distribution conforms to a power law. When the growth rate is slow, the system runs out of free agents in finite time. The second model aims to capture the basic empirical measurements from hospital waiting lists. This model illustrates how the power law distributions found in empirical studies might arise, but also indicates that these distributions are unlikely to be caused by the preferential behaviour of patients or physicians. The third model is a salary comparison model; the salary distributions of most of its variants are power laws. Both mean field and 1-d versions of the model are analysed, and differences between the two versions are identified by looking at the mean absolute difference between the salaries in each version