Thresholds and the structure of sparse random graphs

In this thesis, we obtain approximations to the non-3-colourability threshold of sparse random graphs and we investigate the structure of random graphs near the region where the transition from 3-colourability to non-3-colourability seems to occur. It has been observed that, as for many other properties, the property of non-3-colourability of graphs exhibits a sharp threshold behaviour. It is conjectured that there exists a critical average degree such that when the average degree of a random graph is around this value the probability of the random graph being non-3-colourable changes rapidly from near 0 to near 1. The difficulty in calculating the critical value arises because the number of proper 3-colourings of a random graph is not concentrated: there is a `jackpot' effect. In order to reduce this effect, we focus on a sub-family of proper 3-colourings, which are called rigid 3-colourings. We give precise estimates for their expected number and we deduce that when the average degree of a random graph is bigger than 5, then the graph is asymptotically almost surely not 3-colourable. After that, we investigate the non-$k$-colourability of random regular graphs for any $k \geq 3$. Using a first moment argument, for each $k \geq 3$ we provide a bound so that whenever the degree of the random regular graph is bigger than this, then the random regular graph is asymptotically almost surely not $k$-colourable. Moreover, in a (failed!) attempt to show that almost all 5-regular graphs are not 3-colourable, we analyse the expected number of rigid 3-colourings of a random 5-regular graph. Motivated by the fact that the transition from 3-colourability to non-3-colourability occurs inside the subgraph of the random graph that is called the 3-core, we investigate the structure of this subgraph after its appearance. Indeed, we do this for the $k$-core, for any $k \geq 2$; and by extending existing techniques we obtain the asymptotic behaviour of the proportion of vertices of each fixed degree. Finally, we apply these results in order to obtain a more clear view of the structure of the 2-core (or simply the core) of a random graph after the emergence of its giant component. We determine the asymptotic distributions of the numbers of isolated cycles in the core as well as of those cycles that are not isolated there having any fixed length. Then we focus on its giant component, and in particular we give the asymptotic distributions of the numbers of 2-vertex and 2-edge-connected components

Continuum percolation of polydisperse hyperspheres in infinite dimensions

We analyze the critical connectivity of systems of penetrable $d$-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are tree-like for $d\rightarrow \infty$ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density $\eta_c\rightarrow 2^{-d}$, independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality $z_c$ is smaller than unity, while $z_c\rightarrow 1$ only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a lognormal distribution is no longer universal, and that it can be smaller than $2^{-d}$ for $d\rightarrow\infty$.Comment: 11 pages, 5 figure

Random Graph Models with Hidden Color

We demonstrate how to generalize two of the most well-known random graph models, the classic random graph, and random graphs with a given degree distribution, by the introduction of hidden variables in the form of extra degrees of freedom, color, applied to vertices or stubs (half-edges). The color is assumed unobservable, but is allowed to affect edge probabilities. This serves as a convenient method to define very general classes of models within a common unifying formalism, and allowing for a non-trivial edge correlation structure.Comment: 17 pages, 2 figures; contrib. to the Workshop on Random Geometry in Krakow, May 200

Random Graphs with Hidden Color

We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a nontrivial correlation structure in the resulting graphs. The approach unifies a number of existing random graph ensembles within a common general formalism, and allows for the analytic calculation of observable graph characteristics. In particular, generating function techniques are used to derive the size distribution of connected components (clusters) as well as the location of the percolation threshold where a giant component appears.Comment: 4 pages, no figures, RevTe

Conjoining Speeds up Information Diffusion in Overlaying Social-Physical Networks

We study the diffusion of information in an overlaying social-physical network. Specifically, we consider the following set-up: There is a physical information network where information spreads amongst people through conventional communication media (e.g., face-to-face communication, phone calls), and conjoint to this physical network, there are online social networks where information spreads via web sites such as Facebook, Twitter, FriendFeed, YouTube, etc. We quantify the size and the critical threshold of information epidemics in this conjoint social-physical network by assuming that information diffuses according to the SIR epidemic model. One interesting finding is that even if there is no percolation in the individual networks, percolation (i.e., information epidemics) can take place in the conjoint social-physical network. We also show, both analytically and experimentally, that the fraction of individuals who receive an item of information (started from an arbitrary node) is significantly larger in the conjoint social-physical network case, as compared to the case where the networks are disjoint. These findings reveal that conjoining the physical network with online social networks can have a dramatic impact on the speed and scale of information diffusion.Comment: 14 pages, 4 figure