Related papers: The phase transition in inhomogeneous random inter…
We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ${\mathbb{R}}^d$. The connectivity function is shown to decay superexponentially, and we…
We consider inhomogeneous Erd\H{o}s-R\'enyi graphs. We suppose that the maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the asymptotic behavior of the $n^{1 - o(1)}$ largest eigenvalues of the adjacency matrix and its…
This paper focuses on the problem of the degree sequence for a mixed random graph process which continuously combines the {\it classical} model and the BA model. Note that the number of step added edges for the mixed model is random and…
In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform…
We show the asymptotic degree distribution of the typical vertex of a sparse inhomogeneous random intersection graph.
In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from $(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest…
We present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models, where vertices come in different types, and the probability of realizing an edge depends on the types of its…
We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these…
The energy level statistics of uniform random graphs are studied, by treating the graphs as random tight-binding lattices. The inherent random geometry of the graphs and their dynamical spatial dimensionality, leads to various quantum…
We consider a random walk on a $d$-regular graph $G$ where $d\to\infty$ and $G$ satisfies certain conditions. Our prime example is the $d$-dimensional hypercube, which has $n=2^d$ vertices. We explore the likely component structure of the…
Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that,…
The maximum matching problem on random graphs is studied analytically by the cavity method of statistical physics. When the average vertex degree \mth{c} is larger than \mth{2.7183}, groups of max-matching patterns which differ greatly from…
Understanding what types of phenomena lead to discontinuous phase transitions in the connectivity of random networks is an outstanding challenge. Here we show that a simple stochastic model of graph evolution leads to a discontinuous…
The evolution of the usual Erd\H{o}s-R\'{e}nyi random graph model on n vertices can be described as follows: At time 0 start with the empty graph, with n vertices and no edges. Now at each time k, choose 2 vertices uniformly at random and…
The article considers an inhomogeneous Erd\H{o}s-R\"enyi random graph on $\{1,\ldots, N\}$, where an edge is placed between vertices $i$ and $j$ with probability $\varepsilon_N f(i/N,j/N)$, for $i\le j$, the choice being made independent…
A chord diagram refers to a set of chords with distinct endpoints on a circle. The intersection graph of a chord diagram $\cal C$ is defined by substituting the chords of $\cal C$ with vertices and by adding edges between two vertices…
This work studies fundamental limits for recovering the underlying correspondence among multiple correlated graphs. In the setting of inhomogeneous random graphs, we present and analyze a matching algorithm: first partially match the graphs…
The exponential family of random graphs represents an important and challenging class of network models. Despite their flexibility, conventionally used exponential random graphs have one shortcoming. They cannot directly model weighted…
Consider the interchange process on a connected graph $G=(V,E)$ on $n$ vertices. I.e.\ shuffle a deck of cards by first placing one card at each vertex of $G$ in a fixed order and then at each tick of the clock, picking an edge uniformly at…
In random graph models, the degree distribution of an individual node should be distinguished from the (empirical) degree distribution of the graph that records the fractions of nodes with given degree. We introduce a general framework to…