Related papers: Self-avoiding walks and amenability

Given a quasi-transitive infinite graph $G$ with volume growth rate ${\rm gr}(G),$ a transient biased electric network $(G,\, c_1)$ with bias $\lambda_1\in (0,\,{\rm gr}(G))$ and a recurrent biased one $(G,\, c_2)$ with bias $\lambda_2\in…

Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are…

We study the geometric properties of graphs with non-negative Ollivier-Ricci curvature, a discrete analogue of non-negative Ricci curvature in Riemannian geometry. We prove that for each $d<\infty$ there exists a constant $C_d$ such that if…

We prove that every recurrent graph $G$ quasi-isometric to $\mathbb{R}$ admits an essentially unique Lipschitz harmonic function $h$. If $G$ is vertex-transitive, then the action of $Aut(G)$ preserves $\partial h$ up to a sign, a fact that…

Dynamic graphs have emerged as an appropriate model to capture the changing nature of many modern networks, such as peer-to-peer overlays and mobile ad hoc networks. Most of the recent research on dynamic networks has only addressed the…

For a unimodular random graph $(G,\rho)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,\rho)$, which is the best asymptotic…

We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by…

The join $X\vee Y$ of two graphs $X$ and $Y$ is the graph obtained by joining each vertex of $X$ to each vertex of $Y$. We explore the behaviour of a continuous quantum walk on a weighted join graph having the adjacency matrix or Laplacian…

In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed in some quantum walk algorithms. We find that continuous-time quantum walks can exploit the covering space…

The scaling entropy of a p.m.p. action is a slow-entropy type invariant that characterizes the intermediate growth of entropy in a dynamical system. An amenable group $G$ has a scaling entropy growth gap if the scaling entropy of any its…

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic…

The class A of countable groups that admit a faithful, transitive, amenable -- in the sense that there is an invariant mean -- action on a set has been widely investigated in the past. In this paper, we no longer require the action to be…

Recently, it has been proposed that the natural connectivity can be used to efficiently characterise the robustness of complex networks. Natural connectivity quantifies the redundancy of alternative routes in a network by evaluating the…

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. Hitting times for discrete quantum walks on graphs give an average time before the walk…

In this paper we construct distance-regular graphs admitting a transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$. From the code spanned by…

Let $G$ be an infinite group and let $X$ be a finite generating set for $G$ such that the growth series of $G$ with respect to $X$ is a rational function; in this case $G$ is said to have rational growth with respect to $X$. In this paper a…

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still…

Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$…

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type…

From a graph $G$ with constant valency $v$ and a (non-compact) manifold $C$ with $v$ boundary components, we build a $G$-periodic manifold $M$. This process gives a class of topologically infinite manifolds which generalizes periodic…