# Tagged Questions

**4**

votes

**2**answers

570 views

### Do Random Walks on the Hexagonal Lattice have a limit?

For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn
induces a tiling of ...

**4**

votes

**0**answers

138 views

### Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice.
Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...

**3**

votes

**2**answers

172 views

### Methods to approximate the betweenness centrality on large networks

To calculate the between centrality wiki def:
$g(v) = \sum_{s\neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}$
of a node in a graph/network;$\sigma_{st}$ is the ...

**4**

votes

**2**answers

152 views

### Estimate size of graph by taking random walks

Let $G$ be a connected, finite graph and let $v_0$ be a vertex of $G$. I'm interested in methods of estimating the number of vertices in $G$, based on local exploration only. What I have in mind is:
...

**5**

votes

**1**answer

267 views

### Random path in a graph

Consider a finite graph $G$. I would like to define a random path between two vertices $s$ and $t$ of the graph $G$ by looking at a measure $\mu$ on all spanning trees. Then the probability of a given ...

**1**

vote

**2**answers

190 views

### Random walk on the hypercube

Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number $x$. I think I just ...

**4**

votes

**1**answer

258 views

### Probabilities of a random walk exiting a set

Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as ...

**1**

vote

**1**answer

67 views

### Spanning subgaph with trivial Poisson boundaries

Assume $\Gamma$ is the Cayley graph of an amenable$^{*}$ group and that the simple random walk has non-trivial Poisson boundary$^{**}$. Is there a spanning connected subgraph $\Gamma'$ of some ...

**1**

vote

**1**answer

193 views

### Stationary distribution for directed graph

I want to implement the algorithm of graph partitioning of sparse directed graph. In this algorithm after computing the transition matrix ,we should compute the stationary distribution of the random ...

**12**

votes

**3**answers

456 views

### Estimate on currents in Cayley graphs

Take a Cayley graph $\Gamma$ (thought of as an electrical network with all edges having equal resistance) and break one edge $e$ and put a battery there. (Assume the graph has only one end* so that ...

**4**

votes

**2**answers

257 views

### How does a quasi-isometry affect Poisson or Martin boundaries?

Take two graphs (of bounded valency) or manifolds (f bounded geometry) $G$ and $G'$. Assume there is a quasi-isometry $f:G \to G'$, and assume the Poisson or Martin boundary of $G$ is known, what may ...

**1**

vote

**1**answer

172 views

### Distributions induced by (weighted) random walks on the integer lattice

Consider an integer lattice $\mathbb{Z}^2$ where grid points are separated by a distance $h$. Loosely speaking, a random walk of length $k$ is a sequence of lattice points $(x_1,\cdots,x_k)$ ...

**11**

votes

**5**answers

863 views

### Simple Random Walk on a Locally Finite Graph - when is it recurrent?

I'm giving a talk tomorrow about a result in computer science which I recently proved. It's a recurrence-transience result on a random process which is related in spirit to a simple random walk. My ...

**2**

votes

**2**answers

512 views

### probability distribution of hitting nodes on a finite graph random walk

Consider a finite, undirected, scale-free graph $\{G}$, with uniform edge weights. We define a truncated random walk on $\{G}$ as a random walk that continues for exactly $\{k}$ steps. For an ...

**5**

votes

**0**answers

228 views

### Any approximation algorithms for self-avoiding walks?

I've a graph whose edges are weighted by probabilities, perhaps all equal. I would like to compute the overall probability of traveling between vertices x and y in the graph after I delete each edge ...

**1**

vote

**2**answers

728 views

### Probability of first return to starting vertex in Random walk on regular finite graph

Hi, this is related to this earlier question.
Given Random walk on a regular graph $G=(V,E)$. The Random walk is simple so that transition probabilities are $1/\text{deg}(v_i)$, and time is in ...

**9**

votes

**1**answer

651 views

### Probability of return vs. probability of return in minimal number of steps

Consider a random walk on a connected graph $G=(V,E)$. That is, associate to each neighbouring nodes $a,b\in V\ $ transition probabilities $\mathbb{P}(a\rightarrow b), \mathbb{P}(b\rightarrow a) $ ...

**7**

votes

**1**answer

625 views

### Random walk on a simple finite network

Consider a graph $\Delta_N = \lgroup (x,y)\in\mathbb{Z}^2| x+y\leq N-1, x\geq 0,\ y\geq 0 \rgroup$ (set of edges is defined in a natural way): see here ).
Take a random walker that wonders around ...

**1**

vote

**1**answer

257 views

### Winding number bijection on graphs

Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center ...

**4**

votes

**0**answers

382 views

### Monotonic properties of harmonic functions on graphs

I have a question concerning monotonic properties of "generalized harmonic functions" on graphs. I am a physicist and I didn't take any separate courses in neither graph theory nor discrete harmonic ...