Questions tagged [random-walks]
31 questions
16
votes
1 answer
On "The Average Height of Planted Plane Trees" by Knuth, de Bruijn and Rice (1972)
I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" want to prove that the average height $h_n$ of a…
Christian Rinderknecht
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9
votes
2 answers
Algorithm to generate self-avoiding random walk on a lattice
Where can I find some code to generate random self-avoiding walks on 2 and 3-dimensional lattices whose side-lengths are powers of two? The walk should pass through every point on the lattice More specifically, how can I find a random hamiltonian…
Alecto
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6
votes
1 answer
Generating uniform random connected graphs: doubt about Wilson's algorithm
I want to generate a random connected simple labeled graph with $n$ vertices and $m$ edges, selected uniformly over all connected graphs with such $n$ and $m$. I found this approach. It says: build a random spanning tree using a loop-erased random…
Ivan Smirnov
- 964
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4
votes
2 answers
Average vs Worst-Case Hitting Time
Consider a simple random walk on an undirected graph and let $H_{ij}$ be the hitting time from $i$ to $j$. How much bigger can
$$ H_{\rm max} = \max_{i,j} H_{ij}, $$ be compared to
$$ H_{\rm ave} = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n H_{ij}.$$…
Pramod T.
- 75
- 3
4
votes
1 answer
Examples for directed graphs with super polynomial cover time
The cover time of a graph is the expected number of steps in a random walk on the graph until we visit all the nodes.
For undirected graphs the cover time is upperbounded by $O(n^3)$.
What about directed graphs? I'm looking for examples of…
JonyWalk
- 55
- 2
4
votes
1 answer
Quantum algorithms and quantum computation
Is my (very high-level) understanding correct here regarding quantum algorithms —
Quantum computers can process a massive amount of operations in parallel to the nature of qubits and their ability to have states that are superpositions of…
sonicboom
- 147
- 5
4
votes
1 answer
Probability of reaching a state in asymmetric random walk
Consider the following random walk:
Namely, if $S_i$ is the state at time $i$, then $\Pr(S_{i+1}=1|S_i=0)=1$, and for every $s>0$ we have
$$S_{i+1}|S_i=s=
\begin{cases}
s+1 & \text{w.p. }1-p\\
s-1 & \text{w.p. }p
\end{cases}.
$$
In my application,…
AvidLearner
- 239
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3
votes
1 answer
Generating a random path in a grid without deadlock
I want to write an algorithm that takes an $n \times n$ grid and a number $L$, generate a random walk of length $L$ on the grid that doesn't visit the same cell twice.
One simple solution would be keep randomly choosing a neighbour of the last cell…
thbl2012
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3
votes
1 answer
How many random walks to start from each node?
Assume that we are given a real life graph, DBLP network in my case, where degree distribution of nodes follows a power law (many nodes have 1, 2 neighbors, and only a few nodes have hundreds of neighbors).
A random walk ends when it returns to the…
Cuneyt
- 31
- 1
3
votes
1 answer
Random Walk on the Integer Line
Suppose we are doing a random walk on the infinite integer line and that we take $2n$ total steps. At every step of this walk, the position of the walker is an integer point on this line. For the next step of this walk, the walker moves to one of…
user95224
- 309
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3
votes
0 answers
Lower bound of continuous random walk
Let $X_1,...X_T$ be i.i.d. random variables supported on the $[-1,1]$ segment, with an expected value of 0 and positive variance.
Let $H_t = | \sum_{i=1}^t X_i|$, and let $H = \max_{1\leq t\leq T} H_t$.
I'm interested in lower bounds on $\mathbb…
AvidLearner
- 239
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3
votes
1 answer
Expected length of a random walk on a line
I am given the following randomized algorithm for SAT,
Input: A satisfiable CNF-formula $\varphi$.
Output: An assignment $\rho$, such that $\rho \models \varphi$.
The algorithm works as follow:
Choose an arbitrary assignment $\rho$.
As long as…
Narek Bojikian
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2
votes
0 answers
Graph conductance - program/code/library
Technical question: is there any open source program/code/library which can compute (minimal) conductance of a given graph, probably by some simulated annealing?
Think it is quite well-known problem, but I cannot find anything like I mentioned…
Wojtek
2
votes
1 answer
Increasing entropy of random walk
Let $P$ be a transition matrix of a random walk in an undirected (may not regular) graph $G$. Let $\pi$ be a distribution on $V(G)$. The Shannon entropy of $\pi$ is defined by
$$H(\pi)=-\sum_{v \in V(G)}\pi_v\cdot\log(\pi_v).$$
How do we prove that…
eig
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2
votes
1 answer
2D random walk. Should both dimensions be independent?
My assignment is to compare several probability distributions in random walk algorithm. I'd like to analyse it in 2D linear space to make the concept more intuitive.
What is the correct approach in selecting consecutive positions of the particle? Do…
matt-pielat
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