Questions tagged [markov-chains]

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Pagerank update upon vertex removal

Assume we have computed the Pagerank of the vertices of a given graph. Then, remove a vertex from this graph, with all its edges. How to efficiently compute the Pagerank of remaining vertices in the ...
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1 vote
1 answer
176 views

Effect of self loops on mixing time?

Consider 2 graphs G1 and G2. G1: Any non-regular graph. G2: Same graph but with added self-loops such that degree of each node is the same (either some $\Delta$, or maximum '$n$', where $n$ is the ...
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39 views

Two Questions on the paper Near-optimal Regret Bounds for Reinforcement Learning

I am reading the classic paper Near-optimal Regret Bounds for Reinforcement Learning. I have two questions: How to combine all MDPs in $ \mathcal{M} $ to get a single MDP with extended action space? ...
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  • 21
2 votes
1 answer
133 views

Why Asymptotic Equipartition Property theorem proofs assume the source is memoryless?

I do not understand the assumption $X_1, X_2, \cdots$ are i.i.d. ~p(x) in the AEP proofs I have seen. I have read some different sources for understanding the Asymptotic Equipartition Property. Using ...
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  • 133
1 vote
1 answer
1k views

Difference between CTMC, DTMC, and MDP

I've been reading the Handbook of Model Checking recently; I'm especially interested in probabilistic model checking, so have been led to the PRISM model checker. For background, I am very familiar ...
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  • 151
3 votes
1 answer
111 views

Reconstruction of a sequence generated by a Markov chain - reference request

Let S be a finite sequence of symbols from a finite alphabet, with gaps - that is on some known locations an unknown number of symbols are missing. Assuming that the sequence , including the symbols ...
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2 votes
0 answers
84 views

What is the computational complexity of determining the mixing time of a Cayley graph?

Bayer and Diaconis famously proved that a deck of fifty-two cards will be mixed after only seven dovetail shuffles. Numberphile has a nice series of videos of Diaconis explaining the proof. I ...
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163 views

Crime prevention using graph theory and machine learning

I am looking for a way to the model the incidence of crime among a network of individuals. Part of it will use machine learning, and part of it will have to resort to some graph theoretic ...
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  • 149
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0 answers
62 views

Probabilistic linebreaking algorithm

I'm currently trying to implement this paper: Bouckaert, Remco R., A probabilistic line breaking algorithm, Gedeon, Tamás D. (ed.) et al., AI 2003: Advances in Artificial Intelligence. 16th ...
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3 votes
1 answer
86 views

Reference Request: soundness in a ZKP achieved by walking along a doubly-stochastic Markov chain?

Consider the following variant of a zero-knowledge proof that two graphs, $G_1$ and $G_2$, given by adjacency matrices $M_1$ and $M_2$, respectively, are not isomorphic. Here Peggy the prover wants ...
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6 votes
1 answer
245 views

A random walk that moves to less-visited nodes

Consider a random walk on an undirected graph that keeps track of how many times it has visited every node. At each step, it moves to the node among its neighbors which has been visited the least ...
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1 vote
1 answer
207 views

Simulate a process of state change with transition probability dependent on proportion in state in previous time

I was thinking about a reverse single transferable vote type situation (i.e. most votes is eliminated) where the process continues until there is only on state left and at each new round there really ...
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1 vote
0 answers
168 views

Off-policy Monte Carlo Control

The off-policy Monte Carlo control algorithm to learn the optimal state-value function $V^*$ is given as follows, which is obtained from Sutton's book. I have three questions concerning this ...
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4 votes
1 answer
147 views

What automorphisms on a Markov Chain imply a uniform limiting distribution?

Consider an irreducible aperiodic Markov chain $M$, modeled as a connected directed graph with weighted edges. The existence of certain (graph) automorphisms on this Markov chain imply various ...
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  • 389
1 vote
1 answer
249 views

How can I rank paths through an HMM? [closed]

I have a profile hidden Markov model that I use to identify all instances of a user-defined pattern of symbols in a long sequence of symbols. I use the Viterbi algorithm to find the most probable path ...
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6 votes
1 answer
400 views

How much larger than the relaxation time can the mixing time be?

The notation is mostly taken from the book "Markov chains and mixing times" by Levin, Peres, and Wilmer. Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite ...
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  • 453
4 votes
1 answer
90 views

Data Mining of self-replicators

My current (very limited) understanding of the creative process that leads to the design of self-replicators is that any particular self-replicator, like Universal Constructor, Langton's loop or ...
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  • 185
-4 votes
1 answer
78 views

Probabilistic protocols [closed]

I want to model a probabilistic protocol using a model checker, but a lot of protocols are already implemented (e.g. Randomised Dining Philosophers, Dining cryptographers, Synchronous leader election ...
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1 vote
0 answers
925 views

Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...
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1 vote
1 answer
220 views

Policy Adjustment in Markov Decision Process

I was using MDP on my work to make optimal decision. I used discrete time, finite state MDP. I assumed that I will have an initial parameters, like the Reward/Cost, state transition probabilities and ...
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6 votes
0 answers
77 views

Fast convergence of a contagion process in special graphs

The process: Given is a clique $C_n$ of size $n$. Consider the following synchronous process, also known as the (synchronous) voter model (e.g., Even-Dar and Shapira): Define an indicator variable $...
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  • 531
3 votes
0 answers
117 views

Concentration of Stationary Distribution on Random Directed Graphs

We consider a random directed graph with fixed out-degree $d$. Each vertex chooses $d$ neighbors with replacement, uniformly and independently. Self-loops and multiple arcs are allowed in this model. ...
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  • 103
12 votes
5 answers
447 views

Motivation for volume estimation

What are some concrete and compelling applications for estimating the volume of convex polyhedra of the sort considered in the more recent papers on random walk methods? These papers on volume ...
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5 votes
1 answer
146 views

$\omega$-regular properties of a 2-state Markov Chain

Let $X$ be a Markov Chain on a state space $\{0,1\}$ with a transition matrix $$ P = \left( \begin{align} 1-p & &p \\ q & &1-q \end{align} \right) $$ with both $p,q \in (0,1)$ so in ...
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  • 407
5 votes
2 answers
181 views

Behaviour of Labelled Markov Processes

Labelled Markov Processes (LMP) seem to be a generalization of Probabilistic Automata (PA) studied by Segala to the case of the general state space. Namely, any LMP is given by a be a finite set of ...
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  • 407
4 votes
1 answer
229 views

Boundedness of expected reward Markov chain

This is a repost of a question I asked on math.SE. The problem: I have an infinite Markov chain $M$ over the natural numbers, with transition probabilities $$P(n,m)=\sum_{i=0}^{min(m,n)} {n\choose i}...
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  • 341
5 votes
0 answers
166 views

The regularity of Markov chains with a threshold

(This question has been asked on math.se, with no response.) I am studying Paz's "Introduction to Probabilistic Automata" and there is an exercise I cannot solve: Ex. 11, p. 170: Let $\Sigma = \{a\...
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6 votes
0 answers
143 views

Complexity of DTMC subsystems

A discrete-time Markov chain (DTMC) is a tuple $M=(S,s_{init},P)$ where $S$ is a finite set of states, $s_{init}\in S$ the initial state, and $P:S\times S\to[0,1]$ the one-step transition probability ...
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  • 81
1 vote
0 answers
112 views

Techniques to get nodes in the best Markov Cluster?

I was using Markov Clustering to cluster nodes in my bidirectional graph, and overall the results were great. However, there were a couple instances where a weakly connected node would attract a node ...
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  • 111
11 votes
0 answers
444 views

Cheeger's inequality for directed graphs?

Cheeger's inequality can be used to relate the size of the worst cut in the graph to the eigenvalue gap of a simple random walk on that graph. I am wondering if it possible to extend this result to ...
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1 vote
1 answer
118 views

Inferring optimal utility values from a decision process

I've been able to model a particular decision problem as a Markov Decision Process, where the optimal policy (i.e. what decision should be taken at each step) is defined in order to optimize a given ...
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  • 295
1 vote
0 answers
93 views

Belief Propagation on MRF with complex cliques

Is there a belief propagation algorithm for exact inference on a MRF with complex clique structures (i.e. ones involving more than 2 neighbours)? For MRF's with cliques that only involve pairwise ...
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-2 votes
1 answer
173 views

Dual of a Reversible Markov Chain [closed]

Let a reversible Markov process $m_{t+1}=m_t P$, where $t$ is time that has a stationary distribution $\pi$. I saw in a paper that the dual system was defined as $x_{t+1}=P x_t$. Can anyone give me ...
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7 votes
0 answers
265 views

Complexity of reachability in Markov Chains

Is anything known about the complexity of the following problem beyond membership in PTIME: Given a finite Markov chain $M$, an initial state $q_0$ and a set $F$ of (absorbing) states, is the ...
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1 vote
0 answers
382 views

Implementation of a Logical Hierarchical Hidden Markov Model

Is anyone aware of any implementations of algorithms for learning and/or processing a Logical Hierarchical Hidden Markov Model, as described in this paper? I've found dozens of papers about Logical ...
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  • 127
30 votes
2 answers
1k views

Drunken birds vs drunken ants: random walks between two and three dimensions

It's well known that a random walk in the two dimensional grid will return to the origin with probability 1. It's also known that the same random walk in THREE dimensions has a probability strictly ...
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25 votes
1 answer
765 views

Random self-avoiding lattice cycle within a given bounding box

In connection with the Slither Link puzzle, I've been wondering: Suppose that I have an $n\times n$ grid of square cells, and I want to find a simple cycle of grid edges, uniformly at random among all ...
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-1 votes
1 answer
317 views

Duration Viterbi Algorithm

I am searching for some good resources to understand the Duration Viterbi algorithm. Does anyone knows a good resource to understand and learn how to model a Duration Viterbi Hidden Markov Chain ...
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  • 127
17 votes
2 answers
1k views

Cover Time of Directed Graphs

Given a random walk on a graph the cover time is the first time (expected number of steps) that every vertex has been hit (covered) by the walk. For connected undirected graphs, the cover time is ...
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16 votes
2 answers
541 views

Avalanche like stochastic process

Consider the following process: There are $n$ bins arranged from top to bottom. Initially, each bin contains one ball. In every step, we pick a ball $b$ uniformly at random and move all the balls ...
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  • 1,638
13 votes
2 answers
647 views

One-shot quantum hitting times

In the paper Quantum Random Walks Hit Exponentially Faster (arXiv:quant-ph/0205083) Kempe gives a notion of hitting time for quantum walks (in the hypercube) that is not very popular in the quantum ...
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19 votes
1 answer
708 views

Rapidly mixing Markov chains on 3-colorings of a cycle

The Glauber dynamics is a Markov chain on the colorings of a graph in which at each step one attempts to recolor a randomly chosen vertex with a random color. It does not mix for the 3-colorings of a ...
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11 votes
1 answer
263 views

Can someone suggest a recent survey on product form Markov chains?

I'm especially interested in their use in model checking applications. I have Open, Closed and Mixed Networks of Queues with Different Classes of Customers by Baskett et al. Any other suggestions ...
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