How to show that q-coloring graph is ergodic
$begingroup$
Informal: I want to show that a q-coloring of graph $G$ is ergodic (i.e. strongly connected and aperiodic)
Formally:
For a given graph $G(V,E)$ where $|V|=n$ with maximum degree $Deltageq1$.
Also, let $q=4Delta$.
Denote a q-coloring of $G$ as a function $c:Vrightarrow[q]$ s.t for every edge $(u,v)in E mid c(u)neq c(v)$.
Define $Omega=${The state space of all the legal q-coloring of $G$} and consider the following Markov chain:
Say the current state is $X_0 = cin Omega$
Sample $(v, i)in V times [q]$ uniformly at random and independently from previous choices.
Define $hat{c}:Vrightarrow[q]$ by setting $hat{c}=c(u)quad forall, uneq v$ $hat{c}(v)=i$.$quad$
So a step in the MC defined as:
$$
x_{t+1} = left.
begin{cases}
hat{c}, & text{if } cin Omega \
x_{t}, & text{else }
end{cases}
right}
$$
In words: If $hat{c}$ is a valid coloring,
(which happens if color $i$ isn't used by coloring $c$ on any neighbor of $v$) set $X_{t+1} = hat{c}$. Otherwise
$X_{t+1} = c$.
How can I show (formally) that the MC is ergodic?
probability-theory markov-chains random-walk ergodic-theory coloring
asked Dec 28 '18 at 14:37
self studyself study
1815
$endgroup$
add a comment |
$begingroup$
Informal: I want to show that a q-coloring of graph $G$ is ergodic (i.e. strongly connected and aperiodic)
Formally:
For a given graph $G(V,E)$ where $|V|=n$ with maximum degree $Deltageq1$.
Also, let $q=4Delta$.
Denote a q-coloring of $G$ as a function $c:Vrightarrow[q]$ s.t for every edge $(u,v)in E mid c(u)neq c(v)$.
Define $Omega=${The state space of all the legal q-coloring of $G$} and consider the following Markov chain:
Say the current state is $X_0 = cin Omega$
Sample $(v, i)in V times [q]$ uniformly at random and independently from previous choices.
Define $hat{c}:Vrightarrow[q]$ by setting $hat{c}=c(u)quad forall, uneq v$ $hat{c}(v)=i$.$quad$
So a step in the MC defined as:
$$
x_{t+1} = left.
begin{cases}
hat{c}, & text{if } cin Omega \
x_{t}, & text{else }
end{cases}
right}
$$
In words: If $hat{c}$ is a valid coloring,
(which happens if color $i$ isn't used by coloring $c$ on any neighbor of $v$) set $X_{t+1} = hat{c}$. Otherwise
$X_{t+1} = c$.
How can I show (formally) that the MC is ergodic?
probability-theory markov-chains random-walk ergodic-theory coloring
asked Dec 28 '18 at 14:37
self studyself study
1815
$endgroup$
$begingroup$
Where specifically do you have a problem?
$endgroup$
– Blackbird
Jan 21 at 15:57
$begingroup$
@Blackbird as I wrote in my question - I want to prove that the described MC is ergodic and that the $pi=(frac{1}{|Omega|},...,frac{1}{|Omega|})$ is the stationary distribution.
$endgroup$
– self study
Jan 26 at 17:00
add a comment |
$begingroup$
Informal: I want to show that a q-coloring of graph $G$ is ergodic (i.e. strongly connected and aperiodic)
Formally:
For a given graph $G(V,E)$ where $|V|=n$ with maximum degree $Deltageq1$.
Also, let $q=4Delta$.
Denote a q-coloring of $G$ as a function $c:Vrightarrow[q]$ s.t for every edge $(u,v)in E mid c(u)neq c(v)$.
Define $Omega=${The state space of all the legal q-coloring of $G$} and consider the following Markov chain:
Say the current state is $X_0 = cin Omega$
Sample $(v, i)in V times [q]$ uniformly at random and independently from previous choices.
Define $hat{c}:Vrightarrow[q]$ by setting $hat{c}=c(u)quad forall, uneq v$ $hat{c}(v)=i$.$quad$
So a step in the MC defined as:
$$
x_{t+1} = left.
begin{cases}
hat{c}, & text{if } cin Omega \
x_{t}, & text{else }
end{cases}
right}
$$
In words: If $hat{c}$ is a valid coloring,
(which happens if color $i$ isn't used by coloring $c$ on any neighbor of $v$) set $X_{t+1} = hat{c}$. Otherwise
$X_{t+1} = c$.
How can I show (formally) that the MC is ergodic?
probability-theory markov-chains random-walk ergodic-theory coloring
asked Dec 28 '18 at 14:37
self studyself study
1815
$endgroup$
Informal: I want to show that a q-coloring of graph $G$ is ergodic (i.e. strongly connected and aperiodic)
Formally:
For a given graph $G(V,E)$ where $|V|=n$ with maximum degree $Deltageq1$.
Also, let $q=4Delta$.
Denote a q-coloring of $G$ as a function $c:Vrightarrow[q]$ s.t for every edge $(u,v)in E mid c(u)neq c(v)$.
Define $Omega=${The state space of all the legal q-coloring of $G$} and consider the following Markov chain:
Say the current state is $X_0 = cin Omega$
Sample $(v, i)in V times [q]$ uniformly at random and independently from previous choices.
Define $hat{c}:Vrightarrow[q]$ by setting $hat{c}=c(u)quad forall, uneq v$ $hat{c}(v)=i$.$quad$
So a step in the MC defined as:
$$
x_{t+1} = left.
begin{cases}
hat{c}, & text{if } cin Omega \
x_{t}, & text{else }
end{cases}
right}
$$
In words: If $hat{c}$ is a valid coloring,
(which happens if color $i$ isn't used by coloring $c$ on any neighbor of $v$) set $X_{t+1} = hat{c}$. Otherwise
$X_{t+1} = c$.
How can I show (formally) that the MC is ergodic?
probability-theory markov-chains random-walk ergodic-theory coloring
probability-theory markov-chains random-walk ergodic-theory coloring
asked Dec 28 '18 at 14:37
self studyself study
1815
asked Dec 28 '18 at 14:37
self studyself study
1815
edited Jan 26 at 17:03
self study
asked Dec 28 '18 at 14:37
self studyself study
1815
asked Dec 28 '18 at 14:37
self studyself study
1815
asked Dec 28 '18 at 14:37
self studyself study
1815
1815
$begingroup$
Where specifically do you have a problem?
$endgroup$
– Blackbird
Jan 21 at 15:57
$begingroup$
@Blackbird as I wrote in my question - I want to prove that the described MC is ergodic and that the $pi=(frac{1}{|Omega|},...,frac{1}{|Omega|})$ is the stationary distribution.
$endgroup$
– self study
Jan 26 at 17:00
add a comment |
$begingroup$
Where specifically do you have a problem?
$endgroup$
– Blackbird
Jan 21 at 15:57
$begingroup$
@Blackbird as I wrote in my question - I want to prove that the described MC is ergodic and that the $pi=(frac{1}{|Omega|},...,frac{1}{|Omega|})$ is the stationary distribution.
$endgroup$
– self study
Jan 26 at 17:00
$begingroup$
Where specifically do you have a problem?
$endgroup$
– Blackbird
Jan 21 at 15:57
$begingroup$
Where specifically do you have a problem?
$endgroup$
– Blackbird
Jan 21 at 15:57
$begingroup$
@Blackbird as I wrote in my question - I want to prove that the described MC is ergodic and that the $pi=(frac{1}{|Omega|},...,frac{1}{|Omega|})$ is the stationary distribution.
$endgroup$
– self study
Jan 26 at 17:00
$begingroup$
@Blackbird as I wrote in my question - I want to prove that the described MC is ergodic and that the $pi=(frac{1}{|Omega|},...,frac{1}{|Omega|})$ is the stationary distribution.
$endgroup$
– self study
Jan 26 at 17:00
add a comment |
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$begingroup$
Where specifically do you have a problem?
$endgroup$
– Blackbird
Jan 21 at 15:57
$begingroup$
@Blackbird as I wrote in my question - I want to prove that the described MC is ergodic and that the $pi=(frac{1}{|Omega|},...,frac{1}{|Omega|})$ is the stationary distribution.
$endgroup$
– self study
Jan 26 at 17:00