How to show that q-coloring graph is ergodic
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 defines 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
185
add a comment |
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 defines 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
185
add a comment |
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 defines 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
185
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 defines 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
185
asked Dec 28 '18 at 14:37
self studyself study
185
edited Dec 28 '18 at 14:51
self study
asked Dec 28 '18 at 14:37
self studyself study
185
asked Dec 28 '18 at 14:37
self studyself study
185
asked Dec 28 '18 at 14:37
self studyself study
185
185
add a comment |
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