constructing a directed graph where any node can have highest rank
Given a graph $G = langle V = {0, 1, 2..., |V|-1}, Erangle$ and a parameter $alpha in [0, 1]$, we define the follwoing matrix
$$A = alpha M + (1-alpha) U$$ where
$M$ is the stochastic adjacency matrix of $G$. That is, for every $i, j in V$, if $langle i, jrangle in E$, then $M_{j, i} = frac{1}{deg(i)}$. Otherwise, $M_{j, i} = 0$.
$U$ is an $|V|times |V|$ matrix with the value $1/|V|$ at each entry.
The NodeRank of a node $iin V$ is defined as the $i$'th coordinate of the vector $r$ that satisfies $r = Acdot r$. That is, $r$ is the eigenvector of $A$ with eigenvalue 1. Computing $r$ is known as the problem of computing the PageRank of a network of pages.
I am trying to construct graph $G = langle V, Erangle$ with a subset $Csubseteq V$ of size 3 (i.e., $|C|=3$) such that, depending on the parameter $alpha$, any of the nodes in $C$ can have the maximum NodeRank over all the nodes in $C$.
I tried the following, define a graph that consists of a clique of size 3 and a node $v$, such that for every node in the clique there is an edge to $v$ and there are no outgoing edges from $v$. Intuitively, all the nodes in the clique must have the same NodeRank. I'm not sure if this is a good direction and how to proceed from here.
A help would be appreciated,
thanks
random-walk directed-graphs
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Given a graph $G = langle V = {0, 1, 2..., |V|-1}, Erangle$ and a parameter $alpha in [0, 1]$, we define the follwoing matrix
$$A = alpha M + (1-alpha) U$$ where
$M$ is the stochastic adjacency matrix of $G$. That is, for every $i, j in V$, if $langle i, jrangle in E$, then $M_{j, i} = frac{1}{deg(i)}$. Otherwise, $M_{j, i} = 0$.
$U$ is an $|V|times |V|$ matrix with the value $1/|V|$ at each entry.
The NodeRank of a node $iin V$ is defined as the $i$'th coordinate of the vector $r$ that satisfies $r = Acdot r$. That is, $r$ is the eigenvector of $A$ with eigenvalue 1. Computing $r$ is known as the problem of computing the PageRank of a network of pages.
I am trying to construct graph $G = langle V, Erangle$ with a subset $Csubseteq V$ of size 3 (i.e., $|C|=3$) such that, depending on the parameter $alpha$, any of the nodes in $C$ can have the maximum NodeRank over all the nodes in $C$.
I tried the following, define a graph that consists of a clique of size 3 and a node $v$, such that for every node in the clique there is an edge to $v$ and there are no outgoing edges from $v$. Intuitively, all the nodes in the clique must have the same NodeRank. I'm not sure if this is a good direction and how to proceed from here.
A help would be appreciated,
thanks
random-walk directed-graphs
asked yesterday
bbb3321
1
New contributor
bbb3321 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Given a graph $G = langle V = {0, 1, 2..., |V|-1}, Erangle$ and a parameter $alpha in [0, 1]$, we define the follwoing matrix
$$A = alpha M + (1-alpha) U$$ where
$M$ is the stochastic adjacency matrix of $G$. That is, for every $i, j in V$, if $langle i, jrangle in E$, then $M_{j, i} = frac{1}{deg(i)}$. Otherwise, $M_{j, i} = 0$.
$U$ is an $|V|times |V|$ matrix with the value $1/|V|$ at each entry.
The NodeRank of a node $iin V$ is defined as the $i$'th coordinate of the vector $r$ that satisfies $r = Acdot r$. That is, $r$ is the eigenvector of $A$ with eigenvalue 1. Computing $r$ is known as the problem of computing the PageRank of a network of pages.
I am trying to construct graph $G = langle V, Erangle$ with a subset $Csubseteq V$ of size 3 (i.e., $|C|=3$) such that, depending on the parameter $alpha$, any of the nodes in $C$ can have the maximum NodeRank over all the nodes in $C$.
I tried the following, define a graph that consists of a clique of size 3 and a node $v$, such that for every node in the clique there is an edge to $v$ and there are no outgoing edges from $v$. Intuitively, all the nodes in the clique must have the same NodeRank. I'm not sure if this is a good direction and how to proceed from here.
A help would be appreciated,
thanks
random-walk directed-graphs
asked yesterday
bbb3321
1
New contributor
bbb3321 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Given a graph $G = langle V = {0, 1, 2..., |V|-1}, Erangle$ and a parameter $alpha in [0, 1]$, we define the follwoing matrix
$$A = alpha M + (1-alpha) U$$ where
$M$ is the stochastic adjacency matrix of $G$. That is, for every $i, j in V$, if $langle i, jrangle in E$, then $M_{j, i} = frac{1}{deg(i)}$. Otherwise, $M_{j, i} = 0$.
$U$ is an $|V|times |V|$ matrix with the value $1/|V|$ at each entry.
The NodeRank of a node $iin V$ is defined as the $i$'th coordinate of the vector $r$ that satisfies $r = Acdot r$. That is, $r$ is the eigenvector of $A$ with eigenvalue 1. Computing $r$ is known as the problem of computing the PageRank of a network of pages.
I am trying to construct graph $G = langle V, Erangle$ with a subset $Csubseteq V$ of size 3 (i.e., $|C|=3$) such that, depending on the parameter $alpha$, any of the nodes in $C$ can have the maximum NodeRank over all the nodes in $C$.
I tried the following, define a graph that consists of a clique of size 3 and a node $v$, such that for every node in the clique there is an edge to $v$ and there are no outgoing edges from $v$. Intuitively, all the nodes in the clique must have the same NodeRank. I'm not sure if this is a good direction and how to proceed from here.
A help would be appreciated,
thanks
random-walk directed-graphs
random-walk directed-graphs
asked yesterday
bbb3321
1
New contributor
bbb3321 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
bbb3321
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asked yesterday
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asked yesterday
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asked yesterday
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