It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.4K
- Chat 505
- Study Groups 21
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 2
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- Baez ACT 2019: Online Course 339
- Baez ACT 2019: Lectures 79
- Baez ACT 2019: Exercises 149
- Baez ACT 2019: Chat 50
- UCR ACT Seminar 4
- General 75
- Azimuth Code Project 111
- Statistical methods 4
- Drafts 10
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 148
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 719

Options

## Comments

$$ \begin{array}{c | c c c} \Phi & x & y & z \\ \hline A & 17 & 20 & 20 \\ B & 11 & 14 & 14 \\ C & 14 & 17 & 17 \\ D & 12 & 9 & 15 \end{array} $$ One way to compute this is to form a cost matrix (the same as an adjacency matrix, except with the costs in each entry where the identity counts as 0 and no link counts as \(\infty\)). Taking matrix products using tropical mathematics (sum instead of product and min instead of sum), one can find the least cost to go between two points in a specified number of link traversals. After finitely many powers, the matrix ceases to change on taking higher powers. At this point, one has a matrix containing the minimum cost to travel between two nodes regardless of the number of links traversed.

For this particular system, two properties are helpful. One is that there are no links from \(\mathcal{Y}\) to \(\mathcal{X}\), so the corresponding part of the matrix is always \(\infty\). The other is that there is a minimum non-zero entry in the matrix (i.e., a minimum cost for traversing a link). This means that once an entry is less than or equal to the minimum cost times the number of powers of the matrix taken, it cannot change under taking higher powers.

`\[ \begin{array}{c | c c c} \Phi & x & y & z \\\\ \hline A & 17 & 20 & 20 \\\\ B & 11 & 14 & 14 \\\\ C & 14 & 17 & 17 \\\\ D & 12 & 9 & 15 \end{array} \] One way to compute this is to form a cost matrix (the same as an adjacency matrix, except with the costs in each entry where the identity counts as 0 and no link counts as \\(\infty\\)). Taking matrix products using tropical mathematics (sum instead of product and min instead of sum), one can find the least cost to go between two points in a specified number of link traversals. After finitely many powers, the matrix ceases to change on taking higher powers. At this point, one has a matrix containing the minimum cost to travel between two nodes regardless of the number of links traversed. For this particular system, two properties are helpful. One is that there are no links from \\(\mathcal{Y}\\) to \\(\mathcal{X}\\), so the corresponding part of the matrix is always \\(\infty\\). The other is that there is a minimum non-zero entry in the matrix (i.e., a minimum cost for traversing a link). This means that once an entry is less than or equal to the minimum cost times the number of powers of the matrix taken, it cannot change under taking higher powers.`