| Title:
|
Random walk centrality and a partition of Kemeny's constant (English) |
| Author:
|
Kirkland, Steve |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
757-775 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider an accessibility index for the states of a discrete-time, ergodic, homogeneous Markov chain on a finite state space; this index is naturally associated with the random walk centrality introduced by Noh and Reiger (2004) for a random walk on a connected graph. We observe that the vector of accessibility indices provides a partition of Kemeny's constant for the Markov chain. We provide three characterizations of this accessibility index: one in terms of the first return time to the state in question, and two in terms of the transition matrix associated with the Markov chain. Several bounds are provided on the accessibility index in terms of the eigenvalues of the transition matrix and the stationary vector, and the bounds are shown to be tight. The behaviour of the accessibility index under perturbation of the transition matrix is investigated, and examples exhibiting some counter-intuitive behaviour are presented. Finally, we characterize the situation in which the accessibility indices for all states coincide. (English) |
| Keyword:
|
stochastic matrix |
| Keyword:
|
random walk centrality |
| Keyword:
|
Kemeny's constant |
| MSC:
|
15B51 |
| MSC:
|
60J10 |
| idZBL:
|
Zbl 06644032 |
| idMR:
|
MR3556866 |
| DOI:
|
10.1007/s10587-016-0291-9 |
| . |
| Date available:
|
2016-10-01T15:22:22Z |
| Last updated:
|
2023-10-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145870 |
| . |
| Reference:
|
[1] Campbell, S., Meyer, C. D.: Generalized Inverses of Linear Transformations.Classics in Applied Mathematics 56 SIAM, Philadelphia (2009). Zbl 1158.15301, MR 3396208 |
| Reference:
|
[2] Chartrand, G., Lesniak, L.: Graphs and Digraphs.Chapman and Hall, Boca Raton (2005). Zbl 1057.05001, MR 2107429 |
| Reference:
|
[3] Kirkland, S. J.: On a question concerning condition numbers for Markov chains.SIAM J. Matrix Anal. Appl. 23 (2002), 1109-1119. Zbl 1013.15005, MR 1920936, 10.1137/S0895479801390947 |
| Reference:
|
[4] Kirkland, S. J., Neumann, M.: Group Inverses of M-matrices and Their Applications.CRC Press, Boca Raton (2013). Zbl 1267.15004, MR 3185162 |
| Reference:
|
[5] Kirkland, S. J., Neumann, M., Shader, B. L.: Bounds on the subdominant eigenvalue involving group inverses with applications to graphs.Czech. Math. J. 47 (1998), 1-20. Zbl 0931.15012, MR 1614056, 10.1023/A:1022455208972 |
| Reference:
|
[6] Levene, M., Loizou, G.: Kemeny's constant and the random surfer.Am. Math. Mon. 109 (2002), 741-745. Zbl 1023.60061, MR 1927624, 10.2307/3072398 |
| Reference:
|
[7] Meyer, C. D.: Sensitivity of the stationary distribution of a Markov chain.SIAM J. Matrix Anal. Appl. 15 (1994), 715-728. Zbl 0809.65143, MR 1282690, 10.1137/S0895479892228900 |
| Reference:
|
[8] Noh, J., Reiger, H.: Random walks on complex networks.Physical Review Letters 92 (2004), 118701, 5 pages. |
| Reference:
|
[9] Seneta, E.: Non-Negative Matrices and Markov Chains.Springer Series in Statistics Springer, New York (1981). Zbl 0471.60001, MR 2209438 |
| . |
