# Katz Centrality

#### Definition

The Katz centrality for node

Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors [A HAGBERG, D. S., 2008].

Ogura, M. and Preciado, V.M., 2017, May. Katz centrality of Markovian temporal networks: Analysis and optimization. In American Control Conference (ACC), 2017 (pp. 5001-5006). IEEE.

Yin C, Riedy J. Concurrent Katz Centrality for Streaming Graphs. In2019 IEEE High Performance Extreme Computing Conference (HPEC) 2019 Sep 24 (pp. 1-6). IEEE.

*i*is: where A is the adjacency matrix of the graph G with eigenvalues λ. The parameter β controls the initial centrality and α < 1/λ_{max}.Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors [A HAGBERG, D. S., 2008].

**Katz centrality of markovian temporal networks**Ogura, M. and Preciado, V.M., 2017, May. Katz centrality of Markovian temporal networks: Analysis and optimization. In American Control Conference (ACC), 2017 (pp. 5001-5006). IEEE.

**Concurrent Katz Centrality for Streaming Graphs**Yin C, Riedy J. Concurrent Katz Centrality for Streaming Graphs. In2019 IEEE High Performance Extreme Computing Conference (HPEC) 2019 Sep 24 (pp. 1-6). IEEE.

#### Requirements

Require loop free network.

#### Software

#### References

- L. Katz. A new status index derived from sociometric analysis. Psychometrika, 18(1):39–43, 1953.
- A HAGBERG, D. S., P SWART. Exploring Network Structure, Dynamics, and Function using NetworkX. In: G VAROQUAUX, T. V., J MILLMAN, ed. Proceedings of the 7th Python in Science conference (SciPy 2008), 2008. 11-15.
- GRÄßLER, J., KOSCHÜTZKI, D. & SCHREIBER, F. 2012. CentiLib: comprehensive analysis and exploration of network centralities. Bioinformatics, 28, 1178-1179.
- JUNKER, B. H., KOSCHÜTZKI, D. & SCHREIBER, F. 2006. Exploration of biological network centralities with CentiBiN. BMC bioinformatics, 7, 219.
- NEWMAN, M. 2010. Networks: an introduction, Oxford University Press.
- Ogura, M. and Preciado, V.M., 2017, May. Katz centrality of Markovian temporal networks: Analysis and optimization. In American Control Conference (ACC), 2017 (pp. 5001-5006). IEEE.
- Nathan, E. and Bader, D.A., A Dynamic Algorithm for Updating Katz Centrality in Graphs. In Proceedings of the 2017 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, Sydney, Australia (Vol. 31).