CompleNet 2018: Complex Networks IX pp 3-13

# On the Eccentricity Function in Graphs

• Hend Alrasheed
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)

## Abstract

Given a graph $$G=(V,E)$$, the eccentricity of a vertex u is the distance from u to a vertex farthest from u. The set of vertices that minimizes the maximum distance to every other vertex (has minimum eccentricity) constitutes the center of the graph. The minimum eccentricity value represents the graph’s radius. The eccentricity function of a graph can be unimodal or non-unimodal. A graph with unimodal eccentricity function has the property that the eccentricity of every vertex equals its distance to the center plus the radius. A graph with non-unimodal eccentricity function lacks this property. In this work, we characterize each type of eccentricity function and study the impact of each type on the intersection of shortest paths among distant vertex pairs with the center. A shortest path intersects the center if it includes at least one vertex that belongs to the center. In particular, we show that if the eccentricity function is unimodal, all shortest paths among distant vertex pairs intersect the graph’s center. We also discuss when those paths do not intersect the center in graphs with non-unimodal eccentricity functions.

## References

1. 1.
Adcock, A., Sullivan, B., Mahoney, M.: Tree-Like structure in large social and information networks. In: ICDM (2013)Google Scholar
2. 2.
Alrasheed, H., Dragan, F.F.: Core-periphery models for graphs based on their $$\delta$$-hyperbolicity: an example using biological networks. Complex Networks VI, pp. 65–77. Springer International Publishing, Berlin (2015)Google Scholar
3. 3.
Batagelj, V., Mrvar, A.: Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/ (2006)
4. 4.
Dragan, F.F.: Conditions for coincidence of local and global minima for the eccentricity function on graphs and the Helly property. Studies in Applied Mathematics and Information Science, pp. 49–56 (1990)Google Scholar
5. 5.
Holme, P.: Core-periphery organization of complex networks. Phys. Rev. E 72(4), 046111 (2005)
6. 6.
7. 7.
8. 8.
Jeong, H., et al.: Lethality and centrality in protein networks. Nature 411, 41–42 (2001)
9. 9.
Leskovec, J., Mcauley, J.: Learning to discover social circles in ego networks. Advances in Neural Information Processing Systems, pp. 539–547 (2012)Google Scholar
10. 10.
Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: densification and shrinking diameters. ACM TKDD 1(1) (2007)Google Scholar
11. 11.
Leskovec, J., et al.: Community structure in large networks: natural cluster sizes and the absence of large well-defined clusters. Internet Math. 6(1), 29–123 (2009)
12. 12.
Narayan, O., Saniee, I.: The large scale curvature of networks. Phys. Rev. E 84(6), 066108 (2011)
13. 13.
Stark, C., et al.: BioGRID: a general repository for interaction datasets. Nucleic Acids Res. (2006)Google Scholar
14. 14.
University of Oregon route-views project. http://www.routeviews.org/
15. 15.
Watts, D., Strogatz, S.: Collective dynamics of small-world networks. Nature 393(6684), 440–442 (1998)