# Scaling of Random Walk Betweenness in Networks

## Abstract

We study the betweenness centrality (BC) of vertices of a graph using random walk paths. Random walk BC (RWBC) provides an alternative measure to the shortest path centrality of each vertex in a graph as it aggregates contributions from possibly all vertex-pairs in the graph and not just from those vertex-pairs on whose geodesic path the vertex lies. As such, RWBC is more relevant in the context of information diffusion in virtual networks, such as spread of news or rumors in online social media. We derive a closed-form analytical expression for RWBC using eigenfunctions of the graph Laplacian. We then show the distribution of RWBC scores of the vertices of a graph exhibits a scaling collapse with no adjustable parameters as the graph size *N* is varied. This means that the distribution of RWBC over all the nodes in a large graph can be obtained in terms of the distribution of RWBC for a prototypical or small graph that is generated using the same model. The exact distribution itself depends on the graph model. A normalized random walk betweenness (NRWBC), that counts each walk passing through a vertex only once, is also defined. This measure is argued to be more useful and robust and is seen to have simpler scaling behavior.

## Notes

### Acknowldegments

This work was supported by grants FA9550-11-1-0278 and 60NANB10D128 from AFOSR and NIST, respectively.

## References

- 1.Freeman, L.C.: Centrality in social networks: conceptual clarification. Soc. Netw.
**1**, 215–239 (1979)Google Scholar - 2.Newman, M.E.J.: A measure of betweenness centrality based on random walks. Soc. Netw.
**27**, 39–54 (2005)Google Scholar - 3.Kivimäki, I., Lebichot, B., Saramäki, J., Saerens, M.: Two betweenness centrality measures based on randomized shortest paths. Nat. Sci. Rep.
**6**, 19668 (2016)Google Scholar - 4.Samukhin, A.N., Dorogovtsev, S.N., Mendes, J.F.F.: Laplacian spectra of and random walks on, complex networks: are scale-free architectures really important? Phys. Rev. E
**77**, 036115 (2008)Google Scholar - 5.Noh, J.D., Rieger, H.: Random walks on complex networks. Phys. Rev. Lett.
**92**, 118701 (2004)Google Scholar - 6.Narayan O., Saniee I. and Marbukh V.: Congestion due to random walk routing (2013). https://arxiv.org/abs/1309.0066
- 7.Newman, M.E.J.: The structure and function of complex networks. SIAM Rev.
**45**, 167–256 (2003)Google Scholar - 8.Erdös, P., Rényi, A.: On random graphs. Publ. Math.
**6**, 290–297 (1959)Google Scholar - 9.Barabasi, A.-L., Albert, R.: Emergence of scaling in random networks. Science
**286**, 509–512 (1999)Google Scholar - 10.Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.: Structure of growing networks with preferential linking. Phys. Rev. Lett.
**85**, 4633–4636 (2000)Google Scholar