Cliques in High-Dimensional Random Geometric Graphs
Random geometric graphs are good examples of random graphs with a tendency to demonstrate community structure. Vertices of such a graph are represented by points in Euclid space \(\mathbb R^d\), and edge appearance depends on the distance between the points. Random geometric graphs were extensively explored and many of their basic properties are revealed. However, in the case of growing dimension \(d\rightarrow \infty \) practically nothing is known; this regime corresponds to the case of data with many features, a case commonly appearing in practice. In this paper, we focus on the cliques of these graphs in the situation when average vertex degree grows significantly slower than the number of vertices n with \(n\rightarrow \infty \) and \(d\rightarrow \infty \). We show that under these conditions random geometric graphs do not contain cliques of size 4 a.s. As for the size 3, we will present new bounds on the expected number of triangles in the case \(\log ^2 n \ll d \ll \log ^3 n\) that improve previously known results.
KeywordsRandom geometric graphs High dimension Clique number Triangles
- 1.Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. Ser. A 5(1), 17–60 (1960)Google Scholar
- 4.Preciado, V.M., Jadbabaie, A.: Spectral analysis of virus spreading in random geometric networks. In: Proceedings of the 48th IEEE Conference on Decision and Control (CDC) held Jointly with 2009 28th Chinese Control Conference, pp. 4802–4807 (2009). https://doi.org/10.1109/CDC.2009.5400615
- 8.Xiao, H., Yeh, E.M.: Cascading link failure in the power grid: a percolation-based analysis. In: 2011 IEEE International Conference on Communications Workshops (ICC), pp. 1–6 (2011). https://doi.org/10.1109/iccw.2011.5963573
- 12.Penrose, M.D.: On \(k\)-connectivity for a geometric random graph. Random Struct. Algorithms 15(2), 145–164 (1999). https://doi.org/10.1002/(SICI)1098-2418(199909)15:2%3C145::AID-RSA2%3E3.0.CO;2-GMathSciNetCrossRefGoogle Scholar
- 19.Lee, Y., Kim, W.C.: Concise formulas for the surface area of the intersection of two hyperspherical caps. KAIST Technical Report (2014)Google Scholar