Social Network Analysis of Clustering in Random Geometric Graphs Christine Marshall 1 , James Cruickshank 2 and Colm O’Riordan 1 1 Department of Information Technology 2 School of Mathematics, Statistics and Applied Mathematics National University of Ireland, Galway [ Introduction On the WWW, a huge body of data is linked and interlinked in complex ways. The way that information spreads through a network can be examined by creating different graph models and comparing their structural properties. Figure 1. Model of a Social Network [Hannon Digital 2014] In network analysis, the Erdös Rényi model of random graphs has frequently been used to model real world systems; this model has connections between nodes created with a defined probability parameter. Figure 2. Erdős–Rényi model of random graph Random geometric graphs model situations where physical distance between agents is a factor, such as the spread of disease. Random geometric graphs are created by forming a link wherever the distance between any pair of nodes is less than a specified distance parameter [Penrose 2003]. Figure 3. Random geometric graph model. Transitivity measures the links between triples of nodes. A fully transitive relationship is represented by a closed triple, where the “friends of my friends are my friends'' [Snijders 2011]. The network average clustering coefficient measures links in the neighbourhood of a node [Watts and Strogatz 1998]. Results The two models display different patterns of degree distribution at all probability parameters. Figure 4. Erdős Rényi degree distribution at edge probability of 0.6 The Erdös Rényi model has a Poisson distribution, where the degree of most nodes is closely distributed around the mean; the random geometric graph has a wider range of values. Figure 5. Geometric graph degree distribution at edge probability of 0.6 The geometric graph displays markedly higher values for both network average clustering coefficient and transitivity than the Erdös Rényi model. In the latter the two measures coincide, whereas there is significant variance in the geometric model. Figure 6. Comparison of clustering coefficients in both models Method Random geometric graphs were created in the unit square, by generating uniformly distributed random points (x, y) and creating an edge between pairs of nodes wherever the distance between them was less than a specified distance. Erdös Rényi random graphs were created with defined edge probability parameters. In order to directly compare these models, the geometric graph was converted to a probability model by calculating the distribution of distances within the unit square [Weisstein 2013]. This ensured that at each probability step, both models had identical probability of a connection existing and therefore equal likelihood of similar average degree. Conclusion Although both models have been created with equal edge probability, the way that the node degrees are distributed throughout the graph is different. The degree of all nodes in the Erdös Rényi model is closely distributed around the mean, whereas the Random geometric graph has a wider range of values, with a resultant effect on the patterns of node clustering. The most striking result is that the transitivity and network average clustering coefficients differ within the geometric graphs. This was unexpected, as these measures coincide in the Erdös Rényi model. This raises potential for using these models in network analysis and in game theory applications. Acknowledgements I would like to thank my supervisors Dr. Colm O’ Riordan and Dr. James Cruickshank for their support in this research. References Hannon Digital: Social_Web_Development http://hannondigital.com/2014/01/02/social-web- development/ (2014) Penrose, M.: Random Geometric Graphs. Oxford University Press, New York, USA ( 2003) Snijders, T.A.B.: Statistical Models for Social Networks. Annual Review of Sociology, 37, 131–153(2011) Watts, D., Strogatz, S.: Collective dynamics of 'small- world' networks. Nature, 393, 440-442( 1998) Weisstein, Eric W.: Square Line Picking, http://mathworld.wolfram.com/SquareLinePicking .html. (2014) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Clustering Coefficient Probability Clustering Coefficients Erdos Renyi Model Network Average Clustering Erdos Renyi Model Transitivity Geometric Model Network Average Clustering Geometric Model Transitivity