The effect of clustering in random graphs

The goal of this thesis is to extend the configuration model to arbitrarily sized clusters and furthermore to a graph after bond percolation. To achieve this we first explain the configuration model and the model for graphs with single edges and triangles. Where in the latter model we already see some results of the effect of clustering on the size of the giant component. This is then extended to a generic model for arbitrary size clusters. We see that graphs that show more clustering have a smaller giant component which grows slower when we increase the amount of neighbours. If we use different sorts of graphs we see that their growth speeds differ a lot. We also find that the giant component differs in graphs with the same clustering coefficient. Thereafter we study graphs after percolation. We do this first for the configuration model and work towards a random graph with arbitrarily sized clusters. If we use percolation on graphs with single edges and triangles we already see that a graph with more clustering is more resilient to percolation. When we look at this effect for random graphs with bigger clusters we see that this reasoning does not hold for increasingly larger clusters. The most resilient graph, out of random graphs containing complete graphs only, turns out to be the graph with only complete graphs on 4 vertices. The explanation for this is that the size of the giant component also plays a role in when the giant component start to emerge. Since graphs with more clustering have a smaller giant component than graphs with less clustering their giant components emerge later. There can be done a lot more research on the behaviour on percolation of certain types of graphs after percolation. Our model with only 2 variables (two types of complete graphs in a random graph) could be extend to a model with an arbitrary amount of different complete graphs. Our model of percolation seems plausible but the simulation and theoretical results have not been verified.