Wolfgang Schlauch

• This thesis is concerned with different null-models that are used in network analysis. Whenever it is of interest whether a real-world graph is exceptional regarding a particular measure, graphs from a null-model can be used to compare the real-world graph to. By analyzing an appropriate null-model, a researcher may find whether the results of the measure on the real-world graph is exceptional or not. Deciding which null-model to use is hard and sometimes the difference between the null-models is not even considered. In this thesis, there are several results presented: First, based on simple global measures, undirected graphs are analyzed. The results for these measures indicates that it is not important which null-model is used, thus, the fastest algorithm of a null-model may be used. Next, local measures are investigated. The fastest algorithm proves to be the most complicated to analyze. The model includes multigraphs which do not meet the conditions of all the measures, thus, the measures themselves have to be altered to take care of multigraphs as well. After careful consideration, the conditions are met and the analysis shows, that the fastest is not always the best. The same applies for directed graphs, as is shown in the last part. There, another more complex measure on graphs is introduced. I continue testing the applicability of several null-models; in the end, a set of equations proves to be fast and good enough as long as conditions regarding the degree sequence are met.