Empirical analysis of random graph generators

Sammanfattning

Many real-world systems can be represented as networks, where vertices correspond to entities and edges describe interactions or relations between them. Such networks often exhibit structural properties such as clustering, short paths, and heterogeneous degree distributions. This thesis investigates a finite age-based spatial attachment model for generating directed random graphs, with the aim of studying how the model parameters influence the resulting network structure. The model combines two mechanisms: age-based attachment and spatial locality. Vertices are assigned birth times and positions on the two-dimensional unit torus, and directed edges are formed from younger vertices to older vertices according to a hard cut-off connection rule. The empirical analysis focuses on the parameters $\beta$, which controls the overall density, and $\gamma$, which controls the strength of the age effect. Through repeated simulations, generated graphs are evaluated using edge counts, average local clustering, global clustering, average shortest-path length, and directed degree distributions. The results show that increasing $\beta$ mainly increases graph density and shifts both indegree and outdegree distributions toward larger values. In contrast, increasing $\gamma$ produces stronger age-driven heterogeneity, leading to heavier-tailed indegree distributions and more pronounced hub-like behavior. The simulations also show that average local clustering and global clustering behave differently: average local clustering increases with $\gamma$, while global clustering decreases for larger $\gamma$. A fixed density size-scaling experiment gives empirical support for the theoretical global clustering threshold $\gamma=1/2$. Overall, the study demonstrates that finite simulations can reproduce several qualitative features predicted by the asymptotic theory. The findings support the interpretation of $\beta$ as primarily controlling density, while $\gamma$ governs age-driven structural heterogeneity and plays an important role in clustering and degree behavior.