## 6 Conclusions and Future Work  
In this paper, we presented Agwan, a generative model for random graphs with discrete labels and weighted edges. We included a fitting algorithm to learn a model of edge weights from real-world graphs, and an algorithm to generate random labelled, weighted graphs with similar characteristics to the input graph.  
We measured the closeness of fit of our generated graphs to the input graph over a range of graph statistics, and compared our approach to the state-of-the-art in random graph generative algorithms. Our results and findings are summarised in Table 1.  
Our results demonstrate that Agwan produces an accurate model of the weighted input graphs. The Agwan graphs reproduce many of the properties of the input graphs, including formation of a giant component, heavy-tailed degree distribution, Small World property, Densification Power Law and shrinking diameter effect. Vertex strength and spectral properties were modelled very closely, indicating that these properties are indeed dependent on vertex labels and edge weights. For clustering and triad participation, it appears that these properties are independent of the edge weight distribution. While there  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Statistic** & **Vertex Labels** & **Edge Weights** & **Agwan** & **MAG** \\ \hline Vertex Strength & Dependent & Dependent & Accurate & Less accurate \\ Singular Values & Dependent & Dependent & Accurate & Less accurate \\ Primary Singular Vector & Partially & Dependent & Accurate & Poor \\  & Dependent & & & \\ Clustering Coefficients & Conditionally & Independent & Less accurate when using real attributes \\  & Independent\({}^{*}\) & & More accurate with synthetic attributes \\ Triad Participation & Conditionally & Independent & Less accurate when using real attributes \\  & Independent\({}^{*}\) & & More accurate with synthetic attributes \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of results and findings: dependencies between graph labels and weights and structural properties (“hypothesized); relative accuracy of the properties of weighted graphs generated using Agwan and MAG models.  
Figure 15: Shrinking Diameter Effect. (**a**) Undirected, unweighted; (**b**) Undirected, weighted; (**c**) Directed, unweighted; (**d**) Directed, weighted.  
appears to be a relationship with the vertex label distribution, we suggest that this may be an artefact of the true process giving rise to these properties, triadic closure. Further research is required into the relationship between vertex attributes and triangle formation in graphs.  
We investigated the accuracy of modelling edge weights using BMMs and compared to a previous approach which used GMMs. The edge weights often follow a power-law distribution, which the BMM approach fits more closely, particularly in the lower range of edge weight values. In general, BMMs are more suitable for bounded data; probability mass can only be assigned to the valid range of weight values. For the GMM model, we compensated for this by truncating the GMM during graph generation. We expected that Agwan (BMM) would consistently outperform Agwan (GMM), but the truncated GMM performed surprisingly well. We propose to conduct experiments on a wider range of datasets to investigate this further. It would also be interesting to compare with Poisson Mixture Models to model discrete edge weights.