## 4 Experimental Section  
We evaluate our approach by comparing Agwan with the state-of-the-art in labelled graph generation, represented by the MAG model [9, 14]. Agwan and MAG parameters are learned from real-world graphs. We generate random graphs from each model and calculate a series of statistics on each graph. As MAG does not generate weighted graphs, we fixed the weight of the edges in the generated graphs to the mean edge weight from the input graphs. This ensures that statistics such as average vertex strength are maintained. We used two datasets for the experiments. The first is a graph of \"who exercises with whom\" from a behavioural health study (Figure 2a, \(|V|=279,|E|=1308\)) [29]. Vertices represent participants and are labelled with 28 attributes denoting demographic information and health markers obtained from questionnaire data. Participants were tracked during periods of physical activity; when two people frequently register at the same sensor at the same time, an undirected edge is created, weighted with the number of mutual coincidences. Our second dataset is the \"who communicates with whom\" graph of the Enron e-mail corpus (Figure 2b, \(|V|=159,|E|=2667\)) [11]. Vertices are labelled with the job role of the employee. Edges are weighted with the number of e-mails exchanged between sender and recipient. As e-mail communications are not symmetric, edges in the Enron graph are directed. Both graphs exhibit a core-periphery structure which is typical of many real-world networks [7]. We evaluated Agwan against the following models:  
Erdos-Renyi random graph (ER): The ER model \(G(n,p)\) has two parameters. We set the number of vertices \(n\) and the edge probability \(p\) to match the input graphs as closely as possible. We do not expect a very close fit, but the ER model provides a useful baseline. MAG model with real attributes (MAG-R1) The MAG model with a single real attribute has a set of binary edge probabilities, \(\Theta=\{p^{ij}\}\) instead of a set of BMMs \(\Theta=\{\Omega^{ij}\}\). MAG model with latent attributes (MAG-L\(x\)): The MAG model also allows for modelling the graph structure using latent attributes. The discrete labels provided in the input graph are ignored; instead  
Figure 2: Input Graph Datasets, from (**a**) a health study and (**b**) the Enron e-mail corpus. (**a**) Undirected graph of who exercised with whom; (**b**) Directed graph of who e-mailed whom. MagFit[14] learns the values of a set of latent attributes to describe the graph structure. We vary the number of synthetic attributes \(x\) to investigate the relative contributions of vertex labels and edge weights to graph structure. Agwan model with truncated GMMs: We compare our BMM model to an alternative approach using GMMs [20]. One drawback of using GMMs is that it is possible to draw edge weights \(w_{uv}<0\). To avoid negative edge weights, we implement a _tabula rasa_ rule during graph generation, drawing new values from \(\Omega^{ij}\) until \(w_{uv}\geq 0\). To evaluate the closeness of fit of each model, we use the following statistics:  
Vertex Strength: For an unweighted graph, one of the most important measures is the degree distribution (the number of in-edges and out-edges of each vertex). Real-world graphs tend to have heavy-tailed power-law or log-normal degree distributions [2,5]. For a weighted graph, we generalise the concept of vertex degree to vertex strength [30]:  
\[s_{u}=\sum_{v\
eq u}w_{uv} \tag{12}\]  
We compare using the Complementary Cumulative Distribution Function (CCDF) of the strength of each vertex (both in-strength and out-strength in the case of the directed graph). For Cumulative Distribution Function (CDF) \(F(x)=P(X\leq x)\), the CCDF is defined as \(\bar{F}=P(X>x)=1-F(x)\). We show the unnormalised CCDFs in our plots; the normalised value can be obtained by integrating the area under the curve to 1. The CCDF of a power-law function will appear linear when plotted on a log-log scale, while the CCDF of a log-normal function will appear parabolic.