

![0_image_0.png](0_image_0.png) 

An example used by Good et al. [59], to illustrate the resolution limit is the **ring of 24**
cliques9**, each of which has 5 nodes and is connected to its neighboring clique by a single**
edge. The value of modularity is maximized by a 12-cluster partition that merges pairs of cliques together. A more natural clustering is to consider each clique as an individual cluster. Figure 2b shows the curvature plot for the case of the ring of cliques. In addition to the peak of the curvature that is correctly located at 24 clusters, it is interesting to note that the other local peaks of the curvature are also meaningful; e.g., the peak at 12 clusters corresponds to the clustering that maximizes modularity.

To see how the curvature indicates the true number of clusters for networks with more complex structures, we consider the LFR benchmarks. We fix the LFR parameters for a 1,000-node graph and explore how the curvature changes as µ **is varied. Figure 3 indicates**
that the peak of the curvature plot correctly identifies the true number of clusters for µ in the range 0.1 - 0.5. As µ increases, the main peak becomes less distinct; for µ = 0.5, the second highest peak is almost as large as the primary peak. As the clusters become increasingly interconnected, the curvature plot provides a less clear indication of the true