for a given value of k.

For comparing two partitions on the same graph, we use the Jaccard index [64] which for two partitions, X and Y**, is defined as**

$$J I=\frac{a}{(a+b+c)},\eqno(21)$$

where a, b, and c **are, respectively, the total pair of nodes that are assigned to the same** cluster in both X and Y, the same cluster in X and different clusters in Y**, and the same** cluster in Y and different clusters in X **. When two partitions are identical, the Jaccard index** evaluates to one, and it decreases as the partitions deviate from each other.

Figures 1b show the closeness of the partitionings identified by the algorithms and the ground truth in terms of the Jaccard index. The algorithm by Blondel et al. deviates dramatically from the true clustering (Figure 1b). The reason is that the minimum and maximum cluster sizes are fixed for all the networks in addition to the **average node degree.**
Hence the average number of edges per cluster remains the same making the proportion of the intra-cluster edges decrease. Due to the resolution limit of modularity, as the proportion of intra-cluster edges is decreased, modularity maximization algorithms tend to group several clusters into a single cluster. If we reduce n **to 1000, the Blondel et al. algorithm leads to**
closer results to the ground truth. The algorithm by Dhillon et al. does not perform as well as other algorithms that strive to maximize NAssoc and the local search results in negligible improvement in terms of the Jaccard index.

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

## 5.3. Model Order Selection: The Curvature Metric

We first provide an illustrative example of the use of the curvature metric for selecting the number of clusters in the partitioning. This example highlights the **difference in behavior** compared to the modularity metric used by modularity maximization algorithms [56].