# Local Community Detection Based on Small Cliques
## 4 Experiments
### Overlapping Communities  
Our parameter set for overlapping communities is inspired by the highly overlapping parameter set used in [8]. All nodes belong to the same number of communities \(O_{m}\) which is in the range \([1,5]\). We modified the parameters slightly by not setting the average degree but the minimum degree to \(18\cdot O_{m}\). This has the effect that the minimum internal degree of a community stays the same with increasing values of \(O_{m}\). Using NetworkKit, we calculated the expected average degree for this minimum degree and used these as parameters for the LFR generator. Figure 3 shows the results for these networks. Note that here we do not evaluate if an algorithm is able to discover all or a specific community of a node, we compare to the community of the seed node that is best matched. The performance of LocalT is extremely poor, it cannot detect any overlapping communities and without a clique as initialization, it cannot even detect the non-overlapping communities. PageRank-Nibble is also unable to discover any communities starting for \(O_{m}\geq 2\), which is not really surprising as the PageRank approximation has no way to decide for one of the communities of the seed node. The other algorithms perform better, LTE being best when starting without a clique. The largest clique of the seed node as initialization again improves the performance, and, quite interestingly, when starting with a clique, the density-based algorithms GCE M, GCE L and LFM perform better than the triangle-based algorithms LTE and TCE. A possible explanation for this could be that triangles are less helpful in the case of overlapping communities as a node can have triangles with neighbors of any community. Infomap completely fails to discover the overlapping community structure, but this is expected as we only used the variant for disjoint communities. Figure 3: Results for overlapping communities with LFR graphs with parameter set “Overlapping” as specified in Table 1.  
Weighted Graphs  
For the weighted graphs, we use the parameter set \"Weighted\" in Table 1, those are the parameters also used by [PERSON] et al. [11]. This means that we have a single community size range, \([20,100]\). We further choose three values for the topological mixing parameter \(\mu_{t}=0.3\), 0.5, 0.8. The weight mixing parameter is varied from 0.1 to 0.9 in each of these sets. For each \(\mu_{w}\) we generate 20 instances and run the algorithms from 20 randomly chosen seed nodes. For the weighted graphs, we do not compare LocalT as it only works on unweighted graphs and it is not clear how to generalize it to weighted graphs, the authors discuss this as future work. In Figure 4 we show the results for the other algorithms. For each value of the topological mixing parameter \(\mu_{t}\) we plot the weight mixing parameter \(\mu_{w}\) on the \(x\)-axis and the resulting average \(F_{1}\) score on the \(y\)-axis. Comparing the two plots at \(\mu_{t}=0.3\) and \(\mu_{t}=0.5\), we can see that while most algorithms even improve their performance at \(\mu_{t}=0.5\), LTE has a worse performance. A possible explanation for this is that its score of the community is based on the presence of triangles. While TCE is also based  
Figure 4: Avg. \(F_{1}\)-scores on the LFR benchmark with the parameter set Weighted, as specified in Table 1.  
on triangles for the decision which node to add next, its score of the community is solely based on the density and should thus be independent of the structure of the graph as long as the weights are correctly distributed. This can be seen by its performance which is even better for \(\mu_{t}=0.5\). When initialized with a clique, LTE has fewer problems though and even outperforms Infomap for high values of \(\mu_{w}\). This is probably due to its community score being based on the structure of the graph and thus taking weights less into account. Two-Phase L performs worse than our baseline BFS, apparently the two-phase process is not so suited for this kind of weighted graph.