# Deep Link Entropy for Quantifying Edge Significance in Social Networks
## 3 Results  
Following the flowchart illustrated in Figure 1 and the approach presented in the Methods section, we rank the significance of edges of various networks according to LE and DLE and removing the most significant edge in each step, we test the performance of these two approaches in detecting the faster disintegration. Because the basic improvement of DLE over LE is to take into account the adjacent nodes as well, if the network consists of more than two groups (or communities), there appears the chance that the adjacent nodes might belong to some other groups, which leads to the prediction that DLE can perform significantly better than LE particularly in multi-group networks, regardless of our assumption on the number of groups in the simulation. We test our approach on three typical networks, the first being the one presented by [PERSON] et al. (2019) in their community detecting algorithm. The second and third networks are chosen to be popular benchmark networks in the literature, which are relatively small- and large-scale networks, respectively, so that we can measure the performance of the approach within a wide range. The first network consists of four interacting groups (communities). Because [PERSON] et al. claimed that no matter how many groups the network consists of, considering only two groups perform slightly better and also the simulation runs faster than considering the actual number of groups, we quantify and rank the edges considering that there are two groups (\(k=2\)), first, and then four groups (\(k=4\)) in the network. The adjacency graph of the network in consideration is given in Figure 2. We run the simulations for LE and DLE for \(k=2\) and \(k=4\). Removing the most significant edge according to LE and according to DLE, re-discovering the groups and re-quantifying the edges in each step, we find that DLE significantly outperforms LE as presented in Figure 3.  
Figure 2: Adjacency graph of the network presented in the community discovery work of [PERSON] et al. (2019), consisting of four communities with overlapping nodes such as \(n_{5}\) and \(n_{6}\). Next, we analyze the Zachary's Karate Club network [25] with the adjacency graph presented in Figure 4, which is usually considered to be consisting of two groups. As shown in Figure 5, for the same factor of adjacent nodes, i.e., \(\chi=-0.12\), we find that while at first (until around 20 edges are removed) LE performs better, DLE starts and keeps performing better than LE for the rest of the edge removal process. Figure 4: Adjacency graph of Zachary’s Karate Club network [25]. Figure 5: Testing Deep Link Entropy (DLE) on Zachary’s Karate Club network [25] (adjacency graph illustrated in Figure 4) showing the fraction of nodes of the largest connected group, \(R_{GC}\), as a result of removing the most significant edge in each step after discovering the communities and updating the edge significance ranking. Being outperformed in the first quarter of the process, DLE with \(\chi=-0.12\) outperforms LE in the rest of the process. Figure 3: Testing Deep Link Entropy (DLE) on the network presented by [PERSON] et al. [24] (adjacency graph illustrated in Figure 2) showing the fraction of nodes of the largest connected group, \(R_{GC}\), as a result of removing the most significant edge in each step after discovering the communities and updating the edge significance ranking. DLE with \(\chi=-0.12\) outperforms Link Entropy (LE), i.e., the best strategy so far. Having analyzed two small-scale networks consisting of four and two groups, respectively, we continue with the main target of the present approach, i.e., a large-scale complex network consisting of several groups (communities). We take the American Football Network as an exemplary network, which has 115 nodes and 613 edges as shown in Figure 6. In the literature, different number of communities are detected by various strategies.