# Deep Link Entropy for Quantifying Edge Significance in Social Networks
## 4 Discussion
Here, the key question is which pair of nodes, i.e., which edge to protect. Looking at the adjacency graph of network in Figure 2 suggests that unlike \(e_{7-8}\), \(e_{4-10}\) is a bridging edge between two fragmented communities. So, it would be more effective to protect \(e_{4-10}\) than to protect \(e_{7-8}\) for keeping those two communities connected and the increased interactions would contribute to weakening the echo-chambers and polarization within each community. That is, \(e_{4-10}\) is expected to be more significant than \(e_{7-8}\). Hence, a more precise quantification of edge significance in a given complex social network from theperspective of global connectivity and diffusion of ideologies would have a positive impact on political communications, indicating the link between our results to the objective of this work. For the second set, the key question is to pick which two nodes to suggest connecting. In the same network, a new edge \(e_{14-17}\), or \(e_{5-23}\) would be more effective than a new edge \(e_{27-29}\). Being beyond the present work, we consider another method for determining new edges to suggest. For a given network, set of pairs of unconnected nodes (corresponding to elements in the adjacency matrix which are equal to 0), is constructed. One pair of nodes is picked and it is considered that they are connected (setting the corresponding element of adjacency matrix to 1). Running the edge significance algorithm, significance of this new edge is quantified. Repeating this step for every pair in the set, significance of each candidate edge is determined and SNS can suggest pairs of nodes with greatest edge significance to connect, so that the chance to create bridges between fragmented communities is increased. In addition to social networks, our approach can be used for detecting the significant, or critical edges in a wide range of applications from biology to electrical networks. Moreover, considering von Neumann entropy of quantum particles, i.e., the nodes and considering the non-classical correlations between the particles as the edges, our approach can be adopted to quantum domain. In quantum networks based on multipartite entanglement, significance of edges are identical in typical states such as maximally entangled GHZ or W states [30; 31]. However, for states with more than three particles, generic graph (or cluster) states emerge, enabling measurement-based one-way quantum computation [32], where groups of particles realizing the logic gates of the abstract gate-model can be considered as the _communities_ of the network. In such generic networks, the edges, hence their significance are not identical. Hence, considering the scenario with a limited resource to protect only a set of edges against decoherence, or to apply error-correction operations in a quantum network, the edge-significance quantification problem becomes crucial for selecting the edges to act on. Hence, quantum extensions of our approach can find applications in entanglement percolation [33] and one-way quantum computers [32]. The extra terms in Equation (11) we added to design our DLE method based on LE method are found not by a brute force attack to provide an improved performance, but rather following our hypothesis that the uncertainties of adjacent nodes of \(n_{i}\) and \(n_{j}\) of belonging to each possible group contributes to quantifying the significance of edge \(e_{i-j}\). Only the coefficient \(\chi\) is to be determined which we easily did in our experiments. The drawback of adding these extra terms is the increasing computational complexity. Regardless of number of nodes in a given network, for each edge, LE computes four entropic functions. However, if \(n_{i}\) has \(I-1\) adjacent nodes (excluding \(n_{j}\)) and \(n_{j}\) has \(J-1\) adjacent nodes (excluding \(n_{i}\)) DLE computes additional \(I+J-2\) entropic functions for each edge.