# Deep Link Entropy for Quantifying Edge Significance in Social Networks
## 1 Introduction
Studying the anti-migrant sentiments on Twitter, [PERSON] showed that even among like-minded groups reached consensus on a particular political issue, the reasons leading to the same ideological line can be sharply fragmented [14] and also detected deliberative enclaves [15; 16] and sharp polarization [16; 17] among fragmented groups on Twitter. Communications between different-minded groups are enabled mainly by individuals who constitute a social bridge between the groups, each individual interacting with both groups. Regarding the global connectivity and network diffusion and especially for weakening echo chambers and political polarization, the edges connecting such pairs of individuals are more significant than the edges connecting individuals who are in interaction within their groups only. Several methods were proposed to quantify the edge significance, such as edge betweenness centrality [7; 18; 19], k-path centrality [20], degree product [21] and bridgeness [22]. The recent Link Entropy (LE) proposal of [PERSON] et al. which is based on the divergence of probability distributions of pairs of nodes, outperforms these measures [23]. To calculate the significance of an edge connecting the nodes \(n_{i}\) and \(n_{j}\), [PERSON] et al. calculates the entropies \(H(X_{i})\) and \(H(X_{j})\) of the nodes based on the probabilities \(X_{i}\) and \(X_{j}\) that \(n_{i}\) and \(n_{j}\) belong to each possible group, respectively, and the Jensen-Shannon divergence \(JSD(X_{i}\parallel X_{j})\). The objective of this work is to quantify the significance of edges of a network more precisely and we ask whether it is possible to extract more information from the structure of the network. To answer this question, we consider that two different scenarios where the other adjacent nodes of \(n_{i}\) or \(n_{j}\), have a small, or a great uncertainty of belonging to one group might lead to significantly different results. Finding such a difference in the results would help shrinking the gap between the intrinsic information of a network's structure and our knowledge on it, contributing to the literature. Hence, aiming to provide a more precise edge significance quantification approach, we propose to improve the LE approach by taking into account also the entropies of the adjacent nodes of \(n_{i}\) and \(n_{j}\) and we call our approach as Deep Link Entropy (DLE). Denoting the sum of entropic functions of node \(n_{i}\) with \(D(n_{i})\) and the Link Entropy of the edge connecting the nodes \(n_{i}\) and \(n_{j}\) with \(LE_{ij}\), DLE can be considered roughly as \(DLE_{ij}=LE_{ij}+\chi\big{[}D(n_{i})+D(n_{j})\big{]}\), where \(\chi\) is a small factor to be determined for each unique network. Note that, failing to find a non-zero \(\chi\) for a given network, DLE reduces to LE by setting \(\chi=0\). Hence, one can safely claim that DLE performs at least as good at LE. Nevertheless, we were able to find a non-zero \(\chi\) easily in our experiments on three typical networks, namely \((i)\) the network presented by [PERSON] et al. [24] for discovering groups (communities) in a network, \((ii)\) Zachary's Karate Club Network [25] and \((iii)\) American College Football Network [7]. Considering the edge-significance within the same context of Ref. [23], i.e., the global connectivity and diffusion dynamics, our tests for evaluating the performance of DLE follow the popular approach also used by [PERSON] et al. [23] in their LE approach. We remove the edges one by one in descending order of significance found by each approach and in each step dividing the number of nodes in the largest component by the total number of nodes in the entire network, usually denoted by \(R_{\text{GC}}\). Until the first disintegration, \(R_{\text{GC}}=1\) and as all the edges are removed, \(R_{\text{GC}}\to 0\) for large \(n\), number of nodes.