# Deep Link Entropy for Quantifying Edge Significance in Social Networks
## 1 Introduction  
In a network with interacting communities consisting of nodes and edges each connecting a pair of nodes, some edges are of greater importance than others from a particular perspective. Consider a social network with communities fragmented along ideological lines, for example. Breaking of an edge connecting two nodes within a community and breaking of an edge connecting two nodes each belonging to another community lead to significantly different consequences in interrupting the communications between communities. The edge in the latter case is more _significant_ than the edge in the former case in the sense that it is constituting a social bridge between two communities, enabling their communities' exposure to different ideological lines with a potential weakening of political echo-chambers and polarization. Therefore, quantifying the significance of nodes and edges of a natural or a human-made network is of crucial importance in various fields, especially in analyzing the connectivity and diffusion dynamics. Regarding the spread of computer viruses, for example, analyzing real-world data and through a dynamical model, [PERSON] and [PERSON] found that there is no epidemic threshold in scale-free networks that viruses can spread besides an arbitrary spreading rate [1]. [PERSON] studied the structure of epidemic disease networks and showed that various models can be solved exactly on a wide variety of networks [2]. Based on real data in South Korea, [PERSON] et al. has recently analyzed the social network of the spread of COVID-19 and foundthat the size of the infection network could be significantly decreased if top nodes on the out-degree significantly are removed [3]. To reduce the energy consumption for wireless sensor networks, finding the redundant nodes based on a similarity measure, [PERSON] et al. developed an efficient sleep scheduling mechanism [4] and [PERSON] et al. proposed a clustering approach [5]. [PERSON] et al. designed multi-objective genetic algorithms to detect the edges of a realistic electrical network, which are critical for the structure and the flow [6]. Due to their social and political impacts, social networks are among the most popular structures in this context. [PERSON] and [PERSON]'s seminal work shed light on the community structure in social and biological networks [7]. To find the critical nodes of a sparse graph which can represent a social network with many nodes but a few connections, [PERSON] et al. developed a memetic search algorithm [8]. Based on the topological structure, particularly the cliques and paths, [PERSON] et al. developed an edge-ranking algorithm [9]. Ranking the significance of edges is a core problem in the context of online political communications. Because the emergence of the Internet and social networks enabled faster and more efficient interactions between individuals. On the one hand, this is a great opportunity for societies that there is now an unprecedented chance for the diffusion of ideas among like- or different-minded groups, or communities. Through such a great opportunity for exposure to different ideologies, one might expect that the political polarization among like-minded groups weaken over time. On the other hand, however, there is also an unprecedented chance for gathering around the same ideological lines and for a deliberate enclave as well and this would inevitably contribute to forming and cementing echo chambers in fragmented groups [10]. That is, communications within communities become dominant over the communications among communities, paving the way for a debilitating polarization [11]. Along this line, [PERSON] et al. detected echo chambers on Facebook pages [12] and [PERSON] and [PERSON] detected political polarization among political microblogs (_polblogs_) around the 2004 US presidential elections [13]. The _polblogs_ network they studied became one of the standard networks to test various social network analysis approaches.