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تسجيل مستخدم جديدSpectral Detection of Simplicial Communities via Hodge Laplacians
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While the study of graphs has been very popular, simplicial complexes are relatively new in the network science community. Despite being are a source of rich information, graphs are limited to pairwise interactions. However, several real world networks such as social networks, neuronal networks etc. involve simultaneous interactions between more than two nodes. Simplicial complexes provide a powerful mathematical way to model such interactions. Now, the spectrum of the graph Laplacian is known to be indicative of community structure, with nonzero eigenvectors encoding the identity of communities. Here, we propose that the spectrum of the Hodge Laplacian, a higher-order Laplacian applied to simplicial complexes, encodes simplicial communities. We formulate an algorithm to extract simplicial communities (of arbitrary dimension). We apply this algorithm on simplicial complex benchmarks and on real data including social networks and language-networks, where higher-order relationships are intrinsic. Additionally, datasets for simplicial complexes are scarce. Hence, we introduce a method of optimally generating a simplicial complex from its network backbone through estimating the textit{true} higher-order relationships when its community structure is known. We do so by using the adjusted mutual information to identify the configuration that best matches the expected data partition. Lastly, we demonstrate an example of persistent simplicial communities inspired by the field of persistence homology.
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