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Graph is a ubiquitous data structure in data science that is widely applied in social networks, knowledge representation graphs, recommendation systems, etc. When given a graph dataset consisting of one graph or more graphs, where the graphs are weighted in general, the first step is often to find clusters in the graphs. In this paper, we present some ongoing research results on graph clustering algorithms for clustering weighted graph datasets, which we name as Weighted Graph Node Clustering via Gumbel Softmax (WGCGS for short). We apply WGCGS on the Karate club weighted network dataset. Our experiments demonstrate that WGCGS can efficiently and effectively find clusters in the Karate club weighted network dataset. Our algorithm's effectiveness is demonstrated by (1) comparing the clustering result obtained from our algorithm and the given labels of the dataset; and (2) comparing various metrics between our clustering algorithm and other state-of-the-art graph clustering algorithms.

Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs., Comment: This paper is a new version of arXiv:2001.09094. We need to resubmit it as a replacement. Otherwise, I can not finish the resubmission to the journal "Discrete Mathematics and Theoretical Computer Science"

Recently, in many systems such as speech recognition and visual processing, deep learning has been widely implemented. In this research, we are exploring the possibility of using deep learning in community detection among the graph datasets. Graphs have gained growing traction in different fields, including social networks, information graphs, the recommender system, and also life sciences. In this paper, we propose a method of community detection clustering the nodes of various graph datasets. We cluster different category datasets that belong to Affiliation networks, Animal networks, Human contact networks, Human social networks, Miscellaneous networks. The deep learning role in modeling the interaction between nodes in a network allows a revolution in the field of science relevant to graph network analysis. In this paper, we extend the gumbel softmax approach to graph network clustering. The experimental findings on specific graph datasets reveal that the new approach outperforms traditional clustering significantly, which strongly shows the efficacy of deep learning in graph community detection clustering. We do a series of experiments on our graph clustering algorithm, using various datasets: Zachary karate club, Highland Tribe, Train bombing, American Revolution, Dolphins, Zebra, Windsurfers, Les Mis\'erables, Political books., Comment: 9 Pages, previous title was Clustering for Graph Datasets via Gumbel Softmax

Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs., Comment: 14 pages

Graph Neural Networks (GNNs) have been a latest hot research topic in data science, due to the fact that they use the ubiquitous data structure graphs as the underlying elements for constructing and training neural networks. In a GNN, each node has numerous features associated with it. The entire task (for example, classification, or clustering) utilizes the features of the nodes to make decisions, at node level or graph level. In this paper, (1) we extend the feature selection algorithm presented in via Gumbel Softmax to GNNs. We conduct a series of experiments on our feature selection algorithms, using various benchmark datasets: Cora, Citeseer and Pubmed. (2) We implement a mechanism to rank the extracted features. We demonstrate the effectiveness of our algorithms, for both feature selection and ranking. For the Cora dataset, (1) we use the algorithm to select 225 features out of 1433 features. Our experimental results demonstrate their effectiveness for the same classification problem. (2) We extract features such that they are linear combinations of the original features, where the coefficients for each extracted features are non-negative and sum up to one. We propose an algorithm to rank the extracted features in the sense that when using them for the same classification problem, the accuracy goes down gradually for the extracted features within the rank 1 - 50, 51 - 100, 100 - 150, and 151 - 200.

In this paper, we propose a data structure, a quadruple neighbor list (QN-list, for short), to support real time queries of all longest increasing subsequence (LIS) and LIS with constraints over sequential data streams. The QN-List built by our algorithm requires $O(w)$ space, where $w$ is the time window size. The running time for building the initial QN-List takes $O(w\log w)$ time. Applying the QN-List, insertion of the new item takes $O(\log w)$ time and deletion of the first item takes $O(w)$ time. To the best of our knowledge, this is the first work to support both LIS enumeration and LIS with constraints computation by using a single uniform data structure for real time sequential data streams. Our method outperforms the state-of-the-art methods in both time and space cost, not only theoretically, but also empirically., Comment: 20 pages (12+8)