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Network representation learning plays an important role in the field of network data mining. By embedding network structures and other features into the representation vector space of low dimensions, network representation learning algorithms can provide high-quality feature input for subsequent tasks, such as network link prediction, network vertex classification, and network visualization. The existing network representation learning algorithms can be trained based on the structural features, vertex texts, vertex tags, community information, etc. However, there exists a lack of algorithm of using the future evolution results of the networks to guide the network representation learning. Therefore, this paper aims at modeling the future network evolution results of the networks based on the link prediction algorithm, introducing the future link probabilities between vertices without edges into the network representation learning tasks. In order to make the network representation vectors contain more feature factors, the text features of the vertices are also embedded into the network representation vectors. Based on the above two optimization approaches, we propose a novel network representation learning algorithm, Network Representation learning algorithm based on the joint optimization of Three Features (TFNR). Based on Inductive Matrix Completion (IMC), TFNR algorithm introduces the future probabilities between vertices without edges and text features into the procedure of modeling network structures, which can avoid the problem of the network structure sparse. Experimental results show that the proposed TFNR algorithm performs well in network vertex classification and visualization tasks on three real citation network datasets.

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A hypergraph H = (V, ε) is a pair consisting of a vertex set V, and a set ε of subsets (the hyperedges of H) of V. A hypergraph H is r-uniform if all the hyperedges of H have the same cardinality r. Let H be an r-uniform hypergraph, we generalize the concept of trees for r-uniform hypergraphs. We say that an r-uniform hypergraph H is a generalized hypertree (GHT) if H is disconnected after removing any hyperedge E, and the number of components of GHT − E is a fixed value k (2 ≤ k ≤ r). We focus on the case that GHT − E has exactly two components. An edge-minimal GHT is a GHT whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r-uniform GHT on n vertices has at least 2n/(r + 1) edges and it has at most n − r + 1 edges if r ≥ 3 and n ≥ 3, and the lower and upper bounds on the edge number are sharp. We then discuss the case that GHT − E has exactly k (2 ≤ k ≤ r − 1) components.

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In this paper, we mainly study the exact controllability of several types of extended-path molecular graph networks. Based on the construction of extended-path molecular graph networks with C 3 , C 4 , or K 4 , using the determinant operation of the matrix and the recursive method, the exact characteristic polynomial of these networks is deduced. According to the definition of minimum driver node number N D , the exact controllability of these networks is obtained. Moreover, we give the minimum driver node sets of partial small networks.