Hyperbolic hierarchical graph attention network for knowledge graph completion

Utilizing graph neural networks for knowledge embedding to accomplish the task of knowledge graph completion(KGC)has become an important research area in knowledge graph completion.However,the number of nodes in the knowledge graph increases exponentially with the depth of the tree,whereas the distances of nodes in Euclidean space are second-order polynomial distances,whereby knowledge embedding using graph neural networks in Euclidean space will not represent the distances between nodes well.This paper introduces a novel approach called hyperbolic hierarchical graph attention network(H2GAT)to rectify this limitation.Firstly,the paper conducts knowledge representation in the hyperbolic space,effectively mitigating the issue of exponential growth of nodes with tree depth and consequent information loss.Secondly,it introduces a hierarchical graph atten-tion mechanism specifically designed for the hyperbolic space,allowing for enhanced capture of the network structure inherent in the knowledge graph.Finally,the efficacy of the proposed H2GAT model is evaluated on benchmark datasets,namely WN18RR and FB15K-237,thereby validating its effectiveness.The H2GAT model achieved 0.445,0.515,and 0.586 in the Hits@1,Hits@3 and Hits@10 metrics respectively on the WN18RR dataset and 0.243,0.367 and 0.518 on the FB15K-237 dataset.By incorporating hyperbolic space embedding and hierarchical graph attention,the H2GAT model successfully addresses the limitations of existing hyperbolic knowledge embedding models,exhibiting its competence in knowledge graph completion tasks.

Aquatic Medicine Knowledge Graph Completion Based on Hybrid Convolution

Aquatic medicine knowledge graph is an effective means to realize intelligent aquaculture.Graph completion technology is key to improving the quality of knowledge graph construction.However,the difficulty of semantic discrimination among similar entities and inconspicuous semantic features result in low accuracy when completing aquatic medicine knowledge graph with complex relationships.In this study,an aquatic medicine knowledge graph completion method(TransH+HConvAM)is proposed.Firstly,TransH is applied to split the vector plane between entities and relations,ameliorating the poor completion effect caused by low semantic resolution of entities.Then,hybrid convolution is introduced to obtain the global interaction of triples based on the complete interaction between head/tail entities and relations,which improves the semantic features of triples and enhances the completion effect of complex relationships in the graph.Experiments are conducted to verify the performance of the proposed method.The MR,MRR and Hit@10 of the TransH+HConvAM are found to be 674,0.339,and 0.361,respectively.This study shows that the model effectively overcomes the poor completion effect of complex relationships and improves the construction quality of the aquatic medicine knowledge graph,providing technical support for intelligent aquaculture.

Joint learning based on multi-shaped filters for knowledge graph completion

To solve the problem of missing many valid triples in knowledge graphs(KGs),a novel model based on a convolutional neural network(CNN)called ConvKG is proposed,which employs a joint learning strategy for knowledge graph completion(KGC).Related research work has shown the superiority of convolutional neural networks(CNNs)in extracting semantic features of triple embeddings.However,these researches use only one single-shaped filter and fail to extract semantic features of different granularity.To solve this problem,ConvKG exploits multi-shaped filters to co-convolute on the triple embeddings,joint learning semantic features of different granularity.Different shaped filters cover different sizes on the triple embeddings and capture pairwise interactions of different granularity among triple elements.Experimental results confirm the strength of joint learning,and compared with state-of-the-art CNN-based KGC models,ConvKG achieves the better mean rank(MR)and Hits@10 metrics on dataset WN18 RR,and the better MR on dataset FB15k-237.

The Interval Graph Completion Problem on Split Graphs

The interval graph completion problem on a graph G is to find an added edge set F such that G + F is an interval supergraph with the smallest possible number of edges. The problem has important applications to numerical algebra, V LSI-layout and algorithm graph theory etc; And it has been known to be N P-complete on general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the interval graph completion problem on split graphs is investigated.

The Interval Graph Completion Problem for the Complete Multipartite Graphs

The interval graph completion problem of a graph G includes two class problems： the profile problem and the pathwidth problem, denoted as P（G） and PW（G） respectively, where the profile problem is to find an interval supergraph with the smallest possible number of edges; the pathwidth problem is to find an interval supergraph with the smallest possible cliquesize. These two class problems have important applications to numerical algebra, VLSI- layout and algorithm graph theory respectively; And they are known to be NP-complete for general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the exact solutions of the profile and the pathwidth of the complete multipartite graph Kn1,n2,...nr （r≥ 2） are determined.

Coupling Relation Strength with Graph Convolutional Networks for Knowledge Graph Completion

In the link prediction task of knowledge graph completion,Graph Neural Network(GNN)-based knowledge graph completion models have been shown by previous studies to produce large improvements in prediction results.However,many of the previous efforts were limited to aggregating the information given by neighboring nodes and did not take advantage of the information provided by the edges represented by relations.To address the problem,Coupling Relation Strength with Graph Convolutional Networks(RS-GCN)is proposed,which is a model with an encoder-decoder framework to realize the embedding of entities and relations in the vector space.On the encoder side,RS-GCN captures graph structure and neighborhood information while aggregating the information given by neighboring nodes.On the decoder side,RotatE is utilized to model and infer various relational patterns.The models are evaluated on standard FB15k,WN18,FB15k-237 and WN18RR datasets,and the experiments show that RS-GCN achieves better results than the current state-of-the-art classical models on the above knowledge graph datasets.

How to implement a knowledge graph completeness assessment with the guidance of user requirements

In the context of big data, many large-scale knowledge graphs have emerged to effectively organize the explosive growth of web data on the Internet. To select suitable knowledge graphs for use from many knowledge graphs, quality assessment is particularly important. As an important thing of quality assessment, completeness assessment generally refers to the ratio of the current data volume to the total data volume.When evaluating the completeness of a knowledge graph, it is often necessary to refine the completeness dimension by setting different completeness metrics to produce more complete and understandable evaluation results for the knowledge graph.However, lack of awareness of requirements is the most problematic quality issue. In the actual evaluation process, the existing completeness metrics need to consider the actual application. Therefore, to accurately recommend suitable knowledge graphs to many users, it is particularly important to develop relevant measurement metrics and formulate measurement schemes for completeness. In this paper, we will first clarify the concept of completeness, establish each metric of completeness, and finally design a measurement proposal for the completeness of knowledge graphs.

Semantic-aware graph convolution network on multi-hop paths for link prediction

Knowledge graph(KG) link prediction aims to address the problem of missing multiple valid triples in KGs. Existing approaches either struggle to efficiently model the message passing process of multi-hop paths or lack transparency of model prediction principles. In this paper,a new graph convolutional network path semantic-aware graph convolution network(PSGCN) is proposed to achieve modeling the semantic information of multi-hop paths. PSGCN first uses a random walk strategy to obtain all-hop paths in KGs,then captures the semantics of the paths by Word2Sec and long shortterm memory(LSTM) models,and finally converts them into a potential representation for the graph convolution network(GCN) messaging process. PSGCN combines path-based inference methods and graph neural networks to achieve better interpretability and scalability. In addition,to ensure the robustness of the model,the value of the path thresholdKis experimented on the FB15K-237 and WN18RR datasets,and the final results prove the effectiveness of the model.

Cycle Multiplicity of Total Graph of Complete Bipartite Graph

Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we determine the cycle multiplicity of and then obtain the formula of cycle multiplicity of total graph of complete bipartite graph, this generalizes the result for, which is given by M.M. Akbar Ali in [1].

Maximum Number of Edges of (rK_t)-Free Graphs

With positive integers r,t and n,where n≥rt and t≥2,the maximum number of edges of a simple graph of order n is estimated,which does not contain r disjoint copies of K_r for r=2 and 3.

Asymptotic upper bounds for wheel:complete graph Ramsey numbers

It is shown that r(W_m, K_n)≤(1+o(1))C_1n log n 2m-2m-2 for fixed even m≥4 and n→∞, and r(W_m, K_n)≤(1+o(1))C_2n 2mm+1 log n m+1m-1 for fixed odd m≥5 and n→∞, where C_1=C_1(m)>0 and C_2=C_2(m)>0, in particular, C_2=12 if m=5 . It is obtained by the analytic method and using the function f_m(x)=∫ 1 _ 0 (1-t) 1m dtm+(x-m)t , x≥0 , m≥1 on the base of the asymptotic upper bounds for r(C_m, K_n) which were given by Caro, et al. Also, cn log n 52 ≤r(K_4, K_n)≤(1+o(1)) n 3 ( log n) 2 (as n→∞ ). Moreover, we give r(K_k+C_m, K_n)≤(1+o(1))C_5(m)n log n k+mm-2 for fixed even m≥4 and r(K_k+C_m, K_n)≤(1+o(1))C_6(m)n 2+(k+1)(m-1)2+k(m-1) log n k+2m-1 for fixed odd m≥3 (as n→∞ ).

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K 7,n when7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C（x） denote the set of colors of vertex x and of the edges incident with x, we call C（x） the color set of x. If C（u） ≠ C（v） for any two different vertices u and v of V （G）, then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.

Signless Laplacian Characteristic Polynomials of Complete Multipartite Graphs

For a simple graph G,let matrix Q(G)=D(G) + A(G) be it's signless Laplacian matrix and Q G (λ)=det(λI Q) it's signless Laplacian characteristic polynomial,where D(G) denotes the diagonal matrix of vertex degrees of G,A(G) denotes its adjacency matrix of G.If all eigenvalues of Q G (λ) are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K(n_1,n_2,···,n_t).We prove that the complete t-partite graphs K(n,n,···,n)t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K(m,···,m)s(n,···,n)t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K(m,···,m,)s1(n,···,n,)s2(l,···,l)s3.

Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K8,n

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.

Distance Integral Complete Multipartite Graphs with s=5, 6

Let D(G) =(d_(ij))_(n×n) denote the distance matrix of a connected graph G with order n, where d_(ij) is equal to the distance between vertices viand vjin G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a sufficient and necessary condition for complete r-partite graphs K_(p1,p2,···,pr)=K_(a1·p1,a2·p2,···,as···ps) to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs K_(a1·p1,a2·p2,···,as·ps) with s > 4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with s = 5, 6. The problem of the existence of such distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with arbitrarily large number s remains open.

Minus total k-subdomination in graphs

Let G = (V,E) be a simple graph without isolated vertices. For positive integer k, a 3-valued function f : V → {-1,0,1} is said to be a minus total k-subdominating function (MTkSF) if sum from (u∈N(v)) to f(u)≥1 for at least k vertices v in G, where N(v) is the open neighborhood of v. The minus total k-subdomination number γkt(G) equals the minimum weight of an MTkSF on G. In this paper, the values on the minus total k-subdomination number of some special graphs are investigated. Several lower bounds on γkt of general graphs and trees are obtained.

Quantum search of many vertices on the joined complete graph

The quantum search on the graph is a very important topic.In this work,we develop a theoretic method on searching of single vertex on the graph[Phys.Rev.Lett.114110503(2015)],and systematically study the search of many vertices on one low-connectivity graph,the joined complete graph.Our results reveal that,with the optimal jumping rate obtained from the theoretical method,we can find such target vertices at the time O(√N),where N is the number of total vertices.Therefore,the search of many vertices on the joined complete graph possessing quantum advantage has been achieved.

Bondage and Reinforcement Number of γ_f for Complete Multipartite Graph

The bondage number of γ f, b f(G) , is defined to be the minimum cardinality of a set of edges whose removal from G results in a graph G′ satisfying γ f(G′)> γ f(G) . The reinforcement number of γ f, r f(G) , is defined to be the minimum cardinality of a set of edges which when added to G results in a graph G′ satisfying γ f(G′)< γ f(G) . G.S.Domke and R.C.Laskar initiated the study of them and gave exact values of b f(G) and r f(G) for some classes of graphs. Exact values of b f(G) and r f(G) for complete multipartite graphs are given and some results are extended.

COMPLETE MULTIPARTITE DECOMPOSITIONS OF COMPLETE GRAPHS AND COMPLETE n-PARTITE GRAPHS

In this paper,a new concept of an optimal complete multipartite decomposition of type 1 (type 2) of a complete n-partite graph Q n is proposed and another new concept of a normal complete multipartite decomposition of K n is introduced.It is showed that an optimal complete multipartite decomposition of type 1 of K n is a normal complete multipartite decomposition.As for any complete multipartite decomposition of K n,there is a derived complete multipartite decomposition for Q n.It is also showed that any optimal complete multipartite decomposition of type 1 of Q n is a derived decomposition of an optimal complete multipartite decomposition of type 1 of K n.Besides,some structural properties of an optimal complete multipartite decomposition of type 1 of K n are given.