Set Of Subgraphs Research Articles

Temporal subgraph matching method for multi-connected temporal graph

Temporal graph matching is used to find a matching subgraph set that satisfies temporal and topological information on the temporal graph. It is widely used in real scenarios and provides important support for big data mining and analysis. Most existing matching methods take the single-connected graph as the matching graph, which has a high computational cost. Simultaneously, the construction and matching complexity of its index on multi-connected temporal subgraph matching is high. To solve the above problems, we propose a new temporal subgraph matching problem for multi-connected temporal graphs and proposes the corresponding efficient filtering-selecting-matching three-stage multi-connected temporal subgraph matching method. In the filtering stage, a vertex-based filter is proposed to generate a candidate temporal subgraph set. In the select phase, a set of temporal subgraphs satisfying the vertex mapping is found. In the matching stage, the single-filter temporal subgraph matching method and temporal subgraph matching based on the temporal information tree index method are proposed to solve the problems that the edges of multi-connected temporal subgraphs cannot be mapped and the single graph matching efficiency is low, respectively. The experimental results show that the proposed method improves the matching accuracy, and the corresponding index size is reduced.

TED: Towards Discovering Top-k Edge-Diversified Patterns in a Graph Database

With an exponentially growing number of graphs from disparate repositories, there is a strong need to analyze a graph database containing an extensive collection of small- or medium-sized data graphs (e.g., chemical compounds). Although subgraph enumeration and subgraph mining have been proposed to bring insights into a graph database by a set of subgraph structures, they often end up with similar or homogenous topologies, which is undesirable in many graph applications. To address this limitation, we propose the Top-k Edge-Diversified Patterns Discovery problem to retrieve a set of subgraphs that cover the maximum number of edges in a database. To efficiently process such query, we present a generic and extensible framework called Ted which achieves a guaranteed approximation ratio to the optimal result. Two optimization strategies are further developed to improve the performance. Experimental studies on real-world datasets demonstrate the superiority of Ted to traditional techniques.

The Path-Structure Connectivity of Augmentedk-aryn-cubes

AbstractFor connected graphs $G$ and $H$, the $H$-structure connectivity $\kappa (G; H)$ (resp. $H$-substructure connectivity $\kappa ^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $\mathcal{F}$ of $G$ such that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-\mathcal{F}$ is disconnected or singleton. In this paper, we consider $P_t$-structure connectivity and $P_t$-substructure connectivity of augmented $k$-ary $n$-cubes $AQ_{n,k}$ for $n\geq 2$, $k\geq 3$ and $1\leq t\leq 4n-2$. We obtain that $\kappa (AQ_{n,k}; P_t)=\kappa ^s(AQ_{n,k}; P_t)=\frac{4n-2}{t}+1$ for $t\mid 4n-2$, $t\nmid 2n-1$, $t>6$, $n\geq 3$ and $k\geq 4$; $\kappa (AQ_{n,k}; P_t)=\kappa ^s(AQ_{n,k}; P_t)=\lceil \frac{4n-2}{t}\rceil $, in other cases.

Topological Structures on Vertex Set of Digraphs

Relation on a set is a simple mathematical model to which many real-life data can be connected. A binary relation on a set can always be represented by a digraph. Topology on a set can be generated by binary relations on the set . In this direction, the study will consider different classical categories of topological spaces whose topology is defined by the binary relations adjacency and reachability on the vertex set of a directed graph. This paper analyses some properties of these topologies and studies the properties of closure and interior of the vertex set of subgraphs of a digraph. Further, some applications of topology generated by digraphs in the study of biological systems are cited.

RDF Subgraph Query Based on Common Subgraph in Distributed Environment

With the gradual development of the network, RDF graphs have become more and more complex as the scale of data increases; how to perform more effective query for massive RDF graphs is a hot topic of continuous research. The traditional methods of graph query and graph traversal produce great redundancy of intermediate results, and processing subgraph collection queries in stand-alone mode cannot perform efficient matching when the amount of data is extremely large. Moreover, when processing subgraph collection queries, it is necessary to iterate the query graph multiple times in the query of the common subgraph, and the execution efficiency is not high. In response to the above problems, a distributed query strategy of RDF subgraph set based on composite relation tree is proposed. Firstly, a corresponding composite relationship is established for RDF subgraph set, then the composite relation graph is clipped, and the redundant nodes and edges of the composite relation graph are deleted to obtain the composite relation tree. Finally, using the composite relation tree, a MapReduce-based RDF subgraph set query method is proposed, which can use parallel in the computing environment, the distributed query batch processing is performed on the RDF subgraph set, and the query result of the RDF subgraph set is obtained by traversing the composite relation tree. The experimental results show that the algorithm proposed in this paper can improve the query efficiency of RDF subgraph set.

Combinatorial aspects of sandpile models on wheel and fan graphs

We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model’s recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with 2n vertices, the number of recurrent states with level n is given by the first differences of the central Delannoy numbers. Finally, using similar tools, we exhibit a bijection between the set of recurrent configurations of the ASM on fan graphs and the set of subgraphs of the path graph containing the right-most vertex of the path. We show that these sets are also equinumerous with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.

Towards Obtaining a 3-Decomposition From a Perfect Matching

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list.
 In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching $M$ from a cubic graph $G$ leaves a disjoint union of cycles. Contracting these cycles yields a new graph $G_M$. The graph $G$ is star-like if $G_M$ is a star for some perfect matching $M$, making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.

Star-structure connectivity of folded hypercubes and augmented cubes

The connectivity is an important parameter to evaluate the robustness of a network. As a generalization, structure connectivity and substructure connectivity of graphs were proposed. For connected graphs $G$ and $H$, the $H$-structure connectivity $\kappa(G; H)$ (resp. $H$-substructure connectivity $\kappa^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-F$ is disconnected or the singleton. As popular variants of hypercubes, the $n$-dimensional folded hypercubes $FQ_{n}$ and augmented cubes $AQ_{n}$ are attractive interconnected network prototypes for multiple processor systems. In this paper, we obtain that $\kappa(FQ_{n};K_{1,m})=\kappa^{s}(FQ_{n};K_{1,m})=\lceil\frac{n+1}{2}\rceil$ for $2\leqslant m\leqslant n-1$, $n\geqslant 7$, and $\kappa(AQ_{n};K_{1,m})=\kappa^{s}(AQ_{n};K_{1,m})=\lceil\frac{n-1}{2}\rceil$ for $4\leqslant m\leqslant \frac{3n-15}{4}$.

Incorporating multi‐interest into recommendation with graph convolution networks

In recent years, the appearance of graph convolutional networks (GCNs) provides a new idea for graph structure data processing. Because of that, they can learn excellent user and item embedding by using cooperative signals of high-order neighbors, and the GCNs technique shows great potential in the recommendation. The common problem with the bulk of GCN-based models is that it appears the situation of performance degradation during the stacking of network layers. The recently proposed IMP-GCN alleviates this problem to some extent. It aims to avoid the influence of downside information from high-order propagation on embedding learning. However, we consider that it ignores the multi-interest factor, in which users may have different interests. In this paper, we present a multi-interest GCN(MI-GCN) model for a recommendation, and it conducts high-order graph convolution operations in three sets of subgraphs. Users with similar interests and the corresponding interaction items belong to the identical subgraph. As for the formation of the subgraph, we adopt two varied clustering methods and the user feature to form a subgraph generation mechanism. This mechanism can generate three groups of differential subgraphs to divide users into multi-interest groups and make subgraph division more reasonable. We carry out massive experiments on three real-world datasets, demonstrating the effectiveness of our model. Experimental results confirm that our presented MI-GCN outperforms the state-of-the-art GCN-based recommendation models.

Differentially Describing Groups of Graphs

How does neural connectivity in autistic children differ from neural connectivity in healthy children or autistic youths? What patterns in global trade networks are shared across classes of goods, and how do these patterns change over time? Answering questions like these requires us to differentially describe groups of graphs: Given a set of graphs and a partition of these graphs into groups, discover what graphs in one group have in common, how they systematically differ from graphs in other groups, and how multiple groups of graphs are related. We refer to this task as graph group analysis, which seeks to describe similarities and differences between graph groups by means of statistically significant subgraphs. To perform graph group analysis, we introduce Gragra, which uses maximum entropy modeling to identify a non-redundant set of subgraphs with statistically significant associations to one or more graph groups. Through an extensive set of experiments on a wide range of synthetic and real-world graph groups, we confirm that Gragra works well in practice.

Guessing Numbers and Extremal Graph Theory

For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the colors of the vertices in its neighborhood. This quantity is related to problems in network coding, circuit complexity and graph entropy. We study the guessing number of graphs as a graph property in the context of classic extremal questions, and its relationship to the forbidden subgraph property. We find the extremal number with respect to the property of having guessing number $\leq a$, for fixed $a$. Furthermore, we find an upper bound on the saturation number for this property, and a method to construct further saturated graphs that lie between these two extremes. We show that, for a fixed number of colors, bounding the guessing number is equivalent to forbidding a finite set of subgraphs.

Local topological features of robust supply networks

The design of robust supply and distribution systems is one of the fundamental challenges at the interface of network science and logistics. Given the multitude of performance criteria, real-world constraints, and external influences acting upon such a system, even formulating an appropriate research question to address this topic is non-trivial. Here we present an abstraction of a supply and distribution system leading to a minimal model, which only retains stylized facts of the systemic function and, in this way, allows us to investigate the generic properties of robust supply networks. On this level of abstraction, a supply and distribution system is the strategic use of transportation to eliminate mismatches between production patterns (i.e., the amounts of goods produced at each production site of a company) and demand patterns (i.e., the amount of goods consumed at each location). When creating networks based on this paradigm and furthermore requiring the robustness of the system with respect to the loss of transportation routes (edge of the network) we see that robust networks are built from specific sets of subgraphs, while vulnerable networks display a markedly different subgraph composition. Our findings confirm a long-standing hypothesis in the field of network science, namely, that network motifs—statistically over-represented small subgraphs—are informative about the robust functioning of a network. Also, our findings offer a blueprint for enhancing the robustness of real-world supply and distribution systems.

BSTProv: Blockchain-Based Secure and Trustworthy Data Provenance Sharing

In the Big Data era, data provenance has become an important concern for enhancing the trustworthiness of key data that are rapidly generated and shared across organizations. Prevailing solutions employ authoritative centers to efficiently manage and share massive data. They are not suitable for secure and trustworthy decentralized data provenance sharing due to the inevitable dishonesty or failure of trusted centers. With the advent of the blockchain technology, embedding data provenance in immutable blocks is believed to be a promising solution. However, a provenance file, usually a directed acyclic graph, cannot be embedded in blocks as a whole because its size may exceed the limit of a block, and may include various sensitive information that can be legally accessed by different users. To this end, this paper proposed the BSTProv, a blockchain-based system for secure and trustworthy decentralized data provenance sharing. It enables secure and trustworthy provenance sharing by partitioning a large provenance graph into multiple small subgraphs and embedding the encrypted subgraphs instead of raw subgraphs or their hash values into immutable blocks of a consortium blockchain; it enables decentralized and flexible authorization by allowing each peer to define appropriate permissions for selectively sharing some sets of subgraphs to specific requesters; and it enables efficient cross-domain provenance composition and tracing by maintaining a high-level dependency structure among provenance graphs from different domains in smart contracts, and by locally storing, decrypting, and composing subgraphs obtained from the blockchain. Finally, a prototype is implemented on top of an Ethereum-based consortium blockchain and experiment results show the advantages of our approach.

The cycle-structure connectivity of crossed cubes

Crossed cube is a kind of network structure as a basis for distributed memory parallel computer architecture. Connectivity plays an important role in measuring the fault-tolerance of an interconnection network. As a generalization, Lin et al. [6] introduced the concept of structure connectivity in 2016. For connected graphs G and T, the T-structure connectivity κ(G;T) of G is the minimum cardinality of a set of subgraphs F of G that each member is isomorphic to T so that G−F is a disconnected or a trivial graph. In this paper, we study cycle-structure connectivity of n-dimensional crossed cubes CQn. We show that κ(CQn;C2k−1)=⌈nk−1⌉ for 4⩽k⩽n and κ(CQn;C2k)=⌊nk⌋+1 for even k with 4⩽k⩽n. Furthermore, we also obtain that κ(CQn;C6)=⌊n4⌋+⌈n−3⌊n4⌋2⌉ for n⩾4.

Polychromatic colorings of 1-regular and 2-regular subgraphs of complete graphs

If G is a graph and H is a set of subgraphs of G, we say that an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted by polyH(G), is the largest number of colors in an H-polychromatic coloring. In this paper we determine polyH(G) exactly when G is a complete graph on n vertices, q is a fixed nonnegative integer, and H is one of three families: the family of all matchings spanning n−q vertices, the family of all 2-regular graphs spanning at least n−q vertices, and the family of all cycles of length precisely n−q. There are connections with an extension of results on Ramsey numbers for cycles in a graph.

Structure fault tolerance of WK-recursive networks

The WK-recursive networks K(d,t) have many good properties, which are a kind of recursively constructed interconnection networks. In this paper, we will study two kinds of network connectivities, which are called H-structure-connectivity and H-substructure-connectivity respectively. These two kinds of network connectivities measure a network's reliability based on structure faults. Let F be a set of subgraphs of graph G, where the deletion of all the elements of F from G makes G disconnected or trivial. Then F is an H-structure-cut if each element in F is isomorphic to H, where H is a subgraph of G. The H-structure-connectivity of G is the minimum cardinality of all the H-structure-cuts of G, denoted by κ(G;H). If each element of F is isomorphic to a connected subgraph of H, F is an H-substructure-cut of G. The H-substructure-connectivity of G is the minimum cardinality of all H-substructure-cuts of G, denoted by κs(G;H). We study κ(K(d,t);H) and κs(K(d,t);H) of the WK-recursive networks, where H∈{K1,K1,N,C4} and N∈{1,2,3}.

Mining subgraph coverage patterns from graph transactions

Pattern mining from graph transactional data (GTD) is an active area of research with applications in the domains of bioinformatics, chemical informatics and social networks. Existing works address the problem of mining frequent subgraphs from GTD. However, the knowledge concerning the coverage aspect of a set of subgraphs is also valuable for improving the performance of several applications. In this regard, we introduce the notion of subgraph coverage patterns (SCPs). Given a GTD, a subgraph coverage pattern is a set of subgraphs subject to relative frequency, coverage and overlap constraints provided by the user. We propose the Subgraph ID-based Flat Transactional (SIFT) framework for the efficient extraction of SCPs from a given GTD. Our performance evaluation using three real datasets demonstrates that our proposed SIFT framework is indeed capable of efficiently extracting SCPs from GTD. Furthermore, we demonstrate the effectiveness of SIFT through a case study in computer-aided drug design.

Graph Analysis Using Fast Fourier Transform Applied on Grayscale Bitmap Images

There is spiking interest in graph analysis, mainly sparked by social network analysis done for various purposes. With social network graphs often achieving very large size, there is a need for capable tools to perform such an analysis. In this article, we contribute to this area by presenting an original approach to calculating various graph morphisms, designed with overall performance and scalability as the primary concern. The proposed method generates a list of candidates for further analysis by first decomposing a complex network into a set of sub-graphs, transforming sub-graphs into intermediary structures, which are then used to generate grey-scaled bitmap images, and, eventually, performing image comparison using Fast Fourier Transform. The paper discusses the proof-of-concept implementation of the method and provides experimental results achieved on sub-graphs in different sizes randomly chosen from a reference dataset. Planned future developments and key considered areas of application are also described.

The Star-Structure Connectivity and Star-Substructure Connectivity of Hypercubes and Folded Hypercubes

Abstract As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $\kappa (G; T)$ (resp. $T$-substructure connectivity $\kappa ^{s}(G; T)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube $Q_{n}$, Lin et al. showed $\kappa (Q_{n};K_{1,1})=\kappa ^{s}(Q_{n};K_{1,1})=n-1$ and $\kappa (Q_{n};K_{1,r})=\kappa ^{s}(Q_{n};K_{1,r})=\lceil \frac{n}{2}\rceil $ for $2\leq r\leq 3$ and $n\geq 3$ (Lin, C.-K., Zhang, L.-L., Fan, J.-X. and Wang, D.-J. (2016) Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107). Sabir et al. obtained that $\kappa (Q_{n};K_{1,4})=\kappa ^{s}(Q_{n};K_{1,4})= \lceil \frac{n}{2}\rceil $ for $n\geq 6$ and for $n$-dimensional folded hypercube $FQ_{n}$, $\kappa (FQ_{n};K_{1,1})=\kappa ^{s}(FQ_{n};K_{1,1})=n$, $\kappa (FQ_{n};K_{1,r})=\kappa ^{s}(FQ_{n};K_{1,r})= \lceil \frac{n+1}{2}\rceil $ with $2\leq r\leq 3$ and $n\geq 7$ (Sabir, E. and Meng, J.(2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor. Comput. Sci., 711, 44–55). They proposed an open problem of determining $K_{1,r}$-structure connectivity of $Q_n$ and $FQ_n$ for general $r$. In this paper, we obtain that for each integer $r\geq 2$, $\kappa (Q_{n};K_{1,r})$ $=\kappa ^{s}(Q_{n};K_{1,r})$ $=\lceil \frac{n}{2}\rceil $ and $\kappa (FQ_{n};K_{1,r})=\kappa ^{s}(FQ_{n};K_{1,r})= \lceil \frac{n+1}{2}\rceil $ for all integers $n$ larger than $r$ in quare scale. For $4\leq r\leq 6$, we separately confirm the above result holds for $Q_n$ in the remaining cases.

Star structure connectivities of pancake graphs and burnt pancake graphs

Let H be a connected subgraph of a graph G. The H-structure connectivity of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to H. Similarly, the H-substructure connectivity of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to a connected subgraph of H. Structure connectivity and substructure connectivity generalise the classic connectivity. Let and be the n-dimensional pancake graph and n-dimensional burnt pancake graph, respectively. In this paper we show , and .