Interactive multigraph visualization and exploration with a two-phase strategy

While it is very reasonable to use a multigraph consisting of multiple edges between vertices to represent various relationships, the multigraph has not drawn much attention in research. To visualize such a multigraph, a clear layout representing a global structure is of great importance, and interactive visual analysis which allows the multiple edges to be adjusted in appropriate ways for detailed presentation is also essential. A novel interactive two-phase approach to visualizing and exploring multigraph is proposed. The approach consists of two phases： the first phase improves the previous popular works on force-directed methods to produce a brief drawing for the aggregation graph of the input multigraph, while the second phase proposes two interactive strategies, the magnifier model and the thematic-oriented subgraph model. The former highlights the internal details of an aggregation edge which is selected interactively by user, and draws the details in a magnifying view by cubic Bezier curves; the latter highlights only the thematic subgraph consisting of the selected multiple edges that the user concerns. The efficiency of the proposed approach is demonstrated with a real-world multigraph dataset and how it is used effectively is discussed for various potential applications.

THE MAXIMUM AND MINIMUM DEGREES OF RANDOM BIPARTITE MULTIGRAPHS

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_（ai）,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_（ai）,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_（c,d,A）, the number of vertices in A with degree between c and d of G_（n,m）∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_（n,m）∈ζ（n,m;{pk}） has maximum degree D（n）in A？ under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A？

On K_(1,k)-factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.

{Ck, Pk, Sk} -Decompositions of Balanced Complete Bipartite Multigraphs

Let be a family of subgraphs of a graph G. An L-decomposition of G is an edge-disjoint decomposition of G into positive integer copies of Hi, where . Let Ck, Pk and Sk denote a cycle, a path and a star with k edges, respectively. For an integer , we prove that a balanced complete bipartite multigraph has a -decomposition if and only if k is even, and .

LOWER BOUNDS ON THE MAXIMUM GENUS OF LOOPLESS MULTIGRAPHS

The lower bounds on the maximum genus of loopless graphs are obtained according to the connectivity of these graphs. This not only answers a question of Chen, Archdeacon and Gross, but also generalizes the previous known results. Thus, a picture of the lower bounds on the maximum genus of loopless multigraphs is presented.

Star-factorization of the Complete Bipartite Multigraphs

LetλK_(m,n)be a complete bipartite multigraph with two partite sets having m and n vertices,respectively.A K_(p,q)-factorization ofλK_(m,n)is a set of K_(p,q)-factors ofλK_(m,n)which partition the set of edges ofλK_(m,n).Whenλ=1,Martin,in[Complete bipartite factorizations by complete bipartite graphs,Discrete Math.,167/168(1997),461–480],gave simple necessary conditions for such a factorization to exist,and conjectured those conditions are always sufficient.In this paper,we will study the K_(p,q)-factorization ofλK_(m,n)for p=1,to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.

P4κ-1-factorization of bipartite multigraphs

LetλKm,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A Pv-factorization of λKm,n is a set of edge-disjoint Pv-factors of λKm,n which partition the set of edges of λKm,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pv-factorization of λKm,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v= 3. In this paper we will show that the conjecture is true when v= 4k- 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of λKm,n is (1) (2κ - 1)m ≤ 2kn, (2) (2k - 1)n ≤ 2km, (3) m + n ≡0 (mod 4κ - 1), (4) λ(4κ - 1)mn/[2(2κ - 1)(m + n)] is an integer.

The spectrum of path factorization of bipartite multigraphs

LetλKm,nbe a bipartite multigraph with two partite sets having m and n vertices, respectively.A Pv-factorization ofλKm,nis a set of edge-disjoint Pv-factors ofλKm,nwhich partition the set of edges ofλKm,n.When v is an even number,Ushio,Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pv-factorization ofλKm,n.When v is an odd number,we have proposed a conjecture.Very recently,we have proved that the conjecture is true when v=4k-1.In this paper we shall show that the conjecture is true when v = 4k + 1,and then the conjecture is true.That is,we will prove that the necessary and sufficient conditions for the existence of a P4k+1-factorization ofλKm,nare（1）2km≤（2k+1）n,（2）2kn≤（2k+1）m,（3）m+n≡0（mod 4k+1）,（4）λ（4k+1）mn/[4k（m+n）]is an integer.

The Degree Distribution of the Random Multigraphs

In this paper, as a generalization of the binomial random graph model, we define the model of multigraphs as follows： let G（n; {pk}） be the probability space of all the labelled loopless multigraphs with vertex set V = {v1, v2, ..., vn }, in which the distribution of tvi,vj, the number of the edges between any two vertices vi and vj is P{tvi,vj =k}=Pk, k=0, 1,2,...and they are independent of each other. Denote by Xd = Xd（G）,Yd = Yd（G）, Zd = Zd（G） and Zcd = Zcd（G） the number of vertices of G with degree d, at least d, at most d and between c and d. In this paper, we discuss the distribution of Xd, Yd, Zd and Zcd in the probability space G（n; （Pk））.

Multigraphic Degree Sequences and Hamiltonian-connected Line Graphs

Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-trail.The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected.In this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥3.Applying this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.

Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

Graph theory has a significant impact and is crucial in the structure of many real-life situations.To simulate uncertainty and ambiguity,many extensions of graph theoretical notions were created.Planar graphs play a vital role in modelling which has the property of non-crossing edges.Although crossing edges benefit,they have some drawbacks,which paved the way for the introduction of planar graphs.The overall purpose of the study is to contribute to the conceptual development of the Pythagorean Neutrosophic graph.The basic methodology of our research is the incorporation of the analogous concepts of planar graphs in the Pythagorean Neutrosophic graphs.The significant finding of our research is the introduction of Pythagorean Neutrosophic Planar graphs,a conceptual blending of Pythagorean Neutro-sophic and Planar graphs.The idea of Pythagorean Neutrosophic multigraphs and dual graphs are also introduced to deal with the ambiguous situations.This paper investigates the Pythagorean Neutrosophic planar values,which form the edges of the Pythagorean neutrosophic graphs.The concept of Pythagorean Neutrosophic dual graphs,isomorphism,co-weak and weak isomorphism have also been explored for Pythagorean Neutrosophic planar graphs.A decision-making algorithm was proposed with a numerical illustra-tion by using the Pythagorean Neutrosophic fuzzy graph.

多图融合约束半非负矩阵分解的动作分割方法

基于聚类的无监督动作分割方法主要利用序列中相邻帧之间的结构相似性来提高动作分割的准确性。这在实现动作片段内部一致划分的同时给不同动作边界的准确分割带来隐患。为此提出了一种基于多图融合约束矩阵分解的动作分割方法。通过融合序列中的结构相似性和度量相似性信息构造多图融合约束项,融入到半非负矩阵分解中获得序列的低维表示,进而获得序列的k近邻图并利用图割的方法实现准确分割。在两类动作序列上的实验表明,所提方法在保持动作内部一致划分的同时能够准确划分动作边界,明显提升了分割准确性,时间效率也明显提升。

组织结构的立体多核网络模型

对立体多核网络模型进行了新的阐释,将原来的概念模型向操作层次推进了一步,该模型是将组织作为一个由相互作用的元素构成的、动态的复杂系统来考虑的,它高度概括了组织结构的包括关系多样性、要素多样性、动态特性和层次交错性等方面在内的复杂性和网络中节点的异质性和差异性,是为研究复杂组织而进行仿真设计的前期建模工作.从要素的多样性和关系的多样性出发,初步勾画出立体多核网络的图论描述框架,对以后的仿真设计和计算机编程提出了粗略的设想.最后用一个学术群体的网络研究案例对立体多核网络模型在组织行为宏观处理方面的优点进行了示例性的说明.

多导睡眠图特征提取研究

多导睡眠图是睡眠过程监护和睡眠生理研究的重要工具。本文在6例整夜睡眠监护实验的基础上提出了利用脑电复杂性及多导睡眠图功率谱特征的方法客观反映睡眠的进程,为睡眠的自动分阶分析及睡眠质量的评估研究打下了基础。尤其是脑电复杂性测度与睡眠深度的关系研究对于睡眠生理的研究具有理论意义。

半传递重图的限制性边连通度（英文）

设G=(V,E)是一个重图(包含重边,但不含环).图G的边连通度,记为λ(G),是G的最小边割的基数.我们称G是极大边连通的如果λ(G)=δ(G);称图G是超边连通的如果每个最小边割都是某个点的邻边集合.图G的限制性边连通度,记为λ(G),是图G的最小限制性边割的基数.如果λ(G)达到限制性边连通度的上界,我们称G是λ-最优的.一个二部重图是半传递的如果它作用在每个部分上都是传递的.在本文中,我们将刻画极大边连通的、超边连通的、λ-最优的半传递重图.

基于BP网络的睡眠分阶方法及睡眠质量评估研究

我们利用不同睡眠期脑电复杂性特征与睡眠深度的关系及多道睡眠图功率谱特征,基于3层BP网络进行了睡眠自动分阶的研究,并提出了能部分反映睡眠质量的睡眠时间、浅睡时间、深睡时间、REM时间、觉睡比、醒转次数等参数。通过6例全睡眠监护实验说明,该方法可为睡眠质量的评价提供途径。

粗糙集属性约简的图论方法

通过研究粗糙集与图论的关系,提出了以集合为权的加权多重完全多部图的概念,定义了加权多重完全多部图的邻接矩阵,得到了加权完全多部图与决策表的映射关系;给出了粗糙集决策表信息系统的图论形式和决策表信息系统属性约简的图论方法,并根据图论理论对算法进行了优化;得到了在决策表信息系统中,属性的集合不可以约简的充分必要条件;并进一步提出了基于属性置信度的计算方法和多决策属性的处理方法。编程实验结果证明该方法能有效地降低时间和空间复杂度。

连续属性决策表离散化的图论方法

通过研究粗糙集与图论的关系,提出了以集合为权的加权多重完全多部图的概念,得到了加权完全多部图与连续属性决策表的映射关系,给出了断点重要性和断点效率的一种新的量化定义并得到了相关性质;提出了连续属性决策表信息系统的图论形式和连续属性决策表离散化的图论方法。编程实验结果证明,应用此方法可以确保在离散化后决策表相容的前提下得到无剩余属性值的较小的断点集合。

基于Turán数的广义多图Ramsey数上下界(英文)

将多图Ramsey数推广为广义多图Ramsey数.利用完全图的Turán数,给出一些多图Ramsey数的上界和构造性下界,进而确定出它们的准确值.

睡眠自动分阶方法及睡眠质量评估的研究

目的 :进行睡眠自动分阶的研究 ,并提出了能部分反映睡眠质量的参数。方法 :本文利用不同睡眠期脑电复杂性特征与睡眠深度的关系及多道睡眠图功率谱特征 ,基于 3层BP网络 ,通过 6例全睡眠监护实验 ,并对 2例全睡眠实验的参数进行比较研究。结果 :该方法可进行自动全睡眠分阶分析。结论