On the Vertex-Connectivity of an Uncertain Random Graph

In many practical problems, randomness and uncertainty simultaneously appear in one complex system or network. When graph theory is applied to these problems, these complex systems or networks are usually represented by uncertain random graphs, in which some edges exist with degrees in probability measure, and some other edges exist with degrees in uncertain measure. In this paper, we focus on the connectivity of uncertain random graphs with respect to vertices. We propose the concepts of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-vertex-connectivity index and the connectivity of an uncertain random graph. The former is the chance measure that an uncertain random graph is <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-vertex-connected, and the latter is an uncertain random variable, which characterizes the connectivity of the uncertain random graph with respect to vertices. We discuss some properties of these concepts. Methods and formulas are also presented for calculating the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-vertex-connectivity index, and the distribution and expected value of the connectivity of an uncertain random graph.

[1]  Hua Ke,et al.  Uncertain random multilevel programming with application to production control problem , 2015, Soft Comput..

[2]  Hui Li,et al.  ON THE MATCHING NUMBER OF AN UNCERTAIN GRAPH , 2017 .

[3]  Yuan Gao,et al.  Connectedness Index of uncertain Graph , 2013, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[4]  Yuhan Liu,et al.  Risk Index in Uncertain Random Risk Analysis , 2014, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[5]  Zhongfeng Qin,et al.  Uncertain random goal programming , 2018, Fuzzy Optim. Decis. Mak..

[6]  Denes König,et al.  Theory of finite and infinite graphs , 1990 .

[7]  Fan Yang,et al.  An Inverse Shortest Path Problem on an Uncertain Graph , 2014, J. Networks.

[8]  Baoding Liu.  Uncertain Random Graph and Uncertain Random Network Baoding , 2014 .

[9]  Baoding Liu,et al.  Uncertainty Theory - A Branch of Mathematics for Modeling Human Uncertainty , 2011, Studies in Computational Intelligence.

[10]  S. L. Wong,et al.  Towards a proteome-scale map of the human protein–protein interaction network , 2005, Nature.

[11]  Yuan Gao,et al.  On Computing the Edge-Connectivity of an Uncertain Graph , 2016, IEEE Transactions on Fuzzy Systems.

[12]  Lin Chen,et al.  Cycle index of uncertain random graph , 2018, J. Intell. Fuzzy Syst..

[13]  Yuhan Liu,et al.  Uncertain random variables: a mixture of uncertainty and randomness , 2013, Soft Comput..

[14]  T. S. N. Asih,et al.  Uncertain Programming Models for Assignment Problem with Restriction of Qualification 2 Preliminaries 2 . 1 Uncertainty Theory , 2019 .

[15]  Yongchao Hou,et al.  DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES , 2014 .

[16]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[17]  Yuhan Liu,et al.  Uncertain random programming with applications , 2013, Fuzzy Optim. Decis. Mak..

[18]  Lin Chen,et al.  An uncertain chromatic number of an uncertain graph based on $$\alpha $$α-cut coloring , 2018, Fuzzy Optim. Decis. Mak..

[19]  Lixing Yang,et al.  On distribution function of the diameter in uncertain graph , 2015, Inf. Sci..

[20]  Danah Boyd,et al.  Social Network Sites: Definition, History, and Scholarship , 2007, J. Comput. Mediat. Commun..

[21]  Xiulian Gao,et al.  Tree index of uncertain graphs , 2015, Soft Computing.

[22]  Bo Zhang,et al.  Euler index of uncertain random graph: concepts and properties , 2017, Int. J. Comput. Math..

[23]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[24]  Rui Kang,et al.  Reliability analysis in uncertain random system , 2016, Fuzzy Optimization and Decision Making.

[25]  Bo Zhang,et al.  Connectedness strength of two vertices in an uncertain graph , 2013, Int. J. Comput. Math..

[26]  Bo Zhang,et al.  Euler index in uncertain graph , 2012, Appl. Math. Comput..

[27]  Xiulian Gao,et al.  Regularity index of uncertain graph , 2014, J. Intell. Fuzzy Syst..