The Structure of Geographical Threshold Graphs

We analyze the structure of random graphs generated by the geographical threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. We show how the degree distribution, percolation and connectivity transitions, clustering coefficient, and diameter relate to the threshold value and weight distribution. We give bounds on the threshold value guaranteeing the presence or absence of a giant component, connectivity and disconnectivity of the graph, and small diameter. Finally, we consider the clustering coefficient for nodes with a given degree l, finding that its scaling is very close to 1/l when the node weights are exponentially distributed.

[1]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Margherita Pagani,et al.  Second Edition , 2004 .

[3]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.

[4]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[5]  Jon M. Kleinberg,et al.  The Web as a Graph: Measurements, Models, and Methods , 1999, COCOON.

[6]  N. Mahadev,et al.  Threshold graphs and related topics , 1995 .

[7]  Verzekeren Naar Sparen,et al.  Cambridge , 1969, Humphrey Burton: In My Own Time.

[8]  Anthony Bonato,et al.  A Survey of Models of the Web Graph , 2004, CAAN.

[9]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[10]  Panos M. Pardalos,et al.  Handbook of Massive Data Sets , 2002, Massive Computing.

[11]  N. Konno,et al.  Geographical threshold graphs with small-world and scale-free properties. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Fan Chung Graham,et al.  The Volume of the Giant Component of a Random Graph with Given Expected Degrees , 2006, SIAM J. Discret. Math..

[13]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[14]  Béla Bollobás,et al.  The degree sequence of a scale‐free random graph process , 2001, Random Struct. Algorithms.

[15]  Alan M. Frieze,et al.  A General Model of Undirected Web Graphs , 2001, ESA.

[16]  Eli Upfal,et al.  Stochastic models for the Web graph , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[17]  Aric A. Hagberg,et al.  Giant Component and Connectivity in Geographical Threshold Graphs , 2007, WAW.

[18]  A. Hagberg,et al.  Designing threshold networks with given structural and dynamical properties. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Edmund M. Yeh,et al.  Analytical Lower Bounds on the Critical Density in Continuum Percolation , 2007, 2007 5th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks and Workshops.

[20]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[21]  B. Bollobás.  The evolution of random graphs , 1984 .

[22]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[23]  Béla Bollobás,et al.  Random Graphs , 1985 .

[24]  R. Durrett.  Random Graph Dynamics: References , 2006 .

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

[26]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[27]  P. R. Kumar,et al.  Critical power for asymptotic connectivity , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[28]  J. S. Kong,et al.  Wireless Ad Hoc Networks with Tunable Topology , 2007 .