Crossing Numbers and Stress of Random Graphs

We define a variant of the crossing number for an embedding of a graphGinto ℝ3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the rectilinear space crossing numbers of pseudo-random graphs.

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Random Graphs

10.1017/cbo9780511721342 ◽

1998 ◽

Cited By ~ 37

Author(s):

V. F. Kolchin

Keyword(s):

Random Graphs

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Random graphs

Physics Subject Headings (PhySH) ◽

10.29172/f38e82e956714568a9151b4d5355f4d8 ◽

2018 ◽

Keyword(s):

Random Graphs

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Why Does Collaborative Filtering Work? Recommendation Model Validation and Selection By Analyzing Bipartite Random Graphs

SSRN Electronic Journal ◽

10.2139/ssrn.894029 ◽

2005 ◽

Cited By ~ 2

Author(s):

Zan Huang ◽

Daniel D. Zeng

Keyword(s):

Collaborative Filtering ◽

Model Validation ◽

Random Graphs

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Applications of random graphs

10.1093/oso/9780198709893.003.0011 ◽

2017 ◽

Author(s):

A.C.C. Coolen ◽

A. Annibale ◽

E.S. Roberts

Keyword(s):

Social Networks ◽

Random Graphs ◽

Interaction Network ◽

Power Grids ◽

Protein Protein Interaction ◽

Topological Features ◽

First Case ◽

The Stability ◽

Protein Protein Interaction Network

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.

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Random graphs

10.1093/oso/9780198805090.003.0011 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Random Graph ◽

Random Graphs ◽

Large Scale ◽

Random Network ◽

Graph Model ◽

Clustering Coefficient ◽

Giant Component ◽

Random Graph Model ◽

Definition Of ◽

Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

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