A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
 Frank Ball,
 Tom Britton,
 David Sirl
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Abstract
A random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is defined. The model is an extension of the configuration model, in which stubs (halfedges) are paired to form a network. Clustering is obtained by forming small completely connected subgroups, and positive (negative) degree correlation is obtained by connecting a fraction of the stubs with stubs of similar (dissimilar) degree. An SIR (Susceptible $\rightarrow $ Infective $\rightarrow $ Recovered) epidemic model is defined on this network. Asymptotic properties of both the network and the epidemic, as the population size tends to infinity, are derived: the degree distribution, degree correlation and clustering coefficient, as well as a reproduction number $R_*$ , the probability of a major outbreak and the relative size of such an outbreak. The theory is illustrated by Monte Carlo simulations and numerical examples. The main findings are that (1) clustering tends to decrease the spread of disease, (2) the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and (3) disease spread broadly increases with degree correlation $\rho $ when $R_*$ is just above its threshold value of one and decreases with $\rho $ when $R_*$ is well above one.
Inside
Within this Article
 Introduction
 The network model and the epidemic
 Properties of the network model
 Epidemics on network without rewiring
 Epidemics on rewired networks
 Numerical examples
 Discussion
 References
 References
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 Title
 A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
 Journal

Journal of Mathematical Biology
Volume 66, Issue 45 , pp 9791019
 Cover Date
 20130301
 DOI
 10.1007/s0028501206097
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Branching process
 Configuration model
 Epidemic size
 Random graph
 SIR epidemic
 Threshold behaviour
 92D30
 05C80
 60J80
 Industry Sectors
 Authors

 Frank Ball ^{(1)}
 Tom Britton ^{(2)}
 David Sirl ^{(3)}
 Author Affiliations

 1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
 2. Department of Mathematics, Stockholm University, Stockholm, 106 91, Sweden
 3. Mathematics Education Centre, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK