A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
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 (half-edges) 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 -> Infective -> 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 clustering tends to decrease the spread of disease, the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and 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.
💡 Research Summary
The paper introduces a flexible random‑graph model that simultaneously controls three fundamental network characteristics: degree distribution, clustering, and degree correlation. Building on the classic configuration model, the authors first assign each vertex a prescribed number of half‑edges (stubs) according to a target degree distribution. Then, they perform two distinct pairing procedures. A fraction of stubs is grouped into small, fully connected sub‑graphs (cliques) of size (k_c); the proportion of such cliques, (p_c), determines the global clustering coefficient (C). The remaining stubs are matched according to a degree‑similarity rule: stubs are partitioned into degree classes, and a tunable parameter (\alpha) governs the probability of connecting stubs within the same class (producing positive degree correlation (\rho>0)), while a complementary parameter (\beta) controls cross‑class connections (producing negative correlation (\rho<0)). By adjusting (p_c), (k_c), (\alpha) and (\beta), the model can generate networks with any desired combination of (C), (\rho) and a given degree distribution.
The authors then embed a standard Susceptible–Infective–Recovered (SIR) epidemic on this network. An infective individual transmits the disease to each neighbor independently with probability (\beta) during an infectious period of mean length (\tau). In the limit of infinite population size, the early spread can be approximated by a multitype branching process whose offspring distribution is the “effective degree” (D_{\text{eff}}). This effective degree accounts for the fact that edges inside a clique may lead to redundant transmissions, thereby reducing the number of truly new infections. The basic reproduction number for the network is derived as
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