Large-scale properties of clustered networks: implications for disease dynamics.

Abstract

We consider previously proposed procedures for generating clustered networks and investigate how these procedures lead to differences in network properties other than clustering. We interpret our findings in terms of the effect of the network structure on the disease outbreak threshold and disease dynamics. To generate null-model networks for comparison, we implement an assortativity-conserving rewiring algorithm that alters the level of clustering while causing minimal impact on other properties. We show that many theoretical network models used to generate networks with a particular property often lead to significant changes in network properties other than that of interest. For high levels of clustering, different procedures lead to networks that differ in degree heterogeneity and assortativity, and in broader scale measures such as R(0) and the distribution of shortest path lengths. Hence, care must be taken when investigating the implications of network properties for disease transmission or other dynamic process that the network supports.

Full text

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Large-scale properties of clustered networks:
Implications for disease dynamics
Darren M. Greena∗& Istvan Z. Kissb
September 21, 2009
aInstitute of Aquaculture, University of Stirling, Stirling, Stirlingshire FK9 4LA, UK.
bDepartment of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, UK.
∗Tel: +44 1786 467872; Fax: +44 1786 472133.
E-mail address: darren.green@stir.ac.uk (D.M. Green).
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Large-scale properties of clustered networks
Abstract
We consider previously proposed procedures for generating clustered networks and
investigate how these procedures lead to differences in network properties other than
clustering. We interpret our findings in terms of the effect of the network structure on
disease outbreak threshold and disease dynamics. To generate null-model networks for
comparison, we implement an assortativity-conserving rewiring algorithm that alters the
level of clustering while causing minimal impact on other properties. We show that
many theoretical network models used to generate networks with a particular property
often lead to significant changes in network properties other than that of interest. For
high levels of clustering, different procedures lead to networks that differ in degree
heterogeneity and assortativity, and in broader-scale measures such as R0and the
distribution of shortest path lengths. Hence, care must be taken when investigating the
implications of network properties for disease transmission or other dynamic process
that the network supports.
Keywords: Networks, Clustering, Epidemic Dynamics, Percolation
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Large-scale properties of clustered networks
1Introduction
Contact networks are a frequently used tool in epidemiological modelling: Each
epidemiological unit (be it a person, animal, self-contained sub-population) is considered as
a network node, with potentially infectious contact between nodes represented by
directionless edges or directed arcs. The power of the approach is that by explicitly
considering the pairwise interactions between units, one can extend the results obtained
from compartmental, mean-field, spatial, and metapopulation or household-based models.
Direction, strength, and (potentially) timing of contact can all be accounted for. In STI
models (Anderson & Garnett, 2000), contact heterogeneity and patterns of connectivity can
be accommodated in a straightforward way. They also have the benefit of being able to use
epidemiological data directly, as opposed to modelling using summary parameters (e.g.
variance in sexual partner count) abstracted from the data.
The principal parameter estimated in epidemiological modelling is that of the basic
reproduction number R0. This may be defined as
the average number of secondary infections produced when one infected
individual is introduced into a [homogeneously mixed,] wholly susceptible host
population at equilibrium (Anderson & May, 1991).
However, though for simple models such as the mean field, R0is well defined, in general no
analytic formula is available for R0. Moreover, one must consider whether the concept of a
single R0value is even appropriate in a complex population (Green et al., 2009).
Nevertheless, models of R0and final epidemic size are of utility in considering the
risk of infectious disease between populations with different structure . Various authors
have found that epidemic spread is encouraged or hindered by different network properties.
(For an overview, see Shirley & Rushton, 2005.) A node’s degree – its number of contacts k –
is of key importance, as is the distribution of contacts: networks with a higher variance of
degree enjoy a higher R0for the same between-node disease transmission rate τ (Anderson
& May, 1991). Mixing patterns of contact between nodes are also important. Under
proportionate mixing, nodes with contact rates u and v account for a fraction uv of contacts.
Deviations from this occur where mixing is preferential, for example assortative mixing in
homosexual and disassortative in heterosexual contact networks. Assortativity increases R0
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but decreases final epidemic size (Ghani et al. 1997; Gupta et al. 1989; Anderson et al.
1990; Newman, 2003a).
In this paper, the network property of most concern is that of clustering. Clustering
measures the degree to which ‘any friend of yours is a friend of mine’. In clustered
networks, if edges (a,b) and (a,c) exist, then connections (b,c) are more likely to exist than
would be expected by chance alone in random networks. This is a form of non-random
mixing associated with both assortativity and spatial structure. Clustered networks have a
lower density of nodes within two steps of a focal node compared with random networks,
limiting the spread of disease and reducing R0(Keeling, 1999), since nodes infected by the
focal node are competing for further neighbouring nodes to infect.
Ideally when comparing networks, one would like to be able to vary one parameter
of interest, while keeping all other parameters constant. In this case, we are sure that any
differences in network properties are due to that parameter. In practice, this proves difficult.
To explore the dependency of epidemic dynamics upon network structure imposed by
clustering, various authors have designed different algorithms to generate clustered
networks (e.g. Watts & Strogatz, 1998; Newman, 2003b; Read & Keeling, 2003; Eames,
2007; Kiss & Green, 2008). However, these algorithms come from quite different start
points, and occasionally there are notable side-effects of clustering upon other network
properties (Kiss & Green, 2008).
In this paper, we consider a set of previously used algorithms for generating clustered
networks (Newman, 2003b; Read & Keeling, 2003; Eames, 2007) and investigate in what
ways these networks differ with otherwise similar degree and clustering coefficients. Here,
we are primarily interested in parameters of epidemiological interest, in terms of
transmission threshold for epidemic spread, potential epidemic size, and the time-course of
disease; though one must also consider the effect of network structure on the effectiveness
of control strategies (Kiss et al. 2005; Kiss et al. 2008). Some of these properties will be
disease or disease model dependent, however properties such as the distribution of shortest
path lengths or departures from proportionate mixing are related. We employ rewiring
algorithms to change the clustering coefficients of networks while maintaining other
selected network properties constant (Kiss & Green, 2008). Particularly, we wish to preserve
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the mean degree, degree distribution, and levels of assortative or disassortative mixing.
2 Method
2.1Network construction
A network is described in terms of its number of nodes N and an adjacency matrix Aij,
elements of which are 1 where an edge (i,j) exists, and zero otherwise. The number of
edges from a single node i is given by ki=?
with N = 10000 nodes, with mean node degree of either ?k? = 5 or ?k? = 10 and clustering
coefficients chosen from C = 0.2, 0.4, 0.6 or no clustering 0.0. A set of 100 networks were
generated for each parameter set. The clustering coefficient used is the ratio of triangles to
jAij. All networks were undirected, generated
triples, where triples are permutations of three nodes u,v,w with edges (u,v) and (u,w) and
triangles are those where an additional edge (v,w) exists. Other definitions of clustering
exist (Watts & Strogatz, 1998), but this measure is easy to calculate and epidemiologically
useful. A selection of different network types were then generated, either using algorithms
reported in the literature, or by rewiring of other networks. These algorithms are described
below.
Fixed degree Each node has the same number of edges k, distributed at random by
applying 50 × N rewiring operations to a lattice network, where in each rewiring
operation four unique nodes with edges (a,b) and (c,d) are rewired to give edges (a,d)
and (b,c).
Poisson Random Poisson networks were generated by assigning edges (a,b) for each pair of
nodes a < b with a single constant probability.
Iterative An iterative algorithm suggested by Eames (2007) was implemented. This
algorithm procedes by repeating two steps. In the first step, n1triples are generated by
connecting unique nodes a,b,c with edges (a,b) and (a,c). In the second step, n2
triangles are generated by selecting a node u with at least two neighbours at random,
choosing two random neighbours v,w, and forming a link (v,w). Both steps are
subject to the constraint that no node may have more than k connections, and
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