The social circuit model

While Wasserman and Pattison (1996) popularized the ERGM for social network modeling using the Markov dependence assumption, the resulting ERGM model specications often lead to unreasonable parameter estimates of θ. The Markov model is plagued by a phenomenon called model degeneracy as discussed in Hand- cock (2003a) and Handcock (2003b). It causes non-convergence of the MCMC-ML scheme used to estimate ERGM parameters, see section 2.5.4. A degenerate model will provide useless parameter estimates ˆθ and nonsensical network simulations gi- ven those parameter estimates. Simulating data from p(y|ˆθ) the resulting networks tend to be mixtures of empty graphs with not a single tie present and full graphs with all possible ties present. Using network simulations model degeneracy will be illustrated in section 2.5.4.

What rst was considered an algorithmic problem in optimizing the likelihood function was found to be a problem of the network statistics used in the Markov mo-

Figure 2.5: A directed 4-cycle (right panel) may be represented as EP2(y)or as DP2(y)

(left panel), depending on whether the dashed tie exists.

del, see Snijders et al. (2006) and Schweinberger (2011). The Markov assumption is still too restrictive and unrealistic to capture tie variable dependence of observed social networks. To ameliorate this problem Snijders et al. (2006) develop a new ERGM specication to capture complex patterns of transitivity. This development stage of the ERGM is called the social circuit model. New parameters capturing network transitivity are introduced which require a less restrictive assumption of network dependency. The assumption of the Markov graph by Frank and Strauss (1986) is replaced with the partial conditional dependence assumption introduced by Pattison and Robins (2002): two dyads are dependent if they share a node or if they are part of a 4-cycle. Imagine the situation of married couples: if the two husbands are friends, the wives are also more likely to know each other. Clustering and transitivity may not only occur in the form of triangular subgraphs like T (y) but also in 4-cycles with potential diagonal tie variables. Note that a 4-cycle may also be represented by a shared partner statistic like the 2-triangle EP2(y) or the

2-2-path DP2(y), see gure 2.5. The partial conditional dependence assumption

is more realistic than the Markov assumption but also requires more complex net- work statistics based on k-triangles and k-2-paths. Snijders et al. (2006) develop statistics such as the alternating k-triangle and the alternating k-2-path:5 _{a sin-}

gle statistic represents the whole distribution of possible k-triangles and k-2-paths with k ranging from 1 to n − 2 shared partners. The sign of subsequent k-statistics is toggled, so if k − 1 has a positive sign, k will be negative, k + 1 again positive

5_{Snijders et al. (2006) also propose a new specication for k-stars capturing network}

hierarchy which do not require the partial conditional dependence assumption. In this work the discussion is restricted to network statistics capturing transitivity.

and so on. This helps to prevent the avalanche eect causing model degeneracy, see section 2.5.4.

Hunter (2007) propose a slight modication of the statistics introduced by Sni- jders et al. (2006) using geometrically weighted EPk(y) and DPk(y) statistics. A

dampening parameter regulates the inuence of higher degrees of shared partners to the respective change statistics. The most important conguration for mo- deling transitivity in the social circuit model is the geometrically weighted edge-wise shared partner statistic (GWESP): the distribution of EPk with (k = 1, . . . , n − 2)

is represented using a geometric series resulting in the GW ESP (y) statistic

GW ESP (y) = eαE n−2 X k=1  1 − 1 − e−αE
k  EPk(y). (2.31)

EPk(y)is the number of transitive k-triangles in the network, see section 2.2. The

geometric series

1 − 1 − e−αE
k

helps to prevent the avalanche eect of model degeneracy described in Snijders et al. (2006), see also section 2.5.4. The tuning parameter αE controls the weight

of the contribution of higher degrees of shared partners to the change statistics used in the log linear ERGM formulation. Goodreau et al. (2009) give a detailed illustration of how αE aects the change statistics. Their work is also a rich source

on patterns of transitivity and how they can be modeled using the GW ESP (y) statistic. Tuning of αE may be used to focus on smaller or larger clusters of nodes,

e.g. αE = 0.1 puts most weight on small clusters with k ≤ 2, whereas αE = 1.5

puts substantial weight on larger cluster with k ≥ 10. This is important for two reasons: First, it is possible to tune (2.31) for a good representation of the observed network as αE = 10would not make any sense in a tiny network of only ve nodes.

Second, tuning of αE can be used to prevent model degeneracy. It may happen

that a particular value of αE is plausible for the observed network which might

show substantial transitivity, but the parameter estimate ˆθ lead to a degenerate distribution S(y|ˆθ, αE) of the relevant network statistics. While reducing αE will

reduce the potential of the ERGM to completely explain the observed transitivity, this reduction might prevent model degeneracy. A more limited model might be better than a model which cannot be estimated.

weighted dyad-wise shared parter (GWDSP) statistic GW DSP (y) = eαD n−2 X k=1  1 − 1 − e−αD
k_DP k(y). (2.32)

Using DPk(y) this results in a dierent interpretation than (2.31): (2.32) repre-

sents structural holes and the behavior to skip other nodes in the network. This behaviour is the opposite of transitive triad closure so GW DSP (y) is some kind of counterpart to GW DSP (y). Again, the tuning parameter αD is used to focus on

a certain range of shared partners. The same geometric series as in (2.31) is used to prevent model degeneracy. (2.31) and (2.32) may be used together in a social circuit ERGM where αE and αD do not have to be equal. Typically, the parame-

ter estimate for GW ESP (y) is positive representing the transitive pattern of `a friend of a friend is a friend'. The GW DSP (y) parameter is typically negative if a single dense cluster is observed and may be positive if separate clustered cliques of nodes exist. A positive parameter for GW DSP (y) and a negative parameter for GW ESP (y)is untypical for friendship networks.

Morris et al. (2008) give an overview of possible network statistics in s(y) which may also include geometrically weighted degree distributions (or geome- trically weighted stars) and a variety of networks statistics involving exogenous covariates x. Hunter et al. (2013) show how almost any sucient ERGM statistic may be constructed and implemented in the ergm suite (Hunter et al., 2008a).