Mixing Patterns - Mixing By Scalar or Topological Characteristics

Mixing By Scalar or Topological Characteristics

Scalar characteristics of a node are those that are quantitative. They may be continuous or discrete ordinal variables like counts. Age is perhaps the simplest example, though intelligence and raw income are other obvious possibilities. Some topological features of the network may also be used for examining mixing by scalar properties. Specifically, the degree of a node is often a highly important feature in the mixing patterns of networks. Topological scalar features are very useful, because unlike other measures, they are always available. They are sometimes used as a proxy for real-world "sociability".

For measuring the assortativity of scalar variables, similar to the discrete case (see above) an assortativity coefficient can be defined. One can measure it using the standard Pearson Correlation, as Newman demonstrates. In Fig. 2, for instance, a calculation of the Pearson Correlation Coefficient yields r = 0.574. This indicates a fairly strong association between the age of husbands and wives at the time of marriage.

An alternative coefficient can be computed for measuring the mixing by the degree of the nodes. Newman derives the expression, which is found to be

for an undirected network. In this formula, if refers to the graph's degree distribution (i.e., the probability that a node has degree k) then . This refers to the excess degree of a node, or the number of other edges aside from the currently-examined one. The z refers to the average degree in the network, and is the standard deviation of the distribution . For a directed network the equivalent expression is .

This correlation is positive when nodes are assortative by degree, and negative when the network is disassortative. Thus, the measure captures an overall sense of the mixing patterns of a network. For a more in-depth analysis of this topic, see the article on assortativity.

The method of generating functions is still applicable for this case too, but the functions to be calculated are rarely calculable in closed form. Thus, numerical simulations seem to be the only way to yield results of some interest. The technique used is once again the Monte Carlo one. For the case of networks with a power-law degree-distribution, has a divergent mean, unless, which rarely happens so. Instead, the exponentially truncted power-law distribution yields a distribution for the excess degree of the type . The results for this case are summarized below.

1) The position of the phase transition at which a giant cluster appears moves to higher values of as the value of decreases. That is, the more assortative a network is, the lower the edge density threshold for the giant cluster's appearance will be.

2) The size of the giant cluster in the limit of large is smaller for the assortatively mixed graph, than for the neutral and disassortative ones.

3) Assortative mixing in the network affects the network's robustness under node removal. For assortative networks, it is required to remove about ten times more than usual (usual means a neutral network) high-degree nodes to destroy the giant cluster, while the opposite is true for disassortative networks, i.e. they are more susceptible than neutral ones under removal of the high-degree nodes.

The fascinating result on the dependence of the network's robustness to its node mixing may be explained as follows. According to their definition, high-degree nodes in assortative networks tend to form a core group among them. Such a core group provides robustness to the network by concentrating all the obvious target nodes together in one portion of the graph. Removing these high-degree nodes is still one of the most effective ways to destroy network connectivity, but it is less effective (compared to neutral networks) because by removing them all from the same portion of the graph we fail to attack other portions. If these other portions are themselves percolating, then a giant cluster will persist even as the highest degree nodes vanish. On the other hand, the disassortatively mixed network is particularly susceptible to removal of high-degree nodes because these nodes are strewn far apart across the network, so that attacking them is like attacking all parts of the network at once.