Thedegree-preserving rewiring operation preserves the degrees of all vertices, and hence
preserves the degree sequence of the original graph. Two distinct edges are chosen, and for each edge, one of the two endpoints is chosen; the endpoints of the edges are then swapped. Although there are four combinations of endpoints, in a simple graph there are only two distinct ways to rewire a pair of edges. Figure 3.11 demonstrates both possible rewirings.
50 CHAPTER 3. METHODOLOGY USING RANDOM GRAPHS
Figure 3.11: The degree-preserving rewiring operation. The centre figure represents
the original edges, the other two figures represent the two possible rewirings.
applying enough degree preserving rewiring operations. The first proof of this theorem was given by Taylor (1980). An alternative proof is given in Chapter 4.
The 1K rewiring model introduced by Gkantsidis et al. (2003) operates similarly to the 0K rewiring model. A sequence of graphs are generated by applying rewiring operations. First, two edges are chosen independently and uniformly at random, then for each edge an endpoint is chosen randomly. If it is possible to swap the two edges without violating the simple graph constraint, then the next graph is generated by executing the rewiring operation. Otherwise the next graph is the same as the previous
graph, this is called a hold operation. Once the sequence of graphs is long enough,
it converges on a uniform sample from the space of all graphs with the same degree sequence as the original graph.
The hold operation is essential to achieving uniform sampling (Artzy-Randrup and Stone, 2005). This is because the number of graphs that can be reached by a sin- gle degree-preserving rewiring operation depends on the graph in question, and thus some graphs are more likely than others to show up during a sequence of rewirings. Artzy-Randrup and Stone suggest that the sequence of graphs converges very slowly to uniform sampling when using the hold operation, and they propose a more compli- cated algorithm. However, this is based on an assumption that most attempted rewiring operations fail, which is not the case on realistic undirected complex networks.
As with 0K randomization, an important concern is how many rewiring operations are necessary that the final graph is independent of it’s starting point. As proved in
Chapter 4, up to m−1 rewiring operations may be required to transform the initial
graph into any other graph with the same degree-sequence. To sample from the entire space uniformly, at least this many rewiring operations will be necessary. Experimental
and theoretical evidence suggests that O(m) rewiring operations are sufficient (Ray
et al., 2012; Gkantsidis et al., 2003) to sample uniformly from the entire space of graphs with the desired degree distribution, and that the constant is typically 10 to 30 (Ray et al., 2012).
In order to further validate the results of Gkantsidis et al. and Ray et al., two exper- iments were performed. Each experiment begins with 200 power-law graphs generated by the Havel-Hakimi algorithm described in Section 3.2.3. These graphs are arbitrarily
3.3. RANDOMIZING MODELS 51 divided into two groups of 100. One group is rewired to have a high assortativity or clustering coefficient, using the algorithm described in Section 3.3.3. The other group is rewired to have a low assortativity or clustering coefficient. These graphs are used as starting points for 1K rewiring.
From each starting point, a single graph is produced by performing 10m, 100m, and
1000mrewiring operations (including holds). Thus, for each group of 100 graphs, there
is a sample of 10m rewired graphs, 100m rewired graphs, and 1000m rewired graphs.
The assortativity coefficients and clustering coefficients are computed on each sample and compared to the values in the original graphs. This experimental design allows one to see how many rewiring operations are required for two topologically distinct samples to converge.
The results for the assortativity coefficient are shown in Figure 3.12, and for the clustering coefficient in Figure 3.13. The values of the coefficients become indistinguish-
able after only 10mrewiring operations, and do not change thereafter. The conclusion
then is that 10m rewiring operations are sufficient to generate a uniform sample. This
is consistent with the results of Gkantsidis et al. (2003) and Ray et al. (2012).
These experiments differ in their methodology from those presented in Gkantsidis et al. (2003) and Ray et al. (2012), which use mostly or exclusively distance-based metrics, which are poor at capturing topology as discussed in Section 3.1.1. The exper- iments of Gkantsidis et al. and Ray et al. generate their samples from a single staring graph, and compute the metrics at regular intervals as the samples are generated. Both sets of experiments compare the running means of the metrics as they are computed, and when the means no longer change, it is deemed that enough operations have been performed.
This methodology has the potential problem that the generated sample may appear uniform, when in fact it only represents a relatively small neighbourhood of graphs clustered around the starting graph. By using multiple starting points, the likelihood of this scenario is lessened.
Since it is known that the 1K rewiring process generates uniform samples of random graphs, it is now possible to empirically estimate the mean assortativity and clustering
coefficients. The experiment is similar to those used in Section 3.2. A sample of
100 random graphs is generated for a range of power-law exponents from 2 to 3 in
increments of 0.2, with each graph having 1000 vertices and 4000 edges. The starting
graphs are generated by the Havel-Hakimi algorithm from Section 3.2.3, using the
largest-first/random strategy. This is repeated for 10m, 100m, and 1000m rewiring
operations, to ensure that the convergence is not affected by the power-law exponent. The means and confidence intervals for the assortativity and clustering coefficients are presented in Figures 3.14 and 3.15 respectively. Comparing with the results from
52 CHAPTER 3. METHODOLOGY USING RANDOM GRAPHS 0.00 -0.05 -0.10 -0.15 -0.20 No Rewiring x10 x100 x1000 Minmal assortativity Maximal assortativity
Assortativity Coecient in Rewired Graphs
Figure 3.12: Assortativity coefficients of random graphs, following 1K randomization. The starting graphs are generated by the Havel-Hakimi algorithm using the largest- first/random strategy, with a power-law exponent of 2.6. They are then randomly rewired to have either maximal or minimal assortativity (see Section 3.3.3). Each bar represents the mean of 100 graphs, with 95% confidence intervals. Each graph has 1000 vertices and 4000 edges.
Section 3.2 yields strong evidence that none of the generative models discussed in that section generate uniform samples of random graphs.
By comparing Figures 3.14 and 3.15 with those for the configuration model (Fig- ures 3.3 and 3.4), it can be seen that the simple graph constraint has a large impact on the topology of the graphs. The mechanisms behind the negative assortativity and low clustering were explained in Section 3.2.2. Of all the generative models, the projected configuration model is qualitatively the most similar to a 1K random graph model, but there are quantitative differences.