Figure 4

The methods Fast Network Motif Detection (FANMOD) (involves edge switching), Diaconis Monte Carlo Importance Sampling (DIA-MCIS) (involves importance sampling), and Weighted-and-Reverse-Swap (WaRSwap) were used to produce 10,000 randomized graphs for each of three examples. The resulting graphs were tabulated according to their unique numerical representation (simply shown as integers on the × axis). The percentage of each graph produced is plotted on the Y axis, and the horizontal dotted line in each plot displays the percentage representing perfect uniformity. (A). Because there are 13 possible graphs using the same in-degrees and out-degrees as the example shown, an algorithm that samples the graph space uniformly would produce each possible type of graph exactly 1/13 of the time. The values shown in red correspond to the single possible graph with no connections between the source vertex of out-degree 3 and the target vertex of in-degree 3, which is the graph instance displayed here. (B) An example with 58 possible graphs. (C) The graph instance at the left is a canonical example of an extremely uneven degree distribution [20]. In this example, all vertices are blue because they are represented here as being both sources and targets (that is, as transcription factors). Graphs with these particular in-degrees and out-degrees can have three different types of topologies, as shown here. The instance at the left is the only possible one with no hub to hub connection, and the sampling outcome for this graph is highlighted in red on the plots for each method. The other two topologies shown correspond to 90 and 10 possible outcomes, respectively. Thus, methods that consider self-loops (for example, Diaconis Monte Carlo Importance Sampling (DIA-MCIS) and Weighted-and-Reverse-Swap (WaRSwap)) have a total of 101 possible labeled outcomes, whereas those that do not consider these (for example, Fast Network Motif Detection (FANMOD)) have a total of 91 possible labeled outcomes. Graphs containing self-loops correspond to the 10 rightmost X-axis labels in the plots for DIA-MCIS and WaRSwap.