Upgrade Report - Department of Informatics - King's College London

7 GRAPH GENERATION 24Figure 10: Implicit Graph Model. α = 1.25,N = 10000,c = 1.8We can observe that the out-degree distribution follows a power-law, as expected by definition. Thein-degree distribution approaches a log distribution but has some anomalies which may be a direct effect ofthe way we determine the target vertices of the model’s edges. The modularity Q = 0.6 which is significantlyhigher than expected however we currently consider this to be an artifact of the edge endpoint selectionin conjunction with the fact that there are several disconnected components within the graph. It is worthnoting however that the graph mainly consists of a single large component which in this case contains over99% of the vertices of the graph while the second largest component consists of only 8 vertices.7.5 Partitioned Preferential AttachmentThis model is essentially a combination of the well known preferential attachment model with an additionalelement of periodic partitioning of the graph. This partitioning may happen at random or fixed intervals andit intuitively simulates the cell reproduction procedure. When the graph being generated reaches a “criticalmass” s then we split it in two parts. The split is performed at random. We treat each partition as a newgraph and continue to add vertices to random partitions and attaching edges to each vertex preferentially.The preferential selection is based on the degrees of the vertices in the current partition. During the splitthe edges which have endpoints belonging to different components are maintained, which maintains theconnected nature of the graph.The resulting degree distribution from this generative process is seen in Figure 11