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

8 EXISTING DATA SET ANALYSIS 28From what we can observe in the Figures 14 is that there is indeed the common characteristic of apower-law degree distribution. The co-efficient of this distribution seems to be a distinct characteristic ofthese networks or categories of networks but in general, the web-graph seems to have a co-efficient of c = 3which agrees with the web-graph model suggested by Cooper et al [13–15]. However some of the socialnetworks seem to have a co-efficient c < 2 which is hard to reproduce with existing generative models.8.2.2 Degree-Based Cut ConductunceFor the following plots it is worth clarifying what is actually drawn. The main idea of the plots is conductunceφ however our main interest is the conductance of between subsets of the graph which are partitioned basedon their degree.Assuming a graph G = {V, E} we denote as G[S] the subgraph induced from a vertex set S ∪ V and allthe edges e{u, v} : u, v ∈ S. In our case we define the following set of partitions:S(a) = {u : d(u) ≥ |V | a }G a = G[S(a)]The plot presented below are a on the x-axis and the conductance φ between the sets S(a) and VS(a) using the same formula defined in Equation 4.6.Figure 15: Degree-Based Cut ConductanceIn Figure 15 we can see the conductance of various real world networks compared to manufacturednetworks. From the plot we can infer that for the areas that each real world network carries a different plotand we may consider this as an indication of how well vertices of specific degrees connect between themselves.For example, the SlashDot Zoo networks when compared to preferential attachement graphs, show a muchbetter (lower) conductance at reasonably high degrees, we do not consider extreme cases of very very highdegrees as those cases usually include one or two isolated vertices. What this means in SlashDot is that(reasonably) high degree vertices are better inter-connected than we can expect in a preferential attachmentnetwork. This may indicate additional features in the generative process of these networks which is arenot limited to preferential attachment. It is of interest to us to perform additional measurements in othergenerated networks to determine how well they compare. However another interesting observation we canmake is networks such as LiveJournal and Google have a bad conductance at reasonably high degrees