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

4Part IIRelated Work4 Network properties and metricsOur particular area of research is relatively new. This is mainly due to the fact that it is only recentlythat these network structures have emerged. It started with the emergence of the WWW, however work ongraph generation models greatly predates the rise of the WWW and was previously done to model socialnetworks in the domain of social sciences.The WWW graph in particular has received a great deal of attention. This graph is the result of modelingthe WWW as a set of pages which are the vertices of the graph, and a set of directed links between the pageswhich are the edges of the graph [32]. There are some very interesting properties that have been discoveredin this graph, for example the degree distribution observed follows a long-tail distribution [11, 49]. This isapparently a common attribute that is present in the WWW graph, as well as the Internet (autonomoussystems) [24], citation graphs [51], OSNs [24] and many others.4.1 Diameter and degree distributionThe diameter of the aforementioned networks seems to be relatively small and a “small world” phenomenonhas been observed. This means that the diameter d, or effective diameter [52] is of the scale of d ∝ log Nwhere N is the size of the graph. However according to a study by J. Leskovec et al [31] it was proposed thatthe diameter is even smaller than this proposed value. It is unclear whether or not this observed characteristicof certain networks is related to another observed characteristic of an edge densification power-law [31] where|E(t)| ∝ |V (t)| α where E(t) and V (t) are the sets of edges and vertices at the epoch t respectively and α isnot trivial (α > 1).The graphs with the above properties are commonly referred to as power-law graphs or scale-free graphs,however the latter term is controversial. For simplicity’s sake we will use the term power-law graphs andscale-free graphs interchangeably to refer to all graphs which have power-law degree distributions, a diameterless than or equal to the expected diameter of a small-world network. Additionally we will require our scalefreegraphs to have a scale invariance where these basic properties remain true even at very different sizesof the graph. According to the work of Dill et al [19] the web graph presents with a self-similar structurewhich may be the cause of the aforementioned scale invariance of the web-graph, it may be reasonable toassume that many other scale-free graphs observed such as OSN graphs share this characteristic with theweb-graph. We believe that this is an interesting research objective for our current work.4.2 Degree correlationsThe term “degree correlations” is used to denote the relationship between the degree of a vertex and thedegree of its neighbors. In the work of Pastor-Satorras et al [48] it has been suggested that there are greaterthan expected degree correlations in real world networks. The metric defined was based on the conditionalprobability P c (k ′ /k) which denotes the probability of a node of degree k is connected to a node of degree k ′ .As it is stated by Pastor, due to the statistical fluctuations, measuring this probability is a rather complextask. He suggested the following alternative:〈k nn 〉 = ∑ k ′ P c (k ′ /k) (4.1)This denotes an average value of this metric.4.3 Expander graphsIn general when we talk about the expansion property of a graph, or if we describe a graph as an expanderwe may mean several different properties may hold for a graph. Expander graphs were first defined by