Graph Visualisation Analysis

The Graph Visualisation Analysis uses graph-drawing techniques to visualise network diagrams consisting of their vertices and edges linking these vertices in two-dimensional Euclidean spaces. Graph Visualisation Analysis is closely related to Graph and Network Theory. Moreover, there is a great research interest in visualising the structure of the Internet. The La Jolla, CA – based Center for Applied Internet Data Analysis has engaged in Internet visualisation efforts since the year 2000 (CAIDA, 2015). A great number of possible network graph visualisations are covered as the ‘Internet Topology Zoo’ at the University of Adelaide (2016). A common framework for visualising and analysing the structure of the Internet graphs are the different granularities that can be adopted, as stated in section 2.3.4 above. Given the visualisation efforts, it is important to have a clear understanding of the graph visualisations that are worth analysing to gain a structural understanding. We consider the Graph Visualisation Analysis as an important step in further exploring our findings from the Descriptive and Complex Network Analysis in Chapter 4. Visualising the mobile broadband operator network graphs might yield additional insights for understanding key structural properties. The following section below discusses the reasoning for choosing certain graph visualisation and simulation algorithms and our choice of distinctive visualisation layouts over others.

Graph Visualisation Algorithms

Kleinberg Small-World Network Model

A Small-World Network model refers to a mathematical representation where most vertices in a given network graph are not neighbours of one another but instead, the neighbours of given network vertices are likely to be neighbours of each other so that most network vertices are reachable from every other vertex given a small number of hops (or steps). This means that relatively short paths exist between any two vertices in a given network (Watts and Strogatz, 1998, p.440). Moreover, the typical distance between randomly chosen vertices in a Small-World Network grows proportionally to the logarithm of the number of all vertices in a given network, while Clustering Coefficients in Small-World Networks are naturally, given the above explained effect, large (Watts and Strogatz, 1998, p.442).

A suitable algorithm to study the Small-World Network effect is Kleinberg’s Small-World

Network model (Kleinberg, 2000). This model uses so-called greedy routing algorithms.

This means, in our context, that an IP address vertex in a given traceroute path could choose the next vertex it believes to be closest to the chosen destination, based on the

Small-World Network effects (Kleinberg, 2000). This effect in the Kleinberg Model is

achieved by adding long-range edges to the network, which tend to favour vertices that are closer in distance (not geo-distance), rather than farther. The Small-World Network phenomena are well visualised in using graph visualisation algorithms such as the

Layered Layout by Kuchar (2012), which visualises vertices in different layers,

Small-World Network graph visualisation

Key

Vertex without label.

Undirected edge, linking a pair of vertices.

Figure 3-10: Small-World Network graph visualisation. Barabási-Albert (BA) Scale-Free Model

Scale-Free Network models are those whose Degree distribution follows Pareto or power-law degree distributions. Albert, Jeong and Barabási (1999) find that the World

Wide Web follows such a power-law degree distribution and hence Scale-Free Network properties. The Barabási-Albert dynamic network algorithm simulates alternative scenarios of network growth emergence (Barabási and Albert, 2002). These growth features are achieved through preferential attachment, referred to as ‘rich-get-richer’ effects given the power-law degree distribution. Barabási Labs (2013) refer to three distinct Barabási-Albert Scale-Free Network models:

• Standard Model with vertex growth and preferential attachment to edges. • Model A with vertex growth and uniform attachment of edges.

• Model B without vertex growth but preferential attachment to edges.

While Onnela et al. (2007) use the Barabási-Albert Scale-Free Network models to uncover the structure and tie strength in mobile communication networks, Faloutsos,

Faloutsos and Faloutsos (1999) believe that the Internet has a power-law degree

distribution, which is criticised by a number of researchers including Willinger, Alderson

and Doyle (2009) and Willinger and Roughan (2013). Nevertheless, we believe that the Barabási-Albert Scale-Free Network models are a good simulator choice to explore scenarios of network growth emergence for traceroute-based connectivity. Hence, each of the three Barabási-Albert Scale-Free Network models were utilised to study the three mobile broadband operator network graph visualisations.

Scale-Free Network graph visualisation

Key

Vertex without label.

Undirected edge, linking a pair of vertices.

Figure 3-11: Scale-Free Network graph visualisation.

Scale-Free Network models are often visualised in using so-called force-directed

Layouts. We chose two of these Layouts given their specific properties (a more comprehensive description of these Layouts is provided in the Literature Review, see section 2.3.4). Here, the Force Atlas 2 Layout is considered suitable for visualising Scale-

Free Networks with between 10 and 10,000 vertices, which well suited to our traceroute

observations (Jacomy et al., 2014). The Force Atlas 2 Layout incorporates a force- directed algorithm, which allows to place vertices in a two-dimensional space without crossing edges too much between the pairs of vertices, capturing structural properties of

a given network. Second, the Fruchterman-Reingold (1991) graph visualisation Layout is, just like the Force Atlas 2 Layout, a force-directed layout algorithm.

k-core decomposition

The k-core decomposition algorithm helps to study hierarchical properties of large scale networks through identifying particular subsets of a given network (Alvarez-Hamelin et al., 2005a, p.22), while being usually employed in biological settings to analyse and predict protein interactions (Seidmann, 1983; Alvarez-Hamelin et al., 2005b). The algorithm divides networks into different subsets, called k-cores. Therefore, the k-core

decomposition focuses on the network regions with increasing centrality and

connectedness. More central k-cores are, therefore, inhabiting more densely interconnected network vertices as Figure 3-12 below illustrates (through k-cores 1 – 3).

k-core decomposition graph visualisation

Key

Vertex without label.

Undirected edge, linking a pair of vertices.

Figure 3-12: k-core decomposition graph visualisation.

According to Alvarez-Hamelin et al. (2005b; 2008), the k-core decomposition allows the finding of connectivity paths with specific Quality of Service (QoS), especially when studying models of the Internet at Autonomous System granularity. Hence, the k-core

address and Autonomous System vertices and hence connectedness regions of potential bottlenecks in the upstream Internet market structure, originating from the three Tamil Nadu mobile broadband operators of interest.