4 C OMPLEX N ETWORK A NALYSIS
4.2 Complex Network Analysis at IP granularity
4.2.6 Graph Visualisation Analysis (IP)
For the purpose of exploring the structural operator network properties, we first project the three networks as a graph onto a two-dimensional Euclidean space. Wang, Latapy and Soria, (2012 p.11) note that a first step in Graph Visualisation Analysis represents a description of the network structure following by a description of its dynamic evolution. Hence, we look at the resulting network graph visualisations from two different angles. First by using the Small-World Network model of Kleinberg (2000), and second by using the Scale-Free Network models by Barabási and Albert (2002). Based on these results, we then elaborate a k-core decomposition using the algorithm of Alvarez-Hamelin et al. (2005b).
Small-World Network Model (IP)
The Small-World Network model by Kleinberg (2000) was used in conjunction with the
Layered Layout by Kuchar (2012), which is considered to be suitable for Small-World Network graph visualisations (see section 2.3.4). To obtain the structural differences of
the three operator networks, we consistently chose the same layout properties (see section 3.5.3). The analysis of the edge-distributions in the plotted graph visualisations in Figure 4-4 below indicates that none of the connectivity graphs of the mobile broadband operator networks followed Small-World Network properties. However, some IP address vertices are displaying strong relationships to other IP addresses within the circle in the centre of the graph visualisation. Therefore, it seems that the operator networks are showing a core of densely connected IP addresses. This is an indicator for hierarchical upstream Internet market structuring with large Internet Service Providers (ISPs) at the core. However, the graph visualisation of Bharti Airtel shows less active connections among a particular set of IP address vertices. This is interesting since it potentially indicates a less hierarchical upstream Internet market structure than the graph visualisations for Aircel and Vodafone.
Moreover, the strongest connected edges are clearly visible on the left-hand side of the graph visualisations (see Figure 4-4 below). From this vertex, the edges seem to leave the visualised Small-World Network circle and reach towards other IP address vertices in the network periphery. Moreover, none of the operators’ graph visualisations show perfectly interconnected Small-World Network effects in the centre of their visualised circles. All graph visualisations seem to build new layers around the centred one, which is most clearly visible for the graph visualisation of 𝐺𝐺™ge•´gdf. This indicates that the networks follow Scale-Free Network properties.
Given the findings of the operator’s graph visualisations using the Small-World Network model with a Layered Layout, the next section covers a more detailed analysis and comparison of these networks using the Scale-Free Barabási-Albert algorithm in a Force
Atlas 2 Layout. This algorithm aims to analyse the existence of Scale-Free Network
𝐺𝐺®2§©fâ
edge-thickness: 0.50.
n – size of lattice: 10.
p – lattice distance to local contacts: 2.
q – long range contacts: 2.
r – clustering exponent: 0.
𝐺𝐺é啧U2 ®2§Ufâ
edge-thickness: 0.50. n – size of lattice: 10.
p – lattice distance to local contacts: 2. q – long range contacts: 2.
r – clustering exponent: 0.
𝐺𝐺™ge•´gdf
edge-thickness: 0.25. n – size of lattice: 10.
p – lattice distance to local contacts: 2. q – long range contacts: 2.
r – clustering exponent: 0.
Figure 4-4: Small-World Network graph visualisations in Layered Layout by mobile broadband operator at IP granularity.
Scale-Free Network Model (IP)
Given the findings above, none of the three operator networks seem to display Small-
World Network properties. Therefore, this section tests the three operator networks by
applying Scale-Free Network properties using the Barabási and Albert (2002) algorithms. Based on the observed set of IP addresses for each of the three mobile broadband operator networks, we follow these dynamic network procedures to simulate alternative scenarios of network growth emergence. The associated network features will be visualised in comparing the possible evolution of the mobile broadband operator networks below. The utilised algorithms are derived from the work of Barabási and Albert (2002) as stated in Barabási Labs (2013):
• 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.
The mobile broadband operator network graphs were visualised using the Force Atlas 2
Layout in the Open Source graph visualisation platform Gephi (2016). This layout is
suitable for exploring Scale-Free Network properties of networks with up to 10,000 vertices (Jacomy et al., 2014), which none of the three Tamil Nadu mobile broadband operator networks exceeded. The visualisation parameters are stated in section 3.5.3 above. We first simulate and compare the alternative scenarios of network growth emergence of the three mobile broadband operator networks using the Barabási Standard Model (vertex growth and preferential attachment), followed by the Model A with uniform attachment (and retained growth of vertices) and the Model B with preferential attachment (‘rich-get-richer’ effect) to edges (but no vertex growth).
Barabási-Albert Standard Model
First, comparing the three mobile broadband operator networks’ graph visualisations at IP granularity using the Barabási-Albert Standard Model indicates structural differences between our three operators of interest. The network simulation considers half of all the networks IP address vertices, being 8,647 simulated vertices in 𝐺𝐺®2§©fâ , 600 for
𝐺𝐺é啧U2 ®2§Ufâ and 7,509 for 𝐺𝐺™ge•´gdf. To assure comparability, these vertices are chosen
based on the total number of vertices in the given operator networks. The Barabási-Albert Standard Model simulation shows that IP address vertices in 𝐺𝐺®2§©fâ and 𝐺𝐺™ge•´gdf are more strongly organised in vertex clusters (groupings of IP address vertices) than the vertices in the graph visualisation of 𝐺𝐺é啧U2 ®2§Ufâ (see Figure 4-5 below). Moreover, the
operator networks’ graph visualisations of 𝐺𝐺®2§©fâ and 𝐺𝐺™ge•´gdf show cores of specific
IP addresses that seem densely internetworked. Especially Vodafone seems to make strong use of the same IP address vertices, as indicated by the large blue area in the core of the network. This suggests their potential upstream connectivity reliance on these IP address vertices. On the contrary, the IP address vertices core of 𝐺𝐺é啧U2 ®2§Ufâ seems not strongly internetworked. This is interesting since it indicates an overall fairer distribution of upstream connectivity among the IP address vertices and hence, less internetworking reliance on certain IP address vertices. Moreover, each of the three operator network graph visualisation shows IP address vertices being situated at the edge of the visualised spaces, likely representing IP addresses in the Internet periphery. However, we consider the simulation of network growth emergence using the Barabási-Albert Standard Model to be somewhat fictive, since the IP address vertices, elaborated based on our Paris
traceroute dataset, represent the unique IP addresses of upstream infrastructure devices
(e.g. routers) for the purpose of establishing internetworking connections. Hence, we would not expect a strong vertex growth in a Real-World Network growth situation.
G𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨
N Number of vertices in network: 8647.
𝑮𝑮𝑩𝑩𝑩𝑩𝑩𝑩𝑨𝑨𝑩𝑩𝑨𝑨 𝑨𝑨𝑨𝑨𝑨𝑨𝑩𝑩𝑨𝑨𝑨𝑨
N Number of vertices in network: 600.
𝑮𝑮𝑽𝑽𝑽𝑽𝑽𝑽𝑩𝑩𝑽𝑽𝑽𝑽𝑽𝑽𝑨𝑨
N Number of nodes in network: 7509.
Figure 4-5: Barabási-Albert Standard Model graph visualisations per mobile broadband operator at IP granularity.
Barabási-Albert Model A
Next, we compare the three Tamil Nadu mobile broadband operator networks’ graph visualisations at IP granularity using the Barabási-Albert Model A. The network growth simulation of this model considers the growth of vertices in the network but no preferential attachment or ‘rich-get-richer’ effects. Again, the network simulation of the Barabási-Albert Model A considers 8,647 vertices for 𝐺𝐺®2§©fâ, 600 for 𝐺𝐺é啧U2 ®2§Ufâ and 7,509 for 𝐺𝐺™ge•´gdf in the simulation. Interestingly, the three generated network graph visualisations with simulated vertex growth indicate somewhat similar cores of strongly connected IP address vertices. Like at the Barabási-Albert Standard Model above, the graph visualisations of 𝐺𝐺®2§©fâ and 𝐺𝐺™ge•´gdf seems to have densely internetworked
cores of specific IP addresses. Again, as indicated by the large blue area in the core of the network in Figure 4-6 on the next page, Vodafone seems to make strong use of the same IP address vertices, showing again their potential upstream connectivity reliance on these IP address vertices. On the contrary, the IP address vertices core of 𝐺𝐺é啧U2 ®2§Ufâ seems
again not strongly internetworked. Interestingly, compared to the network growth simulation using the Barabási-Albert Standard Model, the simulation of the Barabási- Albert Model A does not indicate the existence of strong IP address vertex clusters. The lack of these vertex clusters may be attributed to the missing preferential attachment of edges. Above, we indicated the somewhat fictive nature of the network growth emergence simulation (given the nature of IP addresses for internetworking and hence upstream Internet connectivity purposes). Adding to this, we indicate the value of the network growth emergence simulation using the preferential attachment of edges, representing connectivity recurrence in a somewhat fix set of upstream IP address vertices. Hence, we consider the following simulation using the Barabási-Albert Model B to be most suitable to understand and graphically analyse the network growth emergence in mobile broadband operator networks.
𝑮𝑮𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨
N Number of vertices in network: 8647.
𝑮𝑮𝑩𝑩𝑩𝑩𝑩𝑩𝑨𝑨𝑩𝑩𝑨𝑨 𝑨𝑨𝑨𝑨𝑨𝑨𝑩𝑩𝑨𝑨𝑨𝑨
N Number of vertices in network: 600.
𝑮𝑮𝑽𝑽𝑽𝑽𝑽𝑽𝑩𝑩𝑽𝑽𝑽𝑽𝑽𝑽𝑨𝑨
N Number of nodes in network: 7509.
Figure 4-6: Barabási-Albert Model A graph visualisations per mobile broadband operator at IP granularity.
Barabási-Albert Model B
Given the findings above, we indicated that the Barabási-Albert Model B is the most valuable simulation to analyse and understand network growth emergence and structural network properties. Hence, this simulation best represents the connectivity nature of the upstream Internet market, containing a fix set of internetworking-providing agents (Autonomous Systems managing the IP address ranges) but re-establishments of connections amongst the different upstream IP addresses.
Here, the simulated Barabási-Albert Model B again considers the same number of vertices and edges as above, under preferential attachment. The three-generated operator network graph visualisations with simulated vertex growth again indicate somewhat similar cores of strongly connected IP address vertices. In detail, the graph visualisations
of 𝐺𝐺®2§©fâ and 𝐺𝐺™ge•´gdf have cores of specific IP addresses that are densely
internetworked, which is again indicated by the large blue area representing edges between IP address vertices in the following Figure 4-7. Compared to those of 𝐺𝐺®2§©fâ and
𝐺𝐺™ge•´gdf , the IP address vertices core of 𝐺𝐺é啧U2 ®2§Ufâ are again not strongly
internetworked. Interestingly, each of the three operator network graph visualisations using the Barabási-Albert Model B reveals a bi-partitioning of the graph visualisations. This indicates the importance of some IP address vertices that ‘bridge’ connections between the bi-partite parts of the operator networks for internetworking purposes towards the Internet periphery. Here, especially the graph visualisation of 𝐺𝐺é啧U2 ®2§Ufâ indicates the existence of very few of these important vertices. The other two Tamil Nadu mobile broadband operator networks, 𝐺𝐺®2§©fâ and 𝐺𝐺™ge•´gdf bridge the apparent bi- partionioning with a multitude of IP addresses, preventing connectivity issues. Moreover, this structuring also indicates a structuring where a few IP address vertices (potentially belonging to larger Internet Service Providers) would receive most of the upstream internetworking connectivity, representing connectivity-crucial structural bottlenecks.
𝑮𝑮𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨
N Number of vertices in network: 8647. M Number of edges in network: 11411.
𝑮𝑮𝑩𝑩𝑩𝑩𝑩𝑩𝑨𝑨𝑩𝑩𝑨𝑨 𝑨𝑨𝑨𝑨𝑨𝑨𝑩𝑩𝑨𝑨𝑨𝑨
N Number of vertices in network: 600. M Number of edges in network: 803.
𝑮𝑮𝑽𝑽𝑽𝑽𝐝𝐝𝑩𝑩𝑽𝑽𝑽𝑽𝑽𝑽𝑨𝑨
N Number of nodes in network: 7509.
M Number of edges in network: 10390.
Figure 4-7: Barabási-Albert Model B graph visualisations per mobile broadband operator at IP granularity.
Summarising, this section showed that the Barabási-Albert Model B represents the most valuable simulation to study and understand network growth emergence for traceroute- based mobile broadband operator networks, given the nature of the upstream Internet infrastructure. The three respective graph simulations using the Barabási-Albert Model B then revealed a bi-partitioning of the network graphs. This exposed the structural bottlenecks of certain IP address vertices with an internetworking importance for the three mobile broadband operators, forming a densely-connected core of the operator networks. Given the findings above, the following section aims to reveal the nature and identity of these influential IP addresses, using the k-core decomposition used by Alvarez-Hamelin et al. (2005b) and Busch, Béiro and Alvarez-Hamelin (2011). This will demonstrate the mobile broadband operator’s hierarchical upstream Internet market structure.
k-core decomposition (IP)
In this section, we will use the k-core decomposition spectral analysis to identify the set of the most densely connected IP address vertices for each of the graphs generated for the three Tamil Nadu mobile broadband operator’ networks. Referring to the work of Alvarez-Hamelin et al. (2005b), this k-core decomposition reveals the specific roles and relevance of the vertices located in the periphery and core of a network. This method is frequently used for the analysis of Internet structures, such as work of CAIDA shows. Using a k-core decomposition algorithm, as introduced by Seidmann (1983), allows for the division of graph visualisation into densely connected network subsets, called k-cores. Hence, these k-cores represent connectedness properties for the IP address vertices in a given network, where a higher k-core indicates a set of more densely connected IP address vertices (see section 2.3.4). The most densely-connected IP address vertices in the network core provide both internetworking connectivity features amongst themselves and between this central core and those IP addresses located in the overall network periphery. Given the identified k-cores, this method allows for a clear identification and visualisation of some key hierarchical network properties. Below we start with the k-core
decomposition for Aircel, followed by the k-core decomposition for Bharti Airtel and
lastly for Vodafone.
Aircel
When looking at the k-core decomposition for the Aircel graph 𝐺𝐺®2§©fâ in Figure 4-8
below, we observed 179 k-cores. The highest k-core is inhabited by three IP address vertices located in the centre of the graph visualisation, indicating the densest connections
amongst these IP address vertices. Using the Maxmind (2015) Geo IP2 database and associating these IP addresses with their Autonomous System Number in Table 4-8 below, we show that all of these three central core IP addresses are associated with Tata Communications (America) Inc. (AS6453), revealing that this Autonomous System plays a key role in providing internetworking connectivity to Aircel to reach the IP address vertices located in the network periphery of the graph generated by the Aircel observations. This supports the previous results that emerged above from exploring the Network metrics. Moreover, we can also identify some vertices that while inhabiting a lower hierarchical k-core position, are still providing key connectivity to the periphery.
𝑮𝑮𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨
Highest core IP address vertices, visualised as red vertices in the centre (grey edges):
179 cores: ‘180.87.39.25’
179 cores: ‘80.231.154.17’
179 cores: ‘80.231.217.17’
Figure 4-8: Aircel graph visualisation k-core decomposition at IP granularity. Bharti Airtel
Next, when looking at the k-core decomposition for the Bharti Airtel graph 𝐺𝐺é啧U2 ®2§Ufâ
in the following Figure 4-9, we observed 40 k-cores. The highest k-core is inhabited by two IP address vertices located in the centre of the graph visualisation. This shows the densest connections amongst these IP address vertices, followed by one IP address vertex in the 39th k-core. Using the Maxmind (2015) Geo IP2 database and associating these IP
addresses with their Autonomous System Number in using the Maxmind (2015) Geo IP2 database and associating these IP addresses with their Autonomous System Number in Table 4-8 below, we show that all of these three central core IP addresses are associated with Bharti Airtel Ltd. (AS45609), Level 3 Communications Inc. (AS3356) and Bharti Airtel Ltd. (AS9498). These Autonomous Systems are playing a key role in providing
Bharti Airtel’s internetworking connectivity to reach the IP address vertices located in the network periphery of the graph, generated by the Bharti Airtel observations. This again supports the previous results that emerged above from exploring the Network metrics. Additionally, we can also identify vertices that, while inhabiting a lower hierarchical k-core position, are still providing key connectivity to the periphery.
𝑮𝑮𝑩𝑩𝑩𝑩𝑩𝑩𝑨𝑨𝑩𝑩𝑨𝑨 𝑨𝑨𝑨𝑨𝑨𝑨𝑩𝑩𝑨𝑨𝑨𝑨
Highest core IP address vertices, visualised as red vertices in the centre (grey edges):
40 cores: ‘223.224.40.92’ 40 cores: ‘10.155.84.218’ 39 cores: ‘59.144.180.69”’
Figure 4-9: Bharti Airtel graph visualisation k-core decomposition at IP granularity. Vodafone
Last, when looking at the k-core decomposition for the Vodafone graph 𝐺𝐺™ge•´gdf, as
visualised in Figure 4-10 below, we identified 1973 k-cores. The highest k-core is inhabited by three IP address vertices located in the centre of the graph visualisation, indicating the densest connections amongst these IP address vertices, followed by some IP address vertices in the slightly lower k-cores. Using the Maxmind (2015) Geo IP2 database and associating these IP addresses with their Autonomous System Number in using the Maxmind (2015) Geo IP2 database and associating these IP addresses with their Autonomous System Number again in Table 4-8, we show that all of these three central core IP addresses are associated with Vodafone India Ltd. (AS55410). The IP address vertices in the slightly lower central core are associated with the China Education and Research Network Center (AS4538) and Cable and Wireless Worldwide Plc. (AS1273). Similarly, compared to the previous k-core decompositions, this again supports the indicated results that emerged above from exploring the Network metrics. Additionally, we can also identify vertices that, while inhabiting a lower hierarchical k-core position,
are still providing key connectivity to the periphery.
𝑮𝑮𝑽𝑽𝑽𝑽𝑽𝑽𝑩𝑩𝑽𝑽𝑽𝑽𝑽𝑽𝑨𝑨
Highest core IP address vertices, visualised as red, magenta and purple vertices in the centre (grey edges). 1973 cores: ‘182.19.115.70’ 1973 cores: ‘182.19.114.87’ 1973 cores: ‘182.19.105.88’ 1882 cores: ‘182.19.115.233’ 1622 cores: ‘100.64.0.149’ 1458 cores: ‘166.63.217.41’
Highest k-core IP addresses by operator at AS granularity
Mobile broadband operator graph
k-cores IP address Organisational Name (Autonomous System Number) (Maxmind, 2015)
𝐺𝐺®2§©fâ_®∆
179 180.87.39.25 Tata Communications (America) Inc. (AS6453)
179 80.231.154.17 Tata Communications (America) Inc. (AS6453)
179 80.231.217.17 Tata Communications (America) Inc. (AS6453)
𝐺𝐺é啧U2 ®2§Ufâ_®∆
40 223.224.40.92 Bharti Airtel Ltd. (AS45609) 40 10.155.84.218 Level 3 Communications Inc.
(AS3356)
39 59.144.180.69 Bharti Airtel Ltd. (AS9498)
𝐺𝐺™ge•´gdf_®∆
1,973 182.19.115.70 Vodafone India Ltd. (AS55410) 1,973 182.19.114.87 Vodafone India Ltd. (AS55410) 1,973 182.19.105.88 Vodafone India Ltd. (AS55410) 1,882 182.19.115.233 Vodafone India Ltd. (AS55410) 1,622 100.64.0.149 China Education and Research Network Center (AS4538)
1,458 166.63.217.41 Cable and Wireless Worldwide Plc. (AS1273) Key
AS: Autonomous System. IP: Internet Protocol.
Summary Graph Visualisation Analysis (IP)
The visualisation analysis allowed us to show that none of the three mobile broadband operator networks displayed Small-World Network properties. The Scale-Free Network graph simulations provided initial structural insights about the dynamics of network growth emergence based on the existing connectivity. We note that the Barabási-Albert Model B is the most suitable graph algorithm for simulating the emergent network dynamics driven by the possible new connectivity between IP address vertices in the established network based on the principles of the ‘rich-get-richer’ effect. Next, the