Efficient Visualisation of Large Routing Topologies

Siew Cheong Au1 , Christopher Leckie2 , Ajeet Parhar3 and Gerard Wong1, 1

Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville 3010, Australia 2

ARC Special Research Centre for Ultra-Broadband Information Networks

Department of Computer Science and Software Engineering, The University of Melbourne, Parkville 3010, Australia [email protected] 3

Telstra Research Laboratories, 770 Blackburn Road, Clayton 3168, Australia [email protected]

Abstract. As the Internet grows in size and complexity, network managers face a significant challenge in trying to understand the behaviour of routing protocols in large networks. In this paper, we present a tool called VLNT (Visualising Large Network Topologies), which helps network managers t o analyse complex routing topologies. A key contribution of our system is a novel hybrid layout algorithm, which significantly reduces the computation time required to layout large network topologies i n comparison to conventional layout approaches. In addition our algorithm includes a novel termination criterion that avoids unnecessary iterations when optimising the network layout. We demonstrate how the visualisation features of VLNT can be used to analyse and improve BGP routing topologies, and provide examples using real-life routing data.

Contact author: Dr Christopher Leckie Department of Computer Science and Software Engineering, The University of Melbourne, Parkville 3010, Australia [email protected] Phone +61-3-8344 1413 Fax +61-3-9348 1184

1 Introduction There is a growing need for efficient algorithms to visualise large network topologies in telecommunications. For example, layout algorithms can be used in telecommunication network management to help visualise relationships between large numbers of network elements1, such as the routing relationships between Internet routers2. In practice, network managers need to monitor these networks in real-time so that they can detect problems due to equipment failures, misconfigurations or unusual traffic patterns. This raises the problem of how to scale-up existing graph layout algorithms in order to visualise large network topologies in real-time. In this paper we present a novel hybrid algorithm that significantly reduces the computation time of a popular forcedirected layout algorithm on large networks. We have implemented this algorithm in a tool called VLNT (Visualising Large Network Topologies), which is specifically designed to help network managers analyse large communication networks, such as routing topologies in the Internet. In VLNT we have included two key theoretical contributions that enable us to efficiently generate aesthetically pleasing layouts for networks with over 1000 nodes. Our first contribution is a hybrid layout algorithm that combines the strengths of two previous approaches to the graph layout problem. Force-directed layout algorithms, such as the Kamada-Kawai algorithm3, model the network as a set of springs, and then try to find a layout that minimises the energy in those springs. These techniques produce aesthetically pleasing layouts, but their computational complexity often limits their application to small networks. More recently, a stochastic optimisation approach called the ISOM neural network algorithm4 has been shown to scale well with large networks, but often does not produce results with the same aesthetic quality as force-directed algorithms. We have combined these two approaches in a unique hybrid layout algorithm that draws on their complementary strengths in order to improve computation time without sacrificing quality. Our second contribution is a novel termination criterion that avoids unnecessary iterations at the end of the optimisation. We demonstrate that our algorithm yields significant savings in computation time while achieving the same level of aesthetics as a conventional force-directed approach on real-life maps of Internet routing topologies. In this paper, we present our novel algorithm for visualising large network topologies, and describe how we have implemented this algorithm in the VLNT tool. We also present the key features that we have included in

VLNT to help network managers analyse and improve Internet routing topologies. We begin by introducing the problem of visualising Internet routing topologies. In Section 3, we summarise previous layout algorithms that are the foundations of our approach, and present our hybrid layout algorithm and our improved termination criterion. In Section 4, we demonstrate how our hybrid algorithm achieves significantly better performance than a standard force-directed approach when tested on real-life Internet routing topologies from the Border Gateway Protocol (BGP). In Section 5, we describe how our hybrid algorithm and other features have been incorporated into VLNT, and illustrate the use of these features in a network management scenario.

2 Visualising Internet Routing Topologies Our focus is the problem of visualising Internet routing topologies. The role of routing protocols in networks is to ensure that information can be sent between computers connected to a network. The Internet is a collection of thousands of networks ranging from the largest backbones to small Internet service providers. Routing protocols are run internally to each of these networks (intra-domain routing) as well as between a network and its neighbour (inter-domain routing). Inter-domain routing in the Internet is coordinated by the Border Gateway Protocol (BGP)5 . It provides a mechanism for advertising reachability between networks that are controlled by different organisations. A group of networks that are controlled by the same organisation is called an Autonomous System (AS). Every AS is given a unique number by the IP addressing authorities. Every Internet Service Provider (ISP) has an autonomous system number, as do a number of corporations who employ IP in their private networks. BGP is responsible for inter-domain routing between ASs in the Internet. Over the past few years, there has been increasing interest in BGP, due to the increase in ISPs and the growth in dependence on reliable Internet connections for electronic commerce6,7,8. As the Internet has grown in size and complexity, the interconnection between ASs have been dynamically evolving as ISPs add and eliminate connections to other ASs and companies change ISPs. Furthermore, the contractual agreements between ASs can

change due to ISP merging and restructuring. Thus a graph of the Internet from the perspective of Autonomous System relationships provides a unique insight into the evolving structure of the Internet. The vertices of such a BGP network represent the ASs, and an edge between vertices denotes that traffic is exchanged between a pair of ASs. The BGP graph represents logical relationships between ASs rather than physical connections. We refer to this BGP graph as an AS map. Each AS is represented by a 16-bit number. Not all AS numbers are assigned to administrative domains and some assigned AS numbers are not used. Many ISPs possess several ASs. MCI Worldcom for example owns at least 143 ASs. An AS has its own routers and routing policies and connects to other ASs to exchange traffic with remote hosts. Regular snapshots of BGP routing information are collected by the Route Views Project at the University of Oregon9. The Route Views server establishes BGP peering sessions with 53 ISPs in 37 locations. The BGP routing tables are collected once daily from a route server with BGP connections to multiple geographically distributed target routers in Japan, United States, Europe and Australia. The largest sample of the BGP dataset has more than 20000 elements and this number is growing steadily with time. Laying out the entire network of BGP data even if it were possible would not render the graph usable or readable. For example, Huffaker et al.10 use partial geographical information to layout BGP graphs, and report difficulties in visualizing graphs with more than a few hundred nodes. Thus, our approach to visualising the network involves pruning the network to a viewable size by essentially laying out the higher connected ASs or a subset of the network of interest to observe the structure of interconnections and clustering between these ASs. A network of up to1500 elements is considered easily usable and viewable.

3 Our Approach to Graph Layout Network visualisation aims to construct a visual representation of the structural relationships between a set of elements. In the case of BGP routing data, we are interested in the routing relationships between Autonomous Systems. Network visualisation applications such as this are typically modeled as a graph layout problem, where the network is represented as a set of nodes and a set of binary relationships between nodes.

Our focus is the layout of large undirected networks with straight-line edges. For example, each node represents an AS, and each edge represents the exchange of traffic between two ASs. We are interested in algorithms that display the inherent structure of a given network in a succinct fashion. Ideally, we want to maximise the display of symmetry within the network, and preserve the uniformity of edge lengths between pairs of nodes that are identical in topological distance, where the topological distance corresponds to the minimum number of hops between nodes. Although it is desirable to minimise edge crossings, we consider this to be of lesser importance because planarity is not achievable with every graph, and is highly unlikely in the large graphs with high edge density that we have studied. We have developed a novel approach to the graph layout problem for large networks. Our approach includes two key contributions. First, we have developed a hybrid algorithm that combines the strengths of two previous approaches to the graph layout problem. Second, we introduce a new termination criterion called the edge-tension gradient, which provides an accurate test for deciding when to halt the layout algorithm.

3.2

Previous Approaches

A common approach to this type of problem is to use a force-directed algorithm. Force-directed approaches model the graph drawing problem as a physical system of forces often represented by springs and electrical charges11. Using an analogy to physics, vertices are treated as mutually repulsive charges and edges as springs connecting and attracting the charges. Starting with an arbitrary initial placement of vertices, the algorithm iterates the system in discrete time steps by computing the forces between vertices and updating their position accordingly. Although the algorithm does not explicitly support the detection and display of symmetries, it has been found to display symmetries if any exist. Kamada and Kawai3 refined the model subsequently. They introduced an optimal edge length k and removed the charge association with each vertex. Vertices are updated sequentially by moving only one vertex at each time step. The algorithm performs a gradient descent and converges deterministically to a local minimum. A drawback of Kamada-Kawai is the number of iterations (or “epochs”) required to find a local minimum. Moreover, the cost of each iteration is proportional to the number of nodes in the network.

A more recent approach to this type of layout problem is the Inverted Self-Organising Map (ISOM) by Meyer4. Each node in the network is modelled as a vector of weights which may be embedded as a point in Ndimensional space. If the weights are of dimension 2, a node would correspond to a point in 2 dimensional space. A form of competitive learning is used to update these weights by using a random input vector or stimulus. The weights of the closest node to the stimulus are updated, along with its topological neighbours. By using a restricted update neighbourhood, only a constant number of nodes are updated at each iteration. In addition, Meyer reports experimental results that indicate the number of iterations is linear in the size of the network. However, the final results for complex graphs are often not as symmetric as the results from KamadaKawai. This is because the final layout is a side-effect of the local updating procedure, rather than the minimisation of an explicit cost function. In summary, force-directed approaches generally produce high quality layouts at the expense of greater computational complexity, while ISOM creates an approximate layout with greater computational efficiency.

3.3

Hybrid Layout Algorithm

Our hybrid approach first uses the ISOM algorithm for initial layout, and then applies the Kamada-Kawai algorithm for fine-tuning. The impetus for this approach came through testing and observing the convergence of the Kamada-Kawai algorithm to different initial layouts. The ISOM algorithm on its own produces a reasonable layout of the structure of the entire network, but its weakness in clustering causes nodes to overlap and vanish behind each other, depriving the viewer of a complete view of the entire network. On the other hand, KamadaKawai being a force-directed approach spreads nodes apart evenly with an unclustering check in the algorithm. Thus, when applied to an ISOM initial layout, Kamada-Kawai causes unclustering and converges much quicker to an aesthetic final layout. The reason for this is that the initial layout generated by ISOM already approximates the locally optimal layout. The early iterations of Kamada-Kawai are in effect replaced by the much faster ISOM algorithm. KamadaKawai is reserved for fine-tuning the layout. However, the Kamada-Kawai algorithm should not be allowed to

waste time on trivial refinements of the layout. This has motivated our second contribution, which is a new termination criterion for iterative graph layout algorithms.

3.4

Termination Criterion: Edge-Tension Gradient

The conventional termination criterion for force-directed algorithms is based on the change in energy per iteration, and the algorithm terminates after the change in energy falls below a certain threshold. This approach is stochastic and may cause the algorithm either to run indefinitely because the change in energy does not go below the threshold or to run longer than is required to achieve an output of the same quality. In this section, we propose an alternative termination criterion that is based on the ratio of the average edge length in the layout to the ideal edge length. In the pre-processing stage of the Kamada-Kawai algorithm, the ideal edge length (L0) is calculated once and used in the optimisation calculations throughout the execution of the algorithm. We define the edge-tension (equation 1) of the layout to be the average edge length attained at the current iteration normalised by L0, where L0 is defined by equations 2 and 3. Note that the definition of L1, is arbitrary, but should be representative of the screen size so that the span of the graph is kept within the dimensions of the display.

Edge Tension, ϕ =

Average Edge Length …(1) L0

where

L0 =

L1 Maximum Number of Hops

…(2)

and

L1 =

Screen Width × Screen Height 2

…(3)

The choice of L1, is arbitrary but should be at most equal to the minimum of the screen height or width so that the span of the graph is kept within screen dimensions.

Figure 1: A Diagrammatic Representation of the Various Parameters for Edge-Tension

Through empirical observations, we have found that the gradient of the edge-tension characteristic approaches zero as the graph reaches a local minimum correlating to an aesthetic final layout. Our edge-tension gradient approach proposes that the algorithm should terminate when the gradient of the edge-tension characteristic falls below a certain threshold to reduce the number of iterations (or “epochs”) needed to produce an aesthetic layout, i.e. when

∂ϕ ≤β ∂Epoch

where β is the gradient threshold. Through empirical testing, we have found that values of β between 0.5x10-6 to 7x10-6 produce a final layout that is equal in aesthetics to the conventional energy termination approach (see Table 1).

Number of Elements Recommended Edge Tension Gradient Threshold Recommended Maximum Edge Tension Gradient Threshold

< 500

500 – 1000

1000 – 2000

0.5 x 10-6

2.5 x 10-6

5 x 10-6

1 x 10-6

5 x 10-6

7 x 10-6

Table 1: Recommended gradient threshold settings for various graph sizes

These recommended thresholds were obtained through rigorous empirical testing on 50 Internet routing network topologies of various sizes (see Section 4). The edge-node ratios of these topologies were between 1.6 to 2.2. Each network topology was laid out 10 times from different random initial positions. In each case, we recorded the value of the edge tension gradient after which there was negligible improvement in the quality of the layout. The recommended threshold is the mean of these final gradient values. We found that the variance of these final gradient values was within 20% of the mean. We can also vary the epoch interval over which the

gradient is calculated. We have chosen a default value of 50 epochs, which was derived through empirical testing on graphs of various sizes and configurations.

4 Evaluation In order to evaluate our algorithm, we have conducted three sets of experiments on BGP routing topologies. The aim of the first experiment is to compare the quality of the layouts generated be each of the three algorithms we have considered, i.e., Kamada-Kawai, ISOM and our hybrid algorithm. The aim of the second experiment is to evaluate the convergence rate of the Kamada-Kawai algorithm when the traditional energy change threshold and the new edge tension gradient are used. The aim of the final experiment is to compare the scalability of the three layout algorithms in terms of the computation time required for different size network topologies.

Figure 2: ISOM layout of Network 1

Figure 3: Kamada-Kawai layout of Network 1

Figure 4: Hybrid layout of Network 1

Figure 5: Layout convergence Our first set of results illustrates the final layout produced on a 271 node subset of the BGP data (Network 1) using each of the three layout algorithms. The layout generated by ISOM (Figure 2) manages to isolate nodes of high connectivity from the rest of the graph and distribute them in a way that is similar to final layout produced by Kamada-Kawai (Figure 3). The drawback of the ISOM layout is the unwanted clustering of nodes, which gives an incomplete view of the graph. In contrast, our hybrid algorithm (Figure 4) refines the initial ISOM layout by unclustering the graph and distributing the nodes and edges evenly so as to maximise the display of symmetry and maintain the essential correlation between topological distance and Euclidean distance in the

graph. There is no appreciable difference in the quality of the layouts produced by Kamada-Kawai and our hybrid algorithm. Note that the ISOM layout took 0.3 s elapsed time, the Kamada-Kawai layout took 41.8 s, and the hybrid layout took 23.4 s. All algorithms were implemented on a Java 2 Platform, Standard Edition v1.3.1 running on Microsoft Windows ME. The results were obtained on a 1.1 GHz Athlon AMD Processor with 512MB SDRAM. These results illustrate that our hybrid algorithm is able to produce layouts of comparable quality to those generated by Kamada-Kawai, while requiring significantly less computation time. Note that no detailed theoretical models exist for the computational complexity of either Kamada-Kawai or ISOM. Each has a complexity of O(TN), where T is the number of iterations of the algorithm, and N is the number of nodes in the graph. In Kamada-Kawai, O(N) sets of calculations need to be made at each iteration3. In ISOM, O(N) comparisons are required at each iteration, and only the immediate neighbours of the winning node are updated4.

No accurate

bound has been found to model the number of iterations T that are required by each algorithm. However, we have observed that ISOM generally uses many fewer iterations than Kamada-Kawai, and that the calculations performed at each iteration by ISOM are much simpler. Figure 5 uses the edge-tension characteristic to illustrate the difference between Kamada-Kawai and our hybrid algorithm. The wide fluctuations in this characteristic is present with Kamada-Kawai for approximately 2500 epochs, but the hybrid approach which uses the ISOM initial layout only fine-tunes its configuration without having to go through the wide fluctuations in edge-tensions to search for a local minima. Effectively, the ISOM layout has accelerated the edge-tension characteristic of the graph into the fine-tuning stage. Our second set of results use a typical 133 node network (Network 2) to compare the convergence rate of Kamada-Kawai using the traditional energy change threshold criterion (Figures 6 and 7) and our new edge-tension gradient criterion (Figures 8 and 9). The final layouts are almost indistinguishable. However, the number of epochs required is significantly different. The traditional energy change threshold criterion terminated after 3020 epochs (19.1 s), while our edge-tension gradient criterion terminated after 1800 epochs (11.4 s), which is a 40% reduction in execution time. We found that the edge-tension stabilises after around 1700 epochs, while the minimum energy change takes much longer to settle, and a suitable threshold is difficult to set in advance.

Figure 6: Layout of Network 2 using Kamada-Kawai with conventional energy criterion

Figure 7: Convergence of conventional energy criterion on Network 2

Figure 8: Layout of Network 2 using Kamada-Kawai with edge-tension gradient criterion

Figure 9: Convergence of edge-tension gradient criterion on Network 2 Our third set of results compares the scalability of each algorithm. As mentioned above, no accurate theoretical model exists for the computational complexity of the Kamada-Kawai and ISOM algorithms. Consequently, we have conducted an empirical comparison of the computational complexity of the three layout algorithms, using samples of the BGP datasets. Each sample network was a subset of the complete BGP routing topology. For each sample, the edge-to-node ratio was maintained between 1.65 and 1.75 to ensure consistency in connectivity characteristics and the nature of the tested graphs. The samples were chosen from among the top 20 nodes in terms of connectivity of each original dataset, as these nodes present more information and structure than the rest of the graph. Figure 10: Scalability of ISOM Figure 11: Scalability of Kamada-Kawai and Hybrid layout algorithms

The ISOM algorithm scaled in a manner that was close to linear as the number of nodes increased (Figure 10), and took less than 2 minutes to handle a graph of over 6000 nodes. However, the usability of the resultant layout was debatable. Although the algorithm is likely to be able to handle graphs that are larger than this, we encountered memory problems with Java when running graphs with more than 6000 nodes. The Kamada-Kawai algorithm was much less scalable than ISOM. A graph of around 1500 nodes required almost 3000 s of computation time (Figure 11), compared to approximately 4 s for ISOM.

The results in Figure 11 show that our hybrid approach performs consistently better than the Kamada-Kawai algorithm by an average of 30 % for most graph sizes. Both approaches can handle graphs of up to 1500 nodes but the performance degrades more rapidly beyond 1000 nodes. The quality of the hybrid approach as examined previously is similar and in some situations better than the Kamada-Kawai algorithm alone. In certain situations, the quality of the hybrid layout may surpass Kamada-Kawai because the use of the ISOM algorithm results in an initial layout that is close to a favourable local minimum.

5

Interpreting AS Maps Using VLNT

We have implemented our hybrid layout algorithm in a software tool called VLNT (Visualising Large Network Topologies). VLNT has been specifically designed to help network managers understand complex network topologies, such as AS maps showing BGP routing topologies. It enables managers of an AS to check the connectivity and topological distance of their network to other ASs. In order to increase the connectivity of an AS, network managers can use VLNT to highlight networks that would be promising candidates for a peering relationship. In this section, we present the features provided by VLNT for understanding AS maps, and then illustrate its use in a network management scenario. There are four main functions provided in VLNT for analysing network topologies: (1) layout an arbitrary network topology, (2) colour nodes by connectivity, (3) colour nodes by country, and (4) layout the shortest path tree to a selected node. Although we concentrate on the application of these features to analysing AS maps, they can generally be useful for managing a wide variety of communication networks. Layout an arbitrary network topology – As illustrated by the network in Figure 4, our hybrid layout algorithm generates a layout where nodes that are in close topological proximity tend to be grouped together. We also find that nodes with high connectivity tend to gravitate towards the centre of the graph, while nodes with low connectivity are pushed to the periphery of the display. This form of clustering facilitates the identification of nodes (ASs) that are highly connected. These “hub” nodes may be promising candidates for peering relationships, as they are topologically close to a large portion of the Internet.

Colour nodes by connectivity – Having created a network layout, we can colour nodes by their connectivity. This helps to identify ASs that are popular hubs, as well as those ASs that are stub networks. Colour nodes by country – When selecting an AS for a peering relationship, it is often important to take physical proximity into consideration. In order to help identify candidate peering relationships on the basis of geography, we provide the ability to colour nodes by country. A topological layout loses information about physical proximity that would be available in a geographical display. However, our experience has shown that network connections often reflect some degree of geographical clustering, e.g., Australian ISPs tend to be interconnected, even though they are also connected to carriers in the U.S.A. Consequently, we often find that ASs in the same country tend to be clustered in our topological display. If the nodes are coloured by country, then it is easier to identify relevant ASs for peering relationships.

Figure 12: Walker’s tree algorithm illustrating the shortest paths for AS701

Figure 13: Walker’s tree algorithm illustrating the shortest paths for AS701 (country colour-coded)

Layout the shortest path tree to a selected node – Network managers are often interested in their relative distance to other networks. The shortest path distribution from their AS to the rest of the network can be useful information to them. A visual representation of the shortest path network from their AS, like the one shown in Figure 12, provides a visual depiction of the routes that should be taken by traffic originating from their AS. The topological distance from any node in the network is calculated using Dijkstra’s shortest path algorithm applied to the node of interest. The application of Dijkstra’s shortest path algorithm from a node of interest reduces the network to a planar structure. The reduced network is hierarchical in nature and is best viewed as a tree structure, where the node of interest is the root of the tree. We have used Walker’s tree algorithm12, which computes a rooted tree layout to view this shortest path network. Figures 12 and 13 show the tree structure obtained using Walker’s algorithm for the shortest paths from node AS701. This representation can help identify inappropriate routing policies, such as long paths to major ISPs. We can also extract the hop count

distribution from a node, based on the results of the shortest path calculation. For example, Figure 14 shows the extracted hop count distribution from node AS701 as produced by the VLNT tool. We can easily see that approximately 85% of nodes in this network are within 2 hops of the selected AS.

Figure 14: Plot of hop distribution from AS701 To illustrate the use of these features, consider our test network shown in Figure 15. Using a colour scheme corresponding to country, two distinct clusters appear in the network. The American cluster consisting of ASs from Canada, Argentina and the United States connected to AS 1239 (Sprint) shows a typical customer-provider relationship between ASs. The European cluster appears to the right. AS5459 (LINX - the London Internet Exchange) is one of the major Internet exchange points in Europe. It provides a physical interconnection for its members to exchange Internet traffic. The ASs connected to LINX in the European cluster are likely to be engaging in peering agreements.

Figure 15: Network layout colour-coded by country

Other forms of relationships between autonomous systems include mutual-transit and mutual-backup agreements. The former constitutes an agreement to provide connectivity to the Internet for two administrative domains involved in an agreement. Small ISPs who are located close to each other in geographic terms tend to be engaged in such agreements. For the displayed network, we can see that there is a strong correlation between geographical proximity and topological proximity and this is reflective of the various agreements and relationships between administrative domains and their autonomous systems. Taking a closer look at Figure 16 we a see a scattering of 1-connected autonomous systems (red) with connections to either AS1755 (Ebone, a European Backbone Provider), or AS3300 (AUCS, a global networking provider based in the Netherlands). Visualising this network, we see that AS6705 and AS8820 may find themselves disconnected from the network should their provider AS3300 dropout. It would be futile for AS6705 and AS8820 to establish a peering relationship with each other since they are essentially connected with the same

provider and will face the same reliability issues if AS3300 drops out from the network. A possible approach to improving reliability for AS6705 and AS8820 might be in establishing a peering or mutual transit agreement with AS1273 (a German ISP) since it has 2 interconnections that do not encompass AS3300.

Figure 16: Enlargement of network layout colour-coded by node connectivity The use of visualisation here has enabled the identification of options for ASs looking to improve their connectivity. Such a feature can easily be extrapolated to networks of a similar nature, such as Voice-over-IP networks. In that case, gateways would correspond to the highly connected points in the network, and useful colouring schemes could be employed to reflect details such as packet loss and packet delay to aid the management of such a network.

AS127 3

6 Further Work We have shown that VLNT can efficiently layout networks with over 1000 nodes. However, complete AS maps have over 20,000 nodes. None of the algorithms considered is able to layout networks of this size in a reasonable time frame. Furthermore, even if we can layout networks of this size, it is highly likely that the resulting display will be too congested to be of any practical use. An open problem is how to manage displays of such large networks. A promising approach to this problem is to use some form of clustering. Internet topologies have been found to exhibit a power-law distribution in terms of the connectivity of nodes13, i.e., a small proportion of nodes are highly connected, while a large proportion of nodes have low connectivity. We may be able to exploit this property in order to improve the efficiency and practicality of visualising large networks. We could first cluster nodes with low connectivity in order to simplify the structure of the network, and then begin by laying out the highly connected nodes. Once these nodes have been positioned, we could then position the remaining nodes with low connectivity. Users could view the layout of the clustered network at a high level of abstraction, and then select

a portion of the network to be displayed in detail. In this way, we can control the complexity of both the layout problem and the user’s display. Recently, several graph layout techniques based on clustering have been reported14,15,16,17. For example, Walshaw’s algorithm17 uses hierarchical clustering to simplify a given network, and then optimise the network layout at each level in the clustering hierarchy. We are currently investigating how this algorithm performs on Internet topologies with power-law distributions.

7 Conclusion We have presented a novel algorithm for visualising large network topologies, and described how we have implemented this algorithm in the VLNT tool. We have demonstrated that our hybrid layout algorithm achieves the aesthetics of the Kamada-Kawai algorithm while reducing computation time by 30% on network topologies from BGP routing data. Our impetus for developing the hybrid approach lies in the observation that the initial layout affects the convergence of the deterministic Kamada-Kawai algorithm and that the ISOM layout provides a good approximation to the final layout achieved by Kamada-Kawai. We also proposed a novel edge-tension gradient approach for the efficient termination of our algorithm, and demonstrated its advantages over the conventional approach based on energy change. Based on these technical advances, we have demonstrated how the key features in VLNT can help network managers to analyse and improve BGP routing topologies in their daily operations.

Acknowledgements We wish to thank the Route Views Project at the University of Oregon for providing their BGP routing data. This work was supported by the Australian Research Council.

References 1. R. Becker, S. Eick and A. Wilks, “Visualizing Network Data.” In IEEE Transactions on Visualization and Computer Graphics, vol. 1, pp. 16-28, March 1995. 2. H. Burch and B. Cheswick, “Mapping the Internet.” In IEEE Computer, vol. 32(4), pp. 97-102, April 1999. 3. T. Kamada and S. Kawai, “An Algorithm for Drawing General Undirected Graphs.” In Information Processing Letters, vol 31, pp.7-15, 1989. 4. B. Meyer, “Self-Organizing Graphs A Neural Network Perspective of Graph Layout.” In Proceedings of the 1998 Graph Drawing Symposium (GD’98), Montreal Canada, August 1998. 5. Y. Rekhter, “A Border Gateway Protocol 4 (BGP-4).” RFC 1771, IETF, March 1995. 6. L. Gao and J. Rexford, “Stable Internet Routing Without Global Coordination.” In IEEE/ACM Transactions on Networking, December 2001, pp. 681-692. 7. C. Labovitz, A. Ahuja, R. Wattenhofer and V. Srinivasan, “The Impact of Internet Policy and Topology on Delayed Routing Convergence.” In Proceedings of IEEE INFOCOM 2001, pp.537-546, 2001. 8. H. Tangmunarunkit, R. Govindan, S. Shenker and D. Estrin, “The Impact of Routing Policy on Internet Paths.” In Proceedings of IEEE INFOCOM 2001, pp.736-742, 2001. 9. D. Meyer, “Route Views Project Page.” http://www.antc.uoregon.edu/route-views/. March 2002. 10. B. Huffaker, E. Nemeth and K. Claffy. “Otter: A General-Purpose Network Visualization Tool.” In Proceedings o f the 9th Annual Conference of the Internet Society (INET’99), 1999. 11. P. Eades “A Heuristic for Graph Drawing.” In Congressus Numerantium, vol.42, pp. 149-160, 1984. 12. J. Walker. “A Node-Positioning Algorithm for General Trees.” In Software Practice and Experience, vol. 20 no. 7 , pp. 685-705, 1990. 13. M. Faloutsos, P. Faloutsos and C. Faloutsos, “On Power-Law Relationships of the Internet Topology.” In Proceedings of ACM SIGCOMM 1999, pp. 251-262, 1999. 14. P. Gajer, M. Goodrich, and S. Kobourov, “A Fast Multi- Dimensional Algorithm for Drawing Large Graphs.” In Proceedings of the 8th International Symposium on Graph Drawing, LNCS 1984, pp. 211-221, 2000. 15. D. Harel and Y. Koren. “A Fast Multi-scale Method for Drawing Large Graphs.” In Proceedings of the 8th International Symposium on Graph Drawing, LNCS 1984, pp. 183-196, 2000.

16. A. Quigley and P. Eades. “FADE: Graph Drawing, Clustering, and Visual Abstraction.” In Proceedings of the 8th International Symposium on Graph Drawing, LNCS 1984, pp. 197-210, 2000. 17. C. Walshaw. “A Multilevel Algorithm for Force-Directed Graph Drawing.” In Proceedings of the 8th International Symposium on Graph Drawing, LNCS 1984, pp. 171-182, 2000.

Maximum Number of Hops Lo L1 Figure 1: A Diagrammatic Representation of the Various Parameters for Edge-Tension

Figure 2: ISOM layout of Network 1

Figure 3: Kamada-Kawai layout of Network 1

Figure 4: Hybrid layout of Network 1

Edge Tension

Edge Tension v.s. Epoch 0.6

0.4

0.2 0

500

1000

1500 2000

2500

3000

3500

Epoch Kamada-Kawai

Hybrid Approach

Figure 5: Layout convergence

Figure 6: Layout of Network 2 using Kamada-Kawai with conventional energy criterion

Energy Change

Energy Change vs. Epoch 1.00E+07 8.00E+06 6.00E+06 4.00E+06 2.00E+06 0.00E+00 0

500

1000

1500

2000

2500

3000

3500

4000

Epoch

Figure 7: Convergence of conventional energy criterion on Network 2

Figure 8: Layout of Network 2 using Kamada-Kawai with edge-tension gradient criterion

Edge-Tension vs. Epoch

Edge-Tension

1 0.9 0.8 0.7 0.6 0.5 0.4 0

500

1000

1500

2000

2500

3000

3500

4000

Epoch

Figure 9: Convergence of edge-tension gradient criterion on Network 2

Computation Time v.s. Size of Graph

106.48

Computation Time (sec)

100

80

64.35

60

40

24.35

20

1.75

0 0

1000

4.12 2000

3000

4000

5000

Number of Nodes

ISOM Layout

Figure 10: Scalability of ISOM

6000

7000

Computation Time (sec)

Computation Time v.s. Size of Graph 3000

2990.14

2500 2102.22 2000 1792.22 1500

1358.17

1000 620.93 437.3

500 0 0

200

400

600

800

1000

1200

1400

1600

Number of Nodes

Kamada Kawai Only ISOM & Kamada Kawai (Hybrid Approach)

Figure 11: Scalability of Kamada-Kawai and Hybrid layout algorithms

AS701

Figure 12: Walker’s tree algorithm illustrating the shortest paths for AS701

Figure 13: Walker’s tree algorithm illustrating the shortest paths for AS701 (country colour-coded)

Figure 14: Plot of hop distribution from AS701

AS5459 LINX

European Cluster

American Cluster AS1239 Sprint

Figure 15: Network layout colour-coded by country

AS330 0

AS670 5 AS882 0 AS127 3

Figure 16: Enlargement of network layout colour-coded by node connectivity