BOSAM: A Tool for Visualizing Topological Structures Based on the Bitmap of Sorted Adjacent Matrix

Transcription

1 BOSAM: A Tool for Visualizing Topological Structures Based on the Bitmap of Sorted Adjacent Matrix Yuchun Guo and Changjia Chen School of Electrical and Information Engineering Beijing Jiaotong University Beijing , P. R. China yuchun_guo@yahoo.com, changjiachen@sina.com ABSTRACT Though the adjacent matrix is a representation of a topology, it does not tell the topological structures explicitly. A visualization tool which can represent topological structures explicitly is desired for model validation or measurements comparison. However, displaying nodes and links directly in a space as a routine is ineffective for massive topologies, like that of Internet. Instead, researchers have recourse to various statistical metrics extracted from the adjacent matrix. Such a statistical metric often reflects only a single aspect of the topology but at the cost of burdensome computation. In this paper a novel simple visualization tool based on displaying the Bitmap of Sorted Adjacent Matrix, BOSAM, is proposed to fill the gap. Validations of AS-level topology models and comparisons of topology measurements are provided to show that BOSAM works efficiently. The bitmap displaying can avoid messy cross lines and make the correlation between nodes stand out. The rules of sorting node indexes can be defined flexibly to expose diverse topological structures. Moreover, some fractal features in the AS graphs or generated graphs, which are not easily to be covered by statistical analysis, can be exposed explicitly as well. Parting from the huge burden of statistical computation, BOSAM can be realized easily. Categories and Subject Descriptors C.2.1 [Computer Systems Organization]: Network Architecture and Design Network topology. C.4 [PERFORMANCE OF SYSTEMS]: Measurement techniques, Modeling techniques, Performance attributes Keywords Topological structures, bitmap, topology visualization, validation 1. INTRODUCTION The topological structures are critical for many aspects of Internet research, such as the analysis of routing performance, network resilience to anomalies or attacks, multicast scalability, subscriber behaviors, network protocols, etc. Though many studies have been made on discovery and modeling of topology [14][2][1][3][10][8] over the past decade, several recent works [5][9][6][15][11] have post suspicions on the research outcomes so far obtained. For example, the power laws distribution of node degrees is suspected as an artifact due to the topology discovering This work is supported by This work is sponsored by the National Natural Science Foundation Committee of China under Projects and process [9]. Still needed are further studies on modeling the mechanism of evolution of Internet topology and its effects on routing and other aspects of networking [11]. For evaluating topology models [5][3][15] or measurements [5][11], some validation is needed. Though an adjacent matrix represents the topology, it does not tell the topological structures explicitly. Apparently an effective visualization of topology structures is desired for the validation. However, the existing visualization schemes for AS-level Internet topologies are undesired in that they are based on displaying nodes and links in a geographical space [4]. Graphs from these schemes often suffer a messy of crossing lines and show few explicit topological structures. Thereafter researchers have to extract the statistical metrics from the adjacent matrix, such as degree distribution, joint degree distribution, clustering coefficient, rich club connectivity, distance, eccentricity, betweenness, etc., as measures of topological structures [11]. Such a statistical metric often represents a single aspect of topology structures. Hence for validating the topological characteristics of a model-generated or measured graph completely, more and more metrics have been added to the list of measures for validation. A typical example is the recent work [11], which systematically compares the statistical properties of different topology measurements but unavoidably with huge computation load. In fact, this way of validation faces the antinomy between the simplicity of validation process and the integrity of characterizing topological structures. A novel simple visualization tool for representing topological structures is proposed for solving this problem. We name this tool BOSAM as it is based on the Bitmap of Sorted Adjacent Matrix. BOSAM works with only two operations: sorting of node indexes and displaying bitmap of the indexes-sorted adjacent matrix. The bitmap of adjacent matrix is displayed in a two-dimensional grid with a point plotted at the position (i,j) only if a link exists between node i and j. As the topology of a graph can be represented with adjacent matrix, the bitmap of adjacent matrix is a graphical representation of topology independent with the actual positions of nodes in the physical space. It avoids messy cross lines and makes the correlation between nodes stand out. Change the node indexes according to some sorting rule, the bitmap may be changed significantly and a different structural feature may be discovered while the induced graph is kept isomorphic to the original one. Moreover, with the sorting of node indexes, the difference of AS number assignment and node indexing between model-generated graphs and real AS graph can be eliminated so that the comparison between different graphs make sense. Both sorting and bitmap displaying can be realized easily without heavy computation burden.

2 With BOSAM, the rules of sorting node indexes can be defined flexibly to expose diverse topological structures explicitly. It has been emphasized in [11] that the one-dimensional degree distribution is not enough for characterizing the Internet topology so that the two-dimensional joint distribution of node degrees has to be considered. Unfortunately, the analytical form of the joint distribution of node degrees has not yet been obtained for the Internet AS graph and it is evaded via displaying contour map of the statistical outcomes in [11]. While with BOSAM, these features can be visualized easily, as shown in section 3 and 4. Even structures not easily to be discovered by the statistical analysis, for instance, the fractal features, can be exposed with BOSAM. To show that BOSAM works efficiently, we give an example of validating several typical topology models, the Gnp random graph model [7], BA model [2] and PFP model [15]. Given the number of nodes, Gnp model connects a pair of nodes with probability p (0 p 1). Though Waxman model [14], the first random graph model exclusively designed for Internet topology, introduces the geographical distance between a pair of nodes into the connecting probability, the degree distribution of a Waxman graph is the same with that of a Gnp random graph. Hence we choose the Gnp model as an compared object. BA model is built on the preferential attachment (PA) of nodes and incremental growth of network. Starting with a core with m 0 nodes connecting randomly or completely to each other, BA model adds one new node with m new links to constructed graph each step, and chooses node j as the other end of a new link with the probability d j p j =, (1) d where d k is degree node k. PFP model [15] generates a topology graph with the following steps: a) A new node is connected to a host node 1 with the probability p [0,1], and the host node is connected to a peer node 2 ; b) a new node is connected to a host node with the probability q [ 0,1 p], and the host node is connected to two peer nodes; c) a new node is connected to two host nodes with the probability 1 p q, and one host is connected to a peer. The probability of choosing node j as a host or a peer is 1+ δ log d k k 10 j d j p ( j) =, δ [0,1]. (2) 1+ δ log10 di d i i As reported in [15], setting δ = 0.048,p = 0.3 和 q = 0.1, PFP model match the AS connectivity measurements [4] closely in terms of many statistical properties. Another example of measurement comparison is also provided to compare BOSAM to the recent work [11]. With these two examples, it can be found that BOSAM can be served as an efficient visualization tool for the validations of topology models and the comparisons of different measurements. 1 A node exists in the so-far constructed network. 2 Also a node exists in the so-far constructed network, but distinct with the chosen host node(s). The rest of this paper is organized as follows, section 2 gives a description of our visualization tool BOSAM; section 3 and 4 present some outcomes of the validation of 3 models with the real measurements, and those of the comparisons of 4 measurements of Internet AS topology, respectively; section 5 concludes this paper. 2. BITMAP OF SORTED ADJACENT MATRIX Given a graph G(V, E), V =N, its topology can be represented with the adjacent matrix A = { A( i, j) i, j V}, where i, j are the indexes of nodes. If there is a link between node pair (i,j), then A(i,j) = 1; otherwise A(i,j) = 0. BOSAM visualizes a topology graph via displaying the adjacent matrix A as follows: in a N N grid, a dot is plotted on the position (i, j) only if A( i, j) = 1, i, j V. The figure generated in this way is called the bitmap of matrix A, which explicitly represents the adjacency in a 2-dimensional index space neatly without crossing links. The Internet graph is known as a sparse one so that its adjacent matrix is sparse and the sparse pattern, i.e. the nonzero elements A( i, j) = 1, i, j V, can be visualized with a single command spy(a) which is a predefined function in Matlab. To validate the graph models, we use the real measurement of AS adjacency in January 1st, 2001 provided by NLANR [12] (originated from the Oregon RouteViews project [13]) with 3750 distinct AS numbers. Consider the AS graph as undirected, the averaged degree is Set the number of nodes N = 3570 for the 3 models to be validated, the Gnp, BA and PFP model. For the Gnp random graph model, set the parameter p = 3.94/N = ; for the BA model, set m = 3, m 0 = 3; for the PFP model, set δ = 0.048, p = 0.3, q = 0.1. The generated graphs from Gnp, BA and PFP model and the AS graph are shown with bitmaps in Figure 1, in which both the x- and y-axis in each subgraph denote the node indexes. As shown, the bitmap of the generated graph from the Gnp model is uniformly distributed, which means each pair of nodes are connected equally likely. The BA and PFP models are both growth models so that in the corresponding bitmaps of these two models the nodes with smaller indexes are the ones entering the network earlier and possibly having larger degrees. In these two bitmaps, especially the PFP bitmap, dots are denser in the top-left parts but sparser in the bottom-right parts, which tells that the connectivity between the earlier nodes (with smaller indexes) are more intensive than that between the later nodes (with larger indexes). For simplicity and compactness of representation, the distinct AS numbers appeared in the BGP routing tables collected by Route Views project [13] are renumbered continuously. Then its bitmap of adjacent matrix is also plotted in Figure 1. As shown the AS graph is non-uniformly connected. However, the AS numbers assignment is different from the index assignments of the BA and PFP model so that the direct comparison between the bitmap of AS graph and those of the BA and PFP graphs is not quite significant. To tackle this problem, BOSAM visualizes the bitmap of adjacent matrix with the node indexes sorted so that the sorted bitmaps of different graphs are with the same order of indexes. The sorting rules can be defined flexibly. For example, the node indexes of the bitmaps shown in Figure 2 have been sorted in the decreasing order of node degrees. Except that the bitmap of Gnp graph in Figure 2 looks similar

3 with that in Figure 1, some special structural features appear in the bitmaps of BA and PFP graph as well as the AS graph, which cannot be found in the corresponding bitmaps in Figure 1. This sorting operation to node indexes is equivalent to a rename operation of nodes which keeps the topology unchanged. Adjusting the sorting rule of BOSAM, different structures can be brought out as shown in section 3 and 4. The considerations for devising the sorting rule are discussed in section 3. With the index sorting and bitmap showing, BOSAM can bring out the topological structures clearly for massive graphs without heavy processing load the statistical analysis requires. 3. VALIDATION OF TOPOLOGY MODELS For sorting node indexes, a weight function for node indexes must be defined. Denote the node weight function w( i), i V, which can be defined flexibly for implementing different sorting rules. As this paper focus on showing how the proposed visualization tool works, we choose only 3 parameters which we consider the most relevant to the topological properties of Internet graphs. For node i, these 3 parameters are a) deg(i ), the degree of node i; b) max( deg( )), j Nr( i) and c) min( deg( )), j Nr( i) j, the maximal degree of i s neighbors, j, the minimal degree of i s neighbor. Node degree is the mostly noticeable metric in the Internet topology modeling since it is treated as a sign of the importance of a node and the main element in determining the connecting probability in most topology models designed for Internet. Consider the Internet as a network with the high-degree nodes concentrating in the core and the low-degree nodes scattering peripherally, the maximal neighbor-degrees for a node is a measure of the strongest cohesion force from the network core it receives, and the minimal neighbor-degrees for a node is a measure of the strongest radiation force from the peripheral it receives. The sorting weight for node i is defined as w( i) = a deg( i) + a maxdeg( j) + a min deg( ), (3) j deg( j) deg( j) where Nr(i) is the neighborhood of node i, a 1, a 2 and a 3 are the weight parameters which weight the corresponding measure as a primary or subsidiary key. Generally, the greater the absolute value of a parameter, the more significant its corresponding measure in the weight function. If the node indexes are expected to be sorted in an alphabetical way with more than one measure, some constraints on parameters should be kept. For instance, if the node indexes need to be sorted in the order of node degrees, maximal neighbor-degrees and minimal neighbor-degrees alphabetically, the parameters should be set so that a2 a3 max D and a1 ( a2 + a3 ) max D, where max D is the maximal node degree. The sorting operation is conducted with ascending order of defined node weights. Some results of the validation of 3 topology models with BOSAM under 5 sorting rules are presented below. Sorting rule 1: Descending order of degrees. Set the parameters in weight function (3) as a 1 = 1, a 2 = a 3 = 0, then the weight function contains only one key, the node degree. Sort node indexes with the ascending order of node weights defined in (3) actually equivalent to the descending order of node degrees, the bitmap of adjacent matrix exhibit the degree sequence and the correlation connectivity between nodes as shown in Figure 2. The bitmap of Gnp graph remains almost the same with the unsorted one in Figure 1. The bitmap of the BA graph is apparently different from the corresponding unsorted one, which tells that BA graph is not uniformly connected but with special structures: the high degree nodes (with small indexes) prefer connect to nodes with diverse degrees while low degree nodes (with larger indexes) tend to connect to high and medium degrees nodes instead of low degree nodes. Different from the correlation between nodes in BA graph, the low degree nodes in AS graph connect only to high degree nodes. Among the 3 generated graphs, the bitmap of PFP graph is most similar to that of AS graph. But the low degree nodes in PFP graph also connect to medium degree nodes as well, and some correlation structures similar to that of the BA graph exist for the medium and low degree nodes (the lower-right part). In addition, it needs to be noticed that the BA, PFP and AS graph all have cliques of top high-degree nodes These cliques are the so-called rich clubs in [15], which can be viewed as the dense core of a network. Obviously, the rich-clubs of different graphs are of different sizes, and the one of AS graph is the smallest. Sorting rule 2: Descending order of degrees and maximal neighbor-degrees. Setting the node weights (3) with a 1 = 1E7, a 2 = 1E3, the nodes with same degree can be sorted into a continuous sequence in the order of descending maximal neighbor-degrees, the bitmaps with sorted node indexes are shown in Figure 3. Compared with the bitmaps in Figure 2, more structures have been unveiled: in the Gnp graph, the number of high-degree nodes and the number of low-degree nodes are much less than that of medium-degree nodes, and nodes of any degree have almost the same pattern of degree correlation. In the bitmaps of BA and PFP graph, the lower the node degree, the wider the span of the indexes with this degree, i.e. the more the nodes of this degree, which is consistent with the power law degree distribution in Internet AS topology [8]. On the other hand, the neighbor-degree distribution for the nodes with same degree in the BA graph is quite different from that in AS graph. But the PFP graph is similar to the AS graph in this sense except that its range scales of neighbor-degrees with same degrees are different from that of AS graph. For instance, the neighbor-degree distributions for 1-degree nodes and the correlations between low-degree nodes to medium-degree nodes are quite different between PFP graph and AS graph. Sorting rule 3: Descending order of degrees, maximal neighbor-degrees and minimal neighbor-degrees. Setting the node weights (3) with a 1 = 1E7, a 2 = 1E3, a 3 = 1, bitmaps with node indexes sorted according to rule 3 are presented in Figure 4. With the third key introduced in this rule, the tierce structural property is exposed in the bitmap. With this rule, the nodes will be sorted in the descending order of degrees, maximal neighbor degrees and minimal neighbor degrees alphabetically. It can be found that in the AS graph, there are power laws not only of the degree distribution but also of the neighbor-degree distributions for the nodes with same degree for low-degree nodes, and the neighbor-degree distributions for a single node with high degree. Compared with Figure 3, the clustered lines of dots appear in all the generated graphs of three models, but still the bitmap of PFP graph resembles that of the AS graph most. In fact, the maximal

4 and minimal neighbor-degrees represent the intensity of cohesion and radiation force that node receives, and the difference between them are the ranges of neighbor-degrees for a node. Sorting rule 4: Descending order of degrees and minimal neighbor-degrees. Setting the node weights (3) with a 1 = 1E7, a 2 = 0, a 3 = 1E3, bitmaps with node indexes sorted according to rule 4 are plotted in Figure 5. Sorting rule 5: Descending order of degrees, maximal neighbor-degrees and minimal neighbordegrees. Setting the node weights (3) with a 1 = 1E7, a 2 = 1, a 3 = 1E3, bitmaps with node indexes sorted according to rule 5 are presented in Figure 6. With rule 4 and 5, the connection structures between low-degree nodes can be exposed, and still the PFP model approximates the AS graph the most but with different scales of structures. The inconsistency between BA graph and the AS graph is more serious than the cases of rule 2 and rule 3 (see Figure 3 and 4). Limited to the space, outcomes of other sorting rules are not presented here. The conclusions can be drawn are the same with above experiments: PFP graph is the one having the most proximate topological structures to the AS graph, but still need to be improved, for instance, in the scales of diverse structures, especially the adjacency properties in the low-degree part. Besides, with some sorting rules, like rule 2 to 5, some fractal structures can be found in the generated graphs and AS graphs, as shown in Figures 3~6, which deserve further studies. 4. COMPARISONS OF DIFFERENT AS TOPOLOGY MEASUREMENTS Some results of the comparison of 4 different AS topology measurements with BOSAM are given in Figures 7~12. The measurement sets are Whois (March 7, RIPE), Skitter, BGP updates and BGP tables of March 2004, which are used in the recent work [11] and provided by CAIDA [4]. The sorting rules used here are the same with those used in Section 3. Compared with the work in [11], BOSAM is simple to implement without burden of statistical analysis and can bring out the structures in topology graphs neatly clear. The results from applying BOSAM are consistent with those of [11]: Whois data is far different from the other 3 data sets, Skitter is different from the BGP tables and BGP updates data, while the latter two BGP data sets are almost consistent in essence. 5. CONCLUSIONS Though the adjacent matrix is a representation of a topology, it does not tell the topological structures explicitly. However, displaying nodes and links directly in a space as a routine is ineffective for massive topologies, like that of Internet. The common routine of validating a proposed topology model is via burdensome statistical analysis. In this paper a visualization tool BOSAM has been proposed. By displaying the bitmaps of the adjacent matrices with the node indexes sorted, BOSAM can make the topological structures stand out without messy cross lines and keep the topology integrated. The examples show that BOSAM can be used as an efficient tool for model validation and measurement comparison. The sorting rules can be defined flexibly to dig out different topological structures, even those not covered by the proposed statistical measures. Parting from the huge burden of computation of statistical properties in the traditional statistical scheme, BOSAM can be realized simply and work efficiently for massive topologies. 6. REFERENCES [1] W. Aiello, F. Chung, and L. Lu, A Random Graph Model for Massive Graphs, In Proc. STOC [2] A. L. Barabasi and R. Albert, Emergence of scaling in random networks, Science 286, (1999). [3] Tian Bu and Don Towsley, On distinguishing between Internet power law topology generators, In Proc. Infocom [4] s/ [5] Q. Chen, H. Chang, R. Govindan, S. Jamin, S. J. Shenker, and W. Willinger, The origin of power laws in Internet topologies revisited, In Proc. of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies, IEEE Computer Society, [6] A. Clauset and C. Moore, Traceroute sampling makes random graphs appear to have power law degree distributions, preprint [7] P. Erdös and A. Rényi, On Random Graphs I, Publ. Math., Debrecen 6 (1959), pp [8] M. Faloutsos, P. Faloutsos, and C. Faloutsos. On power-law relationships of the Internet topology. Proc. of ACM SIGCOMM, [9] Anukool Lakhina, John W. Byers, Mark Crovella and Peng Xie, Sampling Biases in IP Topology Measurements, IEEE INFOCOM '03. [10] Lun Li, David Alderson, Walter Willinger and John Doyle, A First-Principles Approach to Understanding the Internet's Router-level Topology, Sigcomm2004. [11] Priya Mahadevan, Dmitri Krioukov, Bradley Huffaker, Xenofontas Dimitropoulos, kc claffy, Amin Vahdat, Comparative Analysis of the Internet AS-Level Topologies Extracted from Different Data Sources, 2005, in progress [12] NLANR, [13] Route Views, [14] B. M. Waxman. Routing of multipoint connections, IEEE Jour. of Selected Areas in Comm., Vol. 6, No. 9, [15] S. Zhou and R. J. Mondragon. Accurately modeling the Internet topology. ArXiv: cs.ni/

5 Figure 1. Bitmaps with unsorted indexes Figure 2. Bitmaps with indexes sorted in descending degrees.

6 Figure 3. Bitmaps with indexes sorted in descending degrees and maximal neighbordegrees. Figure 4. Bitmaps with indexes sorted in descending degrees, maximal neighbor-degrees and minimal neighbor-degrees.

7 Figure 5. Bitmaps with indexes sorted in descending degrees and minimal neighbordegrees. Figure 6. Bitmaps with indexes sorted in descending degrees, minimal neighbor-degrees and maximal neighbor-degrees.

8 Figure 7. Comparison of topology measurements: bitmaps with indexes unsorted. Figure 8. Comparison of topology measurements: bitmaps with indexes sorted in descending degrees.

9 Figure 9. Comparison of topology measurements: bitmaps with indexes sorted in descending degrees and maximal neighbor-degrees. Figure 10. Comparison of topology measurements: bitmaps with indexes sorted in descending degrees, maximal neighbor-degrees and minimal neighbor-degrees.

10 Figure 11. Comparison of topology measurements: bitmaps with indexes sorted in descending degrees and minimal neighbor-degrees. Figure 12. Comparison of topology measurements: bitmaps with indexes sorted in descending degrees, minimal neighbor-degrees and maximal neighbor-degrees.