Spectral Analysis of Backbone Networks Against Targeted Attacks

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1 Spectral Analysis of Backbone Networks Against Targeted Attacks Tristan A. Shatto* and Egemen K. Çetinkaya* *Department of Electrical and Computer Engineering Intelligent Systems Center Missouri University of Science and Technology, Rolla, MO 65409, USA {taszrc, mstconets Abstract Network science has been a central focus to correctly model and study resilience characteristics of communication networks. There have been many metrics used to represent connectivity of graphs; however, they do not suffice to compare networks with different numbers of nodes and links. The normalized Laplacian spectra enables network scientists to analyze network structures beyond what traditional graph metrics lacks. In this paper, we study the normalized Laplacian spectra of five backbone networks against targeted attacks. The physical and logical level of four commercial and one research backbone provider networks is studied. The intelligent attacks are modeled based on important graph centrality metrics of betweenness, closeness, and degree. Our results indicate that spectra of eigenvalues converge to zero after attacks. Moreover, we also identify that while in some scenarios different centrality-based attack strategies yield identical eigenvalue distribution, in other scenarios different attacks yield different eigenvalue distributions. Index Terms Internet, backbone network, Laplacian, eigenvalue, normalized Laplacian spectra, resilience, centrality, betweenness, closeness, degree, attack I. INTRODUCTION AND MOTIVATION Modeling and analysis of the Internet has been one of the focus areas of network science research. Many network models have developed to study mathematical underpinnings [1] [4]. The aim of these studies is to understand the evolution of networks, as well as develop mechanisms so that network services remain uninterrupted during attacks and failures. Despite the efforts and technological progress made, communication networks face a variety of challenges and perturbations that disrupt normal operation [5]. Besides, the Internet that operates by many different protocols and functional layers is a complex system. For example, the Internet can be modeled by different levels such as IP, PoP (Point of Presence), and physical [6], in which different defense mechanisms need to be developed and adapted for disparate threat models for different levels. It is therefore important to: i) investigate communication networks that provide important services to the society, ii) analyze these networks at distinct levels for correct modeling against threats, iii) study mathematical underpinnings to improve its resilience against challenges and perturbations. In network science, from a graph theoretic perspective, despite many efforts, there is still not a clear pathway to evaluate graphs with different order (number of nodes) and size (number of links). Consider the two graphs: one with 10 nodes and 9 links as a star topology and the other with 10 nodes and 9 links as a bus (or linear) topology. While both graphs have the same number of nodes, an attack at the central node in the star topology will eliminate all potential flows between every pair of nodes, whereas in a linear topology, removal of a node will not eliminate the flow of information between every pair of nodes completely. While both the star and linear topologies have the same average node degree of 1.8, an intelligent attack against the star topology is obviously more damaging. This simple example shows a scenario in which it might be very misleading to evaluate connectivity by means of relying on a single graph metric. Furthermore, we can expand this argument to evaluate different order and size graphs. How do we know which graph is more resilient to attacks and challenges: one with 10 nodes 9 links, or the one with 100 nodes and 90 links? Obviously, rigorous analysis and tools are needed to investigate such scenarios that further motivates this research. The graph spectra captures eigenvalues and their multiplicities. The normalized Laplacian spectra (nls), which is a connectivity matrix normalized according to the degree of nodes, is a mathematical data structure that can help to evaluate resilience of networks that differ in size and order. Moreover, the nls provide the connectivity information in a spectrum of eigenvalues spread between [0,2] range, rather than a single value. In this paper, we evaluate the normalized Laplacian spectra of five backbone networks against intelligent attack scenarios. Four commercial service provider backbone networks: AT&T, Level 3, Sprint, and TeliaSonera, as well as the Internet2 research backbone network, and their corresponding physicaland logical-level topologies are used to analyze their spectra. The attack strategies are chosen based on well-known graph centrality metrics betweenness, closeness, and degree. With the aim of understanding the nls under attack scenarios, our results indicate that as more network components are removed, eigenvalues converge to a zero value. The nls analysis of these five backbone networks also results that while in some scenarios different attack strategies yield different eigenvalue distribution, in some other cases different attack strategy yield identical eigenvalue distribution.

2 The paper is organized as follows: The background and past work about spectral analysis of communication networks is presented in Section II. The details of the topological dataset is presented in Section III and the description of our methodology is presented in Section IV. In Section V, we present our experimental results of normalized Laplacian spectra to understand the effect of intelligent attacks on backbone networks, then we conclude and showcase some probable future work in Section VI. II. BACKGROUND AND RELATED WORK Let G = (V,E) be an unweighted, undirected graph with n vertices and m edges. The vertex set is denoted by V = {v 0,v 1,...,v n 1 } and the edge set is denoted by E = {e 0,e 1,...,e l 1 } for G. The graph connectivity can be represented by several structures such as an adjacency matrix, incidence matrix, Laplacian matrix, and normalized Laplacian matrix [7], [8]. A(G) is the symmetric adjacency matrix with no self-loops where a ii =0, a ij = a ji =1if there is a link between {v i,v j }, and a ij = a ji =0if there is no link between {v i,v j }. B(G) is the n l incidence matrix in which B ij =1 if vertex v i and edge e j are incident. The Laplacian matrix of G is: L(G) =D(G) A(G) where D(G) is the diagonal matrix of node degrees, d ii = deg(v i ). Given the degree of a node is d i = d(v i ), the normalized Laplacian matrix L(G) is denoted: 8 1, if i = j and d i 6=0 >< 1 L(G)(i, j) = p, if v i and v j are adjacent di d >: j 0, otherwise Let M be a symmetric matrix of order n and I be the identity matrix of order n. Then, eigenvalues ( ) and the eigenvector (x) of M satisfy M x = x for x 6= 0, viz., eigenvalues are the roots of the characteristic polynomial, det(m I) =0. The set of eigenvalues { 1, 2,..., n} together with their multiplicities (number of occurrences of an eigenvalue i) define the spectrum of M. There exists several manuscripts on spectral graph theory covering the topic in depth [7] [11]. The research on graph spectrum with emphasis on communication networks has been progressing primarily in two directions. The first group of research studies modeling and analyzing communication networks using graph spectra, particularly AS-level Internet topologies [4], [12] [18]. The second group of research focuses on connectivity, resilience, robustness of communication networks and how to improve connectedness of communication networks using spectral properties [19] [22]. We summarize the past work chronologically in the proceeding paragraphs. The normalized Laplacian spectrum of AS-level topologies has been shown to differ significantly from that of synthetically generated topologies [12]. The adjacency spectrum of ASlevel topology of the Internet was analyzed and clusters were found to be correlated with the spectrum [13]. The Laplacian spectrum of the IP-level topology of the Internet is compared against synthetically generated topologies, and that spectra differs between measured and synthetic topologies [14]. A weighted spectral distribution metric has been proposed and has shown that synthetically generated AS-level graphs can be generated using spectral properties [15]. The AS-level Internet topology has an evolving structure (e.g. connectivity and clustering properties changing) based on spectral analysis [15], [16]. The normalized Laplacian spectrum of physical- and logical-level backbone networks are compared, and it was identified that spectra of the motorway network is similar to physical-level backbone network [17], [18]. The spectral gap of a normalized Laplacian spectrum is utilized to identify critical nodes in a network [19]. Several spectral-related metrics (i.e. algebraic connectivity, spectral gap, network criticality) has been utilized to analyze robustness and optimize synthetically-generated random graphs [20], [21]. The AS-level topologies are analyzed and optimizations are performed against node and link deletions using normalized Laplacian spectra [22]. In this paper, we analyze the nls of physical- and logical-level five backbone provider networks under targeted attack scenarios. III. TOPOLOGICAL DATASET We utilize backbone networks that are geographically located within the contiguous US to experiment with our spectral analysis methodology. We use four commercial service provider networks: AT&T, Level 3, Sprint, and TeliaSonera, as well as the Internet2 research network. We analyze both the physical-level topologies (e.g. fiber) and logical-level (e.g. PoP-level) topologies of these networks. We also note that our objective is not to compare the performance of different service provider networks, but rather investigate the spectral properties of these realistic backbone networks under challenged conditions. The properties number of nodes, number of links, average node degree, closeness, maximum node betweenness, and maximum link betweenness of these five backbone networks are shown in Table I. Note that L1 represents physical-level and L3 represents logical-level topologies in Table I. TABLE I TOPOLOGICAL PROPERTIES OF BACKBONE NETWORKS Network Node Link Avg. Max. Node Max. Link Clo. Deg. Between. Between. AT&T L AT&T L Level 3 L Level 3 L Sprint L Sprint L TeliaSonera L TeliaSonera L Internet2 L Internet2 L We obtain the data from the KU-Topview Network Topology Tool [23]. An in-depth discussion of structural and visual

3 (a) Internet2 L1 non-degraded (b) Internet2 L1 10 nodes (c) Internet2 L1 20 nodes (d) Internet2 L1 30 nodes (e) Internet2 L1 50 nodes (f) Internet2 L1 56 nodes Fig. 1. Spectra of Internet2 backbone network based on node degree removal properties of these graphs are explained [4], [17], [18]. While the methodology presented in this paper can be applied to other datasets such as the Internet Topology Zoo [24], [25] and Rocketfuel [26], [27], we believe that the dataset we analyze suffices to present the methodology and conclusions. IV. METHODOLOGY A graph-theoretic approach is applied to evaluate the connectivity of backbone networks. The normalized Laplacian spectra (nls) is utilized to measure and observe connectivity. We use the following Python packages for our calculations: the Python NumPy package for numerical calculations [28], Python NetworkX library [29] for simulating network attacks, and Python Matplotlib 2D graphics package for plotting figures [30]. We use three node centrality measures for removal of nodes betweenness, closeness, and degree. For link removals, we use edge-betweenness measure [31]. The betweenness measure informs of the number of shortest paths that flow through a node or a link. Closeness is the reciprocal of farness and computed as the reciprocal of the sum of the distances from the given node to all the other nodes. Degree is calculated as the sum of the number of incident links to a node. While there exist many other measures [32], we want to observe how the backbone networks behave under attack scenarios using the well-known graph metrics. Investigating the nls behavior of graphs under different graph measures will be part of our future work. The sequence of steps involved in connectivity analysis of backbones is as follows. First we calculate the given graph metric for all nodes (i.e. node betweenness, closeness, or degree) or links (link betweenness). We sort the nodes or links in the descending order and remove the highest node or link. We iterate this until we reach to a set number of nodes or links. The removal scenarios are representative of intelligent targeted attacks with topological information. For this study, we set the number of components to be removed at: 10, 20, 30, 50, 100, 200, and 300. Additionally, the maximum number of nodes or links to be removed is limited by the number of nodes or links exist in a backbone network. An example node removal scenario (based on node degree) is shown in Figure 1 for the Internet2 L1 backbone network (Internet2 L1 network has 57 nodes and 65 links as listed in Table I). Figure 1a depicts the network without any node removal; Figure 1b depicts the network snapshot with 10 highest-degree nodes being removed iteratively; Figure 1c depicts the network snapshot with 20 highest-degree nodes being removed iteratively; Figure 1d depicts the network snapshot with 30 highest-degree nodes being removed iteratively; and Figure 1e depicts the network snapshot with 50 highest-degree nodes being removed iteratively. Since the number of nodes in the Internet2 L1 topology is 57 and the attack strategy is based on degree removal, the final step included removal of 56 nodes, as shown in Figure 1f. The final step in evaluating backbone network connectivity is the computation of nls. To better visualize this spectrum and

4 how the network would break down under a targeted attack, we plot the RCF (relative cumulative frequency) as we iteratively and adaptively remove network components. We note that a discussion on RCF vs. RF (relative frequency) representation of deterministic eigenvalues is discussed in our earlier work, and that for multiple lines in a plot the information presented via RF can be noisy; therefore, we use RCF in this paper to represent the nls [18]. The example RCF for the Internet2 L1 backbone network is shown in Figure 2. Each line in Figure 2 represents the RCF of the eigenvalues after a particular number of components have been removed the removal scenario in this case is determined by the degree of nodes. We note important points about inferring conclusions in looking to RCF [18], [33]. First, the number of connected components is captured by the number of zero-valued eigenvalues. For example, when the last node is remaining in the network after 56 nodes are removed, the eigenvalue of that single node is 0. Second, the nls is quasi symmetric around eigenvalue multiplicity of 1. Third, the eigenvalue multiplicity of 1 represents that there are motifs in the graph [33]. An ideally connected full-mesh graph will have all of its eigenvalues around 1 (with the exception of one 0 eigenvalue). Finally, the eigenvalues closer to 2 means graph is closer to being a bipartite graph. (a) Internet2 L1 Fig. 2. Spectra of Internet2 L1 network based on node degree removal V. RESULTS To study the possible effects of potential targeted attack comprehensively, we generate RCF plots while using important graph metrics to systematically remove network components. For the five backbone networks (including physical (L1) and logical (L3) topologies) that we analyze the normalized Laplacian spectra (nls), we have removed components based on node betweenness as shown in Figure 3, node closeness as shown in Figure 4, node degree as shown in Figure 5, and link betweenness as shown in Figure 6. From our analysis of the RCF of backbone networks, we find that as the graph becomes disconnected, an increase in the number of zerovalued eigenvalues can be clearly seen. In other words, as more disconnected components appear in the graph after each step in the removal sequence, the zero-valued eigenvalue multiplicity dominates the spectra. The visible change in the RCF plots is that the quasisymmetric diagonal line converges to be a straight line across the top portion of the spectra plots as more network components are attacked and become inoperable. Furthermore, it is apparent that certain removal algorithms are more effective at disconnecting the graphs than others. This can be seen when comparing Figure 3a (removal based on node betweenness) with Figure 5a (removal based on node degree). Figure 5a shows that almost 30% more zero-valued eigenvalue multiplicity (at 50% RCF) after 100 nodes removed based on nodedegree removal than Figure 3a that shows (at 20% RCF) after the same number of nodes (100 nodes) removed based on betweenness. When reviewing the RCFs of networks across the different removal algorithms, we juxtapose them with one another in order to gain more insight on the properties of these topologies. For AT&T L1, with a nearly diagonal RCF plot before removals, we can see the plot become a horizontal line at 300 nodes removed for betweenness-based removals (Figure 3a), 382 nodes removed for closeness-based (Figure 4a), 200 nodes removed for degree-based (Figure 5a), and 488 links removed for link-betweenness-based removals (Figure 6a). The AT&T L3 topology, originally displaying an approximately 70% RCF of eigenvalue 1 multiplicity, becomes a horizontal line when 20 nodes are attacked based on node betweenness (Figure 3b), 30 nodes are attacked based on node closeness (Figure 4b), 20 nodes attacked based on node degree (Figure 5b), and 140 links attacked based on link betweenness removals (Figure 6b). The Level 3 L1 topology, exhibiting a tight clustering of plot lines around the diagonal for most of the removal algorithms, reaches a horizontal RCF line at 98 nodes removed for both node-betweenness-based removal (Figure 3c) and node-closeness-based removal (Figure 4c), and at 50 nodes removed for degree-based (Figure 5c); indicating that a node-degree based intelligent attack would succeed in fewer number of nodes being taken out of operation. The Level 3 L3 network, which features a large S-curve before removals around multiplicity of eigenvalue 1, becomes a horizontal line at 30 nodes removed for based on node betweenness (Figure 3d), based on node closeness (Figure 4d), and based on node degree removals (Figure 5d); thus attacking the nodes via a different strategy results in an almost identical impact for this 38 node, 376 links, and 19.8 average degree Level 3 L3 topology. For link-betweenness-based removals (Figure 6d), convergence of RCF to 1 occurs when all links are removed, and we can see that the Level 3 L1 topology is the nonbipartite since eigenvalues are farthest from 2. We can observe that the Sprint L1 topology has a tight diagonal clustering of plot lines for all of the algorithms, similar to that of the Level 3 L1 topology. Its RCF plot becomes horizontal when 200 nodes are attacked based on node betweenness (Figure 3e), 263 nodes

5 (a) AT&T L1 (b) AT&T L3 (a) AT&T L1 (b) AT&T L3 (c) Level 3 L1 (d) Level 3 L3 (c) Level 3 L1 (d) Level 3 L3 (e) Sprint L1 (f) Sprint L3 (e) Sprint L1 (f) Sprint L3 (g) TeliaSonera L1 (h) TeliaSonera L3 (g) TeliaSonera L1 (h) TeliaSonera L3 (i) Internet2 L1 (j) Internet2 L3 Fig. 3. Spectra of US backbone networks based on node betweenness removal (i) Internet2 L1 (j) Internet2 L3 Fig. 4. Spectra of US backbone networks based on node closeness removal are attacked based on node closeness (Figure 4e), 200 nodes are attacked based on node degree (Figure 5e), and 300 links nodes are attacked based on link betweenness (Figure 6e). The Sprint L3 topology, which appears to originally contain most eigenvalues valued between approximately 0.4 and 1.6, exhibits a horizontal RCF plot when 20 nodes are attacked based on node betweenness (Figure 3f), node closeness (Figure 4f), and node degree (Figure 5f). The intelligent attacks against the

6 (a) AT&T L1 (b) AT&T L3 (a) AT&T L1 (b) AT&T L3 (c) Level 3 L1 (d) Level 3 L3 (c) Level 3 L1 (d) Level 3 L3 (e) Sprint L1 (f) Sprint L3 (e) Sprint L1 (f) Sprint L3 (g) TeliaSonera L1 (h) TeliaSonera L3 (g) TeliaSonera L1 (h) TeliaSonera L3 (i) Internet2 L1 (j) Internet2 L3 (i) Internet2 L1 (j) Internet2 L3 Fig. 5. Spectra of US backbone networks based on node degree removal Fig. 6. Spectra of US backbone networks based on link betweenness removal TeliaSonera physical- and logical-level topologies require the minimal number of node and link removals (Figures 3g, 4g, 5g, 6g, 3h, 4h, 5h, 6h), since these topologies are relatively smaller compared to the other backbone networks. The Internet2 L1 topology, which can be observed to have a tight clustering of removal plots across the diagonal for most of the removal algorithms (similar to that of AT&T L1, Sprint L1, and Level 3 L1), exhibits a horizontal RCF plot at 56 nodes removed

7 for node-betweenness-based removals (Figure 3i), at 50 nodes removed for node-closeness-based removals (Figure 4i), at 30 nodes removed for node-degree-based removals (Figure 5i). The inferred conclusion is that degree-based attacks cause the most degrading impact in the case of Internet2 L1 topology. When comparing the Internet2 L1 spectra to that of Internet2 L3 spectra (Figures 3j, 4j, 5j, 6j), we observe that due to the smaller size of the Internet2 L3 network, it takes less effort to partition this network. We can observe this trait as one of the main discernible differences between L1 and L3 network structures; the logical topologies (L3) of the discussed networks have fewer nodes than the physical topologies (L1), requiring fewer node removals to cause significant partitioning in the network. Moreover, the star-like structures that many of the logical topologies contain are particularly vulnerable to node removals, due to the termination of numerous edge connections when a single node is removed as opposed to the two or three edge connections commonly seen for nodes in physical topologies. We can also observe that the logical topologies of several networks require fewer edge removals (Figure 6) than their physical topologies in order to cause significant partitioning, with the notable exception being Level 3 L3 (Figure 6d) and TeliaSonera L3 (Figure 6h). The Level 3 L3 (Figure 6d) has approximately 25% RCF of zero-valued eigenvalue multiplicity after 100 edge removals, compared to Level 3 L1 s (Figure 6c) approximately 75% RCF for zero-valued eigenvalue multiplicity after the same amount of edge removals. TeliaSonera L3 (Figure 6h) has approximately 50% RCF of zero-valued eigenvalue multiplicity after 20 edge removals, compared to TeliaSonera L1 s (Figure 6g) 75% RCF for zero-valued eigenvalue multiplicity after the same amount of removals. Comparing and contrasting these topologies and their normalized eigenvalue spectrums provides us with valuable visual information that can be used to understand connectivity behavior. Moreover, insights can help optimize backbone networks against targeted attacks. VI. CONCLUSIONS AND FUTURE WORK Single-valued graph metrics might be misleading to evaluate and compare network performance. The normalized Laplacian spectra (nls) can provide very useful visual information provider in how networks perform under attacks leading to hardening networks. We study the nls of five backbone networks under attack scenarios that are based on graph centrality metrics such as betweenness, closeness, and degree. We observe that the RCF line converges from a quasi-symmetric diagonal line to a straight line atop as more nodes (or links) are removed. Second, in some cases (e.g. Internet2 L1 topology), different attack strategies yield different results (node degree cause more degradation than node betweenness). In other cases (e.g. Level 3 L3 topology), the attack strategy yields identical results for 30 node removal based on betweenness, closeness, and degree. Third, we observe that removing the nodes have a greater degrading impact compared to removing the links, since removal of a node results in removal of all incident edges to that node. Since we utilize the nls, which is normalized based on degree of nodes, the level of connectivity is comparable across networks of different order and size. For our future work, we will investigate different attack strategies using different graph metrics. The bipartiteness and connectivity of the weighted graphs is also worth further investigation. Finally, we will utilize the methodology presented in this paper to harden networks against attacks. ACKNOWLEDGMENT We would like to acknowledge members of the Complex Networks and Systems (CoNetS) group for discussions on this work. Tristan A. Shatto is in part supported by Missouri University of Science and Technology (Missouri S&T) Opportunities for Undergraduate Research Experiences (OURE) Program. REFERENCES [1] J. C. Doyle, D. L. Alderson, L. Li, S. Low, M. Roughan, S. Shalunov, R. Tanaka, and W. Willinger, The robust yet fragile nature of the Internet, Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 41, pp , [2] M. Roughan, W. Willinger, O. Maennel, D. Perouli, and R. Bush, 10 Lessons from 10 Years of Measuring and Modeling the Internet s Autonomous Systems, IEEE Journal on Selected Areas in Communications, vol. 29, no. 9, pp , [3] H. Haddadi, M. Rio, G. Iannaccone, A. Moore, and R. 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