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William Liu 1, Harsha Sirisena 2, Krzysztof Pawlikowski 2 and Andreas Willig 2 1 Auckland University of Technology 2 University of Canterbury New Zealand A New Topological Index for Capacity Allocation Problem in Survivable Networks

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Outline Motivation New Proposed Topological Metric : Algebraic Connectivity/Network mean distance Numerical results Future work 1

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Motivation: Two main design issues of Survivable Networks : survivable routing scheme, and spare capacity allocation strategy. We focus on studies of quantitative structure-performance relationships and try to find the correlation between the total capacity utility and a topological index under the SBPP (Shared Backup Path Protection) scheme. Main result of this paper: Initial studies of selected network topologies confirm that a monotonically decreasing power law correlation between the total capacity and its network structure can be well described by this new topological index. 2

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Why Metric (1/2) Most previous researches on network survivable capacity design have been generally used the average nodal degree (i.e., an arithmetic mean of set of nodal degrees) as network connectivity index In our previous DRCN paper*, we concluded that this metric is only a coarse indicator of how sparse or dense a given topology is. It carries insufficient information on network topological characteristics But our previous results are based on the similar topologies, and do not consider the different sizes and shapes of the topologies How to find a proper topological metric to describe different shapes and sizes of topologies? 3 *William Liu, Harsha Sirisena, Krzysztof Pawlikowski and Allan McInnes, Utility of Algebraic Connectivity Metric in Topology Design of Survivable Networks, 7th International Workshop on the Design of Reliable Communication Networks (DRCN 2009), Washington DC, October 25-28, 2009.

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Why Metric (2/2) Therefore… There is an obvious need for a more informative metric that can capture the topological characteristic of a network : size and shape Algebraic connectivity PLUS Network mean distance introduced in spectral graph theory, are key topological invariants of graphs, which can indicate the connectivity, size and shape of a topology 4

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Algebraic Connectivity Definition Given an unweighted graph G with m links and n nodes, with its adjacency matrix A nxn and degree matrix D nxn Matrix L nxn = D-A is known as the Laplacian matrix of G Spectrum of graph G = the set of n eigenvalues of matrix L The algebraic connectivity of G: λ 2 (G)=the 2nd smallest eigenvalue of L nxn, also called the Fiedler value in spectral graph theory. It provides global information about graphs topological properties: it is larger for stronger connected graphs. 5

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Example- Adjacency Matrix 6 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x1x x2x x3x x4x x5x x6x Important properties: Symmetric matrix Its eigenvectors are real and orthogonal Adjacency matrix (A) a square matrix n x n, where A ij is 1 if vertex x i and x j is connected, otherwise is

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Example- Degree Matrix 7 Degree matrix (D) n x n diagonal matrix where D ii is the nodal degree of vertex x i x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x1x x2x x3x x4x x5x x6x

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Example- Laplacian Matrix Laplacian matrix (L) n x n symmetric matrix L=D-A Important properties: Eigenvalues are non-negative real numbers Eigenvectors are real and orthogonal Eigenvalues and eigenvectors provide an insight into the connectivity of the graph x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x1x x2x x3x x4x x5x x6x

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Properties of The algebraic connectivity is a robust measure of network connectivity: the larger the algebraic connectivity, the better connected the network. Hence more node- and link-disjoint paths exist between pairs of nodes. 9 In addition, the network mean distance here is defined as the average of all hop count between distinct nodes of the graph. This metric can measure the size and shape of the graph.

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Properties of 10 The bounds on the network mean distance can be derived*. Its lower bound is: and its upper bound is: We propose to use to reflect network topological properties such as the connectivity, size and shape of a given topology and it has impacts on the total capacity allocation in network survivability design, * Bojan Mohar, Eigenvalues, Diameter and Mean Distance in graphs, Graphs Combin. (1991), pp. 53–64.

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Simulation Scenario Uniform traffic Shared Backup Path Protection (SBPP) AMPL/CPLEX optimization tool Topologies: 10 different topologies Performance: Total capacity, i.e., working capacity + spare capacity Curve-fitting : the coefficient of determination R 2 (0,1), with a higher value indicating a better fit. 11

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Topologies 12

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13 Experiment results

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14 Total capacities vs. four topological indices

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Experiment results 15 Total capacity vs. average node degree

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Experiment results 16 Total capacity vs. algebraic connectivity/exact mean distance * The exact mean distance can be calculated by the standard graph toolbox in MATLAB

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Experiment results 17 Total capacity vs. algebraic connectivity/lower bound mean distance

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Experiment results 18 Total capacity vs. algebraic connectivity/upper bound mean distance

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Conclusions and Future Work Our initial results show that for the chosen example networks, the new metric is a good quantitative descriptor of network topologies in studies of the capacity allocation problem. We have found a power law relationship between this topological index and total capacity in survivable network designed with the SBPP algorithm. This new finding can be applied into network optimization, for such as the estimation on the total capacity if the topology is given, or on the other hand, the evaluation on the topology structure if the total capacity constraints are given. Future works… An extensive validation on our findings on more complex topologies is underway. Further studies on how to apply this new topological index into network capacity planning& optimization, e.g., one node/link addition problem are needed. 19

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