# USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS

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6 a small-world network is the road network of a country, wherein most of the cities are not neighbors of each other, but can be reached from one another by going through a fewer number of intermediate cities. The simulations for small-world networks are conducted according to the Watts and Strogatz model [10], explained as follows. We start with a regular 1-dimensional ring network comprising of two concentric rings, each with the same number of nodes (say, N; hence, the total number of nodes in the network is referred to as 2N). The nodes are initially organized in a ring network as shown in the left side of Figure 6. The number of neighbors per node is 4. For a node i in the outer ring, its neighbors in the outer ring are chosen as follows: (i + 1) % N and (i N) % N. For a node i in the inner ring, its neighbors in the inner ring are chosen as follows: N + ((i + 1) % N) and N + ((i N) % N). In the initial regular ring network, as each node has 4 neighbors, the average number of neighbors per node is 4; with a total of 2N nodes, the total number of links in the initial regular ring network is (average node degree * total number of nodes / 2) = (4 * 2N / 2) = 4N = 2*total number of nodes in the network. Regular Network (before rewiring) Small-World Network (after rewiring) Figure 6: Transformation of a regular network to small-world network We consider each link in the initial set of links for a possible rewiring with a probability β. For each link (u-v), where u < v (in the order of the numerical IDs), we generate a random number (uniform-randomly distributed from 0 to 1) and if the random number is less than or equal to β, the node with the lower ID u is paired with a randomly chosen node (other than itself, node v and the current neighbors) and the newly created link is not considered for any rewiring. This way, the average number of neighbors per node at any time is still the initial value for the average number of neighbor per node (which is 4 in our simulations); but, variation in the distribution of the node degree starts creeping in with rewiring. The network at the right side of Figure 6 displays a sample network at the end of rewiring. Figure 7: Distribution of variation in node degree during the process of rewiring (total # nodes: 100; rewiring probability: 0.5)

8 observed for the two networks, rather than requiring to compute the standard deviation of the node degrees for the two networks. As explained in Sections 3 and 4, the operating parameters of the theoretical models can be tuned to obtain a desired variation in the degree of the nodes in the network, measured in the form of the spectral radius ratio for node degree. REFERENCES [1] D. Lusseau, K. Schneider, O. J. Boisseau, P. Haase, E. Slooten, and S. M. Dawson, "The Bottlenose Dolphin Community of Doubtful Sound Features a Large Proportion of Long-lasting Associations," Behavioral Ecology and Sociobiology, vol. 54, pp , [2] W. W. Zachary, "An Information Flow Model for Conflict and Fission in Small Groups," Journal of Anthropological Research, vol. 33, no. 4, pp , [3] M. E. J. Newman, "The Structure of Scientific Collaboration Networks," Proceedings of the National Academy of Sciences of the United States of America, vol. 98, no. 2, pp , January [4] M. Newman, Networks: An Introduction, 1st ed., Oxford University Press, May [5] M. Girvan and M. E. J. Newman, "Community Structure in Social and Biological Networks," Proceedings of the National Academy of Sciences of the United States of America, vol. 19, no. 12, pp , April [6] G. Strang, Linear Algebra and its Applications, Cengage Learning, 4th edition, July [7] B. G. Horne, "Lower Bounds for the Spectral Radius of a Matrix," Linear Algebra and its Applications, vol. 263, pp , September [8] A-L. Barabasi and R. Albert, "Emergence of Scaling in Random Networks," Science, vol. 286, pp , October [9] J. Benesty, J. Chen, Y. Huang, I. Cohen, "Pearson Correlation Coefficient," Springer Topics in Signal Processing: Noise Reduction in Speech Processing, vol. 2, pp. 1-4, March [10] D. J. Watts and S. H. Strogatz, "Collective Dynamics of 'Small-World' Networks," Nature, vol. 393, pp , June 1998.

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