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8 Int'l Conf. Foundations of Computer Science | FCS'12 | Proof: Consider the VMTL with components labeled [1, 62, 20, 34, 29, 53], [2, 65, 16, 35, 32, 49], [3, 61, 19, 36, 28, 52], [4, 57, 22, 37, 24, 55], [5, 60, 18, 38, 27, 51], [6, 56, 10, 50, 12, 54], [7, 66, 17, 33, 40, 43], [8, 63, 23, 30, 41, 45], [9, 59, 15, 42, 26, 48], [11, 58, 14, 44, 25, 47], and [13, 64, 21, 31, 46, 39]. This VMTL has a magic constant of 116. The first five components have common differences of 3s. Therefore, by using Lemma 2.3 five times, we have VMTLs of 119C3 with magic constants 113, 110, 107, 104 and 101. By duality, there exist VMTLs with magic constants of 85, 88, 91, 94, 97, and 100. Next, consider the VMTL with components labeled [1, 59, 22, 34, 26, 55], [2, 60, 20, 35, 27, 53], [3, 66, 13, 36, 33, 46], [4, 63, 15, 37, 30, 48], [5, 56, 10, 49, 12, 54], [6, 58, 17, 40, 24, 51], [7, 65, 19, 31, 41, 43], [8, 57, 11, 47, 18, 50], [9, 64, 23, 28, 45, 42], [14, 62, 21, 32, 44, 39], and [16, 61, 29, 25, 52, 38]. This VMTL has a magic constant of 115. The first four components have common differences of 3s. By using Lemma 2.3 four times, there exist VMTLs of 11C3 with magic constants of 112, 109, 106 and 103. By duality, there exist VMTLs with magic constants of 86, 89, 92, 95, and 98. Finally, in [27] it was shown that there exist VMTLs of 11C3 with magic constants with values 117, 114, 111, 108, 105, 102, 99, 96, 93, 90, 87 and 84. References [1] H.R. Andersen and H. Hulgaard, “Boolean Expression Diagrams,” Information and Computation, vol. 179, pp. 194-212, 2002. [2] M. Baca, M. Miller and J.A. MacDougall, “Vertex-magic total labelings of generalized Peterson graphs and convex polytopes,” J. Combin. Math. Combin. Comput., vol. 59, pp. 89-99, 2006. [3] M. Baca, M. Miller and Slamin, “Every generalized Peterson graph has a vertex-magic total labeling,” Int. J. Comp. Math., vol. 79, pp. 1259-1263, 2002. [4] A. Baker and J. Sawada, “Magic Labelings on Cycles and Wheels,” in 2nd Annual International Conference on Combinatorial Optimization and Applications (COCOA ’08), Lecture Notes in Mathematics, vol. 5165, pp. 361-373, 2008. [5] O. Berkman, M. Parnas and Y. Roditty, “All cycles are edge-magic,” Ars Combin., vol. 59, pp. 145-151, 2001. [6] R.E. Bryant, “Symbolic Boolean Expression Manipulation with Ordered Binary Decision Diagrams,” ACM Computing Surveys, Vol. 24, No. 3, pp. 294-318, 1992. [7] S.A. Cook, “The complexity of theorem-proving procedures,” in Proceedings of the third annual ACM symposium on the Theory of computing, ACM, NY, 1971, pp. 151-158. [8] T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein, Introduction to Algorithms, 3rd ed. MIT Press, 2009. [9] M. Davis, G. Logemann, and D. Loveland, “A Machine Program for Theorem Proving,” Communications of the ACM, vol. 3, pp. 394-397, 1962. [10] G. Exoo, A.C.H. Ling, J.P. McSorley, N.C.K. Phillips, W.D. Wallis, “Totally magic graphs,” Discrete Math., vol. 254, pp. 103-113, 2002. [11] D. Froncek, P. Kovar and T. Kovarova, “Vertex magic total labeling of products of cycles,” Australas J. Combin., vol. 33, pp. 169-181, 2005. [12] J.A. Gallian, “A dynamic survey of graph labeling,” Electronic J. Combin., vol. 17, #DS6, 2010. [13] R.D. Godbold and P.J. Slater, “All cycles are edge-magic,” Bull. Inst. Combin. Appl., vol. 22, pp. 93-97, 1998. [14] I. Gray, “New construction methods for vertex-magic total labelings of graphs,” Ph.D. thesis, University of Newcastle, 2006. [15] I. Gray, “Vertex-magic total labellings of regular graphs,” SIAM J. Discrete Math., vol. 21, Issue 1, pp. 170-177, 2007. [16] I.D. Gray and J.A. MacDougall, “Vertex-magic labelings of regular graphs II.,” Discrete Math., vol. 309, pp. 5986-5999, 2009. [17] E.A. Hirsch, and A. Kojevnikov, “UnitWalk: A new SAT solver that uses local search guided by unit clause elimination,” Annals of Mathematics and Artificial Intelligence, vol. 43, pp. 91-111, 2005. [18] J. Holden, D. McQuillan, and J.M. McQuillan, “A conjecture on strong magic labelings of 2-regular graphs,” Discrete Mathematics, vol. 309, pp. 4130-4136, 2009. [19] The International Conferences on Theory and Applications of Satisfiability Testing (SAT). [Online]. Available: http://www.satisfiability.org [20] G. Jäger, “An effective SAT encoding for magic labeling,” in Proceedings of the 9th Cologne Twente Workshop on Graphs and Combinatorial Optimization (CTW 2010), 2010, pp. 97-100. [21] A. Kotzig and A. Rosa, “Magic valuations of finite graphs,” Canad. Math. Bull., vol. 13, pp. 451-461, 1970. [22] P. Kovar, “Vertex magic total labeling of products of regular VMT graphs and regular supermagic graphs,” J. Combin. Math. Combin. Comput., vol. 54, pp. 21-31, 2005. [23] Y. Lin and M. Miller, “Vertex-magic total labelings of complete graphs,” Bull. Inst. Combin. Appl., vol. 33, pp. 68-76, 2001. [24] J.A. MacDougall, M. Miller, Slamin, and W.D. Wallis, “Vertex-magic total labelings of graphs,” Utilitas Mathematica, vol. 61, pp. 3-21, 2002. [25] D. McQuillan, “Edge-magic and vertex-magic total labelings of certain cycles,” Ars Combin., vol. 91, pp. 257-266, 2009. [26] D. McQuillan, “A technique for constructing magic labelings of 2−regular graphs,” JCMCC, vol. 75, pp. 129-135, 2010. [27] D. McQuillan and J.M. McQuillan, “Magic Labelings of Triangles,” Discrete Mathematics, vol. 309, pp. 2755-2762, 2009. [28] D. McQuillan and K. Smith, “Vertex-magic total labeling of multiple complete graphs,” Congr. Numer., vol. 180, pp. 201-205, 2006. [29] D. McQuillan and K. Smith, “Vertex-magic total labeling of odd complete graphs,” Discrete Math., vol. 305, pp. 240-249, 2005. [30] J.P. McSorley and W.D. Wallis, “On the spectra of totally magic labelings,” Bull. Inst. Combin. Appl., vol. 37, pp. 58-62, 2003. [31] Y. Roditty and T. Bachar, “A note on edge-magic cycles,” Bull. Inst. Combin. Appl. vol. 29, pp. 94-96, 2000. [32] B. Selman, H. Kautz, and B. Cohen, “Local Search Strategies for Satisfiability Testing,” in Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, October 11-13, 1993. [33] W.D. Wallis, “Vertex magic labelings of multiple graphs,” Congr. Numer., vol. 152, pp. 81-83, 2001. [34] W.D. Wallis, Magic Graphs. Boston, MA: Birkhauser, 2001. [35] W.D. Wallis, “Two results of Kotzig on magic labelings,” Bull. Inst. Combin. Appl., vol. 36, pp. 23-28, 2002.
Int'l Conf. Foundations of Computer Science | FCS'12 | 9 Assortativity of links in directed networks Mahendra Piraveenan 1 , Kon Shing Kenneth Chung 1 , and Shahadat Uddin 1 1 Centre for Complex Systems Research, Faculty of Engineering and IT, The University of Sydney, NSW 2006, Australia Abstract— Assortativity is the tendency in networks where nodes connect with other nodes similar to themselves. Degree assortativity can be quantified as a Pearson correlation. However, it is insufficient to explain assortative or disassortative tendencies of individual links, which may be contrary to the overall tendency in the network. In this paper we define and analyse link assortativity in the context of directed networks. Using synthesised and real world networks, we show that the overall assortativity of a network may be due to the number of assortative or disassortative links it has, the strength of such links, or a combination of both factors, which may be reinforcing or opposing each other. We also show that in some directed networks, link assortativity can be used to highlight subnetworks which have vastly different topological structures. The quantity we propose is limited to directed networks and complements the earlier proposed metric of node assortativity. Keywords: Complex networks, graph theory, assortativity, social networks, biological networks 1. Introduction In the last few decades, network approaches have been widely used to analyze complex systems in a number of domains, including technical, biological, social and physical domains [1], [2]. Assortativity is a much studied concept in the topological analysis of complex networks [3], [4], [5], [6]. Assortativity has been defined to quantify the tendency in networks where individual nodes connect with other nodes which are similar to themselves [3]. Thus, a social network of people tends to be assortative, since people often prefer to be friends with, or have links to, other people who are like them. A food web could be argued as disassortative, because predator and prey are unlikely to be similar in many respects. However, it is clear that the assortativity of a network needs to be defined in terms of a particular attribute of nodes in that network. A social network could be assortative when the considered attribute is the age of people, because people tend to be friends with other people similar to their age: however, the same network could be disassortative, when the gender of individuals is the considered attribute. Degree assortativity is the most common form of assortativity used in network analysis, whereby similarity between nodes is defined in terms of the number of connections the nodes have. Degree assortativity can be defined and quantified as a Pearson correlation [3], [7]. It has been shown that many technological and biological networks are slightly disassortative, while most social networks, predictably, tend to be assortative in terms of degrees [3], [5]. Recent work defined degree assortativity for directed networks in terms of in-degrees and out-degrees, and showed that an ensemble of definitions are possible in this case [6], [7]. It could be argued, however, that the overall assortativity tendency of a network may not be reflected by individual links of the network. For example, it is possible that in a social network, which is overall assortative, some people may maintain their ‘fan-clubs’. In this situation, people who have a great number of friends may be connected to people who are less famous, or even loners. If there is a link which connects such a popular person with a loner, that link cannot be called an assortative link, even though the network is overall assortative. Similarly, in a network which is overall disassortative many links could arguably be assortative. A good example for this is the so-called rich club scenario in internet AS networks [8]. Even though these networks display overall disassortativity, it has been shown that the hubs among them are strongly connected to each other, forming a ‘rich-club’. The links that connect these hubs, therefore should be considered assortative in nature. These examples highlight that the ‘local assortativity’ of links is a quantity, which, if defined, can throw light on the topological structure of networks. In directed networks, this can be defined separately for out-degree based mixing patterns and in-degree based mixing patterns. Indeed, decomposing the assortativity coefficient of a network into values for network components has already been attempted, and the metric of ‘local assortativity’ has been defined and analysed in detail [4], [6]. However, this decomposition has been done in terms of network nodes, rather than network links. In this work however, we propose to analyze the ‘local assortativity’ of individual links, and will demonstrate that this approach has its advantages, particularly for defining the assortativity of sub networks or regions. We limit our analysis to directed networks. Analyzing the ‘local assortativity’ of links can give us a number of insights about networks. We will be able to identify ‘positive’ and ‘negative’ links in the network in terms of assortativity, and based on these we can see which links help, or hinder, the overall assortative tendency in networks. We can identify if assortativity of a network is primarily determined by the number of a particular type of links, or rather by the strength of such links. If the
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