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ui are arbitrary nodes and c is a (usually small) constant. In the UBG model, nodes are assumed to form a doubling metric space. Two nodes u and v with d(u, v) 1 are adjacent, whereas all other nodes are not. The proof of the following theorem is left to the reader as an exercise. Theorem 4.2.3. Nodes in a two-dimensional Euclidean plane (i.e., the metric space is given by the Euclidean distances) form a doubling metric. A general graph, however, does not. Figure 4.4 shows an example for the Euclidean plane. In this setting, three balls of radius r/2 are enough to cover all nodes in the ball of radius r around node u. To see why a general graph may not form a doubling metric, consider a graph where all nodes have distance 1 to all other nodes. Observe that it is possible to model a UDG with a UBG by using the Euclidean distances of the UDG and connecting those node pairs which have distance at most 1. Moreover, even a QUDG can be modeled with a UBG. We have the following results. Theorem 4.2.4. A UDG is a UBG. Theorem 4.2.5. An undirected QUDG with constant is a UBG. Proof. The idea of the proof is as follows: First, we transform all distances in the QUDG. We then show that during this transformation, all edges are maintained; that is, the resulting graph is isomorphic to the QUDG. Moreover, it can be shown that after the transformation, the graph also ful lls the requirements of a doubling metric space.

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2 LeDuc, J W; Damon, I; Meegan, J M; Relman, D A; Huggins, J; Jahrling, P B Smallpox research activities: US interagency collaboration, 2001 Emerging Infect Dis 2002, 8, 743 745 3 Heymann, D L; Szczeniowski, M; Esteves, K Re-emergence of monkeypox in Africa: a review of the past six years Br Med Bull 1998, 54, 693 702 4 Jezek, Z; Szczeniowski, M; Paluku, K M; Mutombo, M Human monkeypox: clinical features of 282 patients J Infect Dis 1987, 156, 293 298 5 Hooper, J W; Thompson, E; Wilhelmsen, C; Zimmerman, M; Ichou, M Ait; Steffen, S E; Schmaljohn, C S; Schmaljohn, A L; Jahrling, P B Smallpox DNA vaccine protects nonhuman primates against lethal monkeypox J Virol 2004, 78, 4433 4443 6 Bremen, J; Henderson, D A.

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We transform the distances between all pairs of nodes (u, v) in the QUDG as follows. Let dQ (u, v) denote the distance from node u to node v in the QUDG, and let dB (u, v) be the transformed distance in the UBG. Moreover, let > 0 be an arbitrary small number. dQ (u, v)/ 1 dB (u, v) := 1 + dQ (u, v) if if if if dQ (u, v) < dQ (u, v) 1 and v is adjacent to u < dQ (u, v) 1 and v is not adjacent to u dQ (u, v) > 1

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Observe that by this transformation, pairs of nodes that are adjacent in the QUDG are assigned distances of at most 1 and are therefore also adjacent in the UBG. Similarly, nodes that are not adjacent in the QUDG have a distance larger than 1 are therefore not neighboring in the UBG either. Also observe that the transformation increases the distance between two nodes by less than a constant factor of := (1 + )/ , but it never decreases any distances. It remains to show that after the transformation, the nodes indeed form a doubling metric space. In order to form a metric space, the distances between the nodes are to ful ll the following properties: (1) nonnegativity, (2) identity of indiscernibles, (3) symmetry, and (4) triangle inequality. The nonnegativity and the identity of indiscernibles criteria are met trivially. The symmetry criterion, however, might not hold, because the adjacency relation can be directed in a QUDG. Therefore, in the following, we consider undirected QUDGs only. Hence, since our distance transformation maintains symmetry, Property 3 holds as well. It remains to discuss the triangle inequality. Consider two arbitrary nodes u and v. Since in the QUDG, all distances are Euclidean, it holds that w : dQ (u, v) dQ (u, w) + dQ (w, v) (4.1)

Poxvirus dilemmas monkeypox, smallpox and biological terrorism N Engl J Med 1998, 339, 556 559 7 Henderson, D A Bioterrorism as a public health threat Emerging Infect Dis 1998, 4, 488 492 8 Henderson, D A The looming threat of bioterrorism Science 1999, 283, 1279 1282, and references therein 9 Fenner, F; Henderson, D A; Arita, I; Jezek, Z; Ladnyi, I D Smallpox and Its Eradication, World Health Organization, Geneva, 1988 10 Moss, B Poxviridae: the viruses and their replication, in Fields Virology, B N Fields, D M Knipe, and P M Howley (eds) Lippincott-Raven, Philadelphia, 1996, pp 2637 2671 11 Bray, M Filoviridae, in Clinical Virology, Richman, D R, Whitley, R J, Hayden, F G (eds) ASM Press, Washington, Dc, 2002, pp 875 890 12 Sanchez, A, Peter, C J, Rollin, P, Ksiazek, T G, Murphy, F.

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