Artificial Adaptive Systems in Medicine, 79-89 79 Massimo Buscema / Enzo Grossi (Eds.) All rights reserved - © 2009 Bentham Science Publishers Ltd. CHAPTER 9 The Topological Weighted Centroid and the Semantic of the Physical Space – Application Enzo Grossi & , Massimo Buscema § , Tom Jefferson # Abstract: In this chapter several application examples derived from the literature and from the real world show how new elementary mathematics like: Topological Weighted Centroid (TWC); Self Topological Weighted Centroid (STWC); Proximity Scalar Field; Gradient of the Scalar Field Relative Topological Weighted Centroid (TWCi); Paths form Arithmetic Centroid to entities; Paths between entities; Scalar Field of the trajectories, may help decision makers in situations characterized by limited amount of information, and how mathematics of complex system can improve the level of accuracy obtained with classical statistics. In particular the TWC propose itself as a powerful method to identify the source of epidemic spread. The im- pressive results obtained in the example of Russian influenza spreading in Sweden in 1889 and in the colera spreading in London in 1854 are consistent with the idea that the spread of infectious disease is not random but follows a progression which is based on inherent but as yet undiscovered mathematical laws based on probabilistic density function. These methods, which require further for field evaluation and validation, could provide an additional power- ful tool for the investigation of the early stages of an epidemic, and constitute the basis of new simulation methods to understand the process through which a disease is spread. Keywords: Topological Weighted Centroid; Russian influenza; outbreak source Basic example with weak semantic We show as first example a space with a very weak semantic. In fact, in this space, the summation of distances of each entity to the others is quite the same, and consequently the AC and the TWC are located ap- proximately in the same position: Source Map TWC and AC Fig. 1. Using the equation from (29) to (36) (∗) , we can generate the proximity scalar field of the entities: & Enzo Grossi, Bracco Medical Department, Via 25 Aprile 4, 20097 S.Donato Milanese, Milan, Italy. E-mail: enzo.grossi@bracco.com. § Massimo Buscema, Semeion Research Center, Via Sersale 117, 00128 Rome, Italy. E-mail: m.buscema@semeion.it # Tom Jefferson, Cochrane Acute Respiratory Infections Group, Rome, Italy. E-mail: jefferson.tom@gmail.com (∗) All the equations numbers mentioned in this chapter are refered to the previous chapter of this book.