Abstract
In sets with a fractal dimension greater than 1, the average number of neighbors increases with distance. For that reason spherical pulses propagate outward in systems with nearest neighbor interactions. In sets with a negative fractal dimension, such as the set of individual coordinates of a population of a small city, the average number of neighbors decreases with distance in a precise way relating the number of neighbor to the fractal dimension of the set. We study the propagation of diffusive pulses and waves on such sets. We find that on sets with negative fractal dimension, the velocity of pulse peak is negative (i.e. the median radius of circular pulses decreases as a function of time). Eventually the pulse broadens and disappears. We discuss applications in physical systems, such as the spreading of heat and sound, as well as applications in social systems, such as the spread of infectious diseases and the spread of rumors.
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© 2013 ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering
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Hubler, A., Nance, J. (2013). The Dynamics of Propagation Fronts on Sets with a Negative Fractal Dimension. In: Glass, K., Colbaugh, R., Ormerod, P., Tsao, J. (eds) Complex Sciences. Complex 2012. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-03473-7_13
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DOI: https://doi.org/10.1007/978-3-319-03473-7_13
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03472-0
Online ISBN: 978-3-319-03473-7
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