Complex Sciences. Second International Conference, COMPLEX 2012, Santa Fe, NM, USA, December 5-7, 2012, Revised Selected Papers

Research Article

The Dynamics of Propagation Fronts on Sets with a Negative Fractal Dimension

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  • @INPROCEEDINGS{10.1007/978-3-319-03473-7_13,
        author={Alfred Hubler and Josey Nance},
        title={The Dynamics of Propagation Fronts on Sets with a Negative Fractal Dimension},
        proceedings={Complex Sciences. Second International Conference, COMPLEX 2012, Santa Fe, NM, USA, December 5-7, 2012, Revised Selected Papers},
        proceedings_a={COMPLEX},
        year={2013},
        month={11},
        keywords={Negative fractal dimension diffusion propagation front},
        doi={10.1007/978-3-319-03473-7_13}
    }
    
  • Alfred Hubler
    Josey Nance
    Year: 2013
    The Dynamics of Propagation Fronts on Sets with a Negative Fractal Dimension
    COMPLEX
    Springer
    DOI: 10.1007/978-3-319-03473-7_13
Alfred Hubler1,*, Josey Nance1
  • 1: University of Illinois at Urbana-Chamapaign
*Contact email: hubler.alfred@gmail.com

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 ( 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.