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Globally and locally minimal weight spanning tree networks

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  • Kansal, Anuraag R
  • Torquato, Salvatore

Abstract

The competition between local and global driving forces is significant in a wide variety of naturally occurring branched networks. We have investigated the impact of a global minimization criterion versus a local one on the structure of spanning trees. To do so, we consider two spanning tree structures—the generalized minimal spanning tree (GMST) defined by Dror et al. (Eur. J. Oper. Res. 120 (2000) 583) and an analogous structure based on the invasion percolation network, which we term the generalized invasive spanning tree (GIST). In general, these two structures represent extremes of global and local optimality, respectively. Structural characteristics are compared between the GMST and GIST for a fixed lattice. In addition, we demonstrate a method for creating a series of structures which enable one to span the range between these two extremes. Two structural characterizations, the occupied edge density (i.e., the fraction of edges in the graph that are included in the tree) and the tortuosity of the arcs in the trees, are shown to correlate well with the degree to which an intermediate structure resembles the GMST or GIST. Both characterizations are straightforward to determine from an image and are potentially useful tools in the analysis of the formation of network structures.

Suggested Citation

  • Kansal, Anuraag R & Torquato, Salvatore, 2001. "Globally and locally minimal weight spanning tree networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 301(1), pages 601-619.
    • Handle: RePEc:eee:phsmap:v:301:y:2001:i:1:p:601-619
    DOI: 10.1016/S0378-4371(01)00430-7
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    References listed on IDEAS

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    1. Dror, M. & Haouari, M. & Chaouachi, J., 2000. "Generalized spanning trees," European Journal of Operational Research, Elsevier, vol. 120(3), pages 583-592, February.
    2. Kayser, D.R. & Aberle, L.K. & Pochy, R.D. & Lam, L., 1992. "Active walker models: tracks and landscapes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 191(1), pages 17-24.
    3. Geoffrey B. West & James H. Brown & Brian J. Enquist, 1997. "A General Model for the Origin of Allometric Scaling Laws in Biology," Working Papers 97-03-019, Santa Fe Institute.
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    1. M Haouari & J Chaouachi & M Dror, 2005. "Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(4), pages 382-389, April.
    2. Haouari, Mohamed & Chaouachi, Jouhaina Siala, 2006. "Upper and lower bounding strategies for the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 171(2), pages 632-647, June.
    3. Pop, Petrică C., 2020. "The generalized minimum spanning tree problem: An overview of formulations, solution procedures and latest advances," European Journal of Operational Research, Elsevier, vol. 283(1), pages 1-15.
    4. Jennings, Mark & Fisk, David & Shah, Nilay, 2014. "Modelling and optimization of retrofitting residential energy systems at the urban scale," Energy, Elsevier, vol. 64(C), pages 220-233.

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