Abstract
Under an electric field, spherical conducting particles in a dielectric liquid assemble into a dendritic tree in order to dissipate charge. Several topological measures characterize such networks, including degree distributions, Strahler numbers, and total external pathlengths. Here, scaling laws relating these measures to the number of nodes in the system are presented and shown to match diffusion limited aggregation (DLA) structures. Experimental bifurcation and stream-length ratios in this easily reproducible laboratory experiment are found to agree with DLA simulations and Horton's laws for river networks. Certain scaling relations in transportation networks are known to originate from general features of networks. Here we find the experimental structures share properties with hydrological drainage networks.
| Original language | English (US) |
|---|---|
| Article number | 036103 |
| Journal | Journal of Applied Physics |
| Volume | 109 |
| Issue number | 3 |
| DOIs | |
| State | Published - Feb 1 2011 |
| Externally published | Yes |
ASJC Scopus subject areas
- Physics and Astronomy(all)
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In: Journal of Applied Physics, Vol. 109, No. 3, 036103, 01.02.2011.
Research output: Contribution to journal › Article › peer-review
}
TY - JOUR
T1 - Topological similarities in electrical and hydrological drainage networks
AU - Soni, Vishal H.
AU - Ketisch, Pia M.
AU - Rodríguez, Juan D.
AU - Shpunt, Alexander
AU - Hübler, Alfred W.
N1 - Funding Information: Soni Vishal H. 1 Ketisch Pia M. 2 Rodríguez Juan D. 3 Shpunt Alexander 4 Hübler Alfred W. 1,2 a) 1 Department of Physics, Center for Complex Systems Research, University of Illinois at Urbana-Champaign , 1110 W Green Street, Urbana, Illinois 61801, USA 2 The Santa Fe Institute , 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA 3 Institute for Computational Engineering and Sciences, University of Texas at Austin , 1 University Station C0200, Austin, Texas 78712, USA 4 Department of Physics, Massachusetts Institute of Technology , 77 Mass. Avenue, Cambridge, Massachusetts 02139, USA a) Electronic mail: a-hubler@uiuc.edu . 01 02 2011 109 3 036103 29 09 2010 23 11 2010 08 02 2011 2011-02-08T14:10:02 2011 American Institute of Physics 0021-8979/2011/109(3)/036103/3/ $30.00 Under an electric field, spherical conducting particles in a dielectric liquid assemble into a dendritic tree in order to dissipate charge. Several topological measures characterize such networks, including degree distributions, Strahler numbers, and total external pathlengths. Here, scaling laws relating these measures to the number of nodes in the system are presented and shown to match diffusion limited aggregation (DLA) structures. Experimental bifurcation and stream-length ratios in this easily reproducible laboratory experiment are found to agree with DLA simulations and Horton’s laws for river networks. Certain scaling relations in transportation networks are known to originate from general features of networks. Here we find the experimental structures share properties with hydrological drainage networks. NSF NSF PHY 01-40179 NSF NSF DMS 03-25939 ITR Ramified transportation networks are ubiquitous in many fields, 1 ranging from drainage networks in geological sciences, to blood vessel systems, neural nets, and genetic regulatory networks in biology, and highway systems, power grids, and the Internet in engineering. However there have been only a few laboratory experiments on the growth and decay of ramified networks, including the polymerization of organic molecules 2 and the growth of fractal particle agglomerates due to a high-voltage current. 3 The limiting patterns of these fractal particle agglomerates have a box-counting dimension of D b c = 1.67 , minimize the resistance and energy consumption, 4 and have no closed loops. From the underlying physical equations, Dueweke derived that the energy consumption is a Lyapunov function for the dynamics. 5 Marani et al. 6 used a simplified version of the equations to estimate the fractal dimension. However, Marani’s patterns have many closed loops in contrast to the experimental structures and other models, including diffusion limited aggregation (DLA), 7 which are mostly trees. The experiment is set up by placing a single layer of M 0 conducting particles (stainless steal ball bearings of diameter d ≈ 1.6 mm ) into a circular shape in the center of a dish of castor oil (Post Apple Scientific, Inc., Castor Oil Laboratory, part number C2742) of mass m ( m ≈ 30 g ) . The setup consists of an 11.95 cm diameter dish with a grounded electrode, electrode A, of thickness t a ( t a ≈ 1.1 mm ) , and rectangular cross-section built into the boundary. A linear chain of particles links the circular distribution to the grounded electrode (Fig. 1 ). A second, needle-shaped electrode, electrode B, is placed at a height h ( h ≈ 3.81 cm ) above the center of the dish and provides a potential V ( V ≈ 25 kV ) . The voltage is switched on within a fraction of a second and kept on for a time t s ( t s ≈ 30 min ) until stable networks (Fig. 1 ) form. The underlying physical equations are too complex to numerically or analytically derive topological properties of the system. Recently, several researchers have attempted to model the evolution of complex networks with graph theoretical methods. 1 Here, the system is analyzed as a graph where the degree distribution, Horton–Strahler indexing scheme, 8 and total external pathlength 9 are used to characterize the steady state networks. The results are then compared with a DLA model. The positions of the particles are determined with image processing software. The position of particle i is given by r i . An adjacency matrix is then established such that a i j = H ( l c − | r i − r j | ) , where l c is the cutoff distance and H is the Heaviside step function. The cuttoff distance, l c is set to be ( 3 / 2 ) d . Two particles i and j are called neighbors if a i j = 1 . N i j is defined as the j th neighbor of particle i . Each particle is referred to as a node with a unique label and neighboring nodes are connected with edges. The degree of each node, which indicates the number of neighbors it has, is defined as d i = ∑ j = 1 M 0 a i j . The stem of the network is composed of nodes connecting the network to the electrode. This stem is specified by a list of particle indices, S . The sink is defined as the node that is nearest to electrode A, and sources are defined as nodes i with d i = 1 where i is not the sink. The first element in S is the sink, i.e., S 1 = i s , where i s is the label of the sink. The second element is the neighbor of the sink, i.e., S 2 = N S 1 , 1 . This is continued and the stem is defined iteratively as S n + 1 = { N S n , 1 if N S n , 1 ≠ S n − 1 N S n , 2 else } unless d S n + 1 > 2 , in which case the process is ended. In the following analyses, the stem is disregarded, and the number of particles in the network which do not belong to the stem is given by M . The degree of each node is computed. Δ i = ∑ j = 1 M δ i , d j is the number of nodes with degree i . Here, δ is the Kronecker delta function. The number of branching points is B = ∑ i = 3 ∞ Δ i . B and Δ 1 each vary linearly with M for 134 < M < 492 . B is fit as B = α B M , where α B = 0.228 ± 0.003 . Δ 1 is fit as Δ 1 = α 1 M , where α 1 = 0.242 ± 0.003 . These results match those obtained previously for systems with 400 < M < 1200 , where α 1 = 0.252 ± 0.007 and α B = 0.237 ± 0.008 . A directed graph is then defined by assigning each node i another node D i as its direct successor. The connection between the two nodes is then said to be directed from i to D i . Furthermore, i is said to be the direct predecessor of D i . Direct successors are assigned in the following manner: (1)the sink has no direct successor, (2)the direct successor of each neighbor i of the sink is the sink, i.e., D i = i s for all i ∊ { N i s , j ∣ j = 1 , … , d i s } , (3) for each node i with a direct successor, all neighbors j except D i of i are assigned i as their direct successor, i.e., D j = i for all j ∊ { N i s , j ∣ j = 1 , … , d i s , N i s , j ≠ D i } . This last step is repeated until each node besides the sink has a direct successor. Once the directed graph is specified, the Strahler number, s i , of each node i can be recursively defined. First, all sources are given a Strahler number of 1. Then for each remaining node i , the Strahler number is defined as the maximum Strahler number of all of its direct predecessors, s ̂ i = max { s N i , j ∣ ( j = 1 , 2 , … , d i ) , D i ≠ j } , if each direct predecessor has a different Strahler number. If two or more direct predecessors have the maximum Strahler number, the Strahler number of i is defined to be one greater than the maximum Strahler number of all direct predecessors of i . Thus, s i = { 1 if i is a source s ̂ i + 1 if s N i , j = s N i , k = s ̂ , j ≠ k , D i ≠ j or k s ̂ i else . } (1) The distribution of nodes of a given Strahler number is computed. The number of nodes with Strahler number i is referred to as σ i = ∑ j = 1 M δ i , s j . σ 1 , σ 2 , and σ 3 are found to have a linear relationship with M . σ i is fit as σ i = β i M (2) for i = 1 , 2 , 3 . Here, β 1 = 0.455 ± 0.005 , β 2 = 0.275 ± 0.006 , and β 3 = 0.169 ± 0.006 . With the distribution of Strahler numbers one can define a particle bifurcation ratio, 8 R σ i , which is defined as R σ i = σ i / σ i + 1 . Horton’s law of bifurcation ratios states that bifurcation ratios in hydrological drainage networks are similar over a wide range of i . As σ i is given by Eq. (2) , the ratio of successive sigma is given by R σ i = σ i σ i + 1 = β i M β i + 1 M = β i β i + 1 . (3) Taking the ratio of β values of a given order, R σ 1 = 1.65 ± 0.040 and R σ 2 = 1.63 ± 0.068 . Thus, the conducting particle networks align well with Horton’s law of bifurcation ratios. One can also define streams, where a stream of order i is composed of all neighboring particles with Strahler number i . Horton’s law of stream lengths states that the ratios of average stream lengths of successive orders, R l ¯ i = l ¯ i + 1 / l ¯ i , where l ¯ i is the average stream length of order i , are constant over a wide range of i . 8 Here, natural values for river networks range from 1.5–3.0. 8 The average experimental value of l ¯ 1 is 1.93 ± 0.040 , the average of l ¯ 2 is 4.75 ± 0.157 , and the average of l ¯ 3 is 9.34 ± 0.811 . Thus, R l ¯ 1 = 2.46 ± 0.096 and R l ¯ 2 = 1.97 ± 0.183 . So, experimental networks obey Horton’s law within twice the standard error. One can also use streams to define a stream bifurcation ratio R ρ i . Here, ρ i is the number of streams with Strahler number i . ρ is fit with ρ i = η i M for i = 1 , 2 , 3 . Here, η 1 = 0.239 ± 0.003 , η 2 = 0.058 ± 0.001 , and η 3 = 0.017 ± 0.001 . So, we have R ρ i = ρ i / ρ i + 1 = η i M / η i + 1 M = η i / η i + 1 . Taking the ratio of η values of a given order, R ρ 1 = 4.12 ± 0.088 and R ρ 2 = 3.41 ± 0.209 . Using this definition, networks of conducting particles do not match Horton’s law as closely as when using Eq. (3) . However, these results fall into the natural range for river networks, which is 3.0–5.0. 8 Next, the total external path length is computed. The path length p i , the distance between the i th source and the stem, is found in the following way. The procedure begins at the source, where p i is initialized as zero. Then, one is added to p i after moving to the direct successor, D i . This process of moving to D i and adding one is continued until the direct successor belongs to the stem. The path lengths for each source are summed up to obtain the total external path length p e = ∑ i = 1 Δ 1 p i . It is found that the total external path length p e fits a power law with M , where p e = γ 1 M γ 2 , where γ 1 = 0.324 ± 0.128 and γ 2 = 1.54 ± 0.067 (Fig. 2 ). Here, scaling is only shown for M values ranging over about a factor of 5. However, the results in this range fit the power law very well. The experimental results are compared with a DLA model, which has previously been found to follow Horton’s laws. 10 An aggregate of M nodes is produced in the following way. First, one node, referred to as the seed node, is placed in the center of a circle of radius 200. Second, a “walker” particle is placed on a random point on a circle of radius 100 about the seed node. The walker moves along a random path where its position at timestep n , W n , is determined by W n + 1 = W n + ( r 1 , r 2 ) where r 1 and r 2 are uniformly distributed random numbers with − 0.5 < r 1 < 0.5 and − 0.5 < r 2 < 0.5 . If the distance between the walker and a node i is less than 2, then the walker turns into a node, node j . Further i and j become neighbors and D j = i . Each walk results in at most one connection. If the distance between the walker and the center exceeds 200, then the walker is deleted. Step two is repeated until M nodes are created. In DLA structures with 110 < M < 490 , B is fit with the function B = α B , D M , where α B , D = 0.207 ± 0.003 . For the same range of M , Δ 1 is fit with the function Δ 1 = α 1 , D M , where α 1 , D = 0.221 ± 0.003 . Thus, degree distributions for DLA structures are in good agreement with experimental structures. Furthermore, in DLA structures, σ i , i = 1 , 2 , 3 , is fit as σ i = β i , D M , for i = 1 , 2 , 3 . Here, β 1 , D = 0.474 ± 0.007 , β 2 , D = 0.290 ± 0.015 , and β 3 , D = 0.163 ± 0.015 . Therefore, R σ 1 = 1.63 ± 0.088 and R σ 2 = 1.78 ± 0.232 , and Strahler distributions and bifurcation ratios also match experimental results. Also, Horton’s law of bifurcation ratios is exhibited within the standard error. The average DLA value of l ¯ 1 = 2.13 ± .065 , the average of l ¯ 2 = 5.50 ± 0.401 , and the average of l ¯ 3 = 10.50 ± 1.26 . Thus, R l ¯ 1 = 2.58 ± 0.205 and R l ¯ 2 = 1.97 ± 0.277 . So, DLA structures obey Horton’s law of stream lengths within twice the standard error as well. Furthermore, ρ i , i = 1 , 2 , 3 , is fit as ρ i = η i , D M for i = 1 , 2 , 3 . Here, η 1 , D = 0.221 ± 0.003 , η 2 , D = 0.048 ± 0.002 , and η 3 , D = 0.016 ± 0.001 . Therefore, R ρ 1 = 4.60 ± 0.202 and R ρ 2 = 3.00 ± 0.225 , and so DLA structures do not match Horton’s laws as well when using streams. However, the values of R ρ i fall into the natural range for river networks. In addition, p e fits a similar power law with M , where p e = γ 1 , D M γ 2 , D . Here, γ 1 , D = 0.365 ± 0.080 and γ 2 , D = 1.45 ± 0.037 , (Fig. 2 ) agreeing well with experimental values. Again, though M values only range over a factor of about 5, the results fit the power law very well in this range. Indeed, there has been previous connection of DLA and electrostatic scaling. 11,12 A connection between diffusive growth and this system is predicted as the regions where the forces on particles is the largest correspond to regions of high growth probability of DLA clusters. Both the limiting state of the probability density in diffusive growth and the electrostatic system obey Laplace’s equation. Thus some connection in the steady state of each system is expected. It should also be noted that the fractal dimension of DLA structures is 1.67, 7 which is the same as the dimension of the experimental structures mentioned previously. In summary, steady state networks of conducting particles under a high voltage were found to obey several scaling properties which agree with DLA structures. Experimental networks and DLA structures were described using degree distributions, the Strahler numbering, and the total external pathlength. Several systems, including axons of neurons, the coronary arterial tree, and the retinal vascular tree have previously been analyzed using the Horton–Strahler scheme and have been shown to follow Horton’s laws. 13–15 Here, a nonbiological physical system is found to obey the same laws. This material is based upon work supported by the National Science Foundation Grant Nos. NSF PHY 01-40179 and NSF DMS 03-25939 ITR. The authors gratefully acknowledge the support for this work by Defense Advanced Research Projects Agency (DARPA) Physical Intelligence subcontract (HRL9060-000706). FIG. 1. Typical steady state aggregate of 512 particles under a high voltage. FIG. 2. Total external pathlength plotted against number of nodes M in a log-log plot. Data points shown with power law fit, p e = γ 1 M γ 2 , where γ 1 = 0.324 ± 0.128 and γ 2 = 1.54 ± 0.067 . Corresponding data points for DLA structures are shown as squares and best fit dashed line for DLA results is given by p e = γ 1 , D M γ 2 , D . Here, γ 1 , D = 0.365 ± 0.080 and γ 2 , D = 1.45 ± 0.037 .
PY - 2011/2/1
Y1 - 2011/2/1
N2 - Under an electric field, spherical conducting particles in a dielectric liquid assemble into a dendritic tree in order to dissipate charge. Several topological measures characterize such networks, including degree distributions, Strahler numbers, and total external pathlengths. Here, scaling laws relating these measures to the number of nodes in the system are presented and shown to match diffusion limited aggregation (DLA) structures. Experimental bifurcation and stream-length ratios in this easily reproducible laboratory experiment are found to agree with DLA simulations and Horton's laws for river networks. Certain scaling relations in transportation networks are known to originate from general features of networks. Here we find the experimental structures share properties with hydrological drainage networks.
AB - Under an electric field, spherical conducting particles in a dielectric liquid assemble into a dendritic tree in order to dissipate charge. Several topological measures characterize such networks, including degree distributions, Strahler numbers, and total external pathlengths. Here, scaling laws relating these measures to the number of nodes in the system are presented and shown to match diffusion limited aggregation (DLA) structures. Experimental bifurcation and stream-length ratios in this easily reproducible laboratory experiment are found to agree with DLA simulations and Horton's laws for river networks. Certain scaling relations in transportation networks are known to originate from general features of networks. Here we find the experimental structures share properties with hydrological drainage networks.
UR - http://www.scopus.com/inward/record.url?scp=79951826873&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79951826873&partnerID=8YFLogxK
U2 - 10.1063/1.3533389
DO - 10.1063/1.3533389
M3 - Article
AN - SCOPUS:79951826873
SN - 0021-8979
VL - 109
JO - Journal of Applied Physics
JF - Journal of Applied Physics
IS - 3
M1 - 036103
ER -