Predicting the growth of fractal particle agglomeration networks with graph theoretical methods

1
Predicting the growth of fractal particle
agglomeration networks with graph theoretical
methods

• Joseph Jun and Alfred Hübler
• Center for Complex Systems Research
• University of Illinois at Urbana-Champaign

Research supported in part by the National
Science Foundation (PHY-01-40179 and
DMS-03725939 ITR)
2
Growth of a ramified transportation network.
random initial distribution
compact initial distribution

• Experiment Agglomeration of conducting
  particles in an electric field
• 1) We focus on the dynamics of the system
• 2) We explore the topology of the networks using
  graph theory.
• 3) We explore a variety of initial conditions.
• Results
• three growth stages strand formation, boundary
  connection, and geometric expansion.
• networks are open loop
• statistically robust features number of
  termini, number of branch points, resistance,
  initial condition matters somewhat
• 4) Minimum spanning tree growth model predicts
  emerging pattern

3
Description of experimental setup
Basic experiment consists of two electrodes, a
source electrode and a boundary electrode
connected to opposite terminals of a power supply.
source electrode
battery
boundary electrode
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Description of experimental setup
Basic experiment consists of two electrodes, a
source electrode and a boundary electrode
connected to opposite terminals of a power
supply. The boundary electrode lines a dish made
of a dielectric material such as glass or
acrylic. The dish contains particles and a
dielectric medium (oil)
source electrode
battery
particle
boundary electrode
oil
5
Description of experimental setup
20 kV
battery maintains a voltage difference of 20 kV
between boundary and source electrodes
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Description of experimental setup
source electrode sprays charge over oil surface
20 kV
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Description of experimental setup
source electrode sprays charge over oil surface
20 kV
air gap between source electrode and oil surface
approx. 5 cm
8
Description of experimental setup
source electrode sprays charge over oil surface
20 kV
air gap between source electrode and oil surface
approx. 5 cm
boundary electrode has a diameter of 12 cm
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Description of experimental setup
needle electrode sprays charge over oil surface
20 kV
air gap between needle electrode and oil surface
approx. 5 cm
boundary electrode has a diameter of 12 cm
oil height is approximately 3 mm, enough to cover
the particles castor oil is used high viscosity,
low ohmic heating, biodegradable
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Description of experimental setup
needle electrode sprays charge over oil surface
20 kV
air gap between needle electrode and oil surface
approx. 5 cm
ring electrode forms boundary of dish has a
radius of 12 cm
oil height is approximately 3 mm, enough to cover
the particles castor oil is used high viscosity,
low ohmic heating, biodegradable
particles are non-magnetic stainless steel,
diameter D1.6 mm particles sit on the bottom of
the dish
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Phenomenology
The growth of the network proceeds in three
stages I) strand formation II) boundary
connection III) geometric expansion
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Phenomenology Overview

12 cm
stage I strand formation
t0s
10s
5m 13s
14m 7s
13
Phenomenology Overview

12 cm
stage I strand formation
t0s
10s
5m 13s
14m 7s
14m 14s
stage II boundary connection
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Phenomenology Overview

12 cm
stage I strand formation
t0s
10s
5m 13s
14m 7s

14m 14s
14m 41s
15m 28s
stage II boundary connection
stage III geometric expansion
15
Phenomenology Overview

12 cm
stage I strand formation
t0s
10s
5m 13s
14m 7s

14m 14s
14m 41s
15m 28s
77m 27s
stage II boundary connection
stage III geometric expansion
stationary state
16
Motion of the strands
The motion of the lead particles of the six
largest strands from a single experiment.
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Motion of the strands
The motion of the lead particles of the six
largest strands from a single experiment.
Distance of lead particle of a strand correlates
well with number of particles in strand.
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N591
N784
N1044
Comparing for different numbers of particles, N.
The growth of the strands still tend to correlate
for higher N.
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Phenomenology stage II (boundary connection)
Stage II begins when the winning strand
connects to the boundary. It is brief in
duration, and is best characterized by the
particles binding to the boundary.
20
Phenomenology stage III (geometric expansion)
After all the particles bind together, they will
now be like charged and spread apart. This
expansion into the available space is the main
characteristic of stage III.
21
Adjacency defines topological species of each
particle
Termini particles touching only one other
particle Branching points particles touching
three or more other particles Trunks particles
touching only two other particles
Particles become one of the above three types in
stage II and III. This occurs over a relatively
short period of time.
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Graph theory measures for trees
We allow the physical locations of the particles
to define the adjacency.
The particles positions are digitized. Each
particle is considered a node. When the distance
between two particles is shorter than a cutoff
length, they are considered adjacent we put a
link between them.
c5
c3
red circles indicate cutoff length yellow lines
indicate distance between centers of particles
23
Adjacency (number of neighbors)
We can define the average adjacency
mathematically as
ci is the adjacency of particle i T is the
Heaviside step function N is the total number of
particles ri rj are the positions of particles
i j respectively rcut is the cutoff length
Ideally, rcut D, where D is the diameter of a
particle. But because of the noise in digitizing
the position of the particles, we use a slightly
larger value, usually 1.16 rcut /D
1.28. Also ideally, 0 ci 6 we impose this by
hand in the algorithm.
24
Adjacency algorithm
photos from experiment
Digitize the positions of each particle from the
photos.
25
Adjacency algorithm
photos from experiment
digitization of positions
Digitize the positions of each particle from the
photos. Run the adjacency algorithm on the list
of particle positions.
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Adjacency algorithm
photos from experiment
output from algorithm
Digitize the positions of each particle from the
photos. Run the adjacency algorithm on the list
of particle positions. The algorithm picks up how
particles are connected. It identifies holes and
grain boundaries. Graphs from algorithm were
visualized using the Combinatorica package in
Mathematica.
rcut 1.25D
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Visualizing the stages with the adjacency
By looking at ltcgt as a function of time from the
digitization of the photos, we can see this
measure naturally segregates the stages.
The average adjacency versus time.
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Visualizing the stages with the adjacency
By looking at ltcgt as a function of time from the
digitization of the photos, we can see this
measure naturally segregates the stages.
The top dashed lines is an estimate of ltcgt at t0
s, given by (circle)
The bottom dotted line is the value of ltcgt in the
steady-state (single strand)
The average adjacency converges rapidly.
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Visualizing the stages with the adjacency
By looking at ltcgt as a function of time from the
digitization of the photos, we can see this
measure naturally segregates the stages.
The top dashed lines is an estimate of ltcgt at t0
s, given by (circle)
The bottom dotted line is the value of ltcgt in the
steady-state (single strand)
The inset shows the same plot for several values
of the cutoff length.
The average adjacency converges rapidly.
30
Visualizing the stages with the adjacency
A look at the differences in stages between
different particle numbers. The average adjacency
converges rapidly for all cases. We conclude that
the topology of the network establishes in a
relatively short amount of time following stage
II.
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Relative number of each species is robust
Graphs show how the number of termini, T, and
branching points, B, scale with the total number
of particles in the tree.
32
Branching point subspecies
b3
b4
b6
b5
Subspecies b5 and b6 have never been observed in
the experiment.
33
Branching point subspecies
Percentage of branching points that connect to
four other particles as a function of particle
number.
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Most networks are trees.Only a few rare cases
contain loops (cycles).
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Loops (cycles) are unstable
Insets on the left show two particles
artificially placed into a loop separate from one
another. The graph on the right shows the
separation between the two particles as a
function of time.
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Fractal Dimension of Particles
N 792 T 159 B 153
N 791 T 170 B 164
N 794 T 170 B 162
N 784 T 166 B 161
The mass dimension, dm, is defined by S?(r) N
rdm
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Fractal Dimension of Particles
N 792 T 159 B 153
N 791 T 170 B 164
N 794 T 170 B 162
N 784 T 166 B 161
dm 1.74-1.83
dm 1.76-1.82
dm 1.75-1.91
dm 1.79-1.90
The mass dimension, dm, is defined by S?(r) N
rdm
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Fractal Dimension
Particles arrange themselves similarly in
different experiments.
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Spatial distribution in time
The radial distribution of particles for
different times in the experiment. The system
entered stage II after t847s. The fractal
dimension decreases from Dm2 to Dm1.8.
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Spatial distribution of termini is almost
homogeneous, except for small particle numbers
The radial distribution of termini for similar
number of particles and different number of
particles.
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Initial conditions
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Qualitative effects of initial distribution
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Qualitative effects of initial distribution
N 752 T 131 B 85
N 720 T 122 B 106
N 785 T 200 B 187
N 752 T 149 B 146
Initial conditions are a strong constraint on the
final form of tree(s).
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Qualitative effects of initial distribution
?
Will this initial configuration produce a spiral?
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Qualitative effects of initial distribution
No, system is unstable to ramified structures.
46
Perimeter effects (cheat experiments)
Eliminating stage I by artificially placing a
connecting strand to the boundary we call these
cheat experiments.
47
Perimeter effects (cheat experiments)
Eliminating stage I by artificially placing a
connecting strand to the boundary we call these
cheat experiments. In this case, there are no
losing strands that become long termini at the
perimeter.
48
Perimeter effects
Consequently, there are more termini and
branching points for the cheat cases. Initial
conditions directly preceding stage II are
important to determining the relative number of
topological species.
49
Review of experimental results
Growth of trees occurs in three stages.Average
adjacency captures the three stages.Topology of
network forms relatively quickly.Particles
become one of three species.The relative
abundance of each species is statistically
reproducible.Initial conditions are a strong
constraint to formation of networks.
50
Artificially generated networks
How does the state of the system directly
preceding stage II affect the topology of the
trees?Can we predict the final tree at this
stage?
51
Artificially generated networks
Since topology of the networks is established
relatively quickly, particles connect to one
another before they have moved far. Thus, we
attempt to model the connections formed by the
system using only the local information for each
particleits neighborhood.
52
Artificially generated networks
Since topology of the networks is established
relatively quickly, particles connect to one
another before they have moved far. Thus, we
attempt to model the connections formed by the
system using only the local information for each
particleits neighborhood.
We use data from the experiments a snapshot of
the particles directly preceding stage II.
53
Artificially generated networks
Since topology of the networks is established
relatively quickly, particles connect to one
another before they have moved far. Thus, we
attempt to model the connections formed by the
system using only the local information for each
particleits neighborhood.
We take data from the experiments a snapshot of
the particles directly preceding stage II.
Digitize the positions. Run the adjacency
algorithm to obtain a base neighborhood.
cutoff length 3 ? particle diameter
54
Artificially generated networks
loner
From the base neighborhood, we apply algorithms
to generate trees. In other words, particles can
only connect to particles that neighbor it. All
the links shown on the left are potential
connections for the final tree. Algorithms run
until all available particles connect into a
tree. Some particles will not connect to any
others (loners). They commonly appear in
experiments.
loner
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Artificially generated networks
loner
From the base neighborhood, we apply algorithms
to generate trees. In other words, particles can
only connect to particles that neighbor it. All
the links shown on the left are potential
connections for the final tree. Algorithms run
until all available particles connect into a
tree. Some particles will not connect to any
others (loners). They commonly appear in
experiments.
loner
We chose three algorithms to implement 1) random
(RAN) 2) minimum spanning tree (MST) 3)
propagating front model (PFM)
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Random
The random algorithm randomly selects a link from
the neighborhood graph and determines whether to
connect the two particles based on whether the
link maintains or violates a tree structure. In
practice, we do this by tracking a tree label
for each particle. If two particles in a
potential connection have the same label, the
connection would produce a cycle, and
consequently it is rejected.
particle 1
particle 2
summary of RAN connection rule
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RAN
movie of random algorithm
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RAN
Typical connection structure from RAN algorithm.
Distribution of termini produced from 105
permutations run on a single experiment.
Number of termini produced for all experiments,
plotted as a function of N.
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Minimum Spanning Tree
Uses the identical acceptance/rejection criterion
as RAN. The difference between the two is in how
the potential connections are chosen. MST picks
shortest links first (particles that are closest
to one another). Since there are degeneracies in
links, we run the algorithm through 105
permutations of degenerate ordering.
graph (non-tree)
tree (non-minimal)
tree (minimal)
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MST
movie of minimum spanning tree algorithm
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MST
Typical connection structure from MST algorithm.
Distribution of termini produced from 105
permutations run on a single experiment.
Number of termini produced for all experiments,
plotted as a function of N.
62
Propagating Front Model
Since only one strand reaches the boundary, the
connections should propagate from a particular
direction. To capture this, we propose a model
where particles link in order by their geographic
location. Particles can connect only when they
are adjacent to a particle that already belongs
to the boundary.
63
Propagating Front Model
Since only one strand reaches the boundary, the
connections should propagate from a particular
direction. To capture this, we propose a model
where particles link in order by their geographic
location. Particles can connect only when they
are adjacent to a particle that already belongs
to the boundary.
grey thatched particles are already in the
network, connections are shown in black
lines white particles are available to
connect dotted particles are not allowed to
connect because they are not yet adjacent to a
particle in the network
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Propagating Front Model
Since only one strand reaches the boundary, the
connections should propagate from a particular
direction. To capture this, we propose a model
where particles link in order by their geographic
location. Particles can connect only when they
are adjacent to a particle that already belongs
to the boundary.
the grey filled particle was randomly chosen it
must now randomly select one of its neighbors
that are already in the network
65
Propagating Front Model
Since only one strand reaches the boundary, the
connections should propagate from a particular
direction. To capture this, we propose a model
where particles link in order by their geographic
location. Particles can connect only when they
are adjacent to a particle that already belongs
to the boundary.
the chosen particle joined the network any
particles adjacent to it are now added to the
list of particles that may connect
66
Propagating Front Model
Since only one strand reaches the boundary, the
connections should propagate from a particular
direction. To capture this, we propose a model
where particles link in order by their geographic
location. Particles can connect only when they
are adjacent to a particle that already belongs
to the boundary.
the process repeats until all particles join the
boundary.
67
PFM
movie of propagating front model
68
PFM
Typical connection structure from PFM algorithm.
Distribution of termini produced from 105
permutations run on a single experiment.
Number of termini produced for all experiments,
plotted as a function of N.
69
Comparison of all models to experiments
The number of termini and branching points for
all three models and the natural experiments. MST
produces the closest match with experiments.
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Comparison of all models to experiments
cheat initial condition (without stage I) and
natural initial condition
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ltTgt-ltBgt
natural
cheat
ltTgt-ltBgt1b42b53b6 ltTgt-ltBgt is independent of
b3 subspecies Thus, PFM and RAN are not only
generating more branching points, they are
generating higher order branching points.
72
Review of simulations
We applied three algorithms to produce trees
using local connection rules.We found that the
algorithm which uses the interparticle spacing
but neglects the direction of connection produces
the best match to the experiments.
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Predicting the growth of a fractal network.
random initial distribution
compact initial distribution

• Experiment J. Jun, A. Hubler, PNAS 102, 536
  (2005)
• Three growth stages strand formation, boundary
  connection, and geometric expansion
• Networks are open loop
• Statistically robust features number of
  termini, number of branch points, resistance,
  initial condition matters somewhat
• 4) Minimum spanning tree growth model predicts
  emerging pattern.
• 5) To do derive result from first principles,
  random initial condition, predict other
  observables, control network growth
• Applications Hardware implementation of neural
  nets, nano neural nets
• M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian
  Learning in the Agglomeration of Conducting
  Particles, Phys.Rev.E. 59, 3165 (1999)