Predicting the growth of ramified networks

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Predicting the growth of ramified networks

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

Research supported in part by the National
Science Foundation (DMS-03725939 ITR)
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We study Growth of networks in a reproducible
lab experiment, here the structure of materials
with high-voltage currents which quantities a
reproducible? We find Materials produced in a
high-voltage current develop open-loop, fractal
structures which maximize the conductivity for
the applied current. These fractal structures can
be predicted with graph-theoretical models.
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• Potential applications
• -Predict and control the dynamics of networks,
  -i.e. use resonances to efficiently detect,
  grow, nourish, destabilize, disintegrate
  networks
• o ramified chemical absorbers (better
  batteries, better sensors, better purifiers)
• o multi-agent mixed reality systems
• o the rise and fall of social networks
• -Non-equilibrium materials maximum strength in
  a strong gradient
• Atomic neural nets integration processing of
  information in super-brains out of digital
  nano-wires
• M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian
  Learning in the Agglomeration of Conducting
  Particles, Phys.Rev.E. 59, 3165 (1999)

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Experimental Study of Structural Changes in
Materials due to High-voltage CurrentsGrowth of
Fractal Transportation Networks
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 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
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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 termini or three-fold branch
points in stage III. In addition there are a few
loners (less than 1). Loners are not connected
to any other particle. There are no closed loops
in stage III.
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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.
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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
(N Number of Particles, T Number of Termini,
BNumber of Branch Points)
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?
Can we predict the structure of the emerging
transportation network?
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Predicting the Fractal Transporatation Network
Left Initial condition, Right Emergent
transporation network
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Predictions of structural changes in materials
due to a high voltage current Predicting fractal
network growth
loner
Task Digitize stage II structure and predict
stage III transporation network. 1) Determine
neighbors, since particles can only connect to
their neighbors. All the links shown on the left
are potential connections for the final tree. 2)
Use a graph-theoretical algorithms to connect
particles, until all available particles connect
into a tree. Some particles will not connect to
any others (loners). They commonly appear in
experiments.
We test three growth algorithms 1) Random
Growth Randomly select two neighboring particles
connect them, unless a closed loop is formed
(RAN) 2) Minimum Spanning Tree Model Randomly
select pair of very close neighbors connect
them, unless a closed loop is formed (MST) 3)
Propagating Front Model Randomly select pair of
neighbors, where one of them is already connected
connect them, unless a closed loop is formed
(PFM)
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Random Growth Model Randomly select two
neighboring particles
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 Model Randomly select pair
of very close neighbors
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.
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Propagation Front Model Randomly select
connected pair of neighbors
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.
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Comparison of all models to experiments
Main Result The Minimum Spanning Tree (MST)
growth model is the best predictor of the
emerging fractal transportation network
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Structural changes of materials in high voltage
current
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 random initial condition, predict other
  observables, control network growth, study
  fractal structures in systems with a large heat
  flow
• Applications Hardware implementation of neural
  nets, absorbers, batteries
• M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian
  Learning in the Agglomeration of Conducting
  Particles, Phys.Rev.E. 59, 3165 (1999)

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• Illustration of potential applications
• Understanding IED issues
• (a) Survival Armor with non-equilibrium
  materials
• Materials have their maximum strength in a large
  (heat) flow if they are produced in a large
  (heat) flow.
• (b) Detection Nonlinear Resonance Spectroscopy
  Optimal controls complements the dynamics.
• G. Foster, A. Hübler, and K. Dahmen.
  Resonant forcing of multidimensional chaotic map
  dynamics. Phys. Rev. E 75, 036212 (2007).
• (c) Prevention Control of social networks
• Boundary controls are most efficient for social
  networks.
• Nonlinear resonances Optimal controls
  complements the dynamics.
• Adaptation to the edge of chaos Self-adjusting
  systems avoid chaos.
• Leadership principle Predictable systems are
  exploitable.
• Atomic neural nets mixed reality controls

?operating temperature
?heat flow
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Mixed reality Out-of-body experiences with video
feedback
- Subject sees video image of itself with 3D
goggles - Two sticks, one strokes person's chest
for two minutes, second stick moves just
under the camera lenses, as if it were
touching the virtual body. - Synchronous
stroking gt people reported the sense of being
outside their own bodies, looking at themselves
from a distance where the camera is located. -
While people were experiencing the illusion, the
experimenter pretended to smash the virtual body
by waving a hammer just below the cameras.
Immediately, the subjects registered a threat
response as measured by sensors on their skin.
They sweated and their pulses raced. Real system
similar virtual system bi-directional
instant. coupling mixed reality
Blanke O et al.Linking OBEs and self processing
to mental own body imagery at the
temporo-parietal junction. J Neurosci 25550-55
(2006).
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Our work Experimental evidence for mixed reality
states in physical systems
Objective Understand synchronization between
virtual and real systems. Approach - Couple a
real dynamical system to its virtual counterpart
with an instantaneous bi-direction coupling (so
far non-linear pendulum, future network). -
Measure an order parameter of the real and the
virtual systems and then detect synchronization.
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Experimental evidence for mixed reality states in
physical inter-reality systems
Results - Experimental evidence for a phase
transition from dual reality states to mixed
reality states. - Phase diagram of the
inter-reality system is in good agreement with
the phase diagram of the simulated inter-reality
system.
Phase diagram of the inter-reality system
amplitude of the coupling versus the frequency
ratio of the real and the virtual system. The
phase boundary between mixed reality states (I)
and dual reality states (II). The solid, dashed,
and dotted lines indicate the critical points in
the experiment, simulation, and analytic theory,
respectively.
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Resonance Curves of Inter-reality systems
Figure 1. Amplitude X of the real system versus
the frequency ratio for the experimental system
(squares) and for the numerical system
(triangles) Perfect match between real and
virtual system gt largest amplitudes
Figure 2. The opposite of the amplitude of the
real system versus the frequency ratio and
versus the ratio of the third order terms
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Mixed reality states in physical systems Why are
they important?
- Virtual systems match their real counter parts
with ever-increasing accuracy, such as graph
theoretical network predictors. - New hardware
for instantaneous bi-directional coupling, such
as video feedback. - In mixed reality states
there is no clear boundary between the real and
the virtual system. Mixed reality states can be
used to analyze and control real systems with
high precision. And then there is the possibility
for time travel by the virtual system.
Publication The paper "Experimental evidence for
mixed reality states in an inter-reality system"
by Vadas Gintautas and Alfred Hubler, in Phys.
Rev. E 75, 057201 (2007), was selected for the
APS tip sheet http//www.aps.org/about/tipsheets/
tip68.cfm
Photo A. Hubler and V. Gintautas at the
inter-reality system
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Structural changes of materials in high voltage
current
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 random initial condition, predict other
  observables, control network growth, study
  fractal structures in systems with a large heat
  flow
• Applications Hardware implementation of neural
  nets, sensors, batteries, non-equilibrium
  materials, mixed reality systems with
  graph-theoretical models
• M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian
  Learning in the Agglomeration of Conducting
  Particles, Phys.Rev.E. 59, 3165 (1999)

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Thank you!