Predicting the growth of ramified networks

Alfred Hübler

Center for Complex Systems ResearchUniversity of Illinois at Urbana-Champaign

Research supported in part by the National Science Foundation

(DMS-03725939 ITR)

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.

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-wiresM. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

Experimental Study of Structural Changes in Materials due to High-voltage Currents:

Growth of Fractal Transportation Networks

20 kV

needle electrode sprays charge over oil surface

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 D=1.6 mm

particles sit on the bottom of the dish

Phenomenology Overview

12 cm

t=0s 10s 5m 13s 14m 7s

14m 14s 14m 41s 15m 28s 77m 27s

stage I:strand

formation

stage II:boundary

connection stage III: geometric expansion

stationary state

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.

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.

Qualitative effects of initial distribution

N = 752T = 131B = 85

N = 720T = 122B = 106

N = 785T = 200B = 187

N = 752T = 149B = 146

(N = Number of Particles, T = Number of Termini, B=Number of Branch Points)

Can we predict the structure of the emerging transportation network?

?

Predicting the Fractal Transporatation Network

Left: Initial condition, Right: Emergent transporation network

Predictions of structural changes in materials due to a high voltage current: Predicting fractal network growth

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)

loner

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.

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.

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.

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

Structural changes of materials in high voltage current

Experiment: J. Jun, A. Hubler, PNAS 102, 536 (2005)1) Three growth stages: strand formation, boundary connection, and

geometric expansion;2) Networks are open loop;3) 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 flowApplications: Hardware implementation of neural nets, absorbers, batteriesM. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the

Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

random initial distribution compact initial distribution

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

↓heat flow ↓operating temperature

Mixed reality: Out-of-body experiences with video feedback

Blanke O et al.Linking OBEs and self processing to mental own body imagery at the temporo-parietal junction. J Neurosci 25:550-55 (2006).

- 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 => 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

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.

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.

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 => 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

Mixed reality states in physical systems: Why are they important?

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

- 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.

Photo: A. Hubler and V. Gintautas at the inter-reality system

Structural changes of materials in high voltage current

Experiment: J. Jun, A. Hubler, PNAS 102, 536 (2005)1) Three growth stages: strand formation, boundary connection, and

geometric expansion;2) Networks are open loop;3) 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 flowApplications: Hardware implementation of neural nets, sensors, batteries, non-

equilibrium materials, mixed reality systems with graph-theoretical modelsM. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the

Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

random initial distribution compact initial distribution

Thank you!