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## WHAT IS IT?
This simple model demonstrates a phase transition behavior emerging from the interactions among multiple agents in the presence of noise.
At a noise level higher than a certain threshold, the system generates symmetric behavior (e.g. vapor or melt of magnetization) or disagreement; whereas at a noise level lower than the threshold, the system exhibits spontaneous symmetry breaking (solid or magnetization) or consensus.
This provides some simple intuitions for understanding the impact of noise in communication networks on network behavior and the impact of noise (temperature) in phase transitions.
NOTE: For this simulator, the following terms are used interchangeably:
## HOW IT WORKS
A group of nonmobile agents form a network, and each agent, taking bipolar value either +1 or 1, updates its value according to its previous value and the noisy measurements of the values of the agents connected to it.
Update rule: At each moment, each turtle receives all values from all turtles connected to it (including the turtle itself); these values are then averaged at the turtle and perturbed by a noise. The noise is uniformly distributed in interval [z, z], where z is called the noise level. If the perturbed average value is positive, then the turtle will take value +1, otherwise it will take value 1.
In other words, each turtle first aggregates the opinions of all its connections and itself (subject to communication/observation noise perturbation), and then follows the aggregated opinion.
## HOW TO USE IT
To simulate, press SETUP to create the turtles and links, and then press GO to see the network evoluation or GO ONCE to go one step.
The following may be set before SETUP:
POPULATION: Use the slider to set the value for the number of turtles in the simulation. A very high POPULATION may slow down the simulation.
NOISELEVEL: Use the slider to set the noise level.
RANDOMCONNECTIONS?: If ON is chosen, then the network forms a random graph process, with probability p (set by PFORRANDOMCONNECTIONS slider) to form a link between any two turtles. If OFF is chosen, then the network is over a fixed graph formed by connecting the nearest turtles within a distance set by VISION slider.
VISION: For the fixed graph case (RANDOMCONNECTIONS?=OFF) only. Two turtles are connected by a link if they are within the distance set by VISION sider.
PFORRANDOMCONNECTIONS: For the random graph case (RANDOMCONNECTIONS?=ON) only. For any turtle pair, the probablity of forming a connection (i.e. a link) is given by PFORRANDOMCONNECTIONS slider.
MIN/MAX NOISE LEVEL GUARANTEEING AGREEMENT: This displays, for a network after SETUP button is pressed, a pair of numbers such that if NOISELEVEL slider chooses any value between the min and max, then agreement is guaranteed.
The following are output:
\#RED, #BLUE: These display the number of red turtles (with state value +1) and the number of blue turtles (with state value 1) at the current time instant.
PREDCTION: This displays the prediction message. After SETUP button is preseed, the simulator makes a prediction of the behavior based on some analytic results. The prediction message may be:
SUM SEQUENCE: Each data point of the sequence corresponds to a tick in time and the sum of all turtles' state values at that time.
SUM OF SUM SEQUENCE: Each data point of the sequence corresponds to a tick in time and the accumulated sum of the SUM SEQUENCE by that time.
AVERAGE OF SUM SEQUENCE: Each data point of the sequence corresponds to a tick in time and the mean value of the SUM SEQUENCE by that time.
## THINGS TO NOTICE
If the noise level set by NOISELEVEL slider is higher than the displayed MAX NOISE LEVEL GUARANTEEING AGREEMENT, the system generates symmetric behavior (e.g. vapor or melt of magnetization) or disagreement; whereas at a noise level lower than MAX NOISE LEVEL GUARANTEEING AGREEMENT but above MIN NOISE LEVEL GUARANTEEING AGREEMENT, the system exhibits symmetry breaking (solid or magnetization) or consensus. This holds for both the fixed graph (RANDOMCONNECTIONS?=OFF) and random graph process (RANDOMCONNECTIONS?=ON).
If agreement is predicted, #RED, #BLUE, SUM SEQUENCE, SUM OF SUM SEQUENCE, and AVERAGE OF SUM SEQUENCE will reach/converge to some constants.
If disagreement is predicted, #RED, #BLUE, and SUM SEQUENCE will oscillate, SUM OF SUM SEQUENCE will exhibit a random walk behavior, and AVERAGE OF SUM SEQUENCE will tend to zero.
## THINGS TO TRY
For the random graph case, sweep the noise level (NOISELEVEL) and connection probability (PFORRANDOMCONNECTIONS), and observe network behavior.
For the fixed graph case, sweep the noise level (NOISELEVEL) and vision (VISION), and observe network behavior.
## EXTENDING THE MODEL
This model assumes no leader yet. What if there is one or more leaders of a same color? What if there are several leaders with different colors?
What if there are more than two states or more than two colors for each turtle?
What if other noise distributions are used?
## NETLOGO FEATURES
## RELATED MODELS
* Flocking
## CREDITS AND REFERENCES
This model is based on:
* J. Liu, V. Yadav, H. Sehgal, J. M. Olson, H. Liu, N. Elia: Phase Transitions on Fixed Connected Graphs and Random Graphs in the Presence of Noise. IEEE Trans. Automat. Contr. 53(8): 18171825 (2008)
* J. Liu, V. Yadav, H. Sehgal, J. M. Olson, H. Liu, N. Elia: Phase Transitions on Fixed Connected Graphs and Random Graphs in the Presence of Noise. Proc. 44th IEEE Conference on Decision and Control (CDC) and 2005 European Control Conference (ECC): 734739 (2005)
## HOW TO CITE
If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:
* J. Liu, V. Yadav, H. Sehgal, J. M. Olson, H. Liu, N. Elia: Phase Transitions on Fixed Connected Graphs and Random Graphs in the Presence of Noise. IEEE Trans. Automat. Contr. 53(8): 18171825 (2008)
* Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and ComputerBased Modeling, Northwestern University, Evanston, IL.
## COPYRIGHT AND LICENSE
Copyright 2013 J. Liu. 
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