Chapter 14 Spatial Threshold Systems

1
Chapter 14Spatial Threshold Systems

• including Self-Organized Criticality

2
Cellular DEVS Models of Spatial Threshold Systems

• DEVS abstractions of continuous systems with
  threshold behavior allow simulations that would
  otherwise be too complex/time consuming to be
  feasible
• An interesting class of such systems are the
  Self Organized Criticality (SOC) natural systems
  which depend on threshold properties of
  continuous systems
• These properties allow using DEVS to model them
  with appropriate abstractions
• We break space into cells and model these with
  DEVS models the cells are placed into 1,2,3 or
  higher dimensional grids and coupled with
  neighborhoods form a cellular space.

3
SOC Earthquake expressed as a One Dimensional
Cellular Space
i-2
i-1
i
i1
Due to compression on tectonic plates, each cell
accumulates energy at a constant rate until a
critical threshold is exceeded. At this point a
fraction of its accumulated energy is equally
distributed among its neighbors. This reduces the
time to achieve their critical levels. If the
additional energy causes a cell to exceed its
threshold, then it will release energy to its
neighbors. The size of an avalanche is the number
of such propagated releases that occur until all
cells are below the threshold. Releases occur
with ta 0, so the size of an avalanche is the
length of a transitory sequences of external
transitions.
Power law distribution of avalanches (1/x)
Per Bak, How Nature Works H. Jensen, Self
Organizing Complexity
4
The Meaning of Power Law Distributions
Distributions such as the Normal (bell shaped
curve) and exponential have well defined means
which determine the scale of the underlying real
world phenomenon. Power law distributions do not
have a well defined mean and therefore suggest
that the underlying behavior is scale-free.
Y 10/X.5
Y 10/X
5
SOC Earthquake expressed as a One Dimensional
Cellular Space
earthquakeCell inherits from oneDimCell. To
insert digraph models as cells they must satisfy
the cell interface
Neighborhood coupling
i-2
i-1
i
i1
public class earthquakeCellSpace extends
oneDimCellSpace earthquakeCell.numCells
numCells for (int i 0iltnumCellsi)
addCell(i,new earthquakeCell(i,energyScale)) hid
eAll() //hides only components so
far doNeighborCoupling(1,"outEnergy","in")

//nearest neighbor doNeighborCoupling(-1,"outEnerg
y","in") coupleAllTo("out",this,"out") coupleOne
ToAll(this,"stop","stop") coupleAllTo("outPair",t
his,"outPair")

• EarthquakeCellSpace inherits from OneDimCellSpace
  which provides methods
• for easily creating a one dimentional cellular
  space by adding
• addCell(index, oneDimCell)
• and coupling oneDImCells
• doNeighborCoupling(neighborIndex,outPort,
  inPort)
• coupleAllTo(sport,devs,dport)
• coupleOneToAll(sport,devs,dport)

Random. earthquakeCellSpace
6
Self Organizing Critical Earthquake Cell
public void deltext(double e,message x) energy
eaccumulationRate
//updating energy Continue(e)
if (somethingOnPort(x,"in")) energy
getRealValueOnPort(x,"in")
//if you are neighbor get energy if
(energy gtenergyThresh) holdIn("active",0)
else holdIn("active, (energyThresh
energy)/accumRate)
//every cell increases at same rate else if
(somethingOnPort(x,"stop")) passivate()
public void deltint() super.deltint() energy
0//release all energy holdIn("active",energyThr
esh /accumRate))
//schedule yourself to reach threshold
public message out() message m
super.out() m.add(makeContent("outEnergy",
new doubleEnt(
alpha(energy sigmaaccumRate))))
// alpha small implies non conservation
// of accumulated energy return m
Random. earthquakeCellSpace
7
Internal Transition /Output Generation -- a cell
reaches its threshold energy and distributes its
energy to its neighbors the cell starts
accumulating energy starting from zero
8
Response to External Input a neighbor cell
receives energy and schedules itself to reach the
threshold earlier than before (or it may exceed
threshold and become critical)
elapsed time, e
Time advance, ta
9
Increase of Energy Distribution with Alpha
10
Using a SOC earthquake model to drive a line
segmentation model
An SOC earthquake model generates fractures
taken as the centers of avalanches that exceed a
critical size. Each fracture bisects the line
segment in which it occurs. The record of the
current fracture distribution is maintained by a
line segmentation model.
center
Earthquake CellSpace
spredCellSpace (Line Segmentation)
break
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Curdling and line segmentation
At left, blue, grey and red contiguous colors
indicate randomly chosen segments and
sub-segments. Generations proceed downward. At
bottom, the landscape that results after
generation 9. To conserve density, subsegments
get a height obtained by dividing that of the
parent by the fraction of the length of the
subsegment to its parent (small sub-segments get
larger heights). The plot is of log(height) since
the range varies over several orders of magnitude.
Random. earthSpread
12
Experimental Frames for SOC
13
The Meaning of Power Law Distributions
Distributions such as the Normal (bell shaped
curve) and exponential have well defined means
which determine the scale of the underlying real
world phenomenon. Power law distributions do not
have a well defined mean and therefore suggest
that the underlying behavior is scale-free.
Y 10/X.5
Y 10/X
14
The meaning of 1-over-f noise (Power Spectrum
Decay)
1/f
long slow decay of autocorrelation function
one-over-f decay of power spectral density
function
15
Time weighted Average - integration of
piecewise constant inputs
in(x1)
in(x2)
in(x3)
input
x1
lastVal
x3
x2
X1e1x2e2
X1e1
X1e1x2e2x3e3
integral
state
clock
e
e1
e3
e2
output
16
Real-time Discrete Event Cross-correlator/autocorr
elator of piecewise constant inputs
Correlator -- is a coupled model with
lagCorrelator components. Each component is
characterized by a lag (delay) and computes the
correlation of a time function on port in, and
the lagged version of the one on port inDelay.
The correlator computes the correlation for the
lags manifest in the coupled model. The output is
a real time evolution of the correlation
funcitonal. For an autocorrelation the same input
is coupled to both ports.
A lagCorrelotor assumes the input events
represent the states of a piecewise constant time
segment. It uses a multiplier that remembers the
last values on each of its respective ports and
outputs the product of the arriving value with
the most recent value on the other port. The
multiplier output is time averaged to produce the
correlation for the given lag.
delayQ is an instance of JobQueue
avg is an instance of averageTime
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Example autocorrelation of lagged white noise
shows linear decay
white noise generator (independent samples from
zero-mean Normal
Derived from JobQueue by summing the values of
the inputs in the pipeline
The autocorrelation can be proved to decay
linearly to zero -- an example of power law
decay.
The autocorrelation is zero for lags greater than
the delay of the lagSum. It takes considerable
time to settle down to zero
18
Example attempted staircase emulation of Self
Ordered Critical Process buildup
t 30

• The output of a Poisson pulse generator is summed
  until reaching a specified threshold, at which
  point the sum is reset to 0.
• The mean squared (limiting autocorrelation) and
  variance (autocorrelation at 0) are easily
  pre-calculated.
• The idea is to try to replicate the recurring
  boom/bust cycle of an avalanche process.
• The decay of the autocorrelation function looks
  exponential and dips negative.

mean period 30
19
Experimental Frame for Avalanche Observation
Distribution transducer keeps running tally of
event (avalanche) sizes
Causality Transducer observes cell id input
stream and segments it into causal chain events
(avalanches). Causality is determined by
neighborhood adjacency.
Minmax keeps track of state distribution
computing running min, max, and avg
Cellular model under investigation
20
Extinction model inspired by that of Per Bak

• Each cell has a fitness initially small and
  randomly assigned.
• Incremental improvements take longer as fitness
  increases (with mean 1/(1-fitness)).
• The vulnerability of a cell increases with its
  fitness, where vulnerability is the probability
  that a change in fitness of a neighbor will cause
  it to go extinct (fitness reduced to small
  initial value).
• Avalanches are neighborhood connected sequences
  of extinctions
• After initial warm-up, fitness values go through
  recurring boom and bust cycles.
• Do avalanche sizes obey power law?
• Does the max. fitness show a slowly decaying
  autocorrelation?

Autocorrelation for max fitness to time 400
Autocorrelation to time 1000
Avalanche distribution is very close to power law
with degree -3.
Cell space at some time
Time evolution of fitness distribution
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Causality detection in cellular space like
tracking of objects by radar
is this celll in some event neighborhood?
(I,j,k) is part of an existing event if it is a
neighbor of some cell In that track - ask all
tracks is this in your neighbor hood? (I,j,k)
starts a new event if it is not part of any
event when does an event end? need further
hints
event avalanche radar track
22
The meaning of 1-over-f noise (Power Spectrum
Decay)
1/f
long slow decay of autocorrelation function
one-over-f decay of power spectral density
function
23
Experimental Frame for Avalanche Observation
Distribution transducer keeps running tally of
event (avalanche) sizes
Causality Transducer observes cell id input
stream and segments it into causal chain events
(avalanches). Causality is determined by
neighborhood adjacency.
Minmax keeps track of state distribution
computing running min, max, and avg
Cellular model under investigation
24
Models for Earthquake SOC
slowly driven threshold-based interactive system
- Truck Riders Analog
Linear discrete time abstraction
Non-linear discrete time abstraction
Linear discrete event abstraction (motion
transmission)
discrete event abstraction (force transmission)
Non-linear discrete event abstraction (motion
transmission)
alpha ( force transmission parameter)
spring constants, internal and driver
25
Riders start from an equilibrium position held in
position by arm strength resisting truck pull on
seat.
Some riders are closer to overcoming arm strength
and a rigidity threshold that prevents movement
Truck riders analog
Riders upper bodies tend to remain in original
position due to inertial and arm muscle strength.
Truck makes a small displacement.

• Riders sitting hand upon shoulder on a moving
  truck try to resist being displaced forward or
  backward.
• However, the forward motion of the truck must
  eventually require that each rider achieves its
  speed.
• Small bumps of the road cause riders to be
  randomly distributed in closeness to truck speed.
  At any time
• some riders are closer to achieving the truck
  speed than others.
• When one such rider is unstuck and lurches
  forward to match truck speed (at which point s/he
  experiences
• no net force), s/he pushes on the rider in
  front and pulls the rider in back, bringing them
  closer to truck speed.
• An avalanche occurs if several neighbors are
  close to criticality (truck speed) and
  relaxations to truck speed
• started by one propagate instantaneously to
  others.
• Avalanches are only possible if the truck pull
  on riders is not too large relative to their arm
  strength. Otherwise
• all riders will be quickly brought up to
  equilibrium position.

26
Total force on rider is F(t)
2K(avgNeighPos(t) pos(t)) Kd (driverPos(t)
pos(t)) where pos(t) is the position of the
rider at time t, and driverPos(t)
initialDriverPos vt (where v is the constant
speed). Note avgNeighPos(t) remains fixed at its
last value until a neighbor moves Total force
increases until it exceeds the static friciion
threshold At this time, t the rider moves to a
new position, pos(t), that satisfies F(t) 0)
i.e., 2K(avgNeighPos(t) pos(t)) Kd
(driverPos(t) pos(t)) 0 where t was the
last update time before t. This new riders
position is sent to the neighbors and alters the
total force on each. Meanwhile, each rider is
going through the above cycle.
27
truck moves forward
28
The derivation alpha K/(2K Kd)
provides a parameter mapping from the truck
rider model to the earthquake model. We verify
that the external drive spring constant cannot
be too large (i.e., alpha too small) in relation
to the internal spring constant K for avalanches
(earthquakes) to be possible. On the other hand,
the largest value of alpha is ½ for which Kd 0.
Here avalanches are also not possible since the
moving drive cannot exert a pulling force on the
riders. Thus, this model suggests limits on the
parameter alpha of the earthquake model.
The time scale is Thresh/(speedKd) which sets
the energy threshold
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Linear DEVS
Nonlinear DEVS
Discrete Time
Basic Relationships
Each cell (rider) experiences a force which is
the total of the spring orces exerted by its
neighbors and the drive (moving truck)
Total Force (myPos, rightPos, leftPos,
drivePos) Equilibrium position eqPos
satisfies Total Force (ePos,
rightPos, leftPos, drivePos) 0 eqPos (rightPos,
leftPos,drivePos) this cells position that
would nullify the force on it a function of
current values of neighbors and drive
positions. Solve by following gradient down from
initial estimate given by linear solution.
Motion
Motion tracking lag
Activity
The discrete event model computes time advance
to reach Threshold, ta satisfies Total Force
(myPos, rightPos, leftPos,drivePostaspeed)


Threshold Solve by doing internal simulation
timeToReachThreshold() e 0 while (Total
Force(,drivePos espeed) lt Threshold) e
edelta return e
The discrete time model -- until (termination
condition) advance each cells time by
delta drivePos drivePos speeddelta compute
the increased force TotalForce() if the force
exceeds threshold then set myPos
eqPos(rightPos, leftPos,drivePos) (jump
immediately to eqPos) send Pos to neighbors
30
Model-Experimental Frame Relationships for
Earthquake SOC Domain
derivability
morphism
31
Sand Pile Avalanche model vs Salt Pile model
Cells communicate to discover the state (color
coded) they are in. The 8 cell Mealy neighborhood
is needed. Since signals travel at different
speeds, classical CA are not (easily) able to do
this.
black - falling grain
gray -track of falling grain
pink - cells with open side
blue - cell about to destabilize open on at
least one side and with weight threshold (here
4)
red - cell walled in on all sides
Avalanche size Number of lateral moves in one
event very close to power law 53/x
Salt grain model grains fill up empty slots,
come to rest on top of a red (walled in) cell.

• Avalanche sand grains fall down center and
  pile up to threshold, as determined by weight on
  bottom cell. This destabilized cell moves and
  destabilizes those above, which fall with equal
  probability to the left or right.
• This process continues with the same rules
  applying to distributed grains.
• Large avalanches occur when a number of adjacent
  columns have cells near the threshold.
• Sand and Salt Piles show distinct avalanche size
  distributions

32
Sand and Salt Piles show distinct avalanche
distributions
Avalanche size Number of lateral moves in one
event very close to power law 53/x
Salt pile
Sand pile
33
Threshold-based watershed
rain
Discrete flow is injected at the top, middle
cell. It randomly diffuses down and to the sides
under threshold constraint. That is, it cannot
transmit the flow until its accumulated water
exceeds a threshold. When it overflows it acts
like a burst dam. The distinct
increasing-with-depth lateral extents may be
related to the threshold value.
Avalanches are the propagated series of dam
bursts difficult to measure.
depth