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: Visualization and Analysis of a Stochastic ProcessSteven R. DunbarDepartment of MathematicsUniversity of Nebraska-LincolnLincoln, Nebraska 68508http://www.math.unl.edu/~sdunbar1
<Text-field style="Heading 1" layout="Heading 1">Worksheet Information</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Mathematics Content</Text-field>Simulation, visualization and analysis of a stochastic process and quantities associated with the process. Analysis of Markov processes using "first-step analysis" and limiting arguments.
<Text-field style="Heading 2" layout="Heading 2">Keywords</Text-field>stochastic process, Markov process, rumor model, visualization, expiration time, rumor duration, linear recurrence equations, Lambert W-function
<Text-field style="Heading 2" layout="Heading 2">Objective</Text-field>Illustrate simulation of a stochastic process. Illustrate visualizations of a stochastic process. Investigate quantities associated with a stochastic process such as duration, ending state, and mean values.
<Text-field style="Heading 2" layout="Heading 2">Audience</Text-field>Anyone interested in using Maple to do simulation, visualization, and analysis of stochastic processes. Students and instructors in courses on stochastic processes, applied mathematics, and mathematical biology.
<Text-field style="Heading 2" layout="Heading 2">Time</Text-field>Approximately 30-60 minutes
<Text-field style="Heading 2" layout="Heading 2">Prerequisite</Text-field>Basic understanding of stochastic processes.
Basic understanding of Markov processes.Understanding of mean of a random variable.
Familiarity with "first-step analysis" of a Markov process.
Familiarity with basic limits of calculus and limiting arguments in general.
<Text-field style="Heading 2" layout="Heading 2">References</Text-field>Hayes, Brian; Rumours and Errours, American Scientist, May-June 2005, pages 207-211.
Belen, Selma, and C. E. M. Pearce, 2004, Rumours with general initial conditions, The ANZIAM Journal, 45, 393-400.
Daley, D. J. and D.G. Kendall, 1965, Stochastic Rumors, Journal of the Institute of Mathematics and its Applications, 1:42-55.
Maki, D. P. and Maynard Thompson, 1973, Mathematical Models and Applications, with Emphasis on the Social, Life and Management Sciences, Englewood Cliffs, N.J., Prentice-Hall, pages 378-387.
<Text-field style="Heading 1" layout="Heading 1">Background</Text-field>"The stochastic theory of rumors, with interacting subpopulations of ignorants, spreaders and stiflers began with the seminal paper of Daley and Kendall. The most striking result in the area -- that if there is only one spreader initially, then the proportion of the population never to hear the rumor converges to a proportion 0.203188 of the population size as the latter tends to infinity -- was first signaled in that article. This result also occurs in the variant model of Maki and Thompson, ... " Quote from the paper by Selma Belen and C. E. M. Pearce, Rumours with general initial conditions, The ANZIAM Journal, 45, 393-400.
"...rumor mongering is fun only if you know the rumor and your audience doesn't; there's no thrill in passing on stale news. In terms of the three subpopulations, people remain spreaders of rumor only as long as they meet ignorants who are eager to receive it, after that the spreaders become stiflers, who still know the rumor but have lost interest in spreading it." Quote from the article by Brian Hayes; Rumours and Errours, American Scientist, May-June 2005.Rules for a Rumor:Select two individuals at random from a fixed population to meet and exchange information at each time in a sequence of discrete times.The population begins with one spreader who knows the rumor. The rest of the population are ignorants who do not yet know the rumor. Whenever a spreader meets an ignorant, the ignorant becomes a spreader, while the original spreader continues spreading.When a spreader meets a stifler, the spreader becomes a stifler.In the case where two spreaders meet in the Daley-Kendall model , both spreaders become stiflers.All other interactions (ignorant-ignorant, ignorant-stifler, stifler-stifler) have no effect on either party.
<Text-field style="Heading 1" layout="Heading 1">Maple procedures for visualizing the rumor (set to Autoexecute)</Text-field>
The following Maple procedures are hidden, in order to focus attention on their use and the visualization and analysis of the rumor stochastic process. These procedures are also initialized when opening the worksheet so that all procedures are available and ready to use when the worksheet is opened.restart;view_population_rumor :=
proc( population, ending_time )
local SEED, roll, i, person, time, initial_spreader, first_interactor, second_interactor,
interaction, ig_rect, sp_rect, st_rect;
SEED := randomize():
roll := rand(1..population):
for i from 1 to population do
person[i] := ["Ignorant", ending_time, ending_time]:
od:
initial_spreader := roll():
person[initial_spreader] := ["Spreader", 0, ending_time];NiI=
for time from 1 to ending_time do
first_interactor := roll():
second_interactor := roll():
while (first_interactor = second_interactor) do
second_interactor := roll():
od:
interaction[time] := [
plottools[point]( [time, first_interactor + 1/4], color=black),
plottools[line]( [time, first_interactor + 1/4], [time, second_interactor + 1/4],
color=black
),
plottools[point]( [time, second_interactor + 1/4], color=black)
]:
# using an if ... elif ... else construction to mimic a case statement
if ( person[first_interactor][1] = "Ignorant"
and person[second_interactor][1] = "Ignorant" ) then
# do nothing
next:
elif ( person[first_interactor][1] = "Ignorant"
and person[second_interactor][1] = "Spreader" ) then
person[first_interactor] := ["Spreader", time, ending_time]:
next:
elif ( person[first_interactor][1] = "Ignorant"
and person[second_interactor][1] = "Stifler" ) then
next:
elif ( person[first_interactor][1] = "Spreader"
and person[second_interactor][1] = "Ignorant" ) then
person[second_interactor] := ["Spreader", time, ending_time]:
next:
elif ( person[first_interactor][1] = "Spreader"
and person[second_interactor][1] = "Spreader" ) then
person[first_interactor] := ["Stifler", person[first_interactor][2], time]:
person[second_interactor] := ["Stifler", person[second_interactor][2], time]:
next:
elif ( person[first_interactor][1] = "Spreader"
and person[second_interactor][1] = "Stifler" ) then
person[first_interactor] := ["Stifler", person[first_interactor][2], time]:
next:
elif ( person[first_interactor][1] = "Stifler"
and person[second_interactor][1] = "Ignorant" )then
# do nothing
next:
elif ( person[first_interactor][1] = "Stifler"
and person[second_interactor][1] = "Spreader" ) then
person[second_interactor] := ["Stifler", person[second_interactor][2], time]:
next:
else ( person[first_interactor][1] = "Stifler"
and person[second_interactor][1] = "Stifler" )
# do nothing
end if:
od:
for i from 1 to population do
ig_rect[i] := plottools[rectangle]( [0,i+1/2], [person[i][2], i], color=green ):
sp_rect[i] := plottools[rectangle]( [person[i][2], i+1/2], [person[i][3], i],
color=yellow ):
st_rect[i] := plottools[rectangle]( [person[i][3], i+1/2], [ending_time, i], color=red):
od:
plots[display](
ListTools[Flatten](
[
seq( [ ig_rect[i], sp_rect[i], st_rect[i] ], i=1..population ),
[ seq( interaction[time], time=1..ending_time ) ]
]
),
tickmarks = [10,0],
labels=["Time", "Population"]
);
end proc:with(Maplets[Elements]):view_rumor :=
Maplet('onstartup'='Action1','reference'='Maplet1',
Plotter('background'="#FFFFFF",'continuous'='true','delay'='100','height'='650',
'reference'='Plotter1','visible'='true','width'='650'),
Label('background'="#D6D3CE",'caption'="Population",'enabled'='true','foreground'="#000000",
'reference'='Label1','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField1','visible'='true','width'='20'),
Label('background'="#D6D3CE",'caption'="Number of Encounters",'enabled'='true',
'foreground'="#000000",'reference'='Label2','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField2','visible'='true','width'='20'),
Action('reference'='clickButton3',
Evaluate('function'='view_population_rumor(TextField1, TextField2)',`option`="value",
'target'='Plotter1','waitforresult'='true')
),
Button('background'="#D6D3CE",'caption'="View Rumor",'enabled'='true',
'foreground'="#000000",'onclick'='clickButton3','reference'='Button3','visible'='true'),
BoxLayout('background'="#D6D3CE",'border'='false','halign'='center','inset'='5',
'reference'='BoxLayout1','valign'='center','vertical'='false','visible'='true',
BoxColumn(
BoxRow(
BoxCell('hscroll'='never','value'='Plotter1','vscroll'='never')
),
BoxRow(
BoxColumn(
BoxCell('hscroll'='never','value'='Label1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='Label2','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField2','vscroll'='never')
)
),
BoxRow(
BoxCell('hscroll'='never','value'='Button3','vscroll'='never')
)
)
),
Window('layout'='BoxLayout1','reference'='Window1','resizable'='true','title'="Maplet"),
Action('reference'='Action1',RunWindow('window'='Window1'))
):view_group_interaction_rumor :=
proc( population, ending_time)
local SEED, roll, i, person, time, initial_spreader,
number_ignorants, number_spreaders, number_stiflers,
first_interactor, second_interactor,
interaction, ignorant_fraction, ignorant_spreader_fraction,
ig_ig, ig_sp, ig_st, sp_ig, sp_sp, sp_st, st_ig, st_sp, st_st:
global p:
SEED := randomize():
roll := rand(1..population):
for i from 1 to population do
person[i] := "Ignorant":
od:
initial_spreader := roll():
person[initial_spreader] := "Spreader":
number_ignorants[0] := population - 1:
number_spreaders[0] := 1:
number_stiflers[0] := 0:
for time from 1 to ending_time do
first_interactor := roll():
second_interactor := roll():
while (first_interactor = second_interactor) do
second_interactor := roll():
od:
if ( person[first_interactor] = "Ignorant"
and person[second_interactor] = "Ignorant" ) then
number_ignorants[time] := number_ignorants[time-1]:
number_spreaders[time] := number_spreaders[time-1]:
number_stiflers[time] := number_stiflers[time-1]:
next:
elif ( person[first_interactor] = "Ignorant"
and person[second_interactor] = "Spreader" ) then
person[first_interactor] := "Spreader";
number_ignorants[time] := number_ignorants[time-1] - 1:
number_spreaders[time] := number_spreaders[time-1] + 1:
number_stiflers[time] := number_stiflers[time-1]:
next:
elif ( person[first_interactor] = "Ignorant"
and person[second_interactor] = "Stifler" ) then
number_ignorants[time] := number_ignorants[time-1]:
number_spreaders[time] := number_spreaders[time-1]:
number_stiflers[time] := number_stiflers[time-1]:
next:
elif ( person[first_interactor] = "Spreader"
and person[second_interactor] = "Ignorant" ) then
person[second_interactor] := "Spreader":
number_ignorants[time] := number_ignorants[time-1] - 1:
number_spreaders[time] := number_spreaders[time-1] + 1:
number_stiflers[time] := number_stiflers[time-1]:
next:
elif ( person[first_interactor] = "Spreader"
and person[second_interactor] = "Spreader" ) then
person[first_interactor] := "Stifler":
person[second_interactor] := "Stifler":
number_ignorants[time] := number_ignorants[time-1]:
number_spreaders[time] := number_spreaders[time-1] - 2:
number_stiflers[time] := number_stiflers[time-1] + 2:
next:
elif ( person[first_interactor] = "Spreader"
and person[second_interactor] = "Stifler" ) then
person[first_interactor] := "Stifler":
number_ignorants[time] := number_ignorants[time-1]:
number_spreaders[time] := number_spreaders[time-1] - 1:
number_stiflers[time] := number_stiflers[time-1] + 1:
next:
elif ( person[first_interactor] = "Stifler"
and person[second_interactor] = "Ignorant" ) then
number_ignorants[time] := number_ignorants[time-1]:
number_spreaders[time] := number_spreaders[time-1]:
number_stiflers[time] := number_stiflers[time-1]:
next:
elif ( person[first_interactor] = "Stifler"
and person[second_interactor] = "Spreader" ) then
person[second_interactor] := "Stifler":
number_ignorants[time] := number_ignorants[time-1]:
number_spreaders[time] := number_spreaders[time-1] - 1:
number_stiflers[time] := number_stiflers[time-1] + 1:
next:
else # ( person[first_interactor] = "Stifler"
# and person[second_interactor] = "Stifler" )
number_ignorants[time] := number_ignorants[time-1]:
number_spreaders[time] := number_spreaders[time-1]:
number_stiflers[time] := number_stiflers[time-1]:
end if:
od; # end of rumor run at ending_time
for i from 0 to ending_time do
ignorant_fraction[i] := number_ignorants[i]/population:
ignorant_spreader_fraction[i] :=
(number_ignorants[i] + number_spreaders[i])/population:
ig_ig[i] := plottools[rectangle]( [ 0, ignorant_fraction[i] ], #upper left
[ ignorant_fraction[i], 0 ], #lower right
color=green
):
ig_sp[i] := plottools[rectangle]( [0, ignorant_spreader_fraction[i] ],
[ignorant_fraction[i], ignorant_fraction[i]],
color=COLOR(HSV, 0.4, 0.6, 1.0) # medium green
):
ig_st[i] := plottools[rectangle]( [0, 1 ],
[ignorant_fraction[i], ignorant_spreader_fraction[i] ],
color=COLOR(HSV, 0.4, 0.3, 1.0) # light green
):
sp_ig[i] := plottools[rectangle]( [ignorant_fraction[i], ignorant_fraction[i] ],
[ignorant_spreader_fraction[i], 0 ],
color=COLOR(HSV, 0.15, 0.6, 1.0) # medium yellow
):
sp_sp[i] := plottools[rectangle]( [ignorant_fraction[i], ignorant_spreader_fraction[i] ], [ignorant_spreader_fraction[i], ignorant_fraction[i] ], color=yellow
):
sp_st[i] := plottools[rectangle]( [ignorant_fraction[i], 1 ],
[ignorant_spreader_fraction[i], ignorant_spreader_fraction[i] ],
color=COLOR(HSV, 0.15, 1.0, 1.0) # light yellow
):
st_ig[i] := plottools[rectangle]( [ignorant_spreader_fraction[i], ignorant_fraction[i] ],
[1, 0 ],
color=COLOR(HSV, 0.0, 0.3, 1.0) # medium red
):
st_sp[i] := plottools[rectangle]( [ignorant_spreader_fraction[i], ignorant_spreader_fraction[i] ],
[1, ignorant_fraction[i] ],
color=COLOR(HSV, 0.0, 0.6, 1.0) # light red
):
st_st[i] := plottools[rectangle]( [ignorant_spreader_fraction[i], 1 ],
[1, ignorant_spreader_fraction[i] ],
color=red
):
p[i] := plots[display]( [ ig_ig[i], ig_sp[i], ig_st[i],
sp_ig[i], sp_sp[i], sp_st[i],
st_ig[i], st_sp[i], st_st[i]
]
);
od: # end of construction of animation at ending_time
plots[display]( [seq( p[i], i=0..100 ) ], insequence=true);
end proc:view_group_interaction :=
Maplet('onstartup'='Action1','reference'='Maplet1',
Plotter( 'continuous'='false', 'height'='650','width'='650',
'reference'='Plotter1'),
# Plotter('background'="#FFFFFF",'continuous'='true','delay'='100','height'='650',
# 'reference'='Plotter1','visible'='true','width'='650'),
Label('background'="#D6D3CE",'caption'="Population",'enabled'='true','foreground'="#000000",
'reference'='Label1','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField1','visible'='true','width'='20'),
Label('background'="#D6D3CE",'caption'="Number of Encounters",'enabled'='true','foreground'="#000000",
'reference'='Label2','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField2','visible'='true','width'='20'),
Action('reference'='clickButton1',
Evaluate('function'='view_group_interaction_rumor(TextField1,TextField2)',`option`="value",
'target'='Plotter1','waitforresult'='true')
),
Button('background'="#D6D3CE",'caption'="View Rumor Interaction",'enabled'='true',
'foreground'="#000000",'onclick'='clickButton1','reference'='Button1','visible'='true'),
BoxLayout('background'="#D6D3CE",'border'='false','halign'='center','inset'='5','reference'='BoxLayout1',
'valign'='center','vertical'='false','visible'='true',
BoxColumn(
BoxRow(
BoxCell('hscroll'='never','value'='Plotter1','vscroll'='never')
),
BoxRow(
BoxColumn(
BoxCell('hscroll'='never','value'='Label1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='Label2','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField2','vscroll'='never')
)
),
BoxRow(
BoxCell('hscroll'='never','value'='Button1','vscroll'='never')
),
BoxRow( 'border'='true',
[ Button("play",SetOption(Plotter1('play')=true)),
Button("stop",SetOption(Plotter1('`stop`')=true)),
Button("pause",SetOption(Plotter1('pause')=true))
],
[ Button("to start",SetOption(Plotter1('to_start')=true)),
Button("to end",SetOption(Plotter1('to_end')=true))
],
[ Button("backwards",SetOption(Plotter1('frame_backwards')=true)),
Button("forward",SetOption(Plotter1('frame_forward')=true))
],
BoxColumn( 'border'='true',
[ "continuous",
CheckBox[CONTINUOUS](value=false,onchange=SetOption(target=Plotter1, `option`='continuous',
Argument(CONTINUOUS))) ]
),
[ "delay",
Slider[DELAY](50..300, 100, majorticks=50,
onchange=SetOption(target=Plotter1, `option`='delay', Argument(DELAY))),
Button("Quit",Shutdown())
]
)
)
),
Window('layout'='BoxLayout1','reference'='Window1','resizable'='true','title'="Maplet"),
Action('reference'='Action1',
RunWindow('window'='Window1')
)
):view_average_rumor_profile :=
proc( population, ending_time, number_iterations )
local SEED, roll, iter, i, person,
number_ignorants, number_spreaders, number_stiflers,
t, initial_spreader,
sum_ignorants, sum_spreaders, sum_stiflers,
first_interactor, second_interactor,
average_ignorants, average_spreaders, average_stiflers,
ig_plot, sp_plot, st_plot:
SEED := randomize():
roll := rand(1..population):
for t from 1 to ending_time do # initialize the sums to hold the iterations for averaging
sum_ignorants[t] := 0:
sum_spreaders[t] := 0:
sum_stiflers[t] := 0:
od:
for iter from 1 to number_iterations do
for i from 1 to population do # initialize the population for each run of the rumor
person[i] := ["Ignorant", ending_time, ending_time]:
od:
number_ignorants[0] := population - 1;
number_spreaders[0] := 1;
number_stiflers[0] := 0;
initial_spreader := roll():
person[initial_spreader] := ["Spreader", 0, ending_time]:
for t from 1 to ending_time do
first_interactor := roll():
second_interactor := roll():
while (first_interactor = second_interactor) do
second_interactor := roll():
od:
if ( person[first_interactor][1] = "Ignorant" and person[second_interactor][1] = "Ignorant" ) then
number_ignorants[t] := number_ignorants[t-1]:
number_spreaders[t] := number_spreaders[t-1]:
number_stiflers[t] := number_stiflers[t-1]:
next:
elif ( person[first_interactor][1] = "Ignorant" and person[second_interactor][1] = "Spreader" ) then
person[first_interactor] := ["Spreader", t, ending_time]:
number_ignorants[t] := number_ignorants[t-1] - 1:
number_spreaders[t] := number_spreaders[t-1] + 1:
number_stiflers[t] := number_stiflers[t-1]:
next:
elif ( person[first_interactor][1] = "Ignorant" and person[second_interactor][1] = "Stifler" ) then
number_ignorants[t] := number_ignorants[t-1]:
number_spreaders[t] := number_spreaders[t-1]:
number_stiflers[t] := number_stiflers[t-1]:
next:
elif ( person[first_interactor][1] = "Spreader" and person[second_interactor][1] = "Ignorant" ) then
person[second_interactor] := ["Spreader", t, ending_time]:
number_ignorants[t] := number_ignorants[t-1] - 1:
number_spreaders[t] := number_spreaders[t-1] + 1:
number_stiflers[t] := number_stiflers[t-1]:
next:
elif ( person[first_interactor][1] = "Spreader" and person[second_interactor][1] = "Spreader" ) then
person[first_interactor] := ["Stifler", person[first_interactor][2], t]:
person[second_interactor] := ["Stifler", person[second_interactor][2], t]:
number_ignorants[t] := number_ignorants[t-1]:
number_spreaders[t] := number_spreaders[t-1] - 2:
number_stiflers[t] := number_stiflers[t-1] + 2:
next:
elif ( person[first_interactor][1] = "Spreader" and person[second_interactor][1] = "Stifler" ) then
person[first_interactor] := ["Stifler", person[first_interactor][2], t]:
number_ignorants[t] := number_ignorants[t-1]:
number_spreaders[t] := number_spreaders[t-1] - 1:
number_stiflers[t] := number_stiflers[t-1] + 1:
next:
elif ( person[first_interactor][1] = "Stifler" and person[second_interactor][1] = "Ignorant" ) then
number_ignorants[t] := number_ignorants[t-1]:
number_spreaders[t] := number_spreaders[t-1]:
number_stiflers[t] := number_stiflers[t-1]:
next:
elif ( person[first_interactor][1] = "Stifler" and person[second_interactor][1] = "Spreader" ) then
person[second_interactor] := ["Stifler", person[second_interactor][2], t]:
number_ignorants[t] := number_ignorants[t-1]:
number_spreaders[t] := number_spreaders[t-1] - 1:
number_stiflers[t] := number_stiflers[t-1] + 1:
next:
else # ( person[first_interactor][1] = "Stifler" and person[second_interactor][1] = "Stifler" )
number_ignorants[t] := number_ignorants[t-1]:
number_spreaders[t] := number_spreaders[t-1]:
number_stiflers[t] := number_stiflers[t-1]:
end if:
od: # end inner loop for the lifetime of one particular rumor
for t from 1 to ending_time do
sum_ignorants[t] := sum_ignorants[t] + number_ignorants[t]:
sum_spreaders[t] := sum_spreaders[t] + number_spreaders[t]:
sum_stiflers[t] := sum_stiflers[t] + number_stiflers[t]:
od;
od: # end outer loop for the number_iterations
for t from 1 to ending_time do
average_ignorants[t] := evalf( sum_ignorants[t]/number_iterations ):
average_spreaders[t] := evalf( sum_spreaders[t]/number_iterations ):
average_stiflers[t] := evalf( sum_stiflers[t]/number_iterations ):
od:
ig_plot := plot( [ seq( [ i, average_ignorants[i] ], i = 1..ending_time )], color=green, thickness = 2):
sp_plot := plot( [ seq( [ i, average_spreaders[i] ], i = 1..ending_time )], color=blue, thickness = 2):
st_plot := plot( [ seq( [ i, average_stiflers[i] ], i = 1..ending_time )], color=red, thickness = 2):
plots[display]( [ig_plot, sp_plot, st_plot], labels=["Time", "Population"]);
end proc:view_average_rumor :=
Maplet('onstartup'='Action1','reference'='Maplet1',
Plotter('background'="#FFFFFF",'continuous'='true','delay'='100','height'='650',
'reference'='Plotter1','visible'='true','width'='650'),
Label('background'="#D6D3CE",'caption'="Population",'enabled'='true','foreground'="#000000",
'reference'='Label1','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField1','visible'='true','width'='20'),
Label('background'="#D6D3CE",'caption'="Number of Encounters",'enabled'='true',
'foreground'="#000000",'reference'='Label2','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField2','visible'='true','width'='20'),
Label('background'="#D6D3CE",'caption'="Number of Rumors",'enabled'='true',
'foreground'="#000000",'reference'='Label3','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField3','visible'='true','width'='20'),
Action('reference'='clickButton3',
Evaluate('function'='view_average_rumor_profile(TextField1, TextField2, TextField3)',`option`="value",
'target'='Plotter1','waitforresult'='true')
),
Button('background'="#D6D3CE",'caption'="View Average Rumor",'enabled'='true',
'foreground'="#000000",'onclick'='clickButton3','reference'='Button3','visible'='true'),
BoxLayout('background'="#D6D3CE",'border'='false','halign'='center','inset'='5',
'reference'='BoxLayout1','valign'='center','vertical'='false','visible'='true',
BoxColumn(
BoxRow(
BoxCell('hscroll'='never','value'='Plotter1','vscroll'='never')
),
BoxRow(
BoxColumn(
BoxCell('hscroll'='never','value'='Label1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='Label2','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField2','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='Label3','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField3','vscroll'='never')
)
),
BoxRow(
BoxCell('hscroll'='never','value'='Button3','vscroll'='never')
)
)
),
Window('layout'='BoxLayout1','reference'='Window1','resizable'='true','title'="Maplet"),
Action('reference'='Action1',RunWindow('window'='Window1'))
):view_rumor_duration_and_success :=
proc(population, number_iterations)
local SEED, roll, iter, ending_time, i,
initial_spreader, person,
number_ignorants, number_spreaders, number_stiflers,
first_interactor, second_interactor,
sum_ignorants, duration:
global average_duration, average_ignorants:
SEED := randomize():
roll := rand(1..population):
for iter from 1 to number_iterations do
ending_time := 0;
for i from 1 to population do
person[i] := "Ignorant":
od:
initial_spreader := roll():
person[initial_spreader] := "Spreader":
number_ignorants := population - 1:
number_spreaders := 1:
number_stiflers := 0:
while (number_spreaders > 0) do
ending_time := ending_time + 1;
first_interactor := roll():
second_interactor := roll():
while (first_interactor = second_interactor) do
second_interactor := roll():
od:
if ( person[first_interactor] = "Ignorant" and person[second_interactor] = "Ignorant" ) then
# do nothing
next:
elif ( person[first_interactor] = "Ignorant" and person[second_interactor] = "Spreader" ) then
person[first_interactor] := "Spreader":
number_ignorants := number_ignorants - 1:
number_spreaders := number_spreaders + 1:
next:
elif ( person[first_interactor] = "Ignorant" and person[second_interactor] = "Stifler" ) then
# do nothing
next:
elif ( person[first_interactor] = "Spreader" and person[second_interactor] = "Ignorant" ) then
person[second_interactor] := "Spreader":
number_ignorants := number_ignorants - 1:
number_spreaders := number_spreaders + 1:
next:
elif ( person[first_interactor] = "Spreader" and person[second_interactor] = "Spreader" ) then
person[first_interactor] := "Stifler":
person[second_interactor] := "Stifler":
number_spreaders := number_spreaders - 2:
number_stiflers := number_stiflers + 2:
next:
elif ( person[first_interactor] = "Spreader" and person[second_interactor] = "Stifler" ) then
person[first_interactor] := "Stifler":
number_spreaders := number_spreaders - 1:
number_stiflers := number_stiflers + 1:
next:
elif ( person[first_interactor] = "Stifler" and person[second_interactor] = "Ignorant" ) then
# do nothing
next:
elif ( person[first_interactor] = "Stifler" and person[second_interactor] = "Spreader" ) then
person[second_interactor] := "Stifler":
number_spreaders := number_spreaders - 1:
number_stiflers := number_stiflers + 1:
next:
else # ( person[first_interactor] = "Stifler" and person[second_interactor] = "Stifler" )
# do nothing
end if: # end of one rumor
od: # end of number_iterations of rumors
duration[iter] := [ending_time, number_ignorants]:
od:
average_duration := evalf( sum( duration[j][1], j=1..number_iterations)/number_iterations);
average_ignorants := evalf( sum( duration[j][2], j=1..number_iterations)/number_iterations);
plots[pointplot]( [ seq(duration[iter], iter=1..number_iterations) ],
view=[0..2*average_duration, 0..population], labels=["Duration", "Ignorants"], symbol=point
);
end proc:view_duration_success :=
Maplet('onstartup'='Action1','reference'='Maplet1',
Plotter('background'="#FFFFFF",'continuous'='true','delay'='100','height'='650',
'reference'='Plotter1','visible'='true','width'='650'),
Label('background'="#D6D3CE",'caption'="Population",'enabled'='true','foreground'="#000000",
'reference'='Label1','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField1','visible'='true','width'='20'),
Label('background'="#D6D3CE",'caption'="Number of Rumors",'enabled'='true','foreground'="#000000",
'reference'='Label2','visible'='true'),
TextField('background'="#FFFFFF",'editable'='true','enabled'='true','foreground'="#000000",
'halign'='left','reference'='TextField2','visible'='true','width'='20'),
Action('reference'='clickButton1',
Evaluate('function'='view_rumor_duration_and_success(TextField1,TextField2)',`option`="value",
'target'='Plotter1','waitforresult'='true')
),
Button('background'="#D6D3CE",'caption'="View Rumor Duration and Success",'enabled'='true',
'foreground'="#000000",'onclick'='clickButton1','reference'='Button1','visible'='true'),
BoxLayout('background'="#D6D3CE",'border'='false','halign'='center','inset'='5','reference'='BoxLayout1',
'valign'='center','vertical'='false','visible'='true',
BoxColumn(
BoxRow(
BoxCell('hscroll'='never','value'='Plotter1','vscroll'='never')
),
BoxRow(
BoxColumn(
BoxCell('hscroll'='never','value'='Label1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField1','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='Label2','vscroll'='never')
),
BoxColumn(
BoxCell('hscroll'='never','value'='TextField2','vscroll'='never')
)
),
BoxRow(
BoxCell('hscroll'='never','value'='Button1','vscroll'='never')
)
)
),
Window('layout'='BoxLayout1','reference'='Window1','resizable'='true','title'="Maplet"),
Action('reference'='Action1',
RunWindow('window'='Window1')
)
):
<Text-field style="Heading 1" layout="Heading 1">Visualizing a Rumor in a Population : By Individuals</Text-field>The stochastic process generated by this set of rules is a Markov chain. One model of this stochastic process uses a state space with KSIiJEkiUEc2Ig== states, where L0kiUEc2IkkrUG9wdWxhdGlvbkdGJA== is the total population. Each rumor state in the Markov chain state space is a vector indexed by the population, and each entry in the vector corresponds to a person in one of the categories "Ignorant", "Spreader", or "Stifler". At each time step, two people in the population meet. For N time steps, there are KS1JKWJpbm9taWFsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJIlBHRigiIiNJIk5HRig= possible paths in the sample space of all paths. For example, if the population is 30 people, in 100 steps there are 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We can visualize this process in the following way: Time is on the horizontal axis, the population is on the vertical axis, one horizontal bar for each person in the population. Each horizontal bar represents the rumor history for that person, green for the period that a person is an ignorant, yellow for when the person is a spreader, and red for when the person is a stifler. Furthermore, at each time point, a vertical line connects the two people in the population who meet and have an interaction at that time. Execute the following Maple function to visualize one sample path. For good results, use a population in the range of 20 to 40 individuals and time length about 4 to 6 times the population size.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Ynview_population_rumor(20,120);Execute the following Maple command to obtain a Maplet that allows one to experiment easily to create more visualizations.
Maplets[Display](view_rumor):
In this view or diagram of the rumor, time is on the horizontal axis. The population is arrayed along the vertical axis. Each horizontal bar represents the history of an individual in the population. While the individual is "ignorant", the bar is green. When the individual hears the rumor and becomes a "spreader", then the individual history turns yellow. When the individual loses interest in spreading the rumor by virtue of meeting another spreader or stifler, the bar becomes red. Additionally at each time, there is a vertical line connecting two of the individuals in the population. This is a representation of the random interaction that took place between those individuals at that time. Carefully tracing these vertical lines, one can see how and when an individual changes state.
A couple of things can be observed from this simulation and visualization (and several repeats of the same simulation): After a long enough time, no horizontal bar representing an individual is colored yellow, the spreaders have been extinguished.Even when all the spreaders have been extinguished, there are still some green bars, that is ignorants who have still not heard the rumor. These remaining green bars typically constitute about 20% of the population.Occasionally there are rumors which are quickly extinguished when the spreaders meet and become stiflers before meeting many ignorants.This is a visualization of a sample path in the Markov chain. In a vertical line above each time is a vector of the population size. Each entry in the vector is a person in one of the three conditions of the rumor: ignorant, spreader or stifler. The progress of the particular rumor is the sample path of the Markov chain through the many states possible, transitioning randomly from one state to one of the next possible states.
However, this view of individual histories of rumors contains too much detailed information. We need a more condensed view of the rumor progress.
<Text-field style="Heading 1" layout="Heading 1">Another View of a Rumor in a Population : By Interactions from Groups</Text-field>
Define the state or composition of the rumor by specifying the number of people in each condition at any time. Thus at each tick of the clock, the composition of the rumor is an ordered triple: (Ig, Sp, St) where Ig + Sp + St = Population.At each encounter, the composition may either stay the same, or make a transition (Ig, Sp, St) \342\206\222 (Ig -1, Sp + 1, St) , or make a transition (Ig, Sp, St) \342\206\222 (Ig, Sp - 2, St + 2) or (Ig, Sp, St) \342\206\222 (Ig, Sp - 1, St + 1). Therefore, this view of the rumor is again a Markov chain. Now the number of states in the Markov chain is the number of ways the population can be partitioned into 3 classes.
This is LUkpYmlub21pYWxHNiI2JCwmSSJQR0YkIiIiIiIkRihGKQ==possible states,which can be large, for example if P = 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 then Ly1JKWJpbm9taWFsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJkkiUEdGKCIiIiIiJEYsRi0kIiVjYSIiIQ==
Not all of these states may be accessible from the initial population composition, but it is probably more convenient to keep them than to weed them out. Once again, the size of the state space is hard to comprehend.Execute the following Maple function to visualize one sample path. For good results, use a population in the range of 20 to 40 individuals and time length about 4 to 6 times the population size.view_group_interaction_rumor(30,150);
Execute the following Maple command to obtain a Maplet that allows one to experiment easily to create more visualizations.
Maplets[Display](view_group_interaction);
This view of the rumor is a dynamic matrix modeling possible interaction events in the rumor model. The width of each column is the fraction of a subpopulation in the total population. So for instance, the width of the green column on the left is the fraction of ignorants in the population at each time. The width of the yellow column in the middle is the fraction of the spreaders in the population. The width of the red column on the right is the fraction of stiflers in the population. The colors correspond to the same colors in the total population versus time view above. The rows in the matrix are likewise the fractions of the population in each class. Then the probability of choosing two individuals at random from the population is equivalent to the probability of choosing a point at random from within the square. At each time step, which one of the nine boxes the a selected pair of individuals lies within determines what happens next, depending on which interaction is chosen. The area of each of the nine boxes gives an indication of the relative probability of each interaction. The animation gives a view of the progress of the rumor, with the probabilities of the various interactions changing over time.
<Text-field style="Heading 1" layout="Heading 1">The Average Profile of a Rumor</Text-field>The overall mean dynamics of a rumor are revealed by averaging the results of many runs of the rumor model. The average number of ignorants is plotted in green, the average number of spreaders is plotted in blue (rather than yellow, which does not show up well) and the average number of stiflers is plotted in red.Execute the following Maple function to visualize the mean of many runs of the rumor model For good results, use a time length at least 6 times the populations. Using more rumor experiments will result in smoother average dynamics, and using fewer rumor experiments will result in more stochastic variability of the dynamics.view_average_rumor_profile( 30, 170, 50 );
Execute the following Maple command to obtain a Maplet that allows one to experiment easily to create more visualizations.
Maplets[Display](view_average_rumor);
Notice that if the ending time is chosen large enough, the spreaders (plotted here in blue) reach a peak and then die out entirely. The final population is then divided between stiflers (plotted in red) and ignorants (plotted in green). Notice that on average, the final number of ignorants is about 20% of the total population.
<Text-field style="Heading 1" layout="Heading 1">The Life Span of a Rumor</Text-field>The longevity of a rumor, that is, the length of time until there are no more spreaders to spread the rumor, is a quantity of interest.
Execute the following Maple function and commands to visualize and investigate the durations and outcomes of many runs of the rumor model. The first parameter is the population size, the second parameter is the number of rumor simulations. view_rumor_duration_and_success(40,100);average_duration;average_ignorants;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnExecute the following Maple command to obtain a Maplet that allows one to experiment easily to create more visualizations.
Maplets[Display](view_duration_success);
Each dot represents a single run of the rumor program, showing both how long the rumor survived and how many people remained ignorant of the rumor at the end. The longevity of a rumor and its success are not strongly correlated. The distribution of ignorants is about the same whether the rumor dies relatively early, or lasts for a long time. Occasionally in the upper left hand corner are a few exceptional cases of rumors that died off early and reached very few people.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">Mathematical Analysis of the Duration and Final Size of the Rumor</Text-field>One of the most important predictions a model can make is how long the rumor will be in circulation. If we know this length of time, we will also be able to predict what fraction of the population has heard the rumor and the fraction of the population that the rumor has not touched, a fraction that we have simulated above.
We will say that the rumor is alive as long as there is at least one spreader. That is, the rumor is dead once the last spreader is converted to a stifler. Thus, we seek some information about the random variable which is the lifetime of the rumor: T = min{ n : number spreaders at time n = 0}We do not need to describe the state of the population at all times. We will describe the change in the population at the times when the composition shifts by virtue of a spreader meeting an ignorant, two spreaders meeting, or a spreader meeting a stifler. We call such times meaningful times. The sequence of meaningful times, 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 actually a sequenceLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Ynof Markov times. This is so because the occurrence of a meaningful time depends only on the composition of the population at that time and not on events in the future. Considering the Markov chain only at the sequence of meaningful times results again in a Markov chain. It is this derived Markov chain at the meaningful times that we will consider.
At such meaningful times, we set up equations describing the change in expected composition using "first-step analysis", or "single-step analysis", which is a simple version of "expectation by conditioning" for Markov chains. Note that if we are at a meaningful time, we must have a spreader involved as one of the interactors. That means we can restrict our attention to the middle row in the visualization of the interaction matrix above. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnSuppose the population size is LCZJIk5HNiIiIiJGJUYl and let the meaningful times be JkkidEc2IjYjSSJrR0Yk and the expected population composition at a meaningful time JkkidEc2IjYjSSJrR0Ykbe NiUmSSJpRzYiNiNJImtHRiUmSSJzR0YlRiYmSSJmR0YlRiY= for ignorants, spreaders, and stiflers respectively. That is, LywoJkkiaUc2IjYjSSJrR0YmIiIiJkkic0dGJkYnRikmSSJmR0YmRidGKSwmSSJOR0YmRilGKUYp. We can recursively define the next population as follows. If the population composition is NiUmSSJpRzYiNiNJImtHRiUmSSJzR0YlRiYmSSJmR0YlRiY= then NiUmSSJpRzYiNiMsJkkia0dGJSIiIkYpRikmSSJzR0YlRiYmSSJmR0YlRiY= must be one ofNiUsJiZJImlHNiI2I0kia0dGJiIiIkYpISIiLCYmSSJzR0YmRidGKUYpRikmSSJmR0YmRic=orNiUmSSJpRzYiNiNJImtHRiUsJiZJInNHRiVGJiIiIiIiIyEiIiwmJkkiZkdGJUYmRitGLEYrorNiUmSSJpRzYiNiNJImtHRiUsJiZJInNHRiVGJiIiIkYrISIiLCYmSSJmR0YlRiZGK0YrRis=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
More precisely, the expected composition JkkicEc2IjYjSSJrR0Yk is defined recursively as:NiUvJkkicEc2IjYjIiIhNiVJIk5HRiYiIiJGKC8mRiU2I0YrNiUsJkYqRitGKyEiIiIiI0YoMy8mRiU2I0YyNiUmSSJpR0YmRjYmSSJzR0YmRjYmSSJmR0YmRjYzL0Y3LUkpJnZlbGxpcDtHRiY2JC0mSScmRW9wZjtHRiY2I0YlNiMmSSJ0R0YmRjYmRiU2I0kia0dGJjMvRkA2JSZGOUZLJkY7RksmRj1GSy9GTyooLUZENiMmRklGS0YrSSZnaXZlbkdGJkYrJkYlNiMsJkZMRitGK0YxRis=print();
Knowing that each iternaction takes place at a meaningful time means that a spreader has to be involved in each interaction. So the probability of an interaction depends only on the type of the other interactor. Consider the expected change in the number of ignorants from JkkidEc2IjYjSSJrR0Yk to JkkidEc2IjYjLCZJImtHRiQiIiJGKEYo . The number of ignorants will decrease by one if an ignorant encounters a spreader, and this will occur with probability 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 and will remain unchanged if the ignorant encounters any one else with probability 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
Therefore, the recurrence relation for the expected number of ignorants at meaningful time JkkidEc2IjYjSSJrR0YkisNiQvJkkiaUc2IjYjLCZJImtHRiYiIiJGKkYqLCYqKCZGJTYjRilGKkkiTkdGJiEiIiwmRi1GKkYqRjBGKkYqKigsJkYvRipGLUYwRipGL0YwRi1GKkYqLyZGJTYjIiIhRi8=restart;ignorant_recurrence := simplify( i(k+1) = (i(k)/N)*( i(k)-1 ) + ( (N-i(k))/N )*i(k) );ignorants := rsolve( { i(k+1) = (i(k)/N)*( i(k)-1 ) + ( (N-i(k))/N )*i(k), i(0) = N}, i(k) );Note that the number of ignorants is a monotone decreasing sequence as SSJrRzYi increases. This is consistent with the visualizations above.
For the spreaders, the change is
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print();spreader_recurrence := s(k+1) = s(k) + 2*( (N-1)/N )^k - 1 - 1/N;spreaders := rsolve( {spreader_recurrence, s(0) = 1}, s(k) );spreaders := expand( spreaders );The expected number of spreaders at the sequence of meaningful times is:
LyZJInNHNiI2I0kia0dGJSwqKigiIiMiIiJJIk5HRiVGKywmRitGKykqJiwmRishIiJGLEYrRitGLEYxRidGMUYrRitGK0YrRidGMSomRixGMUYnRitGMQ==The behavior of the sequence of spreaders is not as obvious as it is for the sequence of ignorants. As SSJrRzYi increases, the expression LCYiIiJGIykqJiwmRiMhIiJJIk5HNiJGI0YjRihGJ0kia0dGKUYn increases, but LCZJImtHNiIhIiIqJkkiTkdGJEYlRiMiIiJGJQ== decreases, and so the number of spreaders is depends on the relative sizes of the two quantities. plot( subs( N = 30, spreaders), k = 0..50);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjovJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGK0Y1RistRiM2WC1GLDYlUSVGcm9tRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUY2NiZGOC9GPFEmMC41ZW1GJ0Y+L0ZBUSVhdXRvRictRiw2JVEoc2V2ZXJhbEYnRkhGSkZMLUYsNiVRJ2dyYXBoc0YnRkhGSkZMLUYsNiVRJXdpdGhGJ0ZIRkpGTC1GLDYlUSh2YXJ5aW5nRidGSEZKRkwtRiw2JVErcG9wdWxhdGlvbkYnRkhGSkZMLUYsNiVRJnNpemVzRidGSEZKLUkjbW9HRiQ2MFEiLEYnRkovJSZmZW5jZUdGSS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR1EzdmVyeXRoaWNrbWF0aHNwYWNlRicvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RidGTC1GLDYlUSNpdEYnRkhGSkZMLUYsNiVRKGFwcGVhcnNGJ0ZIRkpGTC1GLDYlUSV0aGF0RidGSEZKRkwtRiw2JVEkdGhlRidGSEZKRkwtRiw2JVEpbGlmZXRpbWVGJ0ZIRkotRl9vNjBRMSZJbnZpc2libGVUaW1lcztGJ0ZKRmJvL0Zlb0ZJRmZvRmhvRmpvRlxwRl5wRmBwRmNwL0ZncEZlcEZpcEZccS1GIzYoLUYsNiVRI29mRidGSEZKRkxGaHFGTC1GLDYlUSZydW1vckYnRkhGSkZeci1GLDYlUSVvdmVyRidGSEZKLUZfbzYwRmByRkpGYm9GYXJGZm9GaG9Gam9GXHBGXnBGYHAvRmRwUS90aGlja21hdGhzcGFjZUYnL0ZncEZhc0ZpcEZccUZocUZecy1GLDYlUSttZWFuaW5nZnVsRidGSEZKRl5zLUYsNiVRJnRpbWVzRidGSEZKRkxGK0ZMLUYsNiVRI2lzRidGSEZKRkwtRiw2JVEnYWx3YXlzRidGSEZKRkwtRiw2JVEmYWJvdXRGJ0ZIRkpGTC1JI21uR0YkNiRRJDEuNkYnRkotRl9vNjBRIipGJ0ZKRmJvRmFyRmZvRmhvRmpvRlxwRl5wRmBwL0ZkcFEudGhpbm1hdGhzcGFjZUYnL0ZncEZqdEZpcEZccS1GLDYlUStQb3B1bGF0aW9uRidGSEZKLUZfbzYwUSIuRidGSkZib0ZhckZmb0Zob0Zqb0ZccEZecEZgcEZjcEZickZpcEZccUZMRistRiM2aW5GK0ZMLUYsNiVRI1dlRidGSEZKRkwtRiw2JVEkY2FuRidGSEZKRkwtRiw2JVErZ2VuZXJhbGl6ZUYnRkhGSkZMLUYsNiVRJXRoaXNGJ0ZIRkpGTC1GX282MFEkYW5kRidGSkZib0ZhckZmb0Zob0Zqb0ZccEZecEZgcEZgc0Zic0ZpcEZccUZMLUYsNiVRJW1ha2VGJ0ZIRkpGTEZocUZMLUYsNiVRKm1vdGl2YXRlZEYnRkhGSkZMLUYsNiVRK2Fzc3VtcHRpb25GJ0ZIRkpGTEZlcUZMRmhxRkwtRiw2JVEpZXhwZWN0ZWRGJ0ZIRkpGTC1GLDYlUSlkdXJhdGlvbkYnRkhGSkZMRmVyRkxGaHFGXnNGaHJGTEZpc0ZMLUYsNiVRLXByb3BvcnRpb25hbEYnRkhGSkZMLUZfbzYxUSN0b0YnLyUlYm9sZEdGMS9GM1ElYm9sZEYnRmJvRmFyRmZvRmhvRmpvRlxwRl5wL0ZhcEYuL0ZkcFEwbWVkaXVtbWF0aHNwYWNlRicvRmdwRl54RmlwRlxxRkxGaHFGTEZcdUZMLUYsNiVRJXNpemVGJ0ZIRkpGXm9GTC1GLDYlUSRzYXlGJ0ZIRkpGTC1GLDYlUSJFRidGSEZKLUZfbzYwUSJbRidGSi9GY29GMUZhci9GZ29GMUZob0Zqb0ZccEZecC9GYXBRJ3ByZWZpeEYnRml0Rlt1RmlwRlxxLUYsNiVRIlRGJ0ZIRkotRl9vNjBRIl1GJ0ZKRlx5RmFyRl15RmhvRmpvRlxwRl5wL0ZhcFEocG9zdGZpeEYnRml0L0ZncFEydmVyeXRoaW5tYXRoc3BhY2VGJ0ZpcEZccS1GX282MFEiPUYnRkpGYm9GYXJGZm9GaG9Gam9GXHBGXnBGYHBGYHNGYnNGaXBGXHFGTC1GLDYlUSkmbGFtYmRhO0YnRkhGSkZMLUYsNiVRIk5GJ0ZIRkpGX3VGTEYrRmR1Rl5yLUYsNiVRJXdpbGxGJ0ZIRkotRl9vNjBGYHJGSkZib0ZhckZmb0Zob0Zqb0ZccEZecEZceEZjcEZickZpcEZccS1GLDYlUShhdHRlbXB0RidGSEZKRl5yLUYsNiVGZ3dGSEZKRmZ6LUYsNiVRJWZpbmRGJ0ZIRkpGZnotRiw2JVEjYW5GJ0ZIRkpGZnotRiw2JVEnZW5kaW5nRidGSEZKRmZ6LUYsNiVRJXRpbWVGJ0ZIRkpGXnJGZXJGXnJGaHFGXnItRiw2JVElZm9ybUYnRkhGSkZeci1GLDYlUSombGFtYmRhO05GJ0ZIRkpGX3VGXnJGXnJGKw==Note that this plot has the general characteristics of the blue average curve from the simulation of the average size of the rumor computed above. It increases to a maximum, then decreases until the number of spreaders is zero. However the time scale is compressed, since this curve is computed only on at the meaningful times, not at every encounter. Therefore, the spreaders are extinguished at about 1.6 times the population size instead of about 6 times the population size. This indicates that that there are roughly 4 encounters for every meaningful time.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yntermination_equation := 0 = 2*N* ( 1 - ( (N-1)/N)^(lambda*N)) + 1 - (lambda*N) - (lambda*N)/N;expanded_termination_equation := expand( termination_equation/N);solve( expanded_termination_equation, lambda);This solution is not useful as it exists, so we will make a further limiting approximation for N very large to obtain a more useable expression.limit_termination_equation := expand( limit( expanded_termination_equation, N=infinity) assuming N > 1);limit_termination_equation := lambda = 2*( 1 - exp(-lambda));plot( {lhs( limit_termination_equation), rhs( limit_termination_equation)}, lambda = 0..2);lambda_solutions := solve( limit_termination_equation, lambda);evalf( lambda_solutions);If SSJORzYi is sufficiently large, then \316\273 is approximately 1.594 SSJORzYi and we conclude the rumor dies after there have been 1.594 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The number of ignorants at extinction is 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KS1JIk5HNiI2IyomLCYiIiIhIiJGJEYpRilGJEYqSSombGFtYmRhO05HRiU= \342\211\210 KiZJIk5HNiIiIiIpSSJlR0YkLCRJJ2xhbWJkYUdGJCEiIkYlprint();
The proportion of the ignorants in the population is 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 ( ( (N-1)/N )^(lambda*N), N=infinity);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnThe proportion of ignorants at the end of the rumor is then 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_ignorants_fraction := evalf( exp(-lambda_solutions[1]));Thus we conclude that there is an analytic basis for the observation from the simulations that about 20% of the original ignorant population remains ignorant.Legal Notice: The copyright for this application is owned by the author(s). 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Contact the author for permission if you wish to use this application in for-profit 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