International Workshop on Complex Systems and Networks 2011, Melbourne
ORCHESTRAL CONDUCTING: A PRACTICAL NETWORK
CONSENSUS PROBLEM !
Michael Tse Hong Kong Polytechnic University
Michael Tse, IWCSN 2011
ACKNOWLEDGMENT
Mr Ho-Man Choi Artist-in-Residence, Hong Kong Polytechnic University Conductor & Music Director, Hong Kong Pro Arte Orchestra Resident Conductor, Hong Kong Philharmonic Orchestra !!!Students
Mr Xiaofan Liu, PhD student Mr Bo Yang, MSc student
Michael Tse, IWCSN 2011
CONDUCTING
Conducting is the act of directing a musical performance by way of visible gestures. The primary duties of the conductor are to unify performers, set the tempo, execute clear preparations and beats, and to listen critically and shape the sound of the ensemble. !
!
!
!
!
!
!
!!
Viotti Chamber Orchestra performing the 3rd movement of Mozart’s Divertimento in D Major (K136)
Conductor: Marcello Viotti (1955-2005)http://en.wikipedia.org/wiki/Conducting
Michael Tse, IWCSN 2011
Chamber Orchestra Kremlin Conducted by the Music Director of the orchestra, Misha Rachlevsky. From the Great Hall of
the Moscow Conservatory http://www.youtube.com/watch?v=T9JKCoxn2wc&feature=related
FLIGHT OF THE BUMBLEBEE
Chamber Orchestra Kremlin Conducted by the Music Director of the orchestra, Misha Rachlevsky. From the Great Hall of
the Moscow Conservatory http://www.youtube.com/watch?v=T9JKCoxn2wc&feature=related
Michael Tse, IWCSN 2011
CONDUCTOR CONTROLLING THE PERFORMERS
An extreme case
Merry Christmas Mr Bean
Michael Tse, IWCSN 2011
MUTUAL INTERACTION
ECCO (East Coast Chamber Orchestra) – In 2001, a group of exciting young string players envisioned the creation of a conductor-less chamber orchestra, based upon democratic principles, whose focus is to be purely on music-making. Its members are some of the most talented young chamber musicians, soloists (many of them making appearances with such orchestras as Chicago and Philadelphia), and principals in major American orchestras.
W.A Mozart's Symphony No. 29 K.201-Minuetto http://www.youtube.com/watch?v=OmZfWT9BUDE
Michael Tse, IWCSN 2011
NETWORK
An orchestra is a special network Conductor
Section leaders Group leaders
Tutti players
Michael Tse, IWCSN 2011
NETWORK MODEL
Directed Graph Approach Nodes: Members in the orchestra Edges: Influence of one member to another Weightings of edges represents the influence of each member, i.e., coupling strengths
x1 x2x3
x4x5
w11
w12
w15
w53
w54
w33
w55
w44
w22
Michael Tse, IWCSN 2011
IMITATION MODEL!!!!!!Suppose a state value between 0 and 1 represents a performance state. Let the conductor’s state be 1. All members may start with any number from 0 to 1.
!The conductor’s job is to get everyone to 1 within a finite time. If the conductor is unaffected by others, then the network will evolve, with all other nodes converging toward 1 at some rate. !This is a special consensus problem.
x1 x2x3
x4
x5
w11
w12
w15
w53
w54
w33
w55
w44
Conductor
w22
Michael Tse, IWCSN 2011
DEGROOT MODEL
The DeGroot model Each individual has a state x(t) at time t. Its state will be influenced by other people according to
xi(t + 1) =wiixi(t) +
!wijxj(t)
wii +!
wij
x1 x2x3
x4x5
w11
w12
w15
w53
w54
w33
w55
w44
w22
Michael Tse, IWCSN 2011
CONNECTION MATRIX
Connection matrix T, with normalized weights:
x(t + 1) =
⎡
⎢⎢⎢⎢⎢⎣
w11 w12 · · · w1n
w21 w22 · · · w2n
w31 w32 · · · w3n
... · · ·
. . ....
wn1 wn2 · · · wnn
⎤
⎥⎥⎥⎥⎥⎦
x(t) where x =
⎡
⎢⎢⎢⎣
x1
x2
...xn
⎤
⎥⎥⎥⎦
x1 x2
x3
1
3
1
3
1
3
1
2
1
2
T =
⎡
⎣1/3 1/3 1/30 1/2 1/20 0 1
⎤
⎦
Example 1
Michael Tse, IWCSN 2011
CONSENSUS (CONVENTIONAL)
As the network evolves, the nodes will update themselves and may eventually reach a consensus, where all nodes converge to the the same value. !where x* is the final (consensus) value. !!!!!How practical is this definition when applied to orchestra?
limt→∞
|xi(t) − x∗| = 0 for all i
x1 x2x3
x4x5
w11
w12
w15
w53
w54
w33
w55
w44
w22
Michael Tse, IWCSN 2011
IN PRACTICE
Fixed budget Limited number of rehearsals are allowed
Venue cost, conductor cost, musician cost
|xi(t) − xj(t)| < ϵ for all i, j and t < tf
Tolerance Absolute consensus is normally not required
Final value does not need to be exactly 1.
Consensus considered ‘fail’ if not achieved within the budget time!
Michael Tse, IWCSN 2011
CONVERGENCE PARAMETERS
Convergence rate Supremum of convergence rate Average convergence rate !
Convergence time Time needed for full consensus to be reached Time needed for practical consensus to be reached
Michael Tse, IWCSN 2011
CONVERGENCE RATE
How fast can the orchestra converge?
The connection matrix determines the rate of convergence, which is the second largest eigenvalue λ2 of the connection matrix T, since x(t+1) = T x(t).
!But this definition really gives the supremum of the convergence rate (for all initial conditions), not the average convergence rate. In practice, average convergence rate for specific initial conditions are adequate.
X is the set of all initial conditions excluding the full consensus case
Michael Tse, IWCSN 2011
CONVERGENCE TIME
The time when all x(t) converge close enough to a certain value. !
!
This is like the full consensus time found in simulations or experiments.
Michael Tse, IWCSN 2011
NETWORK STRUCTURE
ConductorSection leader
Section leader
Section leader
Group leader
Group leader
Group leaderTutti player
Tutti player
Tutti playerTutti player
Tutti player
Tutti player
Michael Tse, IWCSN 2011
NETWORK STRUCTURE
Type Conductor to every members
Conductor to group leaders
Connections among members
Cross sectional connections
Stubborn members
1 √ √
2 √ √ √
3 √
4 √ √
5 √ √ √
6 √ √ √
7 √ √ √
Michael Tse, IWCSN 2011
MATLAB SIMULATION
Conductor influences all members with same weights Each section leader influences its members
Magenta (1); Green (0) Full consensus in 7 steps (note the color change)
Conductor=1 and all other nodes start with mean 0.5 and s.t. 0.1, all weights are 1
Michael Tse, IWCSN 2011
MATLAB STATISTICSConductor=1 and all other nodes start with mean 0.5 and s.t. 0.1, all weights are 1
Michael Tse, IWCSN 2011
SIMULATION SAMPLES
Type 1 Type 2
Conductor=1 and all other nodes start with 0, all weights are 1
Every one reached 0.8 0.9
Needs 4 steps 5 steps
Average >0.8 >0.9 1.0
Needs 4 steps 5 steps 16 steps
Variance <0.01 <0.001 0
Needs 2 steps 5 steps 8 steps
Values: the values of all nodes; Average: the average value of all nodes in each step;
Variance: the variance value of all nodes in each step. (Similarly hereinafter)
Type II
Every one reached 0.8 0.9
Needs 9 steps 11 steps
Average >0.8 >0.9 1.0
Needs 6 steps 8 steps 37 steps
Variance <0.01 <0.001 0
Needs 8 steps 13 steps 19 steps
Every one reached 0.8 0.9
Needs 4 steps 5 steps
Average >0.8 >0.9 1.0
Needs 4 steps 5 steps 16 steps
Variance <0.01 <0.001 0
Needs 2 steps 5 steps 8 steps
Values: the values of all nodes; Average: the average value of all nodes in each step;
Variance: the variance value of all nodes in each step. (Similarly hereinafter)
Type II
Every one reached 0.8 0.9
Needs 9 steps 11 steps
Average >0.8 >0.9 1.0
Needs 6 steps 8 steps 37 steps
Variance <0.01 <0.001 0
Needs 8 steps 13 steps 19 steps
Michael Tse, IWCSN 2011
CONVERGENCE RATE
TypeConductor
to every members
Conductor to group leaders
Connections among
members
Cross sectional
connections
Stubborn members
Time constant τ
Full consensus
time
1 √ √ 2.15 5
2 √ √ √ 3.27 11
3 √ 4.62 9
4 √ √ 6.58 22
5 √ √ √ 2.36 7
6 √ √ √ 2.67 9
7 √ √ √ 3.21 25
Michael Tse, IWCSN 2011
0"
5"
10"
15"
20"
25"
30"
0" 1" 2" 3" 4" 5" 6" 7"
Average time constant τ
Full
cons
ensu
s tim
e
tn AND τ
No apparent correlation
Michael Tse, IWCSN 2011
EIGENVALUES
The second largest eigenvalue of the connection matrix does not provide useful clue to the consensus problem
3""
The$second$largest$eigen$value$of$transform$matrix$"
The"transform"matrix"of"the"network"can"be"derived"by"normalizing"each"row"of"the"adjacency"matrix,"or"the"edge"weight"matrix"when"there"is"a"weight"on"the"edge."For"example,"for"the"graph"with"adjacency"matrix"A:"
1" 0" 0"
1" 1" 0"
1" 1" 1"
Will"have"transform"matrix"T:"
1" 0" 0"
.5" .5" 0"
.33" .33" .33"
For"a"set"of"nodes"x"with"transform"matrix"T,"at"time"t,"x(t)"="Ttx(0),"where"x(0)"is"the"initial"condition."
Theory"says"that"the"consensus"rate,"i.e."the"time"t"used"for"Tt"to"approach"limit"T�"depends"on"how"quickly"each"eigen"value"of"Tt"goes"to"0."The"rate"will"generally"be"governed"by"the"(converge"speed"of"the)"second"largest"eigen"value"as"other"eigen"values"will"converge"more"quickly."
Table&2:&consensus&speed&and&eigen&values&
"
performance" eigenvalue"of"transform"matrix"
"
Tau" full"consensus" second"largest" largest"
type"0" 1.49" 4" 0.5" 1"
type"1" 2.15" 5" 0.5" 1"
type"3" 4.62" 9" 0.5" 1"
type"5" 2.36" 7" 0.542" 1"
type"6" 2.67" 9" 0.682" 1"
type"2" 3.27" 11" 0.778" 1"
type"4" 6.58" 22" 0.875" 1"
type"7" 3.21" 25" 0.909" 1"
"
The"table"shows"the"second"largest"eigen"value"of"the"transform"matrix"T."Actually"there"is"not"much"correlation"between"the"second"largest"eigen"value"and"the"two"of"the"performance"measures."Hence"the"second"largest"eigen"value"of"the"transform"matrix"T"actually"has"little"impact"on"the"consensus"rate"in"the"early"stages"of"consensus"forming."
The$network$measures$"
Time constant τ
Full consensus time
Second largest eignevalueType
Michael Tse, IWCSN 2011
EIGENVALUESAppendix(A:(eigen(values(of(different(types(of(orchestra(setting(!
Table&1:&all&eigen&values&of&different&types&of&orchestra&settings&
Type! �1! �2! �3! ...! �42! �43!0! 1.0! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5!1! 1.0! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3!2! 1.0! 0.8! 0.8! 0.7! 0.6! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0!3! 1.0! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5!4! 1.0! 0.9! 0.9! 0.8! 0.8! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0!
5! 1.0! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3!1
0.1!1
0.1!1
0.1! 0.0!
6! 1.0! 0.7! 0.6! 0.6! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.0! 0.0! 0.0!1
0.1!1
0.1!1
0.1!1
0.1!1
0.1!7! 1.0! 0.9! 0.9! 0.8! 0.8! 0.8! 0.8! 0.8! 0.8! 0.8! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3!!
!
!
Figure&1:&eigen&value&distributions&of&different&types&of&orchestra&settings.&From&upper&left&to&lower&right:&Type&0,&Type&1,&...&Type&7.&Vertical&axis:&frequency,&horizontal&axis:&eigen&values.&
Appendix(A:(eigen(values(of(different(types(of(orchestra(setting(!
Table&1:&all&eigen&values&of&different&types&of&orchestra&settings&
Type! �1! �2! �3! ...! �42! �43!0! 1.0! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5!1! 1.0! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3!2! 1.0! 0.8! 0.8! 0.7! 0.6! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0!3! 1.0! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5!4! 1.0! 0.9! 0.9! 0.8! 0.8! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0! 0.0!
5! 1.0! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3!1
0.1!1
0.1!1
0.1! 0.0!
6! 1.0! 0.7! 0.6! 0.6! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.0! 0.0! 0.0!1
0.1!1
0.1!1
0.1!1
0.1!1
0.1!7! 1.0! 0.9! 0.9! 0.8! 0.8! 0.8! 0.8! 0.8! 0.8! 0.8! 0.5! 0.5! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3! 0.3!!
!
!
Figure&1:&eigen&value&distributions&of&different&types&of&orchestra&settings.&From&upper&left&to&lower&right:&Type&0,&Type&1,&...&Type&7.&Vertical&axis:&frequency,&horizontal&axis:&eigen&values.&
Type 0 Type 1 Type 2 Type 3
Type 4 Type 5 Type 6 Type 7
Type 0 : conductor connected to each member
Can we connect consensus performance with suitable network measures? And
what should these measures be?
Michael Tse, IWCSN 2011
CHARACTERISTICS
Mutual connection among players Average path length from conductor Average shortest weighted path length (path resistance) from conductor Weighted diameter (weighted path length from conductor to the most remote node)
Michael Tse, IWCSN 2011
MUTUAL CONNECTION = STUBBORN PLAYER
Mutually connected nodes can be transformed to equivalent separate nodes with amplified self-weights equal to the number of mutually connected nodes. !
Slower convergence of types 2, 4 and 7
=
mutually connected nodes (type 2 and type 4)
stubborn nodes (type 7)
Michael Tse, IWCSN 2011
AVG SHORTEST WEIGHTED PATH LENGTH (PATH RESISTANCE)
Reciprocal of the path weight can be considered as path resistance. Path from conductor to each member is considered. Average path resistance is a measure of conductor’s influence.
0.5
0.33
0.5
0.5
0.33
0.5
0.5
0.5
1
0.33
0.5
e.g., for node X, the path resistance from conductor is 2+2+2 = 6; and for node Y, it is 2+3 = 5.
X
Y
conductor
Michael Tse, IWCSN 2011
RESULTS
4""
Table&3:&consensus&speed&and&network&measures&
"
performance" network"measures"
"
Tau" full"consensus"
average"length"of""
all"binary"paths"
weighted""
shortest"path" weighted"diameter"
type"0" 1.49" 4" 1" 2" 2"
type"1" 2.15" 5" 1.69" 2.86" 3"
type"3" 4.62" 9" 2.36" 4.67" 6"
type"5" 2.36" 7"
"
3.19" 5"
type"6" 2.67" 9"
"
3.60" 6"
type"2" 3.27" 11" 3.51" 5.42" 9"
type"4" 6.58" 22" 5.86" 7.23" 12"
type"7" 3.21" 25"
"
4.74" 12"
"
The"table"shows"the"performance"of"the"eight"network"types"and"their"measures."
The"average"lengths"of"all"binary"paths"are"calculated"for"type"0"to"type"4"network."This"parameter"measures"average"lengths"of"all"paths"from"the"conductor"to"every"member"of"the"orchestra."The"length"of"a"single"hop"is"1."The"average"length"of"all"paths"is"strongly"correlated"with"full"consensus"time."
"Figure&4:&full&consensus&speed&and&average&length&of&all&paths&from&conductor&to&each&individual&
Another"way"to"describe"distance"between"two"connected"nodes"is"to"take"the"reciprocal"of"the"transform"matrix."For"example"transform"matrix"T:"
1" 0" 0"
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30
average&length&of&a
ll&pa
ths
full&consensus
Performance Network Measures
Avg Path Resistance
Avg Path Length
Weighted Diameter
Avg Conv Rate
Full Consensus
Time
!!
Type !
Michael Tse, IWCSN 2011
NETWORK MEASURES AND CONSENSUS PERFORMANCE
Types 3 and 4: conductor does not connect to all members
Types 2, 4 and 7: existence of equivalent stubborn members
0"
1"
2"
3"
4"
5"
6"
7"
8"
0" 2" 4" 6" 8"
average&shortest&weighted&pa
th&length&
average&1me&constant&
T1"
T2"
T5"T6"
T0"
T7"T3"
T4"
0"
2"
4"
6"
8"
10"
12"
14"
0" 5" 10" 15" 20" 25" 30"
weighted(diam
eter(
Full(consensus(
T0"
T2"
T4" T7"
T1"
T3"T5"
T6"
Michael Tse, IWCSN 2011
Special cases
Type V: cross relationships among different sections based on type I.
!"#$%&'(' )*+"*,&'-.'!"#$%&'(' )*+"*,&'-/'!"#$%&'.' )*+"*,&'-0'!"#$%&'.' )*+"*,&'-1'23&&+,&'(' )*+"*,&'-4'23&&+,&'(' )*+"*,&'-5'23&&+,&'.' )*+"*,&'-6'23&&+,&'.' )*+"*,&'-7'
Diagram of this
network:
Same condition =0, Self-confident (except conductor) =1, Conductor Self-
confidence=1,P=0.
Every one reached 0.8 0.9
Needs 5 steps 7 steps
Average >0.8 >0.9
Needs 4 steps 5 steps
Variance <0.01 <0.001
Needs 3 steps 6 steps
CROSS INTERACTION
Type 5 : Basically type 1 with cross sectional mutual links: • Flute 1 — Violin 12 • Flute 1 — Violin 13 • Flute 2 — Violin 14 • Flute 2 — Violin 15 • Bassoon 1 — Violin 16 • Bassoon 1 — Violin 17 • Bassoon 2 — Violin 18 • Bassoon 3 — Violin 19
Michael Tse, IWCSN 2011
CROSS INTERACTION
Type 5 : Basically type 1 with cross sectional mutual links: • Flute 1 — Violin 12 • Flute 1 — Violin 13 • Flute 2 — Violin 14 • Flute 2 — Violin 15 • Bassoon 1 — Violin 16 • Bassoon 1 — Violin 17 • Bassoon 2 — Violin 18 • Bassoon 3 — Violin 19
!
!
The time constant ! is 4 steps, 3 steps and 3 steps separately.
3D Historical Diagram (Overall dynamics can be seen in the gui):
Normal distribution (mean: µ =0.5 and variance !!2 =0.2), Self-confident (except
conductor) =1, Conductor Self-confidence=1, P=0.
!
The time constant ! is 4 steps, 3 steps and 3 steps separately.
3D Historical Diagram (Overall dynamics can be seen in the gui):
Normal distribution (mean: µ =0.5 and variance !!2 =0.2), Self-confident (except
conductor) =1, Conductor Self-confidence=1, P=0.
Michael Tse, IWCSN 2011
MUTUAL INTERACTION WITH CONDUCTOR
hinders consensus / creates compromised consensus
Percentage of nodes with mutual influence with conductor = 100%.
Same condition =0, Self-confident (except conductor) =1, Conductor Self-
confidence=1, P=0.5.
Same condition =0, Self-confident (except conductor) =1, Conductor Self-
confidence=1, P=0.5.
Type 5: basically type 1 with full mutual interactions.
Michael Tse, IWCSN 2011
hinders consensus / creates compromised consensus
Percentage of nodes with mutual influence with conductor = 50%.
Type 5: basically type 1 with 50% mutual interaction with conductor.
Normal distribution (mean: µ =0.5 and variance !!2 =0.2), Self-confident (except
conductor) =1, Conductor Self-confidence=1, P=1.
Normal distribution (mean: µ =0.5 and variance !!2 =0.2), Self-confident (except
conductor) =1, Conductor Self-confidence=1, P=1.
MUTUAL INTERACTION WITH CONDUCTOR
Michael Tse, IWCSN 2011
OBSERVATIONSConvergence rate is greatly influenced by “path resistance”, i.e., weighted path length. Mutual interaction within groups create equivalent stubborn players. Mutual interaction among members in different groups slows down convergence, but does not affect consensus to conductor’s value. Mutual interaction with conductor compromises the final consensus value.
Rainer Hersch conducts the Tasmanian Symphony Orchestra at Centennial Hall, Hobart, Australia
Rainer Hersch conducts the Tasmanian Symphony Orchestra at Centennial Hall, Hobart, Australia
http://www.youtube.com/watch?v=jc7LyERAKc0
Michael Tse, IWCSN 2011
MORE PROBLEMS
The coupling strengths could vary as time goes. Conductor could gain more control as he becomes more familiar with members. This may improve consensus. Members may also get more strongly mutually coupled to hinder consensus. Some existing tools in finite time consensus may be worth exploring.