ORCHESTRAL CONDUCTING - Michael Tsecktse.eie.polyu.edu.hk/IWCSN2011-Orchestral.pdf · ORCHESTRAL CONDUCTING: A PRACTICAL NETWORK CONSENSUS PROBLEM! ... Conducted by the Music Director

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


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


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


CONDUCTOR CONTROLLING THE PERFORMERS

An extreme case

Merry Christmas Mr Bean


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

Orchestra and network model: the system under consideration


ORCHESTRA

http://www.mti.dmu.ac.uk/~ahugill/manual/seating.html


NETWORK

An orchestra is a special network Conductor

Section leaders Group leaders

Tutti players


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


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


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


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

Phenomenon of interest: consensus


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


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!


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


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


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.

Role of network structure


NETWORK STRUCTURE

ConductorSection leader

Section leader

Section leader

Group leader

Group leader

Group leaderTutti player

Tutti player

Tutti playerTutti player

Tutti player

Tutti player


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 √ √ √


Type 1

Type 2

Type 3

Type 4Type 5

Type 6

Type 7


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


MATLAB STATISTICSConductor=1 and all other nodes start with mean 0.5 and s.t. 0.1, all weights are 1


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


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



Average >0.8 >0.9 1.0


Variance <0.01 <0.001 0




Average >0.8 >0.9 1.0


Variance <0.01 <0.001 0


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



Average >0.8 >0.9 1.0


Variance <0.01 <0.001 0



DISTRIBUTIONType 1 Type 2

Type 3 Type 4


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


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


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


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?


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)


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)


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


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 !


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"

Mutual interaction


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.



Average >0.8 >0.9


Variance <0.01 <0.001


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


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):




MUTUAL INTERACTION WITH CONDUCTOR

hinders consensus / creates compromised consensus

Percentage of nodes with mutual influence with conductor = 100%.


confidence=1, P=0.5.


confidence=1, P=0.5.

Type 5: basically type 1 with full mutual interactions.


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.





MUTUAL INTERACTION WITH CONDUCTOR


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


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.

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Hong Kong Polytechnic University http://chaos.eie.polyu.edu.hk

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Documents

ORCHESTRAL CONDUCTING - Michael Tsecktse.eie.polyu.edu.hk/IWCSN2011-Orchestral.pdf · ORCHESTRAL CONDUCTING: A PRACTICAL NETWORK CONSENSUS PROBLEM! ... Conducted by the Music Director