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RESEARCH POSTER PRESENTATION DESIGN © 2015

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Network structure is defined by a labeled graph with as labels for nodes X and Y:

Connection weights XY

Combination functions Y

Speed factors Y

In adaptive networks connection weights X,Y from X to Y are treated in the same way as nodes:

Y(t+t) = Y (t) + Y [cY (X1,Y(t) X1(t), …, Xk,Y(t) Xk(t)) - Y (t)] t

X,Y(t+t) = X,Y (t) + X,Y [cX,Y(X(t), Y(t), X,Y(t)) - X,Y(t)] t

Introduction

Aggregation: Properties of Homophily and Node

Combination Functions

Example Simulations

Initial network

Chapter 13

Relating a Rei#ed Adaptive Network’s Structure

to its Emerging Behaviour for Bonding by Homophily

Relating Adaptive Network Structure to Bonding

Behaviour

Theorem 1 (Relation between equilibrium values for nodes and for connection weights)

Suppose the function c(V1, V2, W) has tipping point for V1 and V2 and the network reaches an

equilibrium state with values X for the nodes X and X,Y for the connection weights X,Y. Then:

|X - Y| < X,Y = 1

X,Y < 1 | X - Y|

|X - Y| > X,Y = 0

Definition 1 (Tipping point of homophily combination functions for connections)

a) The function c(V1, V2, W): [0, 1] [0, 1] [0, 1] [0, 1] has tipping point for V1 and V2

if for all W with 0 < W < 1 and all V1, V2 it holds

(i) c(V1, V2, W) > W |V1 – V2| <

(ii) c(V1, V2, W) = W |V1 – V2| =

(iii) c(V1, V2, W) < W |V1 – V2| >

b) The function c(V1, V2, W) has a strict tipping point if it has tipping point and in

addition it holds:

(i) If |V1 – V2| < then c(V1, V2, 0) > 0

(ii) If |V1 – V2| > then c(V1, V2, 1) < 1

Definition 2 (Properties of node combination functions)

a) A function c(..) is called monotonically increasing if

Ui Vi for all i c(U1, …, Uk) c(V1, …, Vk)

b) A function c(..) is called strictly monotonically increasing if

Ui Vi for all i, and Uj < Vj for at least one j c(U1, …, Uk) < c(V1, …, Vk)

c) A function c(..) is called scalar-free if

c(

V1,…,

Vk) =

c(V1, …, Vk) for all

> 0

Various Examples of Functions

c(V1, V2, W) with Tipping Point

( = 0.2 in the graph, and W = 0.7; D = |V1 – V2|)

X1X2X3X4X5X6X7X8X9X10

X1

X2

X3

X4

X5

X6

X7

X8

X9

X1

0

X1X2X3X4X5X6X7X8X9

X1

0

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

X1X2X3X4X5X6X7X8X9X10

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

X1X2X3X4X5X6X7X8X9X10

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

X1X2X3X4X5X6X7X8X9X10

X1

X2

X3

X4

X5

X6

X7

X8

X9

X1

0

= 1

= 5

= 11

= 11.4

= 15

Theorem 2 (Co-evolution: equilibrium values X,Y all 0 or 1) Suppose the network is weakly

symmetric and normalised, and the combination functions for the nodes are strictly

monotonically increasing and scalar-free. Suppose that c(V1, V2, W) has a tipping point and the

network reaches an equilibrium state. Then:

X Y X,Y = 0

X = Y X,Y = 0 or X,Y = 1

If c(V1, V2, W) has a strict tipping point , then

X = Y X,Y = 1

Theorem 3 (Co-evolution: clusters and equilibrium values of nodes) Suppose the

combination function for the nodes is strictly monotonically increasing and scalar-free, and the

combination function for the connections uses tipping point and is strict and symmetric and

the network reaches an equilibrium state.

a) X = Y for all X and Y in every cluster C.

b) Every cluster is a fully connected graph with weights 1.

c) Any two distinct equilibrium values of nodes X Y have distance .

Therefore there are at most 1 + 1/ equilibrium values X within the interval [0, 1].

Function type Function name Numerical

representation

Values in

Fig. 1

Simple Linear !V V"WW#W $W$D%&

Simple

quadratic '!V V"WW#W $W"(D"%&

Advanced

quadratic '!V V"WW##"$D#"%) *%

Cubic +!V V"

WW# $W $D,-%).

Logistic 1 / !V V"

WW,W# $W0$%

Logistic 2 /"!V V"

W

W#W $W)*$ , #$

D$ %1%*

Sine-based !V V"WW$ $WD$," %"

Tangent-based !V V"WW$ $WD$," %"

Exponential !V V"W $ $WD$%

connections X1X2X3X4X5X6X7X8X9X10

X12 ) )" ) )" ) * ) )"* )"* )

X2)"* 2 )"* )" ) )" ) * )"* )"* )"*

X3) )"* 2 ) )" ) * ) )"* ) ) *

X4)"* ) * )"* 2 ) * )3 )"* ) * )"* )"*

X5)"* )" ) )" 2 )"* )" ) )" ) *

X6)"* ) )"* )"* )"* 2 ) )"* )"* )

X7)" ) )" ) * )" )" 2 )" ) * )"*

X8) )"* ) )"* )* ) * )"* 2 ) )"*

X9)"* ) * )"* ) * )" ) )" ) * 2 ) *

X10 )" )"* )" )" ) )" ) * )3 )" 2

speed

factors ) ) ) ) ) ) ) ) ) )

Clustering process Final network

tipping point = 0.1

speed factors X,Y = 0.4 for connection weights X,Y Y = 0.1 for nodes Y

4V –

V"4

V , V", W