Chapter 13 Relating a Reified Adaptive Network’s Structure to its Emerging Behaviour for Bonding by Homophily

Abstract

In this chapter, it is analysed how emerging behaviour in an adaptive network for bonding based on similarity (homophily) can be related to characteristics of the adaptive network’s structure, which includes the structure of the adaptation principles incorporated. In particular, this is addressed for adaptive social networks for bonding based on homophily and community formation. To this end, relevant characteristics of the reified network structure (including the adaptation principle) have been identified, such as a tipping point for similarity as used in the aggregation for homophily. Applying network reification, the adaptive network characteristics are represented by reification states in the extended network, and adaptation principles are described by characteristics of these reification states, in particular, their connectivity characteristics (their connections) and their aggregation characteristics (in terms of their combination functions). According to this network reification approach, as one of the results, it has been found how the emergence of communities strongly depends on the value of this similarity tipping point. Moreover, it is shown that some characteristics entail that the connection weights all converge to 0 (for persons in different communities) or 1 (for persons within one community). This is Chapter 13 of the book https://www.researchgate.net/publication/334576216.

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Network structure is defined by a labeled graph with as labels for nodes X and Y:
Connection weights XY
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 [cX,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"WW#W $W$D%&
Simple
quadratic '!V V"WW#W $W"(D"%&
Advanced
quadratic '!V V"WW##"$D#"%) *%

Cubic +!V V"
WW# $W $D,-%).
Logistic 1 / !V V"
WW,W# $W0$% 
Logistic 2 /"!V V"
W
W#W $W)*$ , #$
D$ %1%*
Sine-based !V V"WW$ $WD$," %"
Tangent-based !V V"WW$ $WD$," %"
Exponential !V V"W $ $WD$% 
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