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Network-Oriented Modeling for Adaptive Networks:
Designing Higher-Order Adaptive Biological, Mental
and Social Network Models
Jan Treur
Social AI Group, Vrije Universiteit Amsterdam, The Netherlands
Email: j.treur@vu.nl
URL: https://www.researchgate.net/profile/Jan_Treur
Table of Contents
Note that the chapter titles have hyperlinks leading to their abstracts and (eventually) to Powerpoint
presentations.
Part I Introduction
1. On Adaptive Networks and Network Reification
1.1 Introduction
1.2 First and Second-order Adaptation
1.2.1 First-order Adaptation
1.2.2 Second-order Adaptation
1.3 Higher-Order adaptation
1.3.1 What Orders of Adaptation are Addressed?
1.3.2 Examples of Second and Third-order Adaptation from an Evolutionary Context
1.4 Using Network Reification to Model Adaptive Networks
1.4.1 The Hybrid Approach to Model Adaptive Networks
1.4.2 Modeling Adaptive Networks Based on Network Reification
1.5 Modeling Higher-Order Adaptive Networks by Multilevel Network Reification
1.6 Mathematical Analysis of Reified Networks
1.6.1 Mathematical Analysis of a Base Network
1.6.2 Mathematical Analysis Applied to Reification States
1.7 Discussion
2. Ins and Outs of Network-Oriented Modeling
2.1 Introduction
2.2 Network-Oriented Modeling: Conceptual Background
2.2.1 The Unifying Potential of Networks
2.2.2 On the Meaning of Basic Elements in a Network
2.2.3 Meaning as Defined by a Temporal-Causal Network
2.2.4 Biological, Mental and Social Domains Ask for Networks
2.3 Numerical Representation of a Temporal-Causal Network
2.3.1 Numerical-Mathematical Formalisation
2.3.2 Combination Functions as Building Block
2.4 Role Matrices to Specify a Network Model
2.4.1 Role Matrices as a Specification Format
2.4.2 From Network Structure to Network Behaviour: How Role Matrices Define
the Difference and Differential Equations
2.4.3 Simulations for the Example Social Network Model
2.5 Relating Emerging Network Behaviour to Network Structure
2.5.1 Network Emerging Behaviour and Network Structure
2.5.2 Network Structure Properties Relevant for Emerging Behaviour
2.5.3 Relating Network Structure Properties to Emerging Behaviour
2.6 The Wide Applicability of Network-Oriented Modeling
2.6.1 Network-Oriented Modeling Applies Beyond Perceived Networks
2.6.2 Network-Oriented Modeling Applies to Network Adaptation
2.7 Discussion
Part II Modeling Adaptive Networks by Network Reification
3. A Unified Approach to Represent Network Adaptation Principles by Network Reification
3.1 Introduction
3.2 Temporal-Causal Networks: Structure and Dynamics
3.2.1 Conceptual Representation of a Temporal-Causal Network Model
3.2.2 Numerical Representation of a Temporal-Causal Network Model
3.2.3 Basic Combination Functions, their Parameters, and Combining them
3.2.4 Normalisation, Stationary Points and Equilibria for Temporal-Causal Network Models
3.3 Modeling Adaptive Networks by Network Reification
3.4 Incorporating the Impact of Downward Causal Connections for Reification States
3.5 The Universal Combination Function and Difference Equation for Reified Networks
3.5.1 The Universal Combination Function for Combined Reification for all Network Structure
Characteristics
3.5.2 The Universal Difference Equation for Reified Networks and its Differential Equation
Variant
3.6 Using Network Reification for Unified Modeling of Network Adaptation Principles
3.6.1 Network Reification for Adaptation Principles for Hebbian Learning and Bonding by
Homophily
3.6.2 Network Reification for the Triadic Closure Adaptation Principle
3.6.3 Network Reification for the Preferential Attachment Adaptation Principle
3.6.4 Network Reification for the State-Connection Modulation Adaptation Principle
3.6.5 Network Reification for Response Speed Adaptation Principles
3.6.6 Network Reification for Aggregation Adaptation Principles
3.7 A Reified Network Model for Response Speed Adaptation and Aggregation Adaptation
3.7.1 Conceptual Graphical Representation of the Example Reified Network Model
3.7.2 Conceptual Role Matrices Representation of the Example Reified Network Model
3.7.3 Simulation outcomes for the Example Reified Network Model
3.7.4 Analysis of the Equilibria for the Example Reified Network Model
3.8 On the Added Complexity for Network Reification
3.9 Discussion
4. Modeling Higher-Order Network Adaptation by Multilevel Network Reification
4.1 Introduction
4.2 Structure and Dynamics of Temporal-Causal Networks
4.3 Addressing Network Adaptation by Network Reification
4.3.1 Extending the Network by Reification States
4.3.2 The Universal Combination function and Universal Difference Equation for
Reified Networks
4.4 Using Multilevel Network Reification for Higher-Order Adaptive Network Models
4.4.1 Using Multilevel Network Reification for Plasticity and Metaplasticity from
Cognitive Neuroscience
4.4.2 Role Matrices Covering Plasticity and Metaplasticity
4.5 Simulation for a Second-order Reified Network Model for Plasticity and Metaplasticity
4.6 On the Added Complexity for Higher-Order Network Reification
4.7 Discussion
Part III Applications of Higher-Order Adaptive Network Models
5. A Reified Network Model for Adaptive Decision Making Based on the Disconnect-Reconnect
Adaptation Principle
5.1 Introduction
5.2 Neurological Principles
5.3 The Reified Adaptive Network Model
5.3.1 The Base Network
5.3.2 Modeling First and Second-order Adaptation of the Connection Weights
by Reification States
5.4 Combination Functions and Role Matrices for the Reified Network Model
5.4.1 The Combination Functions Used
5.4.2 The Role Matrices
5.5 Example Simulation Scenarios
5.5.1 Scenario 1: First-order adaptation; no adaptive speed of connection weight change
5.5.2 Scenario 2: Second-order adaptation; adaptive speed of connection weight change
5.6 Verification of the Network Model by Mathematical Analysis
5.6.1 Solving the Linear Equilibrium Equations for the Base Network
5.6.2 Addressing the Nonlinear Equilibrium Equations for the Reification States
5.7 Discussion
6. Using Multilevel Network Reification to Model Second-Order Adaptive Bonding by Homophily
6.1 Introduction
6.2 Conceptual Representation of the Second-Order Adaptive Social Network Model
6.2.1 Reification states at the first reification level
6.2.2 Reification states at the second reification level
6.3 Combination Functions Used at the Three Levels
6.3.1 Base level and first reification level combination functions
6.3.2 Second reification level combination functions
6.4 Role Matrices for the Adaptive Social Network Model
6.5 Simulation of the Second-Order Adaptive Social Network Model
6.5.1 Scenario 1: Adaptive Connections for One Person: Homophily Modulation Factor 0.1
6.5.2 Scenario 2: Adaptive Connections for One Person: Homophily Modulation Factor 0.9
6.5.3 Scenario 3: Adaptive Connections for All Persons
6.6. Analysis of the Equilibria of the Reification States
6.7 Discussion
7 Modeling Higher-Order Adaptive Evolutionary Processes by Reified Adaptive Network Models
7.1 Introduction
7.2 Higher-Order Adaptation in Evolutionary Processes
7.3 A Reified Network Model for Fourth-order Adaptive Evolutionary Processes
7.3.1 Adaptive causal modeling of changing causal pathways in evolutionary processes
7.3.2 The reified adaptive network model for the described fourth-order adaptation case
7.4 Simulation Experiments
7.4.1 Simulation for Scenario 1: Occurrence of Pathogens and Defense System
7.4.2 Simulation for Scenario 2: Occurrence of Pregnancy
7.4.3 Simulation for Scenario 3: Occurrence of Disgust
7.5 Mathematical Analysis
7.5.1 General approach to the mathematical analysis of equilibria
7.5.2 Mathematical analysis of the different scenarios
7.5.3 Verification of the reified network model
7.6 Discussion
8 Reified Adaptive Network Models of Higher-Order Modeling a Strange Loop
8.1 Introduction
8.2 The notion of Strange Loop
8.2.1 Strange Loops in Music, Graphic Art and Paradoxes
8.2.2 Strange Loops in the Brain
8.3 A Twelve and Four Level Reified Adaptive Network Model Based on a Strange Loop
8.3.1 A 12 Level Reified Adaptive Network Model Forming a Cycle of Levels
8.3.2 A Simpler 4 Level Reified Example of an Adaptive Network Model for a Strange Loop
8.4 Simulation Example of the Four Level Strange Loop Reified Network Model
8.5 A Two Hands Reified Network Model for Adaptive Decision Making
8.6 An Example Simulation of the Two Hands Reified Network Model
8.7 Discussion
Part IV A Modeling Environment for Reified Networks
9. A Modeling Environment for Reified Temporal-Causal Network Models
9.1 Introduction
9.2 Role Matrices as Specification Format for Reified Network Models
9.3 The Combination Function Library
9.3.1 Different Groups of Combination Functions
9.3.2 The Standard Format for Combination Functions
9.4 The Computational Reified Network Engine
9.4.1 Splitting the Role Matrices and Copying them into Matlab
9.4.2 Retrieving Information from the Role Matrices
9.4.3 The Iteration Step from t to t+t
9.5 Discussion
10 On the Universal Combination Function and the Universal Difference Equation for Reified Temporal-
Causal Network Models
10.1 Introduction
10.2 A Short Route to the Universal Difference and Differential equation
10.3 Downward Causal Connections Defining the Special Effect of Reification States
10.3.1 The overall picture
10.3.2 Downward Causal Connections for role W for Connection Weight Reification
10.3.3 Downward Causal Connections for role H for Speed Factor Reification
10.3.4 Downward Causal Connections for roles C and P for Combination Function Weight
and Parameter Reification
10.4 Deriving the Universal Combination Function and Difference Equation for Reified Networks
10.4.1 Deriving the Universal Combination Function for Reified Networks
10.4.2 Deriving the Universal Difference Equation for Reified Networks
10.5 The Criterion for a Stationary Point for the Universal Difference Equation
10.6 Deriving the Universal Combination Function and Difference Equation from
the Role Matrices
10.7 Compilation of the Universal Difference Equation by Substitution
10.8 Discussion
Part V Mathematical Analysis of How Emerging Network Behaviour Relates to Base
Network Structure
11 Relating Network Emerging Behaviour to Network Structure
11.1 Introduction
11.2 Conceptual and Numerical Representation of a Network
11.3 Examples of a Network’s Emerging Behaviour
11.3.1 The Example Social Network
11.3.2 Three Simulations with Different Emerging Behaviour
11.4 Relevant Properties of Network Structure
11.4.1 Relevant Properties of Combination Functions
11.4.2 Relevant Properties of the Network’s Connectivity
11.5 Results Relating Emerging Behaviour to Network Structure
11.5.1 Basic Definitions and Results
11.5.2 Common Final Values for Acyclic and Strongly Connected Networks
11.5.3 The Effect of Independent States on a Network’s Emerging Behaviour
11.5.4 How the Speed Factors Affect a Common Equilibrium Value
11.5.5 Emerging Behaviour for Fully Connected Networks
11.6 Emerging Behaviour for Euclidean, Scaled Maximum and Scaled Minimum Combination
Functions
11.6.1 Emerging Behaviour for Euclidean Combination Functions of Different Orders
11.6.2 Comparing Equilibrium Values for Euclidean Combination Functions and Scaled
Maximum Combination Functions
11.7 Discussion
12 Analysis of a Network’s Emerging Behaviour via its Structure Involving its Strongly Connected
Components
12.1 Introduction
12.2 Temporal-Causal Networks
12.3 Emerging Behaviour of a Network
12.3.1 Basics on Stationary Points and Equilibria for Temporal-Causal Networks
12.3.2 An Example Network
12.3.3 Simulations for the Example Network
12.4 Network Connectivity and Strongly Connected Components
12.4.1 A Network’s Strongly Connected Components
12.4.2 The Stratified Condensation Graph of a Network
12.5 Relevant Properties of Combination Functions
12.6 Network Emerging Behaviour and Network Structure Characteristics
12.6.1 Network Emerging Behaviour for Special Cases
12.6.2 Network Emerging Behaviour for the General Case
12.7 Further Implications for Example Networks
12.7.1 Further Analysis of the Example Network from Section 8.3.2
12.7.2 Analysis of an Example Mental Network
12.8 Discussion
Part VI Mathematical Analysis of How Emerging Network Behaviour of Adaptive
Networks Relates to Reified Network Structure
13 Relating a Reified Adaptive Network’s Structure to its Emerging Behaviour for Bonding by
Homophily
13.1 Introduction
13.2 Network-Oriented Modeling by Temporal-Causal Networks
13.3 Reified Adaptive Networks for Bonding based on Homophily
13.3.1 Modeling the Bonding by Homophily Adaptation Principle by Reification
13.3.2 Various Examples of Homophily Combination Functions
13.4 Example Simulations for the Coevolution of Social Contagion and Bonding
13.4.1 Simulations for Varying Homophily Modulation Factor
13.4.2 Exploring the Birth of an Extra Cluster
13.5 Relevant Properties of Combination Functions for Bonding by Homophily and for Social
Contagion
13.5.1 Relevant Properties of Combination Functions for Bonding by Homophily
13.5.2 Relevant Properties of Combination Functions for Social Contagion
13.6 Relating Adaptive Network Structure to Emerging Bonding Behaviour
13.6.1 Relating Structure and Emerging Behaviour Independent of Social Contagion
Properties
13.6.2 Relating Structure and Emerging Behaviour for some Social Contagion Properties
13.7 Characterising Emerging Behaviour in Terms of Strongly Connected Components
13.8 Overview of Other Simulation Experiments
13.9 Discussion
14 Relating a Reified Adaptive Network’s Structure to its Emerging Behaviour for Hebbian learning
14.1 Introduction
14.2 Temporal-Causal Networks and Network Reification
14.3 Reified Adaptive Networks for Hebbian Learning
14.3.1 Reification States for Hebbian Learning and their Hebbian
Learning Combination Functions
14.3.2 An Example Reified Network Model with Multiple Hebbian
Learning Reification States
14.4 Relevant Properties of Hebbian Learning Functions and the Implied Emerging Behaviour
14.4.1 Relevant Properties of Hebbian Learning Combination Functions
14.4.2 Functional Relation for the Equilibrium Value of a Hebbian Learning Reification State
14.5 Variable Separation for Hebbian Learning Combination Functions and
the Implied Emerging Behaviour
14.5.1 Hebbian Learning Combination Functions with Variable Separation
14.5.2 Functional Relation for the Equilibrium Value of a Hebbian Learning Reification State:
the Variable Separation Case
14.6 Implications for Different Classes of Hebbian Learning Functions
14.6.1 Hebbian Learning Functions with Variable Separation and Linear Connection Factor
14.6.2 Hebbian Learning Functions with Variable Separation and Quadratic Connection
Factor
14.7 Discussion
Part VII Finalising
15 Mathematical Details of Specific Difference and Differential Equations and Mathematical Analysis of
Emerging Network Behaviour
15.1 Introduction
15.2 Two Different Formulations of Hebbian Learning are Equivalent
15.3 Numerical Representation for an Example Reified Network Model
15.4 The Difference Equations for Combined Hebbian Learning and State-Connection Modulation
15.5 Difference and Differential Equations for Connection Weight Reification States
15.6 Emerging Behaviour for Classes of Combination Functions and Types of Network
Connectivity
15.7 Using Strongly Connected Components to Explore Emerging Behaviour for a Class of
Combination Functions for any Type of Network Connectivity
15.8 Analysis of Emerging Behaviour for Classes of Homophily Functions
15.9 Analysis of Emerging Behaviour for Classes of Hebbian Learning Functions
16 Using Network Reification for Adaptive Networks: Discussion
16.1 Adaptive Networks and Network Reification
16.2 Conceptual Representations of Reified Networks: 3D Pictures and Role Matrices
16.3 The Universal Combination Function, and the Universal Difference and Differential
Equation
16.4 Analysis of How Emerging Reified Network Behaviour Relates to the Reified
Network’s Structure
16.5 Network-Oriented Modeling Based on Reified Networks
16.6 Relations to Longstanding Themes in AI and Beyond