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Modeling Higher-Order Adaptivity of a Network by Multilevel Network Reification

March 2020
Network Science 8(S1):S110-S144

DOI:10.1017/nws.2019.56

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CC BY 4.0

Authors:

Vrije Universiteit Amsterdam

For related videos, see the YouTube channel on Self-Modeling Networks here: https://www.youtube.com/channel/UCCO3i4_Fwi22cEqL8M_PgeA. In network models for real-world domains often network adaptation has to be addressed by incorporating certain network adaptation principles. In some cases, also higher-order adaptation occurs: the adaptation principles themselves also change over time. To model such multilevel adaptation processes it is useful to have some generic architecture. Such an architecture should describe and distinguish the dynamics within the network (base level), but also the dynamics of the network itself by certain adaptation principles (first-order adaptation level), and also the adaptation of these adaptation principles (second-order adaptation level), and maybe still more levels of higher-order adaptation. This paper introduces a multilevel network architecture for this, based on the notion network reification. Reification of a network occurs when a base network is extended by adding explicit states representing the characteristics of the structure of the base network. It will be shown how this construction can be used to explicitly represent network adaptation principles within a network. When the reified network is itself also reified, al-so second-order adaptation principles can be explicitly represented. The multilevel network reification construction introduced here is illustrated for an adaptive adaptation principle from Social Science for bonding based on homophily. This first-order adaptation principle describes how connections are changing, whereas this first-order adaptation principle itself changes over time by a second-order adaptation principle. As a second illustration, it is shown how plasticity and metaplasticity from Cognitive Neuroscience can be modeled.

Network reification for a temporal-causal network within the upper, blue plane reification states H Y for the speed factor of base state Y, C j,Y for the combination functions of Y, and W Xi,Y for the weights of the connections from X i to Y, and P i,j,Y for the combination function parameters. The downward connections from these reification states to state Y in the base network (in the lower, pink plane) indicate their causal effect on Y.

…

Scenario 1: Uppergraph: The adaptive weights W X1,Xi of outgoing connections from X 1 over time, with the thick pink line showing the average weight of them. This average weight initially is 0.344, which is below the desired value 0.6 of the norm ν TPW X 1 ,X , but finally it approximates the value 0.6 of this norm. But before that is achieved, over-controlling reactions cause strong fluctuations of having too intensive connections until time point 300, and again too low intensively of connections between time point 400 and time point 750. Lower graph: The adaptive tipping points TP WX 1 ,X j over time. Initially the tipping point values were much too high, which explains why in the first phase (until time 400) too many connections were strengthened. After the short initial phase the tipping point values have been adapted so that they became very low so that the connections were weakened, and finally they reached some equilibrium values.

…

Scenario 1: Resulting connection weights W X1,Xi at time 1750 compared to their initial values. As by the homophily principle the similarity of state values has an important effect on the dynamics of these connection weights, it can be seen that there is not much correlation between initial and final values of the weights.

…

Scenario 2: Upper graph: The adaptive weights of outgoing connections from X 1 over time, with the thick pink line showing the average weight for X 1 . Compared to Scenario 1, this time it turns out more difficult to reach an equilibrium. Lower graph: The adaptive tipping points TP WX 1 ,X j over time. The tipping point values react in a rather sensitive manner to the fluctuations shown in the upper graph.

…

Scenario 3: Upper graph: The adaptive weights of outgoing connections from X 1 over time, with the thick pink line showing the average weight for X 1 . Here, after time 750 all connection weights become 0 or 1. Middle graph: The adaptive tipping points TP WX 1 ,X j over time. Due to the very low connection weights between 500 and 750 (see upper graph) the tipping point values show a strong temporary increase to enable strengthening of connections. Lower graph: Average connection weights for each of X 1 to X 10 and overall average of all connections over time.

…

Figures - uploaded by Jan Treur

Content may be subject to copyright.

Content uploaded by Jan Treur

Content may be subject to copyright.

Network Science (2020), 1–35

doi:10.1017/nws.2019.56

ORIGINAL ARTICLE

Modeling higher order adaptivity of a network

by multilevel network reiﬁcation

Jan Treur

Social AI Group, Vrije Universiteit Amsterdam, Amsterdam, Netherlands (email: j.treur@vu.nl)

Action Editor: Hocine Cheriﬁ

Abstract

In network models for real-world domains, often network adaptation has to be addressed by incorporating

certain network adaptation principles. In some cases, also higher order adaptation occurs: the adaptation

principles themselves also change over time. To model such multilevel adaptation processes, it is useful to

have some generic architecture. Such an architecture should describe and distinguish the dynamics within

the network (base level), but also the dynamics of the network itself by certain adaptation principles (ﬁrst-

order adaptation level), and also the adaptation of these adaptation principles (second-order adaptation

level), and may be still more levels of higher order adaptation. This paper introduces a multilevel network

architecture for this, based on the notion network reiﬁcation. Reiﬁcation of a network occurs when a base

network is extended by adding explicit states representing the characteristics of the structure of the base

network. It will be shown how this construction can be used to explicitly represent network adaptation

principles within a network. When the reiﬁed network is itself also reiﬁed, also second-order adaptation

principles can be explicitly represented. The multilevel network reiﬁcation construction introduced here

is illustrated for an adaptive adaptation principle from social science for bonding based on homophily and

one for metaplasticity in Cognitive Neuroscience.

Keywords: network reiﬁcation; higher order network adaptation; adaptive social networks; plasticity and metaplasticity

1. Introduction

Within the complex dynamical systems area, adaptive behavior is an interesting and quite relevant

challenge, addressed in various ways, see, for example, Helbing et al. (2015) and Perc & Szolnoki

(2010). In particular for network-oriented dynamic modeling approaches, network models for

real-world domains often show some form of network adaptation based on certain network adap-

tation principles. Such principles describe how certain characteristics of the network structure

change over time, for example, the connection weights in mental networks with Hebbian learning

(Hebb, 1949) or in social networks with bonding based on homophily, for example, Byrne (1986),

McPherson et al. (2001), Pearson et al. (2006), Sharpanskykh & Treur (2014). Sometimes higher

order adaptation also occurs in the sense that the adaptation principles for a network themselves

also change over time. For example, plasticity in mental networks as described, for example, by

Hebbian learning is not a constant feature but usually varies over time, according to what in cog-

nitive neuroscience has been called metaplasticity, for example, Abraham & Bear (1996), Magerl

et al. (2018), Parsons (2018), Schmidt et al. (2013), Sehgal et al. (2013), Zelcer et al. (2006). To

model such multilevel network adaptation processes in a principled manner, it is useful to have

some generic architecture. Such an architecture should be able to distinguish and describe

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reproduction in any medium, provided the original work is properly cited.

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2J.Treur

1. the dynamics within the base network;

2. the dynamics of the base network structure by network adaptation principles (ﬁrst-order

adaptation);

3. the adaptation of these adaptation principles (second-order adaptation);

4. and maybe still more levels of higher order adaptation.

In the current paper, it is shown how such distinctions indeed can be made within a network-

oriented modeling framework using the notion of reiﬁed network architecture.

Reiﬁcation is known from different scientiﬁc areas. It literally means representing something

abstract as a material or concrete thing, or making something abstract more concrete or real

(Merriam Webster and Oxford dictionaries). Reiﬁcation has been shown to provide advantages

in modeling and programming languages in other areas of AI and computer science, for example,

Bowen & Kowalski (1982), Demers & Malenfant (1995), Galton (2006), Hofstadter (1979), Sterling

&Shapiro(1996), Sterling & Beer (1989), Weyhrauch (1980). Such advantages concern, for exam-

ple, enhanced expressive power and more support for modeling adaptivity. Network reiﬁcation

provides similar advantages. In Treur (2018a), it has been shown how network reiﬁcation can be

used to explicitly represent adaptation principles for networks in a transparent and uniﬁed man-

ner. Examples of such adaptation principles are, among others, principles for Hebbian learning

(to model plasticity in the brain) and for bonding based on homophily (to model adaptive social

networks). Using network reiﬁcation, adaptive mental networks and adaptive social networks can

be addressed well, as shown in Treur (2018a).

Including reiﬁcation states for the characteristics of the base network structure (connection

weights, speed factors, and combination functions) in the extended network is one step. The next

step is deﬁning proper temporal–causal relations for them and relations with the other states.

Then, a reiﬁed network is obtained that explicitly represents the characteristics of the base net-

work, and, moreover, how this base network evolves over time based on adaptation principles

that change the causal network relations. In Treur (2018a), it was shown how this can be used for

a variety of adaptation principles known from cognitive neuroscience and social science.

However, this is not the end of the story. Such reiﬁed adaptive networks form again a basic

network structure deﬁned by certain characteristics, such as learning rate or adaptation speed of

change of connections. Adaptation principles may be adaptive themselves too, according to cer-

tain second-order adaptation principles. From recent literature, it has become quite clear that in

real-world domains such characteristics can still change over time. The notion of metaplasticity

or second-order adaptation has become a focus of study in literature such as Arnold et al. (2015),

Chandra & Barkai (2018), Daimon et al. (2017), Magerl et al. (2018), Parsons (2018), Robinson

et al. (2016), Sehgal et al. (2013), Schmidt et al. (2013), Zelcer et al. (2006). This area of higher order

adaptivity is a next challenge to be addressed. To this end, network reiﬁcation can be applied again

on reiﬁed structures, thus obtaining repeated or multilevel reiﬁcation. In the current paper, such

a construction of multilevel reiﬁcation is illustrated for a network-oriented modeling approach

based on temporal–causal networks (Treur, 2016;2019a). This multilevel reiﬁcation construction

introduced here can be used to model higher order adaptivity of any level. The multilevel reiﬁca-

tion architecture has been implemented by the author in Excel (limited version) and in MATLAB

(general version). The homophily context will be used as a ﬁrst application for the social sci-

ence area and the context of plasticity and metaplasticity as a second application for the cognitive

neuroscience area.

The current paper is an extended (by more than 90%) and rewritten version of Treur (2018b).

In the discussion, the differences are discussed in more detail. In Section 2, the network-oriented

modeling approach based on temporal–causal networks is brieﬂy summarized. Section 3provides

an overview of different application domains for higher order adaptation. Next, in Section 4,the

network reiﬁcation concept is introduced, and in Section 5, the more general multilevel network

reiﬁcation construction is deﬁned. Moreover, it is shown how it can model second-order network

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Network Science 3

Tab le 1 . Conceptual and numerical representation of a temporal–causal network structure, adopted from (Treur, 2019a).

Concepts Notation Explanation

States and

connections

X,Y,X→YDescribes the nodes and links of a network structure (e.g., in graphical

or matrix format)

........................................................................................................................................................................................................................................................................

Connection weight ωX,YThe connection weight ωX,Y∈[−1, 1] represents the strength of the

causal impact of state Xon state Ywith X→Y

Aggregating

multiple impacts

cY(..)ForeachstateY,acombination function cY(..) is chosen to combine the

causal impacts of other states on state Y

Timing of the causal

effect

ηYFor each state Y,aspeed factor ηY≥0 is used to represent how fast a

state is changing upon causal impact

Concepts Numerical representation Explanation

State values over

time t

Y(t) At each time point t,eachstateY

in the model has a real number

value in [0, 1]

Single causal impact impactX,Y(t)=ωX,YX(t)Att,stateXwith connection to

state Yhas an impact on Y,using

weight ωX,Y

Aggregating

multiple impacts

aggimpactX,Y(t)=cY(impactX1,Y(t), ..., impactXk,Y(t))

=cY(ωX1,YX1(t), ..., ωXk,YXk(t))

The aggregated impact of multi-

ple states Xion Yat tis determined

using combination function cY(..)

Timing of the causal

effect

Y(t+t)=Y(t)+ηY[aggimpactX,Y(t)−Y(t)]t

=Y(t)+ηY[cY(ωX1,YX1(t), ..., ωXk,YXk(t)) −Y(t)]t

The causal impact on Yis exerted

over time gradually, using speed

factor ηY

adaptivity. This is illustrated by a second-order adaptive network based on a ﬁrst-order adapta-

tion principle for bonding based on homophily and a second-order adaptation principle for the

characteristics of this ﬁrst-order adaptation principle. In Section 6, example simulations for this

multilevel network reiﬁcation example are presented. Section 7shows a mathematical analysis of

this example model. Section 8presents a complexity analysis of network reiﬁcation. In Section 9,

it is shown how the modeling approach based on reiﬁed network models can be applied to plas-

ticity and metaplasticity from cognitive neuroscience. Section 10 discusses a speciﬁcation format

for this method and an implemented modeling environment for it. Section 11 is a discussion.

2. Structure and dynamics of temporal–causal networks

The network structure of a temporal–causal network model can be described conceptually by a

graph with nodes and directed connections and a number of labels for such a graph representing

the network characteristics: connection weights ωX,Y, speed factors ηYof states Y,andcombi-

nation functions cY(..) for states Y;seeTable1, upper part, and Figure 1for an example of a

basic fragment of a network with states X1,X2,andY, and labels ωX1,Yand ωX2,Yfor connec-

tion weights, cY(..)for combination function, and ηYfor speed factor. A library with a number

of standard combination functions is available as option, but also new functions can be added. In

the lower part of Table 1, it is shown how the numerical representation of the network’s dynamics

is deﬁned in terms of the above labels (see also Treur 2016, Chapter 2). Here, X1,...,Xkare

the states that have outgoing connections to state Y. These formulas in the last row in Table 1

deﬁne the detailed dynamic semantics within a temporal–causal network. They can be used for

mathematical analysis and for simulation and can be written in differential equation format as

follows:

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4J.Treur

X2,Y

X1,Y

(..)

Figure 1. Fragment of a temporal–causal networkstructure in a labeled graph representation. The basic elements are nodes

and their connections, with for each node Ya speed factor ηYand a combination function cY(..), and for eachconnection from

Xto Ya connection weight ωX,Y.

dY(t)

dt=ηY[cY(ωX1,YX1(t), ...,ωXk,YXk(t)) −Y(t)]

Examples of combination functions are the identity id(.)for states with impact from only

one other state, the scaled sum ssumλ(..)with scaling factor λ, the scaled minimum function

sminλ(..) and maximum function smaxλ(..),andtheadvanced logistic sum combination function

alogisticσ,τ(..)with steepness σand threshold τ(see also Treur 2016, Chapter 2, Table 2.10):

id(V)=V

ssumλ(V1,...,Vk)=V1+···+Vk

sminλ(V1,...,Vk)=min(V1,...,V

smaxλ(V1,...,Vk)=max(V1,...,Vk)

alogisticσ,τ(V1, ...Vk)=1

1+e−σ(V1+...+Vk−τ)−1

1+eστ (1 +e-στ)

Note that for basic combination functions, two parameters may be considered. Examples are the

scaling factor λ, the steepness σ, and the threshold τabove. These parameters may also be included

as arguments in the function, for example, alogistic(σ,τ,V1, ...Vk)

Examples of combination functions applied in particular for adaptive networks are the

following

• Hebbian learning (see Section 9)

hebbμ(V1,V2,W)=V1V2(1−W)+μW

Here, V1and V2refer to the activation levels of two connected states and Wto their connection

weights; μis a parameter for the persistence factor.

• Simple linear homophily (see Section 5)

slhomoα,τ(V1,V2,W)=W+αW(1 −W)(τ−|V1−V2|)

Here, V1and V2refer to the activation levels of the states of two connected persons and Wto

their connection weight; τis a tipping point parameter, and αis a modulation parameter. This is

applied to model bonding based on homophily (see Sections 5and 6).

The set of already available combination functions forms a combination function library (with

currently 35 functions), which can be chosen during the design of a network model.

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Network Science 5

3. Adaptivity and higher order adaptivity of networks

Adaptivity of a system or model is always relative to a chosen basic structure. Such a structure is

deﬁned by a number of characteristics that for a nonadaptive case are assumed constant but for an

adaptive case (some of them) are assumed to change over time. In principle, any adaptive system

could also be modeled as just some type of complex dynamic system, deﬁned as a state-determined

system in the sense of Ashby (1960). However, from a conceptual viewpoint, the choice of a suit-

able basic structure often has advantages. For example, for a network structure as a basis such

characteristics concern the connections and their weights, the way in which different connections

to the same node are aggregated, and the characteristics indicating how ﬂexible or rigid nodes

are (for the speed by which nodes can change). For the case that a network structure is chosen as

a basis, adaptive means that the network structure is changing; in particular, given the concepts

used to deﬁne a network structure in Section 2, this means that some of the connection weights,

the combination functions to aggregate multiple impacts, and/or the speed factors are changing

over time. In Section 4, it will be shown how adaptation principles deﬁning an adaptive network

can be modeled in a reiﬁed network.

3.1 First-order adaptation principles

There are many well-known examples of ﬁrst-order adaptive networks, for example, related to

or inspired by adaptation principles from neuroscience, cognitive science, or social science. Just

some examples are

• neural networks equipped with (machine) learning mechanisms such as back propagation or

deep learning to adapt connection weights over time;

• mental or neural networks equipped with a Hebbian learning mechanism (Hebb, 1949)to

adapt connection weights over time;

• Social networks equipped with an adaptation mechanism for bonding based on homophily

(McPherson et al., 2001) to adapt connection weights over time.

Although the majority of the known ﬁrst-order adaptive networks consider adaptations of con-

nection weights over time, also other elements of the network structure can be considered to be

adaptive, such as

• adaptive combination functions for aggregation and activation of nodes, for example, their

threshold values to model adaptive intrinsic properties of neurons such as their excitabilty

(e.g., Chandra & Barkai, 2018);

• adaptive speed factors of nodes to model adaptive processing speed, for example, Robinson

et al. (2016): “Adaptation accelerates with increasing stimulus exposure” (p. 2).

As several real-world examples show, adaptation principles may be adaptive themselves too,

according to certain second-order adaptation principles. This will be discussed next.

3.2 Second-order adaptation principles

Second-order adaptivity can occur in many forms and applications. From recent literature,

it has become quite clear that in real-world domains, characteristics representing adaptation

principles can still change over time, and often for good reasons. The notion of metaplasticity

or second-order adaptation has become a focus of study in the domains of cognitive neuroscience

and social sciences. Some examples are brieﬂy discussed here:

• In an adaptive mental network based on Hebbian learning, the learning speed or persistence

factor may change over time, for example, due to age or due to experiences or medicin. Such

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6J.Treur

second-order adaptation has been discussed for plasticity of the brain and in evolutionary

context, for example, see Arnold et al. (2015), Daimon et al. (2017), Robinson et al. (2016).

• In literature such as Arnold et al. (2015), Chandra & Barkai (2018), Daimon et al. (2017),

Magerl et al. (2018), Parsons (2018), Robinson et al. (2016), Sehgal et al. (2013), Schmidt

et al. (2013), Zelcer et al. (2006), various studies are reported which show how adaptation of

synapses as described, for example, by Hebbian learning can be modulated by suppressing the

adaptation process or amplifying them, thus some form of metaplasticity is described. Factors

affecting synaptic plasticity as reported are temporarily enhanced excitability of neurons, for

example, based on previous (learning) experiences or stress.

• From the social science area, in an adaptive social network based on an adaptation principle

for bonding based on homophily (McPherson et al., 2001), the similarity measure determin-

ing how similar two persons are may change over time, for example, due to age or other

varying circumstances. As an example, for somebody who is very busy or already has a lot of

connections, the requirements for being similar might become more strict.

• Also from the social science area is the second-order adaptation concept called “inhibit-

ing adaptation” described in Carley (2002;2006). The idea is that networked organizations

need to be adaptive in order to survive in a dynamic world. However, some types of circum-

stances affect the adaptivity in a negative manner, for example, frequent changes of persons or

(other) resources. Such circumstances can be considered as inhibiting the adaptation capa-

bilities of the organization. Especially, in Carley (2006), it is described in some detail how

such inhibiting of adaptation can be exploited as a strategy to attack organizations that are

considered harmful or dangerous such as terrorist networks, by creating circumstances that

indeed achieve inhibiting adaptation.

The third topic above on adaptive adaptation principles for bonding based on homophily will be

illustrated in more detail in Sections 5and 6; here, the ﬁrst-order adaptation principle, describing

adaptation of connections based on bonding by homophily, is adapted by a second-order adapta-

tion principle that takes into account how many (and how strong) connections a person already

has. For the ﬁrst two topics above, to address metaplasticity, adaptive adaptation principles for

Hebbian learning can be considered, for example, in which the adaptation speed factor (learn-

ing rate) for the ﬁrst-order adaptation principle is changing based on a second-order adaptation

principle.

3.3 Higher order adaptation principles

A reasonable question is whether also adaptation principles of third or even higher order can

make sense. Interestingly, a Google Scholar search (on February 3, 2019) on “adaptation” provides

4 million results, “second order adaptation” 360, “third order adaptation” only 14, and “fourth

order adaptation” only 3. The graphs in Figure 2show the logarithm and double logarithm of

the number of results (hits) for the different orders. The dotted line is a linear trend line with

linear formula in xand yas indicated: the double logarithm seems to approximate the pattern

best, so the number of hits is in the order of a double (negative) exponential pattern e(35.19 e−0.8684n)

as function of the order n. This very strong pattern might suggest that adaptation of order higher

than 2 is not often considered a very useful or applicable notion, to say the least. However, at least

in Fessler et al. (2015), some interesting ideas are put forward on higher order adaptation for the

area of evolutionary adaptive processes.

For example, the S-curve in the human spine reﬂects the determinative inﬂuence of

the original function of the spine as a suspensory beam in a quadrupedal mammal, in

contrast to its current function as a load-bearing pillar: whereas the original design

functioned efﬁciently in a horizontal position, the transition to bipedality required

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Network Science 7

Figure 2. Logarithm (left graph) and double logarithm (right graph) of the number of hits of adaptivity of different orders

(vertical axis) in Google Scholar vs the order of adaptation (horizontal axis). The dotted linear trend line shows that the

double logarithm of the number of hits has an almost linear dependence of the order of adaptivity.

the introduction of bends in the spine to position weight over the pelvis (Lovejoy,

2005). The resulting conﬁguration makes humans prone to lower back injury, illus-

trating how path dependence can both set the stage for kludgy designs and constrain

their optimality. Moreover, the combination of bipedality and pressures favoring

large brain size in humans exacerbates a conﬂict between the biomechanics of loco-

motion (favoring a narrow pelvis) and the need to accommodate a large infant skull

during parturition. This increases the importance of higher-order adaptations such

as relaxin, a hormone that loosens ligaments during pregnancy, allowing the pelvic

bones to separate. (Fessler et al., 2015)

Here, it is suggested that the following types of adaptation can be considered for the human

spine:

• First-order adaptation:

For quadrupedal mammals, a straight horizontal spine is an advantage.

• Second-order adaptation:

Transition to bipedality requires introduction of bends in the spine to position weight over

the pelvis, making humans prone to lower back injury.

Similarly, the following types of adaptation can be considered for the human pelvis:

• First-order adaptation:

Bipedality favors a narrow pelvis.

• Second-order adaptation:

Larger brain size needs a wider pelvis: using relaxin allowing the pelvic bones to separate

during giving birth

In another part of Fessler et al. (2015), the following is put forward:

Also of relevance here, one form of disgust, pathogen disgust, functions in part as

a third-order adaptation, as disease-avoidance responses are upregulated in a man-

ner that compensates for the increases in vulnerability to pathogens that accompany

pregnancy and preparation for implantation—changes that are themselves a second-

order adaptation addressing the conﬂict between maternal immune defenses and the

parasitic behavior of the half-foreign conceptus (Fessler et al., 2005, Jones et al., 2005,

Fleischman & Fessler, 2011). (Fessler et al., 2015)

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8J.Treur

Here, it is suggested that the following types of adaptation can be considered for the human way

of handling pathogens in the external world, in particular during the ﬁrst trimester of pregnancy:

• First-order adaptation:

Immune system attacks pathogens.

• Second-order adaptation:

During pregnancy, the immune system is less active as foreign material has to enter.

• Third-order adaptation:

During pregnancy, disgust makes that potential pathogens are avoided so that less risks are

taken for entering of pathogens while the immune system is low functioning.

It could be argued that this domain of evolutionary development is not exactly comparable to

the types of application domains considered in the current paper. For example, in evolutionary

processes, no organisms in their daily life are considered but species on an evolutionary relevant

long-term timescale. However, at least in a metaphorical sense, this evolutionary domain might

provide an interesting source of inspiration.

4. Addressing network adaptation by network reiﬁcation

Network reiﬁcation is a construction principle by which a base network is extended by extra states

that represent the characteristics of the network structure. The new states added represent the

characteristics for the network structure. They are what are called reiﬁcation states for the char-

acteristics, in other words, the characteristics are reiﬁed by these states. More speciﬁcally, these

reiﬁcation states represent the labels for connection weights,combination functions,andspeed fac-

tors showninTable1. For connection weights ωXi,Yfor the connection from state Xito state Y

and speed factors ηYfor state Y, their reiﬁcation states WXi,Yand HYrepresent the value of them,

and the vector CY=(C1,Y,C2,Y,... .) represents the chosen combination functions for state Y.

In Figure 3, the reiﬁcation states are depicted in the upper (blue) plane, whereas the states of the

base network are in the lower (pink) plane.

Causal relations for these reiﬁed characteristics of a network can be deﬁned within the reiﬁed

network: incoming connections affecting them and outgoing connections from them to base net-

work states. Such connections are the way in which adaptation principles are explicitly represented

within the (reiﬁed) network (Treur, 2018a). The downward pink arrows in Figure 3deﬁne how

the reiﬁcation states contribute to an aggregated impact on the related base network state.

These downward connections in Figure 3and the combination functions for the base states are

deﬁned in a generic manner. The general pattern is that each of the reiﬁcation states WXi,Y,HY,

CY,andPYfor connection weights, speed factors, and combination functions has a causal

connection to state Yin the base network, as they all affect Yin their own way. All depicted

connections get weight 1, and in the reiﬁed network, the speed factors of the base states are set at

1 too. For the base states, new combination functions are needed that will be deﬁned below. The

different components

C1,Y,C1,Y, ...

for CYare explained as follows. During modeling, a sequence of basic combination functions

bcf1(...),...,bcf

m(...)

is chosen from the function library discussed in Section 2(if desired also new functions can be

added to that library), to be used in the speciﬁc application addressed. For example,

bcf1(...)=ssumλ(..)

bcf2(...)=alogisticσ,τ(..)

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Network Science 9

First reification level

Base level

C1,Y

WXi,Y

P1,1,Y

C2,Y

P1,2,Y

Figure 3. Network reiﬁcation for a temporal–causal network within the upper, blue plane reiﬁcation states HYfor the speed

factor of base state Y,Cj,Yfor the combination functions of Y, and WXi,Yfor the weights of the connections from Xito Y, and

Pi,j,Yfor the combination function parameters. The downward connections from these reiﬁcationstates to state Yin the base

network (in the lower, pink plane) indicate their causal effect on Y.

Figure 4. Multilevel reiﬁed network model picture of a second-order adaptive social network based on homophily. At the

ﬁrst reiﬁcation level (middle, blue plane), the reiﬁcation states WX1,Yand WX2,Yrepresent the adaptive connection weights

for Y; they are changing based on a homophily principle. At the second reiﬁcation level (upper, purple plane), the reiﬁed

tipping point states TPWX1,Yand TPWX2,Yrepresent the adaptive tipping point values for the connection adaptation based on

homophily, and HWX1,Yand HWX2,Yrepresent the connection weight adaptation speed factors.

Each basic combination function bcfj(...)is assumed to have two parameters for each state:

π1,j,Yπ2,j,Y.Thesecombination function parameters π1,1,Y,π2,1,Y,...π1,m,Y,π2,m,Yin the m

selected combination functions can also be explicitly represented by parameter value reiﬁcation

states

P1,1,Y,P2,1,Y,...P1,m,Y,P2,m,Y

so that they can also become adaptive. Their values are considered as the ﬁrst arguments in

bcfm(..) and also included as arguments in cY(...). Note that for applications, often more

informative names are used for these parameters πi,j,Yand their reiﬁcation states Pi,j,Y,for

example, reiﬁcation state TPWXi,Yin Figure 4for the tipping point parameter τfor bonding by

homophily and MWsrss,psain Figure 10 for the persistence parameter μfor Hebbian learning.

Inthebasenetwork,foreachstateY,combination function weights γ

i,Yare assumed: numbers

that may change over time such that the combination function cY(..)is expressed by:

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10 J. Treur

Box 1. Derivation of the universal combination function for a reiﬁed network; see also Treur (2018a).

cY(t,π1,1,Y,π2,1,Y, ..., π1,m,Y,π2,m,Y,V1, ..., Vk)=

γ1,Y(t)bcf

1(π1,1,Y,π2,1,Y,V1, ..., Vk)+... +γm,Y(t)bcf

m(π1,m,Y,π2,m,Y,V1,...,Vk)

γ1,Y(t)+... +γm,Y(t)

This describes that for Ya weighted average of basic combination functions is used. But if exactly

one of the γi,Y(t) is nonzero, just one basic combination function is selected for cY(..).This

approach makes it possible, for example, to gradually switch from one combination function bcfi

to another one bcfjover time by decreasing the value of γi,Y(t) and increasing the value of γi,Y(t)

The basic combination function weights γi,Yare represented by the reiﬁcation states Ci,Y.InTreur

(2018a), a universal combination function c∗

Y(..) has been found for any base state Yin the reiﬁed

network (for a derivation, see Box 1):

c∗

Y(H,C1, ... .. , Cm,P1,1,P2,1 , ... , P1,m,P2,m,W1,...,Wk,V1, ..., Vk,V)

=HC1bcf1(P1,1,P2,1 ,W1V1, .., WkVk)+..... +Cmbcfm(P1,m,P2,m,W1V1,..,WkVk)

C1+..... +Cm

+(1−H)V

=HC1bcf1(P1,1,P2,1 ,W1V1, .., WkVk)+..... +Cmbcfm(P1,m,P2,m,W1V1, .., WkVk)

C1+..... +Cm

−V+V

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Network Science 11

where

•Hrefers to the speed factor reiﬁcation HY(t)

•Pi,jto the parameter reiﬁcation value Pi,j,Y(t) of parameter i=1, 2 of basic combination

function j=1, ...,m

•Cito the combination function weight reiﬁcation Ci,Y(t)

•Vito the state value Xi(t)of base state Xi

•Wito the connection weight reiﬁcation WXi,Y(t)

•Vto the state value Y(t) of base state Y.

This combination function c∗

Y(..) makes that the dynamics of any base state Ywithin the reiﬁed

network is described by the following universal difference equation:

Y(t+t)=Y(t)+c∗

Y(HY(t), C1,Y(t), ....., Cm,Y(t), P1,1,Y(t),P2,1,Y(t), ...,

P1,m,Y(t),P2,m,Y(t),WX1,Y(t)....,

WXk,Y(t),X1(t), ..., Xk(t),Y(t)) −Y(t)]t

which can be rewritten into equivalent forms such as

Y(t+t)=Y(t)+

[HY(t)C1,Y(t)bcf

1P1,1,Y(t), P2,1,Y(t), WX1,Y(t)X1(t), ...., WXk,Y(t)Xk(t)+..... +Cm,Y(t)bcf

m(P1,m,Y(t), P2,m,Y(t), WX1,Y(t)X1(t), ...., WXk,Y(t)Xk(t))

C1,Y(t)+..... +Cm,Y(t)

+(1 −HY(t))Y(t)−Y(t)]t

Y(t+t)=Y(t)+

[HY(t)C1,Y(t)bcf

1P1,1,Y(t), P2,1,Y(t), WX1,Y(t)X1(t), ...., WXk,Y(t)Xk(t)+..... +Cm,Y(t)bcf

m(P1,m,Y(t), P2,m,Y(t), WX1,Y(t)X1(t), ...., WXk,Y(t)Xk(t))

C1,Y(t)+..... +Cm,Y(t)

−HY(t))Y(t)−Y(t)]t

Y(t+t)=Y(t)+

HY(t)[ C1,Y(t)bcf

1P1,1,Y(t), P2,1,Y(t), WX1,Y(t)X1(t), ...., WXk,Y(t)Xk(t)+..... +Cm,Y(t)bcf

m(P1,m,Y(t), P2,m,Y(t), WX1,Y(t)X1(t), ...., WXk,Y(t)Xk(t))

C1,Y(t)+..... +Cm,Y(t)

−Y(t)]t

Note that structures added by the reiﬁcation process are not reiﬁed themselves. However, the

structure of the reiﬁed network can also be reiﬁed by a next step: providing what is then called a

second-order reiﬁcation. In next section, it is explored how such second-order reiﬁcation can be

done and how it can be used to model second-order adaptation: adaptive adaptation principles.

5. Multilevel network reiﬁcation

In this section, the multilevel reiﬁcation architecture is introduced that allows modeling of net-

works with arbitrary orders of adaptation. In this architecture, the base network has its own

internal dynamics, but it also evolves through one or more adaptation principles (called ﬁrst-

order adaptation principles). Moreover, these ﬁrst-order adaptation principles themselves can

change based on other adaptation principles (called second-order adaptation principles). So the

architecture offers nreiﬁcation levels for an arbitrary nwhere on reiﬁcation level iadaptation

principles are deﬁned for ith-order adaptation.

In the current section, an example inspired by social science will be used as an illustration

of this for n=2, so level 0 provides the base network and then reiﬁcation level 1 and reiﬁca-

tion level 2 are used to represent ﬁrst- and second-order adaptation principles, respectively. This

example describes the way in which connections between two persons change over time based

on similarity between the persons. This concerns the bonding-by-homophily adaptation principle

as a ﬁrst-order adaptation principle, represented at reiﬁcation level 1 of the multilevel reiﬁcation

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12 J. Treur

architecture. In this adaptation principle, there is an important role for the homophily similarity

tipping point τ. This indicates the value such that

• when the dissimilarity between two persons is less than this value, their connection will

become stronger, and

• when the difference is more, their connection will become weaker.

Such tipping points are usually considered constant, but it may be more realistic when they are

assumed adaptive over time. Here, an adaptive form of the bonding-by-homophily adaptation

principle is used where (for good reasons) the tipping points change over time by a second-order

adaptation principle represented at level 2 in the multilevel reiﬁcation architecture. In addition,

also a second-order adaptation principle is included for the speed factor of the connection weight

adaptation based on the ﬁrst-order adaptation principle.

The architecture of the example multilevel reiﬁed network is shown in Figure 4. The middle

(blue) plane shows how the reiﬁcation states WXi,Yare used for ﬁrst-order reiﬁcation of the

connection weights ωXi,Y. The downward arrows show the network relations of these reiﬁcation

states WXi,Yto the states Xiand Yin the base network. Such network relations (including their

labels, such as combination functions; see below) for reiﬁcation states deﬁne the ﬁrst-order adap-

tation principle based on homophily. Note that for this example, speed factors and combination

functions for the base level states Yare considered constant, so speed factor reiﬁcation states HY

and combination function reiﬁcation states Ci,Yfor the base network states Yhave been left out

of the middle plane.

On top of the ﬁrst-order reiﬁed network, a second reiﬁcation level has been added (the upper,

purple plane), in order to get a second-order reiﬁed network. Here, the following reiﬁcation states

are added:

• reiﬁcation states TPWXi,Yfor the similarity tipping point parameter of the homophily

adaptation principle for the connection weight reiﬁed by state WXi,Y

• reiﬁcation states HWXi,Yfor the speed factor characteristic of the homophily adaptation

principle.

Also, for these reiﬁcation states (upward and downward), connections have been added.

These connections (together with the relevant combination functions discussed below) deﬁne

the second-order adaptation principle based on them. After having deﬁned the overall architec-

ture, the combination functions for the new network are deﬁned. Note that the speed factors and

connection weights in this new network are kept simple: all of them are set at 1.

5.1 Base-level combination functions (the lower plane)

For this level, the combination functions are

c∗

Y(W1, ..., Wk,V1, ..., Vk)=γ1,Ybcf1(W1V1, .., WkVk)+..... +γk,Ybcfm(W1V1, .., WkVk)

γ1,Y+..... +γk,Y

where

•Vistands for Xi(t);

•Wistands for WXi,Y(t).

See also Treur (2018a). Here, for this example, the coefﬁcients γi,Y(t) are assumed constant. For

example, if cY(..)is chosen as the advanced logistic sum function alogisticσ,τ(...),whichisthe

second in the row bcf1(..), bcf2(..)..., then γ1,Y=0, γ2,Y=1, and this obtains

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Network Science 13

c∗

Y(W1, ..., Wk,V1, ..., Vk)=bcf2(W1V1, .., WkVk)=alogisticσ,τ(W1V1, .., WkVk)

=1

1+e−σ(W1V1+...+WkV−τ

k)−1

1+eστ)(1 +e−στ )

5.2 First reiﬁcation level combination functions (the middle plane)

At this ﬁrst reiﬁcation level, the combination function for the homophily adaptation principle at

the (the middle plane) is needed (see Section 2above or Treur, 2016, Chapter 11, Section 11.7):

c∗

WXi,Y(H,V1,V2,T,W)=H(W+αWXi,YW(1 −W)(T−|V1−V2|)) +(1 −H)W

where

•Hrefers to the speed factor reiﬁcation HWXi,Y(t)forWXi,Y

•V1to Xi(t),V2for Y(t)

•Tto homophily tipping point reiﬁcation TPWXi,Y(t)forWXi,Y

•Wto connection weight reiﬁcation WXi,Y(t)

•αWXi,Yis a homophily modulation factor

This combination function (together with connection weights and speed factor 1) deﬁnes the

following difference equation for WXi,Y(see Section 2,Table1):

WXi,Y(t+t)=WXi,Y(t)+

[HWXi,Y(t)(WXi,Y(t)+αWXi,Y(t)(1 −WXi,Y(t))(TPWXi,Y(t)−|Xi(t)−Y(t)|))

+(1 −HWXi,Y(t))WXi,Y(t)−WXi,Y(t)]t

This indeed makes that an increase in connection weight will take place (in a linear fashion) when

the difference in states |Xi(t)−Y(t)|of two persons is less than the tipping point TPWXi,Y(t)and

decrease will take place when this difference is more.

5.3 Second reiﬁcation level combination functions (the upper plane)

At this level, it is deﬁned how the tipping points should be adapted to circumstances. The principle

is used that the tipping point of a person will become higher if the person lacks strong connec-

tions (the person becomes less strict) and will become lower if the person already has strong

connections (the person becomes more strict). This is handled using an average norm weight ν

for connections. This can be considered to relate to the amount of time or energy available for

social contacts. So the effect is

• if the connections of a person are on average stronger than ν, downward regulation takes

place: the tipping point will become lower;

• when the connections of this person are on average weaker than ν, upward regulation takes

place: the tipping point will become higher.

This is expressed in the following combination function for the second reiﬁcation level:

c∗

TPWY,Xi

(W1, ..., Wk,T)=T+αTPWY,XiT(1 −T)(νTPWY,Xi−(W1+... +Wk)/k)

where

•Trefers to the homophily tipping point reiﬁcation value TPWY,Xi(t)forWY,Xi;

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14 J. Treur

•Wjrefers to connection weight reiﬁcation value WY,Xj(t);

•αTPWY,Xiis a modulation factor for the tipping point;

•νTPWY,Xiis a norm for Yfor average connection weight for WY,X1to WY,Xk

Together with connection weights and speed factor 1, this combination function deﬁnes the

following difference equation for TPWY,Xi(t) (see Section 2,Table1):

TPWY,Xi(t+t)=TPWY,Xi(t)+

[TPWY,Xi(t)+αTPWY,XiTPWY,Xi(t)(1 −TPWY,Xi(t))(νTPWY,Xi

−(WY,X1(t)+... +WY,Xk(t))/k)−TPWY,Xi(t)]t

As an alternative, note that as a slightly different variant the division by kcan be left out. Then,

the norm does not concern the average but the cumulative connection weights.

For the opposite connections, similarly, the following combination function can be used:

c∗

TPWXi,Y(W1, ..., Wk,T)=T+αTPWXi,YT(1 −T)(νTPWXi,Y−(W1+... +Wk)/k)

where

•Trefers to the WXi,Yhomophily tipping point reiﬁcation value TPWXi,Y(t);

•Wjrefers to the connection weight reiﬁcation value WXj,Y(t);

•αTPWXi,Yis a modulation factor for TPWXi,Y;

•νTPWXi,Yis a norm for average (incoming) connection weights for Y.

For the adaptive connection adaptation speed factor, the following combination function can be

considered making use of a similar mechanism using a norm for connection weights.

c∗

HWY,Xi

(W1, ..., Wk,S)=S+βHWY,Xi

S(1 −S)(νHWY,Xi−(W1+... +Wk)/k)

where

•Srefers to WY,Xispeed factor reiﬁcation value HWY,Xi(t);

•Wjto connection weight reiﬁcation value WY,Xj(t);

•βHWY,Xiis a modulation factor for HWY,Xi;

•νHWY,Xiis a norm for average of (outgoing) connection weights for Y.

This combination function deﬁnes the following difference equation for HWY,Xi:

HWY,Xi(t+t)=HWY,Xi(t)

+HWY,Xi(t)+βHWY,Xi

HWY,Xi(t)1−HWY,Xi(t)×νHWY,Xi−WY,X1(t)+... +WY,Xk(t)

k−HWY,Xi(t)t

Also, here, an opposite variant is possible:

c∗

HWXi,Y(W1, ..., Wk,S)=S+βHWXi,YS(1 −S)(νHWXi,Y−(W1+... +Wk)/k)

where

•Sis used for WXi,Yspeed factor reiﬁcation value HWXi,Y(t);

•Wjfor connection weight reiﬁcation value WXj,Y(t);

•βHWXi,Yis a modulation factor for HWXi,Y;

•νHWXi,Yis a norm for average of (incoming) connection weights for Y.

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Network Science 15

Tab le 2 . Main network characteristics for Scenario 1/Scenario 2.

Base level First reiﬁcation level Second reiﬁcation level

Contagion alogistic 1 Homophily modulation 1 Tipping point speed factors 1

steepness σXifor Xifactor αWX1,Xifor WX1,XiηTPWX1,Xi

for TPWX1,Xi

Contagion alogistic 1.5 Connection weight speed 1 Tipping point modulation 0.1/0.9

threshold τXifor Xifactor ηWX1,Xifor WX1,Xifactors αTPWX1,Xifor TPWX1,Xi

Speed factor ηXifor 0.5 Tipping point connection 0.6

base state Xinorms νTPWX1,Xifor TPWX1,Xi

Tab le 3 . Scenarios 1 and 2: Initial values for connection weights and tipping points.

Connections X1X2X3X4X5X6X7X8X9X10

X10.5 0.3 0.1 0.2 0.6 0.5 0.2 0.3 0.4

X20.5 0.6 0.3 0.4 0.7 0.7 0.9 0.5

X30.3 0.6 0.7 0.4 0.4 0.6 0.8

X40.6 0.4 0.6 0.4 0.6 0.7 0.8

X50.2 0.5 0.7 0.6 0.4 0.9

X60.6 0.6 0.7 0.5 0.7 0.7 0.5 0.7

X70.2 0.8 0.6 0.7 0.6 0.7 0.7

X80.6 0.5 0.6 0.5 0.4 0.5

X90.6 0.6 0.7 0.4 0.7 0.6

X10 0.6 0.7 0.7 0.4 0.6 0.8

TPWX1,X2TPWX1,X3TPWX1,X4TPWX1,X5TPWX1,X6TPWX1,X7TPWX1,X8TPWX1,X9TPWX1,X10

TP(0)

WX1,Xi0.4 0.35 0.5 0.65 0.2 0.3 0.25 0.55 0.6

Tab le 4 . Scenario 3: Main network characteristics.

Base level First reiﬁcation level Second reiﬁcation level

Contagion alogistic

steepness σXifor Xi

0.8 Homophily modulation

factor αWXj,Xifor WXj,Xi

1 Tipping point speed factors

ηTPWXj,Xi

for TPWXj,Xi

0.5

Contagion alogistic

threshold τXifor Xi

0.15 Connection weight speed

factor ηWXj,Xifor WXj,Xi

1 Tipping point modulation

factors αTPWXj,Xifor TPWXj,Xi

0.4

Speed factor ηXifor

base state Xi

0.5 Tipping point connection

norms νTPWXj,Xifor TPWXj,Xi

0.4

6. Simulation scenarios

The example simulations concern adaptive social network scenarios with 10 persons X1to X10.

The ﬁrst two scenarios address a case in which only the outgoing connections of X1are adaptive

and the other connection weights are kept constant. For all simulations, t=1 was used, and the

focus in all three scenarios was on the homophily adaptation with constant connection weight

speed factor HWXj,Xi=ηWXj,Xi=1. In Table 2, the main network characteristics for Scenarios 1

and 2 can be found, and in Table 4for Scenario 3. In Table 3, the initial values for connection

weights and tipping points are shown for Scenarios 1 and 2.

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16 J. Treur

Figure 5. Scenario 1: Uppergraph: The adaptive weights WX1,Xiof outgoing connections from X1over time, with the thick

pink line showing the average weight of them. This average weight initially is 0.344, which is below the desired value 0.6

of the norm νTPWX1,X, but ﬁnally it approximates the value 0.6 of this norm. But before that is achieved, over-controlling

reactions cause strong ﬂuctuations of having too intensive connections until time point 300, and again too low intensively of

connections between time point 400 and time point 750. Lower graph: The adaptive tipping points TPWX1,Xjover time. Initially

the tipping point values were much too high, which explains why in the ﬁrst phase (until time 400) too many connections

were strengthened. After the short initial phase the tipping point values have been adapted so that they became very low so

that the connections were weakened, and ﬁnally they reached some equilibrium values.

6.1 Scenario 1: Adaptive connections from X1;αTWX1,Xi

=0.1

For this scenario, the initial values for connection weights and tipping points can be found in

Table 3. The average of the initial values of WX1,Xiis 0.344, which is below the norm νTPWX1,Xi

which is 0.6. The example simulation for this scenario shown in Figures 5and 7may look a bit

chaotic where some connections seem to meander between high and low. However, in this sce-

nario, it can be seen that the average connection weight, indicated by the thick pink line converges

to 0.60145 (at time point 1750), which is close to 0.6, which was chosen as the norm νTPWX1,Xifor

the average connection weight. So at least this convergence of the average connection weight to

νTPWX1,Ximakes sense. As can be seen in Figure 5, there is some variation of the connection weights

around the average connection weight 0.60145 at time 1750. Note that the connection weights at

time 1750 do not correlate with the initial connections weights; they are determined by the simi-

larity in states via the homophily principle. With all of these nine persons, X1initially developed

very strong connections (above 0.97) around time 50, but that turned out too much. Therefore, six

of the nine were reduced between time 100 and 500, while three stayed high all the time: WX1,X3,

WX1,X5,andWX1,X9. Two of these six stayed very low: WX1,X8and WX1,X10 .

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Network Science 17

Figure 6. Scenario 1: Resulting connection weights

WX1,Xiat time 1750 compared to their initial values.

As by the homophily principle the similarity of state

values has an important effect on the dynamics of

these connection weights, it can be seen that there is

not much correlation between initial and ﬁnal values

of the weights.

Figure 7. Scenario 2: Upper graph: The adaptive weights of outgoing connections from X1over time, with the thick pink

line showing the average weight for X1. Compared to Scenario 1, this time it turns out more difﬁcult to reach an equilibrium.

Lower graph: The adaptive tipping points TPWX1,Xjover time. The tipping point values react in a rather sensitive manner to

the ﬂuctuations shown in the upper graph.

As with six connections very low, this made the average of connections too low, from these

six, three were increased after time 750, and a fourth one after time 1000. Eventually, two of them,

WX1,X2and WX1,X7, are around 0.6, one, WX1,X4, is around 0.8, and one, WX1,X6, is around 0.35. So

what has emerged is that the person eventually has developed and kept three very good contacts

X3,X5,andX9, has lost two contacts X8and X10, and has kept the other four contacts with an

intermediate type of different strengths. Figure 7lower graph shows the variation in tipping point

reiﬁcation states over time.

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18 J. Treur

Tab le 5 . Scenario 3: Initial connection weights.

Connections X1X2X3X4X5X6X7X8X9X10

X10.5 0.3 0.1 0.2 0.6 0.5 0.2 0.3 0.4

X20.5 0.6 0.3 0.4 0.7 0.7 0.9 0.5

X30.3 0.6 0.7 0.7 0.4 0.4 0.6 0.8

X40.6 0.4 0.6 0.4 0.6 0.7 0.8 0.9

X50.2 0.5 0.7 0.4 0.4 0.9 0.4

X60.6 0.6 0.7 0.5 0.7 0.7 0.5 0.7

X70.2 0.8 0.6 0.7 0.6 0.7 0.7

X80.6 0.5 0.4 0.6 0.5 0.4 0.5

X90.6 0.6 0.7 0.4 0.7 0.6

X10 0.6 0.7 0.7 0.4 0.6 0.8

6.2 Scenario 2: Adaptive connections from X1,αTPWX1,Xi

=0.9

Scenario 1 shown above is actually not one of the most chaotic scenarios; some other scenarios

show a much more chaotic pattern. As an example, when for the tipping point adaptation the

much higher modulation factor αTPWX1,Xi=0.9 is chosen (instead of the 0.1 in Scenario 1; all other

values stay the same), the pattern is still more chaotic, as shown in Figure 7. Yet, on the long term,

in this case, the average connection weight moves around the set point 0.6; but notice that around

time point 1250 it seemed that the process was close to an equilibrium, but that was violated by

what happened later. Moreover, the ﬂuctuating pattern of the tipping points in Figure 7also does

not suggest that it will become stable.

6.3 Scenario 3: All connections adaptive

In Scenario 3, all connections are adaptive with main network characteristics shown in Table 4

and initial connection weight values shown in Table 5. The norm for average connection weight is

0.4 this time.

In Figure 8, the simulation results are shown for Scenario 3. As can be seen in Figure 9,even-

tually, all connection weights converge to 0 or 1. Figure 8upper graph shows in particular the

values of the connection weights from X1and their average, and Figure 8middle graph shows the

corresponding tipping points.

Figure 8shows that in the process eventually the average connection weights per person con-

verge in some unclear manner to a discrete set of values: 0.111111 (X10), 0.222222 (X5), 0.333333

(X3,X9), and 0.555555 (X1,X2,X4,X6,X7,X8), all multiples of 0.111111; the overall average ends

up in 0.433333 (recall that the norm νTPWXi,Xjfor average connection weight for each person was

0.4). Also, in other simulations, this discrete set of multiples of 0.111111 shows up. In Section 6,it

will be analyzed where these values come from.

Figure 9shows that all connection weights converge to 0 or 1. Also, this will be analyzed in

Section 6. The tipping points for all outgoing connections of X1converge to 0 (see also Figure 8)

and for all outgoing connections of the other persons they converge to 1.

7. Analysis of equilibrium values

In this section, the possible values to which certain states in the second-order reiﬁed network may

converge are analyzed.

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Network Science 19

Figure 8. Scenario 3: Upper graph: The adaptive weights of outgoing connections from X1over time, with the thick pink line

showing the average weight for X1. Here, after time 750 all connection weights become 0 or 1. Middle graph: The adaptive

tipping points TPWX1,Xjover time. Due to the very low connection weights between 500 and 750 (see upper graph) the tipping

point values show a strong temporary increase to enable strengthening of connections. Lower graph: Average connection

weights for each of X1to X10 and overall average of all connections over time.

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20 J. Treur

Figure 9. Scenario 3: All connection weights are 0 or 1 at time 1750.

7.1 Deﬁnition (stationary point and equilibrium)

A state Yhas a stationary point at tif dY(t)/dt= 0. The network is in equilibrium at tif every state

Yof the model has a stationary point at t.

Note that this Yalso applies to the reiﬁcation states. Given the differential equation for a

temporal–causal network model, a more speciﬁc criterion can be found.

7.2 Criterion for a stationary point in a temporal–causal network

Let Ybe a state with speed factor ηY>0andX1,...,Xkbe the states with outgoing connections

to state Y.Then,Yhas a stationary point at tiff cY(ωX1,YX1(t), ..., ωXk,YXk(t)) =Y(t).

This can be applied to the states at all levels in the second-order reiﬁed network, in particular

to the ﬁrst and second reiﬁcation levels. As a ﬁrst step, the above criterion applied to tipping point

reiﬁcation states at the second reiﬁcation level is as follows:

c∗

TPWXi,Xj

(W1,...,Wk,T)=T

with T=TPWXi,Xjand Wl=WXi,Xl. This provides the following equation:

T+αTPWXi,XjT(1 −T)(νTPWXi,Xj−(W1+... +Wk)/k)=T

which can be rewritten as follows:

T+αTPWXi,XjT(1 −T)(νTPWXi,Xj−(W1+... +Wk)/k)=0

Assuming that αTPWXi,Xjis nonzero, this equation has three solutions:

T=0orT=1or(W1+... +Wk)/k=νTPWXi,Xj

Similarly, the criterion can be applied to connection weights at the ﬁrst reiﬁcation level:

c∗

WXi,Xj(H,V1,V2,T,W)=W

with

H=HWXi,Xj

Vi=Xi(t)

T=TPWXi,Xj

W=WXi,Xj

This provides the following equation

HW+αWXi,XjW(1−W)(

T−|V1−V2|)+(1−H)W=W

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Network Science 21

Tab le 6 . Overview of the different solutions of the equilibrium equations for WXi,Xjand TPWXi,Xj.

HWXi,Xj(t)=0WXi,Xj(t)=0WXi,Xj(t)=1|Xi(t)−Xj(t)|=TPWXi,Xj(t)

TPWXi,Xj(t)=0HWXi,Xj(t)=0WXi,Xj(t)=0WXi,Xj(t)=1Xi(t)−Xj(t)

TPWXi,Xj(t)=0TPWXi,Xj(t)=0TPWXi,Xj(t)=0TPWXi,Xj(t)=0

TPWXi,Xj(t)=1HWXi,Xj(t)=0WXi,Xj(t)=0WXi,Xj(t)=1Xi(t)=0 and Xj(t)=1orXi=1

TPWXi,Xj(t)=1TPWXi,Xj(t)=1TPWXi,Xj(t)=1 and Xj=0

TPWXi,Xj(t)=1

mWXi,Xm/k=νTPWXi,XjHWXi,Xj(t)=0 XXXXXXX XXXXXXX |Xi(t)−Xj(t)|=TPWXi,Xj(t)

mWXi,Xm/k=νTPWXi,XjmWXi,Xm/k=νTPWXi,Xj

which can be rewritten as follows

HαWXi,XjW(1−W)(

T−|V1−V2|)+W=W

HαWXi,XjW(1−W)(

T−|V1−V2|)=0

Assuming that αWXi,Xjis nonzero, this equation has four solutions:

H=0orW=0orW=1or |V1−V2|=T

In combination with the three solutions for TWXi,Xj(t), the matrix in Table 6can be found.

In cases that HWXi,Xj(t) is nonzero and |Xi(t)−Xj(t| = TPWXi,Xj(t) (which was the case for

the simulations displayed in Section 5), all WXi,Xj(t)are0or1,sowithk=9theaverage

(WXi,X1(t)+.+WXi,Xk(t))/kis a multiple of 1/9. In the simulation case, the norm νTPWXi,Xj=0.4

is not a multiple of 1/9. Therefore, the cases (WXi,X1(t)+.+WXi,Xk(t))/k=νTPWXi,Xjcannot actu-

ally occur, so then TPWXi,Xj(t)=0orTPWXi,Xj(t)=1andWXi,Xj(t)=0orWXi,Xj(t)=1arethe

only solutions. This is also shown by the simulations (e.g., see Figure 8); indeed all averages are

multiples of 1/9=0.111111, as found above (see Figure 8). This explains the discrete set of num-

bers 0.111111, 0.222222, 0.333333, and so on, observed in the simulations; as shown here, this

strongly depends on the number of states. Note that although in general the reiﬁed speed factor

HWXi,Xj(t) may be assumed nonzero, there may also be speciﬁc processes in which it converges

to 0, for example, like the temperature in simulated annealing.

8. On the added complexity

Note that, as for any dynamical system, by adding adaptivity to a network always complexity is

added. In this section, it is discussed how complexity of a network increases when reiﬁcation is

applied: ﬁrst for ﬁrst-order reiﬁcation and next for higher order reiﬁcation.

8.1 Added complexity for ﬁrst-order adaptation

To start with the outcome, network reiﬁcation will increase complexity, but this will at most be

quadratic in the number of nodes Nand linear in the number of connections Mof the original

network. More speciﬁcally, if mis the number of basic combination functions considered, then

the number of nodes in the reiﬁed network is at most N(original nodes) + N(nodes for speed

factors) + N2(nodes for connection weights) + mN (nodes for combination functions), which is

(2 +m+N)N

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22 J. Treur

If not all connections are used but only a number Mof them, the outcome is

(2 +m)N+M

This is linear in number of nodes and connections. The number of connections in the reiﬁed

network is M(original connection weights) + N(speed factors to their states) + Yindegree

(Y)=M(connection weights to their states) + mN (combination function weights to their states),

which is

(m+1)N+2M

Again, this is linear in number of nodes and connections.

8.2 Added complexity for higher order adaptation

If this analysis is applied in an iterative manner for second-order network reiﬁcation, then the

increase in complexity is still polynomial: at most in the fourth power of the number of nodes:

(N2)2=N4

Can this iteration still be continued further, thus obtaining nth-order reiﬁcation for any n?Yes,

theoretically, there is no end in this. But also practically, for example, in the case used as illustra-

tion in the current paper, the parameter νTXi,Xjfor the norm of the average connection weight

for the tipping point adaptation used as characteristic at the second reiﬁcation level still could be

made adaptive (e.g., related to how busy someone is) and reiﬁed at a third reiﬁcation level. For

third-order reiﬁcation, the increase in complexity is still polynomial: at most in the order of

((N2)2)2=N8

If nreiﬁcation levels are added, then it is in the order of

N(2n)

which is still polynomial in N, but double exponential in n. The latter may suggest to limit

the number of reiﬁcation levels in practical applications to just a few, or alternatively, in each

reiﬁcation step add only a few new reiﬁcation states: for each step reiﬁcation can be done in a

partial manner as well. For example, if only speed factors are reiﬁed, the number of states will

only increase in a linear way: one extra state for each existing state. Recall the double negative

exponential pattern of hits in the order of

e(35.19 e−0.8684n)

discussed in Section 3and illustrated in Figure 2. In the current literature, adaptation of order

higher than 2 is extremely rare and of order higher than 3 practically absent. This supports the

idea that for now adaptation of order >3 is not considered interesting enough to be addressed.

As shown above, for adaptation of order 3, the added complexity is in the order of an 8th degree

polynomial and for order 2 a 4th degree polynomial. Note, however, that Section 10.3 points out

how efﬁcient simulation of large-scale reiﬁed networks of thousands or even millions of states can

be achieved by applying a form of precompilation.

9. Application to plasticity and metaplasticity in cognitive neuroscience

In this section, it is shown how the reiﬁed temporal–causal network modeling approach can be

used to model important developments in empirical science, in particular concerning plasticity

and metaplasticity. These are important notions in state-of-the-art research on cognitive neu-

roscience, introduced not from a computational modeling perspective but by purely empirical

researchers to clarify what was found empirically. This section shows how these notions can be

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Network Science 23

Tab le 7 . State names for the plasticity and metaplasticity network model with their explanations.

State number State name Level Explanation

X1sssBase level Sensor state for stimulus s

X2srssBase level Sensory representation state for stimulus s

X3bssBase level Belief state for stimulus s

X4psaBase level Preparation state for response a

X5Wsrss,psaFirst reiﬁcation level Reiﬁed representation state for connection

weight ωsrss,psa

X6TsrssFirst reiﬁcation level Reiﬁed representation state for threshold

parameter τsrssof base state srss

X7TpsaFirst reiﬁcation level Reiﬁed representation state for threshold

parameter τpsaof base state psa

X8HWsrss,psaSecond reiﬁcation level Reiﬁed representation state for speed factor

ηWsrss,psafor reiﬁed representation state Wsrss,psa

X9MWsrss,psaSecond reiﬁcation level Reiﬁed representation state for persistence

factor parameter μWsrss,psafor reiﬁed

representation state Wsrss,psa

connected to the reiﬁed temporal–network modeling approach described in the current paper.

This particular example shows the essential elements but is kept relatively simple; it can easily be

extended by adding more states and connections.

Mental networks equipped with a Hebbian learning mechanism (Hebb, 1949) are able to adapt

connection weights over time and learn or form memories in this way. Within neuroscience, this is

usually called plasticity.Insomecircumstances,itisbettertolearn(andchange)fast,butinother

circumstances, it is better to stay stable and persist what has been learnt in the past. To control

this, a type of (higher order) adaptation called metaplasticity is used. It has become an important

focus of study in neuroscience. In literature such as Abraham & Bear (1996), Chandra & Barkai

(2018), Magerl et al. (2018), Parsons (2018), Robinson et al. (2016), Sehgal et al. (2013), Schmidt

et al. (2013), Sjöström et al. (2008), various studies are reported which show how adaptation of

synapses as described, for example, by Hebbian learning, is modulated by suppressing the adapta-

tion process or amplifying it. Among the reported factors affecting synaptic plasticity are stimulus

exposure, activation, previous experiences, and stress, which can accelerate or decelerate learning,

or induce temporarily enhanced excitability of neurons, which in turn positively affects learning,

for example, Chandra & Barkai (2018)andOhetal.(2003).

9.1 Plasticity and metaplasticity in cognitive neuroscience

The reiﬁﬁed temporal–causal network modeling approach was applied to a case involving both

plasticity and metaplasticity, acquired from the literature mentioned above. Recall the different

types of reiﬁcation states depicted in Figure 3. A network picture of the designed reiﬁed network

model for plasticity and metaplasticity is shown in Figure 10.Table7displays the explanations

of the states. Section 10.1 shows the complete speciﬁcation. Here, the plasticity of the response

connection from srssto psais considered, modeled by Hebbian learning. Note that the two T-

states and the M-state are combination function parameter states here, respectively, for excitability

threshold τof srssand psaand for persistence parameter μfor the Hebbian learning of the con-

nection from srssto psa. The alternative path via the belief state bsssupports this learning by

contributing to the activation of psa, thus relating to the original formulation of the (ﬁrst-order)

Hebbian learning adaptation principle in Hebb (1949):

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24 J. Treur

srs

,ps

srs

,ps

srs

,ps

Second

reiﬁcation

level

First

reiﬁcation

level

Base

level

Figure 10. Overview of the reiﬁed network architecture for plasticity and metaplasticity with base level (lower plane, pink),

ﬁrst reiﬁcation level (middle plane, blue) and second reiﬁcation level (upper plane, purple), and upward causal connections

(blue) and downward causal connections (red) deﬁning interlevel relations. The downward causal connections from the two

T-states affect the excitability of the (presynaptic and postsynaptic) states srssand psa. The downward causal connections

from the H-state and M-state affect the adaptation speed and the persistence factor of the connection weight reiﬁcation

state Wsrss,psa.

When an axon of cell A is near enough to excite B and repeatedly or persistently takes

part in ﬁring it, some growth process or metabolic change takes place in one or both

cells such that A’s efﬁciency, as one of the cells ﬁring B, is increased. (Hebb, 1949)p.62

In principle, this will start to work when the external stimulus sis sensed through sensor state sss.

However, as discussed in Section 1, whether or not and to which extent learning actually takes

place is controlled by a form of metaplasticity, and also depends on factors such as excitability

characteristics of the involved states. To model metaplasticity, the model includes a second reiﬁ-

cation level with states HWsrss,psarepresenting the speed of the learning (learning rate) of ωsrss,psa,

and MWsrss,psarepresenting the persistence μWsrss,psaof the connection weight ωsrss,psa.Theyhave

dynamic values depending on the other states. For example, if at some point in time the value

of HWsrss,psais 0, no learning will take place, and if MWsrss,psahas value 0, no learnt effects will

persist; the values of these second-order reiﬁcation states depend on activation of the presynaptic

and postsynaptic states srssand psa, also see the following second-order adaptation principle in

Robinson et al. (2016):

Adaptation accelerates with increasing stimulus exposure. (Robinson et al., 2016,p.2)

To address dynamic levels of excitability of base states, ﬁrst-order reiﬁcation states Tsrssand

Tpsahave been included that model the intrinsic excitability of the presynaptic and postsynap-

tic states srssand psa, respectively, by the value of the thresholds τsrssand τpsaof their logistic sum

combination functions (also see Chandra & Barkai, 2018):

Learning-related cellular changes can be divided into two general groups: modiﬁca-

tions that occur at synapses and modiﬁcations in the intrinsic properties of the neu-

rons. While it is commonly agreed that changes in strength of connections between

neurons in the relevant networks underlie memory storage, ample evidence suggest

that modiﬁcations in intrinsic neuronal properties may also account for learning-

related behavioral changes. Long-lasting modiﬁcations in intrinsic excitability are

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Network Science 25

First and second order reiﬁcation states

sss

srss

bss

psa

srss,psa

Wsrss,psa

Tsrss

Tpsa

HWsrss,psa

MWsrss,psa

Figure 11. Upper graph: dynamics of base states and the adaptive connection weight ωsrss,psa. Lower graph: dynamics of

the reiﬁcations states including the ﬁrst-order reiﬁcation state Wsrss,psafor the adaptive connection weight, and Tsrssand

Tpsafor the activation threshold for the presynaptic and postsynaptic states srssand psa, and the second-order reiﬁcation

states HWsrss,psaand MWsrss,psafor the adaptation speed and persistence factor of the connection weight reiﬁcation state

Wsrss,psa.

manifested in changes in the neuron’s response to a given extrinsic current (gener-

ated by synaptic activity or applied via the recording electrode). (Chandra & Barkai,

2018, p. 30)

For most states, the combination function used is the alogisticσ,τ(..) function. The only exceptions

are the sensor state ssswhich uses the Euclidean function eucl1,λ(..) and Wsrss,psawhich uses the

Hebbian combination function hebbμ(..).

9.2 Simulation experiments for plasticity and metaplasticity

Following what is reported in the literature on metaplasticity, a number of simulation experiments

have been performed. In particular, a scenario is shown here in which the focus was on the effect

of activation of the postsynaptic state psaon plasticity; the effect of the presynaptic state srsson

reiﬁcation states was blocked (weights of upward links from srsswere set to 0). In Figure 11,the

simulation results are shown. For settings, see the speciﬁcation in Section 5. The upper graph

shows the activation levels of the base states and how the weight of the connection from srssto psa

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26 J. Treur

is learnt. Here, the activation levels and also the exact shape of the learning curve also depend on

controlling factors shown in the lower graph in Figure 11. As can be seen there, following exposure

to stimulus s, the threshold values Tsrssand Tpsafor the activation of srssand psaare decreasing

to low levels, which substantially increases the excitability of srssand psaconform (Chandra &

Barkai, 2018) and therefore gives a boost to the activation levels of these base states, which in

turn strengthens the Hebbian learning. Also, it is shown that following exposure to stimulus sthe

learning speed HWsrss,psastrongly increases, which conforms Robinson et al. (2016).

These controlling measures together result in a quite steep increase of the connection weight

reiﬁcation state. However, after the learnt level of the weight has become high, the thresholds

increase again, and the learning speed decreases again. This makes the excitability of srssand

psalower and stops the boosts on learning; this has a positive effect on stabilizing the situation,

in accordance with what, for example, in Sjöström et al. (2008), is called “The Plasticity Versus

Stability Conundrum” (p. 773).

10. A modeling environment for reiﬁed temporal–causal network models

The modeling environment as developed by the author and described in this section covers a

modeling format for a type of computational architecture (reiﬁed network architecture), which

is implementation-independent (it can be used for simulation by implementations in differ-

ent ways), and an example implementation of a computational reiﬁed temporal–causal network

engine implemented in MATLAB.

Section 10.1 introduces the implementation-independent notions of role matrices and a new

speciﬁcation format for dynamic and adaptive causal network models. This format has a basis

in the implementation-independent mathematical notions of matrix and function. The notion

of role matrix provides a new, alternative, and more compact speciﬁcation format for causal

networks in contrast to connection matrices that are often used to specify networks at an

implementation-independent level.

The basic elements of the speciﬁcation format are declarative: connection weights, combination

functions, speed factors, and role matrices grouping them are declarative mathematical objects.

Together these elements assemble in a standard manner a set of ﬁrst-order difference or differ-

ential equations (as shown in Table 1), which are declarative temporal speciﬁcations. The model’s

behavior is fully determined by these declarative speciﬁcations, given some initial values. The

modeling process is strongly supported by using these declarative building blocks. Very complex

adaptive patterns can be modeled easily, and in (temporal) declarative form.

In Section 10.2, the computational reiﬁed temporal–causal network engine is described that

has been developed in MATLAB. This is the software that takes the declarative speciﬁcation of

Section 10.1 and makes it run. The design of this software is structure-preserving in relation to

the mathematical description and can be easily translated into any other software environment

that supports the mathematical notions of function and matrix.

10.1 Speciﬁcation format for a reiﬁed temporal–causal network model

For networks, matrices are often used to describe them in a way that is easily accessible com-

putationally. Moreover, MATALAB (“Matrix Laboratory”) is available as a software environment

dedicated to handle matrices. Therefore, the choice was made to use speciﬁc types of matrices

(called role matrices) to deﬁne a speciﬁcation format to specify reiﬁed network models, and as a

well-deﬁned basis for implementation in MATLAB or any other environment supporting matri-

ces. In role matrices, it is speciﬁed which other states have impact on a given state (the incoming

arrows in Figures 4and 10), but distinguished according to their role: base or nonbase connections,

from which for the latter a distinction is made for the roles connection weight reiﬁcation,speed

factor reiﬁcation,combination function weight reiﬁcation,andcombination function parameter

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Network Science 27

reiﬁcation (see also Figure 3). During execution of the model, each of the nonbase roles (indi-

cated by the downward connections in the model pictures) deﬁnes a special effect on the target

state. The role speciﬁcation indicates which special effect is exerted by which state. Note that this

makes that the more informative naming of states is not needed to specify such information. For

example, the reiﬁcation state names such as H,W,C, or Pas used in descriptions of the models

are only meant for human understanding and are not used in execution of the model; instead,

in execution, the role matrices deﬁne which are the downward causal connections and which

of the network characteristics they affect. Here, state names used are just the X1,X2, ...based

on numbering of the states. Role matrices enable to apply structure-preserving implementation.

The matrices as ﬁrst dimension have rows according to the states X1,X2,X3, .... numbered by

1, 2, 3, ...

Note that the matrices used in the reiﬁed network model speciﬁcation format have a compact

format and (in contrast to network pictures as in Figure 10) also specify an ordering, which is

important as combination functions used to integrate the impact from multiple connections are

not always symmetric.

A ﬁrst role matrix mb speciﬁes on each row for a given state from which states at the same

or a lower level that state gets incoming connections (see Box 2). They play the role of the base

connectivity. This matrix contains the information depicted in Figures 4and 10 by upward or

leveled arrows plus for each state a numbering of the incoming base connections (see the 1 to 4

in the top row). For example, in the third row, it is indicated that state X3(=bss) only has one

incoming base connection, from state X2(=srss). As another example, the ﬁfth row indicates that

state X5(=Wsrss,psa) has incoming base connections from X2(=srss), X4(=psa), X5(=Wsrss,psa)

itself, and in that order, which is important as the Hebbian combination function hebbμ(.)used

here is not symmetric in its arguments. Note that the second column with more informative state

names in each of the matrices depicted in Box 2is not part of the speciﬁcation but have just been

added for human understanding.

In a similar way, the four types of role matrices for nonbase connectivity (i.e., connectivity

from reiﬁcation states at a higher level of reiﬁcation: the downward arrows in Figures 4and 10)

were deﬁned (see Box 2): role matrices mcw for connection weights, ms for speed factors, mcfw

for combination function weights, and mcfp for combination function parameters. Within each

role matrix, a difference is made between cell entries indicating (in red) a reference to the name

of another state that as a form of reiﬁcation represents in a dynamic manner an adaptive net-

work characteristic, and entries indicating (in green) ﬁxed values for nonadaptive characteristics.

The red cells represent the downward causal connections from the reiﬁcation states in pictures as

shown in Figures 4and 10, with their speciﬁc roles W,H,C,Pindicated by the type of role matrix.

The type of role matrix in which they are represented actually deﬁnes the roles of the reiﬁcation

states so that there is no need to computationally use information from the names W,H,C,Pof

them; they may have any own given names.

For example, in Box 2,thenameX5in the red cell row-column (4, 1) in role matrix mcw

indicates that the value of the connection weight from srssto psacan be found as value of the ﬁfth

state X5.Incontrast,the1ingreencell(5,1)ofmcw indicates the static value of the connection

weight from X2(=srss)toX5(=Wsrss,psa). Similarly, role matrix ms indicates (in red) that X8

represents the adaptive speed factor of X5, and (in green) that the speed factors of all other states

have ﬁxed values.

For a given application, a limited ﬁxed sequence of combination functions is speciﬁed by mcf

= [1 2 3], where the numbers 1, 2, and 3 refer to the numbering in the function library which

currently contains 35 combination functions, the ﬁrst three being eucln,λ(..),alogisticσ,τ(..),and

hebbμ(..).InBox2, the role matrices mcfw and mcfp are shown for combination function weights

and parameters, respectively. Here, the matrix mcfp is a 3D matrix with ﬁrst dimension for the

states, second dimension for the two combination function parameters, and third dimension for

the combination functions.

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28 J. Treur

mb 1 2 3 4

X1sssX1

X2srssX1

X3bssX2

X4psaX2X3

X5Wsrss,psaX2X4X5

X6TsrssX2X4X6

X7TpsaX2X4X7

X8HWsrss,psaX2X4X5X8

X9MWsrss,psaX2X4X5X9

mcfw

1 2 3

eucl alog-

istic hebb

X1sss1

X2srss1

X3bss1

X4psa1

X5Wsrss,psa1

X6Tsrss1

X7Tpsa1

X8HWsrss,psa1

X9MWsrss,psa1

ms 1

X1sss0.5

X2srss0.5

X3bss0.2

X4psa0.5

X5Wsrss,psaX8

X6Tsrss0.3

X7Tpsa0.3

X8HWsrss,psa0.5

X9MWsrss,psa0.1

mcw 1 2 3 4

X1sss1

X2srss1

X3bss1

X4psaX51

X5Wsrss,psa1 1 1

X6Tsrss-0.4 -0.4 1

X7Tpsa-0.4 -0.4 1

X8HWsrss,psa1 1 -0.4 1

X9MWsrss,psa1 1 1 1

function

mcfp

parameter

1 2 3

eucl alog-

istic hebb

1 2 1 2 1 2

X1sss1 1

X2srss5X6

X3bss50.2

X4psa5X7

X5Wsrss,psaX9

X6Tsrss50.7

X7Tpsa50.7

X8HWsrss,psa5 1

X9MWsrss,psa5 1

τμσ

Box 2. Speciﬁcation in role matrices format for the second-order adaptive reiﬁed example network, covering from top to

bottom: role matrix mb for connected base states (= states of same or lower level with outgoing connections to the given

state), role matrix mcw for connection weights, role matrix mcfw for combination function weights, role matrix mcfp (3D)

for combination function parameters, and role matrix ms for speed factors.

10.2 The computational reiﬁed temporal–causal network engine developed in MATLAB

The computational reiﬁed network engine developed takes a speciﬁcation in the format as

described in Section 10.1 and runs it; for more details, see Treur (2019b). It consists of two parts.

First, each role matrix (which can be speciﬁed easily as table in Word or in Excel, for example) is

copied or read in MATLAB in two variants:

•avalues matrix for the static values (adding the letter vto the name) from the green cells with

constant values and

•anadaptivity matrix for the adaptive values represented by reiﬁcation states (adding the letter

ato the name) from the red cells with state names, thereby replacing Xiby the index i.

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Network Science 29

Tab le 8 . Role matrices mcwa and mcwv for the connection weights.

mcfp 123 4

X1sss

.........................................................................................................................................................

X2srss

.........................................................................................................................................................

X3bss

.........................................................................................................................................................

X4psa5

.........................................................................................................................................................

X5Wsrss,psa

.........................................................................................................................................................

X6Tsrss

.........................................................................................................................................................

X7Tpsa

.........................................................................................................................................................

X8HWsrss,psa

.........................................................................................................................................................

X9MWsrss,psa

mcwv 123 4

X1sss1

.........................................................................................................................................................

X2srss1

.........................................................................................................................................................

X3bss1

.........................................................................................................................................................

X4psa1

.........................................................................................................................................................

X5Wsrss,psa111

.........................................................................................................................................................

X6Tsrss−0.4 −0.4 1

.........................................................................................................................................................

X7Tpsa−0.4 −0.4 1

.........................................................................................................................................................

X8HWsrss,psa11−0.4 1

.........................................................................................................................................................

X9MWsrss,psa111 1

Note that states Xjare represented in MATLAB by their index number j. For example, from mcw

two matrices mcwa (adaptive connection weights matrix) and mcwv (connection weight values

matrix) are derived in this way (see Table 8). The numbers in mcwa indicate the state numbers of

the reiﬁcation states where the values can be found, and in mcwv, the numbers indicate the static

values directly.

Empty cells are ﬁlled with NaN (Not a Number) indications. After copying to MATLAB, this

results in the matrices in MATLAB representation as shown in Box 3.

Note that the values in the value matrices can differ per scenario addressed, they represent the

settings of the simulation as usually occur for any type of simulation.

As another example, for the 3D role matrices mcfpa and mcfpv, their representations in

MATLAB are as depicted in Boxes 4and 5.

During a simulation, for each step from kto k+1 (with step size t,inMATLABdenotedby

dt), based on the above role matrices, ﬁrst for each state Xj, the right values (either the ﬁxed value,

or the adaptive value found in the indicated reiﬁcation state) are assigned to:

s(j, k) speed factor for state Xj

b(j, p, k) value for the pth state with outgoing base connection to state Xj

cw(j, p, k) connection weight for the pth state with outgoing base connection to state Xj

cfw(j,m,k)weightforthemth combination function for Xj

cfp(j,p,m,k)thepth parameter value of the mth combination function for Xj

Note that what is called an outgoing base connection to state Xjhere is an outgoing connec-

tion of a base state to state Xjof the software, as speciﬁed in the base matrix mb (Box 2). For the

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30 J. Treur

Box 3. Role matrices mcwa and mcwv for adaptive connection weights (mcwa) and connection weight values (mcwv)as

represented in MATLAB.

Box 4. Role matrix mcfpa for adaptive combination function parameters as represented in MATLAB. This consists of three

2D matrices each with ﬁrst (vertical) dimension for the states and second (horizontal) dimension for the two parameters,

grouped in the vertical direction according to the third dimension for the three combination functions used in this model

(eucl(.),alogistic(.), and hebb(.), respectively). Note that here default values are used for parameter values for cases in

which they actually are not used.

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Network Science 31

Box 5. Role matrix mcfpv for combination function parameter values as represented in MATLAB. These are three 2D matri-

ces each with ﬁrst (vertical) dimension for the states and second (horizontal) dimension for the two parameters, grouped

in the vertical direction according to the third dimension for the three combination functions used in this model (eucl(.),

alogistic(.), and hebb(.), respectively). Note that here default values are used for parameter values for cases in which they

actually are not used.

adaptive network characteristics, this part is processing the downward connections from the reiﬁ-

cation states (indicated in the adaptation matrices, as for these adaptive characteristics the values

can be found there); for the nonadaptive characteristics, it is just assigning the ﬁxed value taken

from the value matrix.

Then, as a second part of the computational reiﬁed network engine, for the step from kto k+1,

the following is applied for each j; here X(j,k) denotes Xj(t)fort=t(k) =kdt; see Box 6.

Note that functions with multiple groups of arguments here in MATLAB get vector arguments

where groups of arguments become vectors of variable length. For example, the basic combina-

tion function bcfi(P1,i,P2,i,W1V1,...,WkVk) as expressed in Section 3becomes bcf(i, p, v) in

MATLAB with vectors p =[P1,i,P2,i] for function parameters and v =[W1V1,...,WkVk]forthe

values of the function arguments. This format bcf(i, p, v) is used as the basis of the combination

function library developed (currently numbered by i = 1–35); for an overview of this basic com-

bination function library, see Treur (2019b). As can be seen, the structure of the code of this

computational reiﬁed network engine is quite compact, based on the universal difference equa-

tion discussed in Section 3and closely resembles the formulae in Table 1: structure-preserving

implementation.

10.3 On simulating large-scale reiﬁed networks

In the above description, the role matrices deﬁning the model are inspected at every simulation

step. There is a second option for implementation by separating this work from simulation time,

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32 J. Treur

Box 6. Main loop in the developed MATLAB implementationof the Reiﬁed Temporal–Causal Network Engine; see also (Treur,

2019b)

in the form of precompiling. Doing so, the one universal difference equation in the code shown

above is instantiated for each of the states, so it is replaced by nspeciﬁc difference equations with

nthe number of states. The resulting set of difference (or differential) equations can be run in any

software environment for differential equation simulation. As there are quite efﬁcient software

environments for this, for large-scale reiﬁed networks of thousands or even millions of states,

such environments can be used for successful simulation.

11. Discussion

The multilevel network reiﬁcation architecture described here has advantages similar to those

found for reiﬁcation in modeling and programming languages in other areas of AI and Computer

Science, for example, Bowen & Kowalski (1982), Demers & Malenfant (1995), Galton (2006),

Hofstadter (1979), Sterling & Shapiro (1996), Sterling & Beer (1989), Weyhrauch (1980). A rei-

ﬁed network enables to model dynamics of the original network by dynamics within the reiﬁed

network, thus representing an adaptive network by a nonadaptive network. Network reiﬁcation

provides a uniﬁed manner of modeling adaptation principles and allows comparison of such prin-

ciples across different domains, as has been illustrated in (Treur, 2018a). In the current paper, it

was shown how a multilevel reiﬁed network architecture enables a structured and transparent

manner to model network adaptation of any order.

This was illustrated ﬁrst for a ﬁrst-order adaptation principle based on bonding-by-homophily

from social science (Byrne, 1986, McPherson et al., 2001, Pearson et al., 2006, Sharpanskykh &

Treur, 2014) represented at the ﬁrst reiﬁcation level, and in addition a second-order adaptation

principle describing change of the characteristics “similarity tipping point” and “speed factor”

of this ﬁrst-order adaptation principle. This second-order adaptive network model for bonding

by homophily was not yet compared to empirical data. First-order adaptive network models for

bonding by homophily that can be considered precursors of the current second-order network

model have been compared to empirical data sets in Beukel et al. (2019), Blankendaal et al. (2016),

Boomgaard et al. (2018). In order to compare the new second-order network model described here

to empirical data, the second-order adaptation effect most likely will require further requirements

for the data sets that are needed. This is left for a future enterprise. Also, in further social science

literature, cases are reported where network adaptation is itself adaptive, for example, in Carley

(2002,2006), the second-order adaptation concept called “inhibiting adaptation” for network

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Network Science 33

organizations is described. For further work, it would be interesting to explore the applicability

of the introduced modeling environment for such domains further.

In this paper, the introduced modeling environment for reiﬁed temporal–causal networks was

also applied to model plasticity and metaplasticity known from the empirical neuroscientiﬁc liter-

ature (see Section 9). Although some speciﬁc computational models for metaplasticity have been

put forward with interesting perspectives for artiﬁcial neural networks, for example, in Marcano-

Cedeno et al. (2011), Andina et al. (2007), Andina et al. (2009), and Fombellida et al. (2017),

the modeling environment proposed here provides a more general architecture. Application may

extend well beyond the neuro-inspired area (as already shown in Sections 5–7).

The causal modeling area has a long history in AI, for example, Kuipers & Kassirer (1983)and

Kuipers (1984). The current paper can be considered a new branch in this causal modeling area. It

adds dynamics to causal models, making them temporal, but the main contribution in the current

paper is that it adds a way to specify (multiorder) adaptivity in causal models, thereby conceptually

using ideas on meta-level architectures that also have a long history in AI, for example, Weyhrauch

(1980), Bowen & Kowalski (1982), Sterling & Beer (1989). So the modeling approach connects

two different areas with a long tradition in AI, thereby strongly extending the applicability of

causal modeling to dynamic and adaptive notions such as plasticity and metaplasticity of any

order, which otherwise are out of reach of causal modeling.

The dedicated modeling environment as described in Section 10 includes a new speciﬁcation

format for reiﬁed networks and comes with a newly implemented dedicated computational reiﬁed

network engine, which can simply run such speciﬁcations. Moreover, a library of currently 35

combination functions is offered, which can be extended easily. Using this software environment,

the development process of a model can focus in a declarative manner on the reiﬁed network

speciﬁcation and therefore is quite efﬁcient, while still all kinds of complex (higher order) adaptive

dynamics are covered without being bothered by implementation details.

This construction can be continued to obtain a network architecture that is adaptive up to

any order n. In Section 3, it was discussed in how far adaptation principles of order 3 or higher

are considered to be useful in the current literature, and a double negative exponential pattern

was found for the number of hits in Google Scholar against the order of adaptation. In an nth-

order reiﬁed network, there still will be network structures introduced in the last step from n-1

to nthat have no reiﬁcation within the nth-order reiﬁed network. From a theoretical perspective,

the construction can be repeated (countable) inﬁnitely many times, for all natural numbers n;then

ω-order reiﬁcation is obtained, where ωis the ordinal for the natural numbers. This is theoretically

well deﬁned as a mathematical structure. All network structures in this ω-order reiﬁed network are

reiﬁed within the network itself, so it is closed under reiﬁcation. Whether or not such an ω-order

construction has a useful application in practice, or can be used to explore theoretical research

questions is still been open question, another subject for future research.

Compared to Treur (2018b), the following have been added, which together makes more than

90% extra:

• Many more references.

• More examples and explanation on combination functions in Section 2.

• A new Section 3was added to relate the work more to the literature on adaptivity and higher

order adaptivity.

• More technical details were added in Section 4.

• In Section 6, two more simulation scenarios were added.

• A new Section 8was added with a more extensive complexity analysis.

• A new Section 9was added to describe applicability to plasticity and metaplasticity in

cognitive neuroscience.

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34 J. Treur

• A new Section 10 was added to describe the speciﬁcation format used to design reiﬁed

network models and the compuational reiﬁed network engine that was implemented in

MATLAB.

Acknowledgments. The author is grateful to the anonymous reviewers with comments which led to improvements in the

text.

Conﬂict of interest. The author has nothing to disclose.

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Multi-order Adaptive Dynamical System Modeling of the Role of Epigenetics in the Development of Hypomania

Conference Paper

Mar 2025

For a presentation video, see https://youtu.be/Sgf3XdVvRZU. Hypomania is a mood state characterized by elevated energy, impul-sivity, and heightened goal-directed activity, which often manifests in the context of Bipolar Disorder Type II and Cyclothymic Disorder, but also presents independently. Recent research suggests that epigenetic mechanisms may play a significant role in the development of hypomania via their influence on dopamine metabolism control and neural plasticity. However, because hypomania as an independent disorder has been in the focus of limited research, the neurobiological mechanisms involved have not been analysed in much depth. This study employs a self-modeling adaptive network model to simulate and computationally analyse the epigenetic mechanisms contributing to hypomania adaptation at six levels, ranging from neurotransmitter activity and protein expression to DNA methyla-tion and epigenetic alteration. The model has a specific focus on the COMT gene that is considered to have a key role in dopamine regulation. Simulation experiments were conducted, demonstrating how environmental stressors such as sleep deprivation, life changes, and charged atmospheres may trigger long-term epige-netic changes, causing dysregulation of prefrontal cortex, amygdala, and dopa-mine pathways. The results highlight how stable epigenetic changes can maintain a degree of dopaminergic hyperactivity that generates and perpetuates hypomanic episodes. The COMT gene was chosen as a lead candidate due to its established role in the metabolism of dopamine. However, further research is needed to establish its specific role in hypomania in more depth. The paper also acknowledges the difficulty in teasing out hypomania from bipolar disorder and substance use effects, which were outside the scope of this paper. By integrating epigenetics, neurobiology, and computational modeling, this research offers a novel framework for the computational analysis of hypomania and a foundation for the exploration of its underlying mechanisms and new therapeutic targets in the future.

Optimizing Speaking up Behavior for Safety and Security Through Cyberspace: A Computational Network Model of a Virtual Coach Supporting Speaking up

Chapter

Jan 2025

Previous reports show that a substantial proportion of (near) medical errors in the operating theatre is attributable to ineffective communication between healthcare professionals. Speaking up about observed medical errors is a safety behaviour which promotes effective communication between health care professionals, consequently optimising patient care by reducing medical error risk. Speaking up by health care professionals (e.g., nurses, residents) remains difficult to execute in practice despite increasing awareness of its importance. Therefore, this chapter introduces a computational model concerning the mechanisms known from psychological, observational, and medical literature which underlie the speaking up behaviour of a health care professional. It also addresses how a doctor may respond to the communicated message. Through several scenarios, we illustrate what pattern of factors causes a healthcare professional to speak up when witnessing a (near) medical error. We moreover demonstrate how introducing an observant agent can facilitate effective communication and helps to ensure patient safety through speaking up when a nurse can not. In conclusion, the current chapter introduces a computational model which predicts speaking up behaviour from the perspective of the speaker and receiver, with the addition of a virtual coach to further optimise patient safety when a patient could be in harm's way.

Multilevel Causality Reification as a Basis for Higher-Order Self-Representation: A Dynamic, Interactive, Adaptive, and Evolutionary Perspective

Conference Paper

Full-text available

Jan 2025

Jan Treur

Key ingredients • Multilevel causality reification • Higher-order self-representation; representation of past/future patterns • Grounding in philosophy, biology, neuroscience, evolutionary principles. Multilevel reification for higher-order approaches often uses a linguistic/logical perspective, with terms indicating reified linguistic expressions from a lower level. In recent years, reification was also applied more directly to temporal-causal networks of brain pathways. Connection weight characteristics of such a network can be represented by reification states and included at a higher level in the network as nodes with their own dynamics. For example, if a person has a strong response connection from hearing something to responding to it, this can be represented by a reification state for the weight of this connection, thus representing the person’s responding strength characteristic (and its adaptation over time). As this state takes part in the causal network, this can be applied iteratively to obtain multilevel causality reification. Multilevel causality reification provides higher-order self-representations, relating it to higher-order approaches to consciousness. For example, a psychiatry case study was analysed by three subsequent levels of reification for self-referencing, self-awareness, and self-interpretation. For representational content, it was more generally acknowledged that a wide perspective is needed. Examples are relational specification of representational content or the interactivist perspective on representational content. Such types of representational content can refer to patterns in past and/or future, between which the considered state acts as a mediating present state. Thus higher-order self-representations of past and future patterns concerning own structure, dynamics and adaptivity are analysed. The higher-order approach based on multilevel causality reification has grounding in philosophy, biology, and neuroscience. A dynamic, temporal perspective of states referring to past and/or future patterns is used as philosophical foundation for dynamical systems of the world or brain: state-determinedness, temporal factorisation and criterial causation. For neuroscience, brain plasticity and metaplasticity: (1) Synapse strength and inherent neuron excitability as world structures for reification of causation criteria representing patterns of (past) synaptic and nonsynaptic learning process and (future) strength of responding, (2) Inherent metaplasticity as world structures representing patterns of (past) meta-learning processes and (future) speed of adaptation. Damasio’s view describes from a neuroscience perspective as a causal network how emotional responding leads to first-order neural maps and second-order neural maps for the related bodily changes and induced subjective feeling and conscious feeling states, respectively. Similarly, Graziano’s view describes how in relation to attention, subjective awareness can be created as a higher-order state. Multilevel causality reification more in general aligns with multiple control levels like in the perspective on mechanistic explanation of biological processes in philosophy. It has also been shown that such multiple control levels provide a biological foundation for social interaction science, addressing by multilevel causal reification how subjective detection of behavioral homophily or synchrony patterns leads to bonding. Finally, multilevel causality reification plays an important role for evolutionary processes, related to principles for evolutionary development and the origin of life like ‘frozen metabolic accident’ and ‘metabolism-first’. Here, causality reification takes place by creating world structure for it and favouring the obtained causal pathways that enable dynamics that provide some evolutionary advantage.

Higher-Order Adaptive Dynamical System Modeling for the Role of Environmental Neurotoxic Pesticide Paraquat on the Epigenetics of Neurodegeneration in N27 Dopaminergic Cells

Chapter

Apr 2025

On the Adaptive Interplay of Mirroring and Bonding by Homophily in Joint Decision Making: A Second-Order Adaptive Network Model

Chapter

Mar 2025

Higher-Order Multi-Adaptive Neural Network Modeling of the Role of Epigenetics in Epilepsy

Conference Paper

Mar 2025

For a presentation video, see https://www.youtube.com/watch?v=GsVleLHWQHw. Epilepsy is a disorder that originates from complex interactions between factors such as genetic, epigenetic, and environment, which result in recurrent, unprovoked seizures that often resist conventional treatments. In this paper, a fifth-order adaptive self-modelling network is introduced to capture the way epigenetic processes (including DNA methylation, histone modifications, and microRNA regulation) can cause a shift in neuronal circuits toward a persistent hyperexcitability, resulting in more frequent seizures. The model focuses on BDNF, LIMK1, and two microRNAs (miR-132 and miR-134), illustrating how the presence of excessive excitatory signals may lock brain networks into a chronic seizure, a prone state under continued stress. The results of the simulation show that sudden increases in stress can push the system beyond its capacity to maintain normal excitability, leading to seizures that persist even after the stress is not present anymore. In a second simulation, an epigenetic therapy aimed at correcting abnormal methylation and histone acetylation successfully restores many of the disrupted processes, allowing the network to get closer to its pre-stress baseline. These findings indicate that the therapeutic interventions that target maladaptive epigenetic marks seem sound in theory, offering a framework for predicting seizure outcomes and guiding future research in personalized epilepsy treatment.

Modeling Multiple Orders of Adaptivity from a Higher-Order Adaptive Dynamical System Perspective

Chapter

Feb 2025

A Second-Order Adaptive Network Model for Bonding Based on Subjectively Perceived Homophily

Chapter

Jan 2025

Jan Treur

In this chapter, the role of subjective elements and control in social network adaptation is analysed computationally. In particular, it is analysed: (1) how the coevolution of social contagion and bonding by homophily may be controlled by the persons involved, and (2) how subjective representation states (e.g., what they know) can play a role in this coevolution and its control. To address this, a second-order adaptive social network model is presented in which persons do have a form of control over the coevolution process, and, in relation to this, their bonding depends on their subjective representation states about themselves and about each other, and social contagion depends on their subjective representation states about their connections.

Organisational Learning for Safety and Security Through Cyberspace: Adaptive Modelling of the Implementation of Environment Health and Safety Standards

Chapter

Jan 2025

This chapter focusses on using a self-modelling network modeling approach to analyse and predict the behavior of people within organizations, specifically on the implementation of EHS standards. The study focusses on a real-world scenario in which EHS standards are implemented in a multi-level organizational learning scenario. To ensure the health and safety of employees worldwide, the company provides standards for every EHS aspect of activities. The standards are fed by state-of-the-art technologies and science. They require local implementation at every site, also incorporating legislation for the country where it is based. The technique used in this study is dynamic computational modelling wielding the self-modelling network modeling approach. In the base scenario, the organization started with the implementation in a pilot phase. After a while the organization acknowledges it needs to take the lessons learned in the pilot and apply them to all other standards, enabling feed-forward and feedback learning within the organisation. Subsequently five alternative scenarios are described in which the organization is challenged with issues complicating the implementation process. The study shows the possibility of using a self-modelling social network in the execution of EHS activities. It demonstrates the importance of cooperation and open communication between the system owner, EHS expert and department. This is needed to be able to learn about processes and efficiently safeguard the organization and its members.

Learning from Mistakes Within Organizations: An Adaptive Network-Oriented Model for a Double Bias Perspective for Safety and Security Through Cyberspace

Chapter

Jan 2025

Although making mistakes is a crucial part of learning, it is still often being avoided in companies as it is considered as a shameful incident. This goes hand in hand with a mindset of a boss who dominantly believes that mistakes usually have negative consequences and therefore avoids them by only accepting simple tasks. Thus, there is no mechanism to learn from mistakes. Employees working for and being influenced by such a boss also strongly believe that mistakes usually have negative consequences but in addition they believe that the boss never makes mistakes, it is often believed that only those who never make mistakes can be bosses and hold power. That's the problem, such kinds of bosses do not learn. So, on the one hand, we have bosses who select simple tasks to be always seen as perfect. Therefore, also they believe they should avoid mistakes. On the other hand, there exists a mindset of a boss who is not limited to simple tasks, he/she accepts more complex tasks and therefore in the end has better general performance by learning from mistakes. This then also affects the mindset and actions of employees in the same direction. This chapter investigates the consequences of both attitudes for the organizations. It does so by computational analysis based on an adaptive dynamical systems modeling approach represented in a network format using the self-modeling network modeling principle.

Design of a Software Architecture for Multilevel Reified Temporal-Causal Networks

Method

Full-text available

Jun 2019

Jan Treur

For more background in network reification and how it can be applied to model adaptive networks of any order of adaptivity, see (Treur, 2018a) and (Treur, 2018b). The presented design fits well with Matlab as a programming environment, but can also be used for any other programming environment that allows the representation of basic mathematical concepts such as functions and matrices. The software architecture is illustrated for the developed Matlab software for reified temporal-causal networks. A version of this software itself can be found as an illustration in this document. Updated versions of the software will be provided later.

An Adaptive Temporal-Causal Network Model for Social Networks Based on the Homophily and More-Becomes-More Principle

Article

Full-text available

Feb 2019
NEUROCOMPUTING

This study describes the use of adaptive temporal-causal networks to model and simulate the development of mutually interacting opinion states and connections between individuals in social networks. The focus is on adaptive networks combining the homophily principle with the more becomes more principle. The model has been used to analyse a data set concerning opinions about the use of alcohol and tobacco, and friendship relations. The achieved results provide insights in the potential of the approach.

Multilevel Network Reification: Representing Higher Order Adaptivity in a Network

Conference Paper

Full-text available

Dec 2018

Jan Treur

Network reification occurs when a base network is extended by adding explicit states representing the characteristics defining the structure of the base network. This can be used to explitly represent network adaptation principles within a network. The adaptation principles may change as well, based on second-order adaptation principles of the network. By reification of the reified network, also such second-order adaptation principles can be explicitly represented. This multilevel network reification construction is introduced and illustrated in the current paper. The illustration focuses on an adaptive adaptation principle from Social Science for bonding based on homophily; here connections are changing by a first-order adaptation principle which itself changes over time by a second-order adaptation principle.

The Ins and Outs of Network-Oriented Modeling: From Biological Networks and Mental Networks to Social Networks and Beyond

Article

Full-text available

Sep 2018

Jan Treur

Network-Oriented Modeling has successfully been applied to obtain network models for a wide range of phenomena, including Biological Networks, Mental Networks, and Social Networks. In this paper it is discussed how the interpretation of a network as a causal network and taking into account dynamics in the form of temporal-causal networks, brings more depth. The basics and the scope of applicability of such a Network-Oriented Modelling approach are discussed and illustrated. This covers, for example, Social Network models for social contagion or information diffusion, adaptive Mental Network models for Hebbian learning and adaptive Social Network models for evolving relationships. From the more fundamental side, it will be discussed how emerging network behavior can be related to network structure. This paper describes the content of my Keynote lecture at the 10th International Conference on Computational Collective Intelligence, ICCCI'18.

Meta-adaptation in the auditory midbrain under cortical influence

Article

Full-text available

Nov 2016

Neural adaptation is central to sensation. Neurons in auditory midbrain, for example, rapidly adapt their firing rates to enhance coding precision of common sound intensities. However, it remains unknown whether this adaptation is fixed, or dynamic and dependent on experience. Here, using guinea pigs as animal models, we report that adaptation accelerates when an environment is re-encountered—in response to a sound environment that repeatedly switches between quiet and loud, midbrain neurons accrue experience to find an efficient code more rapidly. This phenomenon, which we term meta-adaptation, suggests a top–down influence on the midbrain. To test this, we inactivate auditory cortex and find acceleration of adaptation with experience is attenuated, indicating a role for cortex—and its little-understood projections to the midbrain—in modulating meta-adaptation. Given the prevalence of adaptation across organisms and senses, meta-adaptation might be similarly common, with extensive implications for understanding how neurons encode the rapidly changing environments of the real world.

A Temporal-Causal Modelling Approach to Integrated Contagion and Network Change in Social Networks

Conference Paper

Full-text available

Aug 2016

This paper introduces an integrated adaptive temporal-causal network model for dynamics in networks of social interactions addressing contagion between states, and changing connections within these social networks by two principles: the homophily principle and the more-becomes-more principle. The model has been evaluated in three different manners: by simulation experiments, by verification based on mathematical analysis, and by validation against an empirical data set.

The human pain system exhibits higher-order plasticity (metaplasticity)

Article

Apr 2018

The human pain system can be bidirectionally modulated by high-frequency (HFS; 100Hz) and low-frequency (LFS; 1Hz) electrical stimulation of nociceptors leading to long-term potentiation or depression of pain perception (pain-LTP or pain-LTD). Here we show that priming a test site by very low-frequency stimulation (VLFS; 0.05Hz) prevented pain-LTP probably by elevating the threshold (set point) for pain-LTP induction. Conversely, prior HFS-induced pain-LTP was substantially reversed by subsequent VLFS, suggesting that preceding HFS had primed the human nociceptive system for pain-LTD induction by VLFS. In contrast, the pain elicited by the pain-LTP-precipitating conditioning HFS stimulation remained unaffected. In aggregate these experiments demonstrate that the human pain system expresses two forms of higher-order plasticity (metaplasticity) acting in either direction along the pain-LTD to pain-LTP continuum with similar shifts in thresholds for LTD and LTP as in synaptic plasticity, indicating intriguing new mechanisms for the prevention of pain memory and the erasure of hyperalgesia related to an already established pain memory trace. There were no apparent gender differences in either pain-LTP or metaplasticity of pain-LTP. However, individual subjects appeared to present with an individual balance of pain-LTD to pain-LTP (a pain plasticity "fingerprint").

A non-synaptic mechanism of complex learning: Modulation of intrinsic neuronal excitability

Article

Nov 2017

Training rats in a particularly difficult olfactory discrimination task initiates a period of accelerated learning of other odors, manifested as a dramatic increase in the rats' capacity to acquire memories for new odors once they have learned the first discrimination task, implying that rule learning has taken place. At the cellular level, pyramidal neurons in the piriform cortex, hippocampus and bsolateral amygdala of olfactory-discrimination trained rats show enhanced intrinsic neuronal excitability that lasts for several days after rule learning. Such enhanced intrinsic excitability is mediated by long-term reduction in the post-burst after-hyperpolarization (AHP) which is generated by repetitive spike firing, and is maintained by persistent activation of key second messenger systems. Much like late-LTP, the induction of long-term modulation of intrinsic excitability is protein synthesis dependent. Learning-induced modulation of intrinsic excitability can be bi-directional, pending of the valance of the outcome of the learned task. In this review we describe the physiological and molecular mechanisms underlying the rule learning-induced long-term enhancement in neuronal excitability and discuss the functional significance of such a wide spread modulation of the neurons' ability to sustain repetitive spike generation.

Behavioral and neural mechanisms by which prior experience impacts subsequent learning

Article

Nov 2017

Ryan G Parsons

Memory is often thought about in terms of its ability to recollect and store information about the past, but its function likely rests with the fact that it permits adaptation to ongoing and future experience. Thus, the brain circuitry that encodes memory must act as if stored information is likely to be modified by subsequent experience. Considerable progress has been made in identifying the behavioral and neural mechanisms supporting the acquisition and consolidation of memories, but this knowledge comes largely from studies in laboratory animals in which the training experience is presented in isolation from prior experimentally-controlled events. Given that memories are unlikely to be formed upon a clean slate, there is a clear need to understand how learning occurs upon the background of prior experience. This article reviews recent studies from an emerging body of work on metaplasticity, memory allocation, and synaptic tagging and capture, all of which demonstrate that prior experience can have a profound effect on subsequent learning. Special attention will be given to discussion of the neural mechanisms that allow past experience to affect future learning and to the time course by which past learning events can alter subsequent learning. Finally, consideration will be given to the possible significance of a non-synaptic component of the memory trace, which in some cases is likely responsible for the priming of subsequent learning and may be involved in the recovery from amnestic treatments in which the synaptic mechanisms of memory have been impaired.

The emergence of executive functions by the evolution of second-order learning

Article

Sep 2017

Mental representation (MR) is regarded as part of sophisticated cognitive processes. It has been argued that under selection pressure for second-order learning (learning how to learn), first-order learning evolves to facilitate second-order learning within lifetime by capturing inherent structures of changing environment as MR. Two hypotheses derive from this theory: (1) solving MR-dependent tasks should involve second-order plasticity at the neural level. (2) Solving MR-dependent tasks should involve internalization of structural features of environment into corresponding features of the cognitive system. In this paper, constructive approach was taken and the result was analyzed from the viewpoint of these two hypotheses. Executive functions, a collection of cognitive processes necessary for good performance on complex tasks, are the theme of our model. They are considered to be related to theory of mind, which is a typical example of MR. We conducted an evolutionary simulation where agents with recurrent neural networks tackled the Wisconsin card sorting test (WCST), a widely used task to measure abilities of executive functions. The results showed some agents were successfully able to achieve ideal scores in the WCST, hence the emergence of executive functions. In addition, we also discussed the hypotheses based on one of the evolved neural networks.

Modeling Higher-Order Adaptivity of a Network by Multilevel Network Reification

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