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Taking Causal Modeling to a Next Level:
Self-Modeling Networks Adding Adaptivity to Causality
Jan Treur
Social AI Group, Department of Computer Science, Vrije Universiteit Amsterdam
j.treur@vu.nl
Abstract Causal modeling is an intuitive, declarative way of modeling. Due to
the universal character of causality, in principle it applies to practically all disci-
plines. In spite of this seemingly very wide scope of applicability, there are also
serious limitations and challenges that stand in the way of applicability. This con-
cerns in particular cases where dynamics and adaptivity play a role. This paper
addresses these challenges by exploiting the notion of self-modeling network that
has been developed from a Network Science perspective. Adaptivity is obtained
by adding to a given causal base network, a self-model which represents part of
the base network’s causal structure. Moreover, this construction can easily be
iterated so that multiple orders of adaptation can be covered as well. This indeed
takes causal modeling to a next level in more than one way. Therefore, in this
way dynamics and adaptivity are also covered well, which substantially widens
the scope of applicability of causal modeling.
Keywords: causal modeling, self-modeling network, network reification, adap-
tive social network, controlled adaptation
1 Introduction
Causal modelling provides a declarative approach that has a long tradition in Artificial
Intelligence; e.g., [1-4]. One of the challenges, however, is that causal modelling in-
volving cyclic paths in causal graphs poses difficulties; therefore many approaches to
causal modelling limit themselves to Directed Acyclic Graphs (DAG’s). More in gen-
eral, to avoid temporal complexity, dynamics is often not addressed in approaches
based on causal networks, neither for the causal effects on nodes, nor for the network
structure itself. The difficulty to allow cyclic paths in a causal network is one conse-
quence of this form of abstraction from dynamics of the nodes in a causal network.
Another consequence of abstracting from dynamics is that distinctions in timing and
asynchrony of causal effects (i.e., how fast causal effects actually are effectuated) can-
not be made, whereas often such differences in timing and asynchrony are crucial for
real-world processes modelled by a causal network. Finally, within causal models, not
only the nodes but also the causal relations are usually considered static, they cannot
change over time. This excludes many adaptive real-world processes from the scope of
applicability for causal modeling.
In the meantime, working from the perspective of Network Science, new approaches
have been developed that can be used to overcome the above-mentioned limitations of
causal modeling. In particular, in this paper it will be addressed how both within-
2
network dynamics (dynamics of the node states) for causal network models and adap-
tivity of the causal relations can be addressed using the network-oriented modeling ap-
proach developed in [5-7].
Using this approach as introduced for within-network dynamics in [5], the dynamic
perspective is based on a continuous time dimension, represented by real numbers, so
that all nodes have state values (also represented by real numbers) that vary over time.
The added temporal dimension enables modelling by cyclic causal networks as well,
and also timing of causal effects can be modelled in detail and differently per node, so
that also asynchronous processes are covered. Due to this, causal modeling can be used
for causal networks that contain cycles, such as many networks modelling mental or
brain processes, or networks describing social interaction processes (for example, in
social media). Moreover, in [5, 7] it is shown how – supported by a dedicated software
environment – networks with these within-network dynamics can be specified by de-
clarative means, by mathematical relations and functions; the modeler does not need to
address procedural descriptions nor program code.
In addition to these within-network dynamics, another useful element from the net-
work-oriented modeling perspective is the notion of self-modeling network or reified
network introduced in [6-8]. This is a network that includes a self-model for part of its
own network structure in the form of nodes that represent certain network structure
characteristics such as connection weights or excitability thresholds. Any (base) net-
work can be extended by including such a self-model, which can be considered to be at
a next level, compared to the base network; this step is also called network reification;
e.g., [6-8]. This construction for networks in particular was inspired by another long-
standing tradition in AI, namely that of meta-programming and metalevel architectures;
e.g., [9-13]. Having such self-models within a network enables to model adaptation of
the network structure by the within-network dynamics of the self-model representing
this network structure. As the latter can be specified by declarative means in the form
of mathematical relations and functions, also adaptivity of the network structure can be
specified in a similar declarative manner. To support the modeler, a dedicated software
environment (described in [7], Ch 9) is available that also applies to self-modeling net-
works.
In this paper, the perspective pointed out above will be illustrated in more detail.
First in Section 2 the network-oriented modeling approach based on self-modeling net-
works will be briefly introduced. Next, in Section 3 it will be illustrated for an example
of a multilevel second-order adaptive causal (social) network model for bonding by
(faked) homophily, while in Section 4 an example of a simulated scenario for this model
is described. Finally, Section 5 is a discussion.
2 Modeling Adaptivity by Self-Modeling Networks
In this section, the network-oriented modeling approach by self-modeling networks
used is briefly introduced in two steps.
2.1 Network-Oriented Modeling by Temporal-Causal Networks
As in this approach nodes Y in a network have activation values Y(t) that are dynamic
over time t, they serve as state variables and will usually be simply called states. For
these dynamics, the states are considered to affect each other by the connections within
the network; therefore these connections are interpreted here as causal relations. This
has been inspired partly by how in Philosophy of Mind networks of mental states and
their causation relations are described; e.g., [14]. In line with this, following [5, 7], a
basic temporal-causal network structure is characterised by:
• connectivity characteristics
Connections from a state X to a state Y and their weights X,Y
• aggregation characteristics
For any node Y, some combination function cY(..) defines aggregation that is ap-
plied to the impacts on Y from its incoming connections from states
• timing characteristics
Each state Y has a speed factor Y defining how fast it changes for given causal
impact
Here, the states Xi and Y have activation levels Xi(t) and Y(t) that vary (often within the
[0, 1] interval) over time, described by real numbers t. These dynamics are described
by the following difference (or differential) equations that incorporate in a canonical
manner the network characteristics X,Y, cY(..), Y:
[ (1)
for any state Y and where are the states from which Y gets its incoming con-
nections. The equations (1) are useful for simulation purposes and also for analysis of
properties of the emerging behaviour of temporal-causal networks. The overall combi-
nation function cY(..) for state Y is taken as the weighted average of some of the avail-
able basic combination functions cj(..) by specified weights j,Y, and parameters ,
of cj(..), for Y:
cY(V1, …, Vk) =
(2)
Such equations (1), (2) are hidden in the dedicated software environment that can be
used for simulation and analysis; see [7], Ch 9. This software environment is freely
downloadable from URL
https://www.researchgate.net/project/Network-Oriented-Modeling-Software.
Combination functions are similar to the functions used in a static manner in the
deterministic Structural Causal Model perspective described, for example, in [3, 4, 15].
However, in the Network-Oriented Modelling approach described here they are used in
a dynamic manner. For example, Pearl [3], p. 203, denotes nodes by Vi and combination
functions by fi (although he uses a different term for these functions). In the following
quote he points at the issue of underspecification concerning aggregation of multiple
connections, as in the often used graph representations the specification of combination
functions fi for nodes Vi, is lacking:
‘Every causal model M can be associated with a directed graph (…) This
graph merely identifies the endogeneous and background variables that
have a direct influence on each Vi; it does not specify the functional form
of fi.’ [3], p. 203
4
Therefore, in addition to graph representations for connectivity, at least aggregation in
terms of combination functions has to be addressed, as indeed is done for temporal-
causal networks, in order to avoid this problem of underspecification. That is the reason
why aggregation in terms of combination functions is part of the definition of the net-
work structure for temporal-causal networks, in addition to connectivity in terms of
connections and their weights and timing in terms of speed factors.
As part of the software environment, a large number > 35 of useful basic combina-
tion functions are included in a Combination Function Library, and also a facility to
easily indicate any function composition of any available basic combination functions
in the library. One of the combination functions from this library used for states Y in
the example network model described in Section 3 is:
• the Euclidean combination function eucln,(V1, …, Vk) defined by
eucln,(V1, …, Vk) =
(3)
where n is the order and a scaling factor and V1, …, Vk are the impacts from the
states from which the considered state Y gets incoming connections.
In Section 3, it will be explained in more detail how the combination function eu-
cln,(…) is used to model social contagion. Social contagion makes that states of con-
nected persons such as emotions or opinions, causally affect each other; e.g., (Levy and
Nail, 1993).
The above concepts (the characteristics X,Y, j,Y, i,j,Y, Y) enable to design network
models and their dynamics in a declarative manner, based on mathematically defined
functions and relations for them. Note that for each state Y, all characteristics X,Y, j,Y,
i,j,Y, Y mentioned above causally affect the activation level of Y, as also can be seen
from equations (1) and (2). Each of these characteristics do that causing in their own
way from a specific role, either for connectivity, for aggregation or for timing. Below,
this observation will also turn out useful in the context of self-models to address adap-
tivity.
2.2 Using Self-Modeling Networks to Model Adaptive Networks
Realistic network models are usually adaptive: often some of their network character-
istics X,Y, j,Y, i,j,Y, Y change over time. For example, for mental networks often the
connections are assumed to change by hebbian learning [16] and for social networks, it
is often assumed that connections between persons change through a bonding by ho-
mophily principle [17-19].
Adaptive networks are often modeled in a hybrid manner by considering two differ-
ent types of separate models that interact with each other: a network model for the base
network and its within-network dynamics, and a numerical model for the adaptivity of
the network structure characteristics of the base network. The latter dynamic model is
usually specified in a format outside the context of network modeling: in the form of
some adaptation-specific procedural or algorithmic programming specification used to
run the difference or differential equations underlying the network adaptation process.
In contrast, by including self-models, a network-oriented conceptualisation similar
to what was described above, can also be applied to adaptive networks to obtain a de-
clarative description using mathematically defined functions and relations for them as
well; see [6, 7]. This works through the addition of new states to the network (called
self-model states) which represent network characteristics by network states. Then the
causal impacts of these characteristics on a state Y as mentioned above can be modelled
as causal impacts from such self-model states. This brings the causal impacts from these
characteristics on a state Y in the standard form of a causal model where via causal
connections nodes affect other nodes.
More specifically, adding a self-model for a temporal-causal base network is done in
the way that for some of the states Y of the base network and some of the network
structure characteristics for connectivity, aggregation and timing (i.e., some from X,Y,
j,Y, i,j,Y, Y), additional network states WX,Y, Cj,Y, Pi,j,Y, HY (self-model states or reifi-
cation states) are introduced and connected to other states:
a) Connectivity self-model
• Self-model states WX,Y are added representing connectivity characteristics,
in particular connection weights X,Y
b) Aggregation self-model
• Self-model states Cj,Y are added representing aggregation characteristics, in
particular combination function weights j,Y
• Self-model states Pi,j,Y are added representing aggregation characteristics, in
particular combination function parameters i,j,Y
c) Timing self-model
• Self-model states HY are added representing timing characteristics, in partic-
ular speed factors Y
This step of adding a self-model to a base network is also called network reification. If
such self-model states are dynamic, they describe adaptive network characteristics. In
a graphical 3D-format, such self-model states are depicted at a next level (also called
reification level), where the original network is at a base level. As an example, the
weight X,Y of a connection from state X to state Y can be represented (at a next reifi-
cation level) by a self-model state named WX,Y (e.g., for an objective representation) or
RWX,Y (e.g., for a subjective representation).
Having self-model states to model an adaptation principle in a network-oriented
manner is only a first step. To fully model a certain adaptation principle by a self-mod-
eling network, the dynamics of each self-model state itself and its effect on a corre-
sponding target state Y have to be specified in a network-oriented manner by the three
general standard types of network structure characteristics a) connectivity, b) aggrega-
tion, and c) timing:
a) Connectivity for the self-model states in a self-modeling network
For the self-model states, their connectivity in terms of their incoming and outgoing
connections has two different functions:
• Effectuating its special effect from its specific role
The outgoing downward causal connections from the self-model states WX,Y, Cj,Y,
Pi,j,Y, HY to state Y represent the specific causal impact (their special effect from
their specific role) each of these self-model states has on Y. These downward causal
impacts are standard per role, and make that the adaptive values WX,Y(t), Cj,Y(t),
Pi,j,Y(t), HY(t) at t are actually used for the adaptive characteristics of the base net-
work in equations (1) and (2).
• Indicating the input for the adaptation principle as specified in b)
The incoming upward or leveled connections to a self-model state are used to spec-
ify the input needed for the particular adaptation principle that is addressed.
6
b) Aggregation for the self-model states in a self-modeling network
For the self-model states, their aggregation characteristics have one main aim:
• Expressing the adaptation principle by a mathematical function
For the aggregation of the incoming causal impacts for a self-model state, provided
as indicated in a), a specific combination function is chosen to express the adapta-
tion principle in a declarative mathematical manner.
c) Timing for the self-model states in a self-modeling network
For the self-model states, their timing characteristics have one main aim:
• Expressing the adaptation speed for the adaptation principle by a number
Finally, like any other state, self-model states have their own timing in terms of
speed factors. These speed factors are used as the means to express the adaptation
speed.
As a base network extended by including a self-model is also a temporal-causal network
model itself, as has been shown in [7], Ch 10, this self-modeling construction can easily
be applied iteratively to include self-models of multiple (reification) levels. This can
provide higher-order adaptive network models, and has turned out quite useful to
model, for example, within Cognitive Neuroscience plasticity and metaplasticity (e.g.,
[20-23]) in a unified form by a second-order adaptive mental causal network with three
levels, one base level and a first- and a second-order self-model level for causation
concerning plasticity and metaplasticity, respectively, as shown in [7], Ch 4.
In the current paper, the notion of a multi-level self-modeling network will be illus-
trated by a higher-order adaptive social network model. In this model, in addition to the
Euclidean combination function described in Section 2.1, two other combination func-
tions from the library are used:
• the advanced logistic sum combination function de-
fined by:
(1+e-στlog) (4)
where is a steepness parameter and log a threshold parameter and V1, …, Vk are
the impacts from the states from which the considered state Y gets incoming con-
nections
• the simple linear homophily combination function slhomo,hom(V1, V2, W) de-
fined by
slhomo,hom(V1, V2, W) = W + W (1-W) (homo - |V1 - V2|) (5)
where is an amplification parameter and hom a tipping point parameter and V1,
V2 are a person’s representations of the two persons’ states involved and W repre-
sents the weight of their connection
Here, slhomo,hom(V1, V2, W) is used to model bonding based on (faked) homophily
by internal connection weight representations, and to model control
of the bonding. Bonding based on homophily [17-19] is the social network adaptation
principle that is sometimes expressed by
‘Birds of a feather flock together’
This expresses how being ‘birds of a feather’ or ‘being alike’ (modeled by state
values V1 and V2 for the two persons not differing much) causally affects the connection
between two persons. Note that the homophily tipping point hom is the point where the
difference between the states of the two individuals (represented by |V1 - V2|) turns an
increase of bonding (outcome > W) into a decrease (outcome < W), and conversely. In
Section 4 this tipping point is set at 0.25: so in that case a difference |V1 - V2| < 0.25
has as causal effect that the connection will be strengthened (increase of W), whereas a
difference |V1 - V2| > 0.25 has as causal effect that it will be weakened (decrease of W).
This shows an example of how for a social application domain, within a causal net-
work, states can have a causal effect on network connections. By applying a self-mod-
eling network model, this form of causation (for adaptation of connections through
bonding by homophily) together with the causation between states in the base network
(for social contagion) is addressed in a unified manner by one overall (two-level) causal
network model, in contrast to the commonly used hybrid modeling approach to adaptive
networks pointed out above in the second paragraph of this Section 2.2. Moreover, in
Section 3 it will be shown how also a third level for the control of the adaptation process
can be incorporated within such a self-modeling causal network, thus obtaining a three-
level network model unifying within one causal model the base network dynamics with
adaptation of the connections of the base network and the control of that adaptation.
3 A Social Causal Network with Controlled Adaptation
To illustrate the use of self-modeling networks to incorporate in a unified manner both
dynamics and (multi-order) adaptivity in a causal model, this section presents an adap-
tive causal network model for controlled bonding based on homophily by using subjec-
tive representations (some of which are based on fake input). The presented causal net-
work model integrates three types of interacting processes, modeled within the causal
model at three different levels:
• The considered social base network itself with its (within-network) dynamics
for social contagion [24]
• Change of this social network over time based on bonding by homophily
[17-19]: first-order social network adaptation
• Control of the first-order social network adaptation: second-order social net-
work adaptation
In contrast to what is usually done, for example, also in [19], here the bonding is not
assumed to depend on the objective states for the two persons, but on how these states
are perceived and represented by the persons through the formation of subjective state
representation states. By controlling the formation of these subjective state representa-
tion states, indirectly the bonding is affected; contrarely, if you don’t take care to ac-
quire information about the other person, then you miss a good reason for stronger or
weaker bonding. To cover this, the above three types of processes have been modeled
by a second-order adaptive causal network model based on a multi-level self-modeling
network using a first-order self-model (for formation of the subjective state representa-
tion states and for the bonding based on them) and a second-order self-model (for the
control of the formation of the subjective representation states). That offers some room
to model cheating about one’s own properties, as regularly happens in real life: by fak-
ing an own state, the other person will make a false representation for it, which then
will affect that person’s bonding in a false manner.
8
The model’s connectivity is depicted in Fig. 1 by an example for two persons, one
of which is faking his or her properties in order to achieve successful bonding. In this
3D picture, each of the three planes models one of the three types of processes men-
tioned above; for an explanation of the states, see Table 1.
Table 1 Types of states in the introduced controlled adaptive social network model
SA
Objective state Z of person A
SB
Objective state Z of person B
FSB
Objective state of person B faking state Z of person A
RSA,A
Subjective representation of person A for state Z of person A
RSB,B
Subjective representation of person B for state Z of person B
RSA,B
Subjective representation of person B for state Z of person A
RSB,A
Subjective representation of person A for state Z of person B
RFSB,B
Subjective representation of person B for his or her faked state Z
RWA,B
Subjective representation of person A for the weight of the connection
from person A to person B
RWB,A
Subjective representation of person B for the weight of the connection
from person B to person A
CCA,B
Control state for communication from A to B for the state Z of A: repre-
sentation of the weight of the connection from RSA,A to RSA,B
CCB,A
Control state for communication from B to A for the state Z of B: repre-
sentation of the weight of the connection from RSB,B to RSB,A
COA,B
Control state for observation by B for the state Z of A observed by B:
representation of the weight of the connection from SA to RSA,B
COB,A
Control state for observation by A for the state Z of B observed by A:
representation of the weight of the connection from SB to RSB,A
The types of connections used at and between the three levels within this network
model are shown in Table 2. Here Z is a type of state of a person, for example, how
often the person listens to a certain type of music; to keep the notations simple, this
type is left out of them; if needed, the Z could be used as an additional subscript.
At the base level, social contagion is modelled by intralevel connections (depicted
by black arrows in the lower plane in Fig. 1) such as SA → SB, FSB → SA, and SA →
FSB. Here the last connection models B faking by intentionally listening to the same
type of music as A just at the moments that A can observe it. In contrast to FSB, state
SB indicates how much B normally listens to that type of music. In the simulated sce-
nario, SA will have high values and SB low values, whereas by copying SA also FSB gets
high values.
Fig. 1. Overview of the connectivity of the second-order adaptive social network model for bond-
ing by homophily for two persons A and B, where B is faking the homophily for A.
Within the first-order self-model, each person has subjective internal representation
states of other persons’ states Z and the of state Z of her or himself, and also of his or
her connections to others. This first-order self-model is modeled in the middle plane.
For example, person A’s internal representation state for person B having state Z is
modeled by state representation RSB,A, and A’s subjective representation of his or her
connection to B is modeled by connection weight representation RWA,B.
There are two pathways that contribute to formation of state representations such as
RSA,B. First, these representations can be obtained through observation of SA by B. This
is modeled by an upward interlevel connection SA → RSA,B from the base network to
the first-order self-model. As B is faking his or her base state, observation by A is mod-
eled not by a connection SB → RSB,A but by connection FSB → RSB,A.
A second pathway for a person B to get information on person A’s state is through
communication between persons. For example, if A communicates his or her subjective
representation RSA,A of the own state SA to B (e.g., ‘I often play this type of music!’),
this is modeled by an intralevel connection RSA,A → RSA,B within the middle plane for
the first-order self-model. Also in the communication, B is faking; therefore communi-
cation from B to A is not modeled by a connection RSA,B → RSB,A, but by connection
RFSB,B → RSB,A (so that B may falsely communicate ‘What a coincidence, I also often
play that type of music!’).
Second-order self-model:
Control of base network adaptation
First-order self-model:
Base network adaptation
Base network
S
A
SB
FSB
RWA,B
RSB,B
RWB,A
RSA,B
RFSB,B
RSB,A
RSA,A
COB,A
CCB,A
CCA,B
COA,B
10
Table 2 Connections in the controlled adaptive social network model and their explanation
Intralevel connections
SA → SB
Social contagion from A to B for state Z
FSB → SA
Social contagion from B’s faked state for Z to A
SA → FSB
Faking contagion from state Z of A to faked state Z of B
RSA,A → RSA,B
Communication of state Z from A to B
RFSB,B → RSB,A
Communication of faked state Z from B to A
RSA,A → RWA,B
Effect of represented state Z of A by A on the connection from A to
B (bonding by homophily)
RSB,A → RWA,B
Effect of represented state Z of B by A on the connection from A to
B (bonding by homophily)
RFSB,B → RWB,A
Effect of represented faked state Z of B by B on the connection from
B to A (bonding by homophily)
RSA,B → RWB,A
Effect of represented state Z of A by B on the connection from B to
A (bonding by homophily)
Interlevel connections
SA → RSA,A
Impact of observation of A’s state Z by A
on A’s representation of A’s state Z
Upward from
base network to
first-order self-model
SB → RSB,B
Impact of observation of B’s state Z by B
on B’s representation of B’s state Z
SA → RSA,B
Impact of observation of A’s state Z by B
on B’s representation of A’s state Z
FSB → RSB,A
Impact of observation of B’s faked state Z
by A on A’s representation of B’s state Z
RWA,B → SB
Effectuation of base connection weight
for social contagion from state Z of A to
state Z of B
Downward from
first-order self-model to
base network
RWB,A → SA
Effectuation of base connection weight
for social contagion from faked state Z of
B to state Z of A
RSA,A → CCB,A
Communication control monitoring con-
nection for A
Upward from
first-order self-model to
second-order self-model
RSB,B → CCA,B
Communication control monitoring con-
nection for B
RSA,A → COB,A
Observation control monitoring connec-
tion for A
RSB,B → COA,B
Observation control monitoring connec-
tion for B
CCB,A → RSB,A
Effectuation of control of communication
from B by A
Downward from
second-order self-model
to first-order self-model
CCA,B → RSA,B
Effectuation of control of communication
from A by B
COB,A → RSB,A
Effectuation of control of observation of
B by A
COA,B → RSA,B
Effectuation of control of observation of
A by B
As indicated, person A’s representation of her or his connection to person B is mod-
eled by RWA,B. It is assumed that for the bonding by homophily adaptation principle,
the adaptive change of the represented connection for A to B depends on the internal
representation states RSB,A and RSA,A. Therefore, this adaptation is supported by in-
tralevel connections RSA,A → RWA,B and RSB,A → RWA,B within the first-order self-
model. The connection representations by RW-states in turn affect the social contagion
within the social network, which is modeled by downward interlevel connections
RWA,B → SB and RWB,A → SA from the first-order self-model in the middle plane to
the base network.
To control the social network adaptation processes, two types of control actions are
considered in particular:
• controlling the observation of state Z from person A by person B is mod-
eled by control state COA,B and from person B by person A is modeled by
control state COB,A
• controlling the communication about state Z from person A to person B,
modeled by control state CCA,B and the communication about state Z
from person B to person A, is modeled by control state CCB,A
Activation of a communication control state makes that the related connection in the
first-order self-model in the middle plane gets a high value (1 or close to 1); this is
achieved by interlevel connections from control states to RS-states in the first-order
self-model. For example, activation of communication control state CCA,B makes that
the connection RSA,A → RSA,B from A’s state RSA,A to B’s state RSA,B gets a high
value (1 or close to 1) so that the transfer of information by communication happens;
this is modeled by interlevel connection COA,B → RSA,B. This can be considered as B
asking A for the information about him or herself, upon which A communicates this
information. Similarly, activation of an observation control state COA,B makes that the
connection SA → RSA,B from A’s state SA to B’s state RSA,B gets a high value (1 or
close to 1) so that the transfer of information by observation takes place; this is modeled
by connection COA,B → RSA,B. In the case modeled here, control states such as CCA,B
and COA,B themselves may become active depending on B’s state RSB,B; this is mod-
eled by connections RSB,B → CCA,B and RSB,B → COA,B. But this may be addressed in
many other ways as well, including externally determined control, for example, by en-
abling or allowing observation or communication (only) at specific time slots.
To specify a network model according to the approach described in [7], as discussed
in Section 2, three types of network characteristics are to be covered: connectivity, ag-
gregation and timing characteristics. Any state in the network is causally affected by
all of such characteristics, each from its own specific role. Following the role matrices
specification format defined in [7] (pp. 39-41, 89), they are specified by role matrices
as shown in Box 1 which are used as input for the dedicated software environment to
automatically obtain the simulation discussed In Section 4.
More specifically, role matrices indicate in rows successively for all network states,
the factors that causally affect them from the different roles. So in the row for a state Y,
in each column a causal relation is specified affecting state Y for the role described by
that role matrix. In this way, role matrices describe the network model by mathematical
relations and functions.
In the first place, concerning connectivity roles, each state is causally affected by the
other states from which it has incoming connections and by the weights of these con-
nections. In role matrix mb (see Box 1), for each state it is indicated from which other
states it has incoming connections from the same or a lower level. In role matrix mcw,
it is indicated what are the connection weights for the connected states indicated in mb.
If these weights are static, their value is indicated, in green shaded cells (here always
1), but if the connection weight is adaptive, instead of a number the self-model state
12
representing this weight is indicated in role matrix mcw. This can be seen (cells shaded
in a peach-red colour) in mcw for the incoming connections for the first two states X1
and X2, and for the incoming connections for the states X7 and X8. Indicating these adap-
tive value representations, defines the downward connections of Fig. 1. From the timing
role, also its speed factor causally affects a state; they are shown in Box 1 (role matrix
ms, which actually is a vector).
In the lower part of Box 1, showing the aggregation roles causally affecting a state,
it can be seen which states use which combination functions (role matrix mcfw) and
which parameter values for them (role matrix mcfp). In addition to the five role matri-
ces for the different roles of causal impacts, the initial values for the example simulation
are also shown in Box 1, which may be considered as initial causal impacts.
Box 1 Full specification of the adaptive self-modeling causal network model by role matrices for
all (connectivity, aggregation and timing) characteristics causally affecting the network states
4 Simulation: Faking Homophily for Bonding
In this section, a simulation of a simulated example scenario will be discussed to illus-
trate the introduced second-order adaptive causal social network model for faking ho-
mophily. In Fig. 2 the simulation for the example scenario is shown. Here the states SX
are slowly changing whereas the connection representations in the form of the RW-
mb base
connectivity
1
2
3
X1
SA
X3
X2
SB
X1
X3
FSB
X1
X4
RSA,A
X1
X5
RSB.B
X2
X6
RFSB,B
X1
X7
RSA,B
X1
X4
X8
RSB,A
X3
X6
X9
RWA,B
X4
X8
X9
X10
RWB,A
X6
X7
X10
X11
CCA,B
X4
X12
CCB,A
X3
X13
COA,B
X4
X14
COB,A
X3
mcfw combination
function weights
1
eucl
2
slhomo
3
alogistic
X1
SA
1
X2
SB
1
X3
FSB
1
X4
RSA,A
1
X5
RSB.B
1
X6
RFSB,B
1
X7
RSA,B
1
X8
RSB,A
1
X9
RWA,B
1
X10
RWB,A
1
X11
CCA,B
1
X12
CCB,A
1
X13
COA,B
1
X14
COB,A
1
ms speed
factors
1
X1
SA
0.0005
X2
SB
0.0005
X3
FSB
0.8
X4
RSA,A
0.9
X5
RSB.B
0.9
X6
RFSB,B
0.9
X7
RSA,B
0.9
X8
RSB,A
0.9
X9
RWA,B
0.1
X10
RWB,A
0.1
X11
CCA,B
0.2
X12
CCB,A
0.2
X13
COA,B
0.4
X14
COB,A
0.4
iv initial
values
1
X1
SA
0.9
X2
SB
0.2
X3
FSB
0.2
X4
RSA,A
0.7
X5
RSB.B
0.4
X6
RFSB,B
0.5
X7
RSA,B
0.5
X8
RSB,A
0.5
X9
RWA,B
0.5
X10
RWB,A
0.5
X11
CCA,B
0
X12
CCB,A
0
X13
COA,B
0
X14
COB,A
0
mcw connection
weights
1 2 3
X1
SA
X10
X2
SB
X9
X3
FSB
1
X4
RSA,A
1
X5
RSB.B
1
X6
RFSB,B
1
X7
RSA,B
X13
X11
X8
RSB,A
X14
X12
X9
RWA,B
1
1
1
X10
RWB,A
1
1
1
X11
CCA,B
1
X12
CCB,A
1
X13
COA,B
1
X14
COB,A
1
mcfp combi-
nation function
1
eucl
2
slhomo
3
alogistic
parameters
1
n
2
1
2
hom
1
2
log
X1
SA
1
1
X2
SB
1
1
X3
FSB
1
1
X4
RSA,A
1
1
X5
RSB.B
1
1
X6
RFSB,B
1
1
X7
RSA,B
1
2
X8
RSB,A
1
2
X9
RWA,B
3
0.25
X10
RWB,A
3
0.25
X11
CCA,B
50
0.1
X12
CCB,A
50
0.1
X13
COA,B
50
0.1
X14
COB,A
50
0.1
states are changing faster. It indeed can be seen that for A and B both directional con-
nection representations RWA,B and RWB,A start to gradually increase from time point 5
on to reach values above 0.7 which in the long run eventually reach a value (close to)
1. These changes of the connections are a consequence of the homophily principle, as
the values of state SA of A and the faked states FSB and RFSB,B for B quickly get close
to each other; note that the tipping point for similarity set was 0.25, so a difference
between the relevant representation states < 0.25 is strengthening a connection.
In Fig. 2, also the roles that are played by the control states in the form of the CO-
and CC-states and by the RS-states for subjective representations can be seen. The two
lines that start at 0 and get close to 1 around or soon after time 10 indicate the control
states COA,B and COB,A (light green) for observation and CCA,B and CCB,A (light blue)
for communication, respectively. This makes that at that time their mutual observation
and communication channels SA → RSA,B and FSB → RSB,A, and RSA,A → RSA,B and
RFSB,B → RSB,A get weights close to 1. This implies that then they indeed both observe
and communicate to each other about the type of music they usually listen to. These
control states are triggered in this example scenario because each of the persons auto-
matically observes his or herself and therefore they quickly (before time point 4) form
representation states RSA,A and RSB,B of their own S-states concerning music (the red
lines, starting at 0.4 for B and at 0.7 for A).
Fig. 2 Outcomes for the example scenario simulation
Because of these communication and observation actions, the mutual subjective rep-
resentations RSA,B of B about A (the dark green line) and RSB,A of A about B (the
orange line) based on fake information are formed, and around time 20 reach levels
close to 0.9. Only now these subjective representations have been formed in a con-
trolled manner, the homophily principle can start to work, as the bonding works through
the (subjective) representation RS-states, not through the (objective) states SX them-
selves. More specifically, from the moment on that the subjective representations of A
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30
X1 - S-A X2 - S-B X3 - FS-B X4 - RS-AA X5 - RS-BB
X6 - RFS-BB X7 - RS-AB X8- RS-BA X9 - RW-AB X10 - RW-BA
X11 - CC-AB X12 - CC-BA X13 - CO-AB X14 - CO-BA
14
about B and A’s own subjective representation about her- or himself get closer than
0.25 (which is just before time point 5), her/his self-model representation RWA,B of her
connection to B (the pink line) starts to gradually increase. Similarly, the effect of the
subjective representations of B for A and B’s own subjective self-model representation
about him or herself, on the subsequent increase of his representation RWB,A of this
connection to A (the blue line) can be noted. Before that point in time their connections
were not increasing, but instead go slightly downward; this illustrates the effect of the
control via the subjective self-model representation states on the adaptation.
5 Discussion
Causal modeling combines two quite useful properties. In the first place, it is an intui-
tive, declarative way of modeling supported by often used graphical representations.
Secondly, due to the universal character of causality, in principle it should apply to
practically all scientific disciplines. However, limitations for dynamics and adaptivity
stand in the way of applicability in many domains. In this paper it was discussed how
these challenges can be addressed by exploiting the notion of self-modeling network
developed from a Network Science perspective [6, 7]. Self-modeling causal networks
cover dynamics of the states of the nodes as well as adaptivity of these causal relations.
Here adaptivity of a base network is obtained by explicit representations of the charac-
teristics of the causal relations in the form of a self-model added to the base network.
These self-modeling causal networks are specified in a declarative manner by mathe-
matical relations and functions, and provide a causal network addressing the adaptation.
Therefore, this approach indeed takes causal modeling to a next level so that now dy-
namics and adaptivity are also covered by a unified causal perspective. By an illustra-
tion for a controlled adaptive social causal network model, it has been shown how this
widens the scope of applicability of causal modeling.
Another topic that illustrates the applicability of the causal modeling approach based
on self-modeling networks well is plasticity and metaplasticity within Cognitive Neu-
roscience, as described, for example, in empirical literature such as [20-23]. In [7], Ch
4, it is shown how this can be modeled as a self-modeling causal network incorporating
a first-order self-model for plasticity and a second-order self-model for metaplasticity.
Such multi-level self-modeling causal networks incorporate different types of cau-
sation. In the first place this covers causation between base states, as is a familiar form
of causation known from traditional causal models. This is also the form of causation
usually focused on (for mental states) within Philosophy of Mind, such as in [14]. How-
ever, in self-modeling network models there is also causation from these base states to
other types of states representing causal relations, and back. Such forms of causation
have to occur as soon as causal relations can change in the world, as such change should
be caused by something. In turn, such changes causally affect the future processes.
So, for adaptive cases, from a completeness of causation perspective such less fa-
miliar forms of causation cannot be avoided, and have direct relations to what actually
happens in the world. Indeed, for example in empirically focussed Cognitive Neurosci-
ence literature such as [20-23], it is described in some detail how states and processes
addressing plasticity and metaplasticity are realised by specific (changing) brain con-
figurations and causal relations for them. So, self-models are not just artificial modeling
concepts created by some fantasy: they relate to real counterparts of them in the physi-
cal world. In that sense, it may be claimed that self-modeling causal networks actually
exist in the world, at least for this context of Cognitive Neuroscience. A similar illus-
tration for the biological domain can be found in [7], Ch. 7, addressing a five-level self-
modeling causal network model describing different stages in an evolutionary process.
Here the different types of states and causation in the self-modeling causal network
have counterparts in the physical world in the form of (changing) configurations and
processes as described in literature from Biology.
The presented approach allows declarative modeling of dynamic and adaptive be-
haviour of multiple orders of adaptation from a unified causal perspective. Tradition-
ally, declarative modeling approaches are a strong focus of AI. There are two
longstanding themes in AI to which the work presented here relates in particular: causal
modeling as already mentioned [1-4] and metalevel architectures and metaprogram-
ming [9-13]. As discussed, a main contribution to the causal modeling area is that this
is extended by dynamics and adaptivity of the causal modeling, addressing both the
dynamics of the causal effects and the adaptive dynamics of the causal relations them-
selves. A main contribution to the area of metalevel architectures and metaprogram-
ming is that now network models are covered as well in the form of self-modeling net-
works, while traditionally the focus in this area is mainly on logical, functional and
object-oriented modeling or programming approaches; e.g., [10].
In relation to the area of Neural Networks within AI, the network-oriented modeling
approach described here distinguishes itself by a multidisciplinary Network Science
focus on causality and adaptation within empirical natural and human-directed sci-
ences. In contrast, the area of Neural Networks has its main focus on artificial neural
networks to solve optimisation challenges and on their computational efficiency. An-
other important distinction is the notion of self-modeling network which is the main
focus in the current paper. However, there are also some technical elements in common,
for example, the format of the canonical difference equation (1) (see Section 2.1) used
here can be considered a form of socalled recurrent network as also used in the Neural
Networks area. But a difference here is the use of speed factors per node which enables
to model different nodes that are not necessarily synchronous in their dynamics. This
asynchrony is usually needed to model real-world processes as these are not often syn-
chronous and can even involve entirely different time scales. This explicit way to model
differentiated timing is not a common practice in the Neural Networks area within AI.
From a more theoretical side, following Ashby [26] in [25] Section 3.1 it has been
shown that any state-determined dynamical system (as defined in [26] and also used in
[27]) can be described by a set of first-order differential equations, and conversely.
Moreover, in [25], Section 3.2 it has also been shown how any set of first-order differ-
ential equations can be (re)modeled by a temporal-causal network model. It has been
shown in [7], Ch 10 that any self-modeling network obtained by adding a self-model to
a temporal-causal network is iself also a temporal-causal network. Therefore, these
methods can also be applied to adaptive processes: any description of an adaptation
process by a state-determined system or by first-order differential equations can be re-
written as a self-model in temporal-causal network format. This provides evidence from
a more theoretical analysis perspective that the approach discussed here has a wide
scope of applicability.
There are still some more interesting challenges that can be addressed. A first chal-
lenge is to explore other interesting cases of higher-order adaptation and to investigate
whether self-modeling causal networks indeed are suitable to model them. Within Cog-
nitive Neuroscience, from an empirical perspective the notions of plasticity and
16
metaplasticity have been introduced [20-23], relating to first- and second-order adapta-
tion. It has been found how these can be modeled by a second-order self-modeling net-
work; see [6] and [7], Ch 4. Similarly, it has been described how second-order adaptive
social networks for bonding by homophily can be modeled by self-modeling networks;
see [6] and [7], Ch 6. However, in general higher-order adaptation for social networks
has not been addressed well in the literature. As an exception, in [28, 29] the notion of
inhibiting adaptation for networks has been described, which refers to some form of
second-order adaptive social networks. This applies, for example, to terrorist network
organisations. It would be interesting to investigate whether and how such second-order
social networks can also be described as self-modeling causal networks.
Within Biology, some literature can be found on how evolutionary processes can be
described as higher-order adaptation; e.g., [30, 31]. It has been shown in [7], Ch 7, how
one case study concerning pregnancy and disgust can be modeled by a fourth-order
adaptive self-modeling causal network model. It is interesting to address more case
studies in this area. Moreover, in Hofstadter [32] claims that the notion of Strange Loop
underlies human intelligence. This is described in [32] informally as a form of self-
modeling of multiple levels, where for some n, the nth level is equal to the base level,
so that the levels form a cycle. It has been found that this also can be modeled by a self-
modeling network; see [7], Ch 8 for an example for a mental network and [33] for an
example for a social network. However, the notion of Strange Loop could be explored
for more cases.
Finally, as mentioned the research described in the current paper follows the multi-
disciplinary perspective of Network Science. Therefore, the focus is on adaptation prin-
ciples known from nature and described in empirical disciplines such as Biology, Neu-
roscience, Cognitive Science or Social Sciences. In contrast, it may be an interesting
challenge to investigate how some wellknown artificial methods for machine learning
can be modeled by self-modeling networks. As the self-modeling network approach
provides a declarative perspective on modeling adaptation processes, this might provide
more declarative descriptions of such artifical methods, which usually are described in
a procedural manner by algorithms. As pointed out in one of the paragraphs above,
from a theoretical perspective this should be possible. But it would be interesting to see
how this would actually look like for some examples. It might provide a more clear
modeling separation of the conceptual core of a machine learning method and the pro-
cedural optimisation involved. As an example, in this way backpropagation for artificial
neural networks could be modeled in a network-oriented manner with gradient descent
as conceptual core plus an efficient procedure to do the required calculations; e.g., [34],
Ch 7.
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