Multilevel Network Reification: Representing Higher

Multilevel Network Reification: Representing Higher Order Adaptivity in a Network Jan Treur Behavioural Informatics Group, Vrije Universiteit Amsterdam [email protected] Abstract 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.

1 Introduction Reification literally means representing something abstract as a material or concrete thing (Merriam-Webster dictionary), or making something abstract more concrete or real (Oxford dictionaries). It is a notion that is known from different scientific areas; in [13] it has been shown how it can be used to explicitly represent adaptation principles in networks. Reification can also be applied again on reified structures, thus obtaining repeated or multilevel reification. In the current paper such a construction of multilevel reification is applied to networks, illustrated for a Network-Oriented Modeling approach based on temporal-causal networks [11, 12]. By reifying the parameters of the base network structure (connection weights, speed factors, and combination functions) by adding them as states in the extended network, and defining proper temporal-causal relations for themselves and with the other states, a reified network explicitly represents the characteristics of the base network, and how this base network evolves over time based on adaptation principles that change the causal relations. This was illustrated in [13] for a variety of adaptation principles known from Cognitive Neuroscience and Social Science. Adaptation principles may be adaptive themselves too, according to certain second-order adaptation principles. To model such second-order adaptivity, the reified network can be reified itself, thus providing multilevel reification. This multilevel reification construction is introduced here; it can be used to model higher order adaptivity of any level. Higher order adaptivity can occur in many forms and applications. Some examples are the following:  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 second-order adaptation has been discussed for plasticity of the brain and in evolutionary context, for example, see [1, 3, 8].

 In an adaptive social network based on an adaptation principle for bonding based on homophily [7] the similarity measure determining how similar two persons are may change over time, for example, due to age or other varying circumstances The homophily context will be used as illustration in the current paper. In the paper, first in Section 2 the Network-Oriented Modeling approach based on temporal-causal networks is briefly summarized. Next, in Section 3 the idea of reifying the network structure is briefly introduced. Section 4 introduces a multilevel network reification construction and shows how it can model second-order network adaptivity. This is illustrated by a second-order adaptive network based on a firstorder adaptation principle for bonding based on homophily, and a second-order adaptation principle for the characteristics of this first-order adaptation principle. In Section 5 example simulations within a developed software environment for this multilevel network reification are presented. Section 6 is a discussion.

2 Temporal-Causal Networks: Structure and Dynamics A conceptual representation of the network structure of a temporal-causal network model is described by a graph with nodes and directed connections which also includes a number of labels for such a graph: connection weights X,Y, speed factors Y of states Y, and combination functions for states Y. These three notions form the defining network structure of a temporal-causal network model; see Table 1, upper part; see Fig. 1 for an example of a basic fragment of a network with states X1, X2 and Y, and labels X1,Y, X2,Y for connection weights, cY(..) for combination function, and Y for speed factor. Table 1 Conceptual and numerical representation of a temporal-causal network structure Concepts States and connections

Notation

Explanation Describes the nodes and links of a network structure (e.g., in X, Y, XY graphical or matrix format) The connection weight X,Y  [-1, 1] represents the strength Connection weight of the causal impact of state X on state Y through connection X,Y XY For each state Y (a reference to) a combination function cY(..) Aggregating cY(..) is chosen to combine the causal impacts of other states on multiple impacts state Y Timing of the For each state Y a speed factor Y  0 is used to represent Y causal effect how fast a state is changing upon causal impact Concepts Numerical representation Explanation At each time point t each state Y in State values over Y(t) the model has a real number value in time t [0, 1] impactX,Y(t) At t state X with connection to state Y Single causal impact = X,Y X(t) has an impact on Y, using weight X,Y The aggregated causal impact of aggimpactY(t) Aggregating multiple states Xi on Y at t, is = cY(impactX1,Y(t),…, impactXk,Y(t)) multiple impacts determined using combination = cY(X1,YX1(t), …, Xk,YXk(t)) function cY(..) The causal impact on Y is exerted Y(t+t) = Y(t) + Y [aggimpactY(t) - Y(t)] t Timing of the over time gradually, using speed = Y(t) + causal effect factor Y Y [cY(X1,YX1(t), …, Xk,YXk(t)) - Y(t)] t

To provide sufficient flexibility, a library with a number of standard combination functions are available as options, but also own-defined functions can be added. In Table 1, lower part it is shown how a conceptual representation describing a network structure defines a numerical representation of the network’s dynamics; see also [11], Ch. 2. Here X1, …, Xk are the states with outgoing connections to state Y. This defines the detailed dynamic semantics of a temporal-causal network. The difference equations in the last row in Table 2 can be used for simulation and mathematical analysis, and can also be written in differential equation format: dY(t)/dt = Y [cY(X1,YX1(t), …, Xk,YXk(t)) - Y(t)]. Examples of combination functions often used for aggregation are the identity id(.) for states with impact from only one other state, the scaled sum ssum(..) with scaling factor , the minimum function min(..), and the advanced logistic sum combination function alogistic,(..) with steepness  and threshold ; see also [11], Ch 2, Table 2.10: id(V) = V, ssum(V1, …, Vk) = (V1, …, Vk)/, min(V1, …, Vk) = minimum of Vi, and alogistic,(V1, …,Vk) = [ 1/(1+e–σ(V1+ … + Vk -)) – 1/(1+eσ)] (1+e–σ).

X1

X1,Y X2,Y

Y

cY(..)

Y

X2 Fig. 1 A fragment of a temporal-causal network structure in a conceptual labeled graph representation

3 Network Reification Network reification provides a way to extend a base network by extra states that represent the parameters describing the network structure. The new additional states representing the parameter values for the network structure are what are called reification states for the parameters, the parameters are reified by these states. What will be reified in temporal-causal networks in particular are the parameters used to define their network structure: the labels for connection weights, combination functions, and speed factors (see Table 1). For connection weights Xi,Y and speed factors Y, their reification states Xi,Y and  Y represent the value of them. These reification states are depicted in the upper plane in Fig. 2, whereas the states of the base network are displayed in the lower plane. By having explicit states for the reified characteristics of a network within the reified network, causal relations can be added affecting them, and causal connections from them to related base network states, thus explicitly representing adaptation principles within the (reified) network, as shown in [13]. To this end it is defined how the reification states contribute to an aggregated impact on the related base network state (see the downward arrows in Fig. 2).

Y

C3,Y

X1,Y

C2,Y

X2,Y

X1

C1,Y

Y X2

Fig. 2 Network reification for temporal-causal networks: downward connections from reification states to base network states

These downward causal relations and the combination functions will be defined in a generic manner, related to how a specific parameter functions in the overall dynamics as part of the (intended) semantics of a temporal-causal network. More specifically, the general pattern is that each of the reification states Xi,Y, Y and CY for connection weights, speed factors and combination functions has a causal connection to state Y in the base network, as they all affect Y. These are the downward arrows from the reification plane to the base plane in Fig. 2. All depicted (downward and horizontal) connections get weight 1. In the extended network the speed factors of the base states are set at 1 too. As the base states have more incoming connections now, new combination functions for them are needed. The different components C1,Y, C2,Y, … for CY are explained as follows. A countable number of basic combination functions bc1(..), bc2(..), …is assumed. From this sequence of basic combination functions bc1(..), bc2(..), …… for any arbitrary m the finite subsequence up to m can be chosen to be used in a specific application. For example, bc 1(..) = sum(..), bc2(..) = ssum(..) and bc3(..) = alogistic,(..). In the base network for each Y combination function weights are assumed: numbers cw1,Y, cw2,Y,… 0 that may change over time such that the combination function cY(..) is expressed by: cY(t, V1, …, Vk) = [ cw1,Y(t) bc1(V1, …, Vk) + … + cwm,Y(t) bcm(V1, …, Vk) ]/ [cw1,Y(t) + … + cwm,Y(t)] In this way it can be expressed that for Y a weighted average of basic combination functions is used, if more than one of cwi,Y(t) has a nonzero value, or just one basic combination function is selected for cY(..), if exactly one of the cwi,Y(t) is nonzero. This approach makes it possible, for example, to smoothly switch to another combination function over time by decreasing the value of cwi,Y(t) for the earlier chosen basic combination function and increasing the value of cwj,Y(t) for the new choice of combination function. For each basic combination function weight cwi,Y(..) a different reification state Ci,Y is added. The value of that state represents the extent to which that basic combination function bci(..) is applied for state Y. In (Treur,

2018b) it has been found that the combination function for base state Y in the reified network is as follows: c*Y(S, C1, …., Cm, V1, …, Vk, W1, …, Wk, W) = S (C1 bc1(W1V1, …, WkVk) + … + Cm bcm(W1V1, …, WkVk))/(C1 + ….+ Cm) + (1-S) W where S is used for the speed factor reification HY(t), Ci for the combination function weight reification Ci,Y(t), Vi for the state value Xi(t) of base state Xi, Wi for the connection weight reification Xi,Y(t), and W for the state value Y(t) of base state Y. This combination function c*Y(..) makes that the dynamics of Y within the reified network is described by the following difference equation: Y(t+t) = Y(t) + [c*Y(HY(t), C1,Y(t), …, Cm,Y(t), X1(t), …, Xk(t), X1,Y(t), …, Xk,Y(t), Y(t)) - Y(t)] t = Y(t) + [ HY(t) (C1,Y(t) bc1(X1,Y(t) X1(t), …, Xk,Y(t) Xk(t)) + … + Cm,Y(t) bcm(X1,Y(t) X1(t), …, Xk,Y(t) Xk(t)))/(C1,Y(t) + ….+ Cm,Y(t)) + (1- HY(t)) Y(t) - Y(t)] t Note that in the reification process structures are added which are not reified themselves. A next step is to explore whether and how the structure of the reified network also can be reified within a second-order reification. Structures in the firstorder reified network not part the base network are, for example, those used to model specific adaptation principles for evolving networks. In a second-order reified network these adaptation principles can be explicitly represented by states at the second reification level and considered to be adaptive themselves too. In next section it is explored how such second-order reification can be useful to model such adaptive adaptation principles.

4 Multilevel Network Reification In this section it will be shown how network reification can be applied in a repeated manner, and thus provide multilevel reification. It will be shown how multilevel reification can be useful in modeling evolving networks based on adaptation principles that change over time themselves: adaptive adaptation principles. So, the idea is that the base network evolves through an adaptation principle (called firstorder adaptation principle), and that adaptation principle changes itself based on another adaptation principle (called second-order adaptation principle). The firstorder adaptation principle is represented at the first reification level, and the secondorder adaptation principle at the second reification level. The approach will be illustrated through an example inspired by Social Science, in particular concerning the way in which connections between two persons are adapted over time based on similarity between the persons (the homophily adaptation principle as a first-order adaptation principle). One of the notions that plays an important role in the homophily adaptation principle is the homophily similarity tipping point or threshold . This indicates the value such that when the difference 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 often considered constant, but it may be more realistic when they are

considered adaptive over time. This is indeed what is done in the current section. An adaptive form of the homophily adaptation principle is considered where the tipping points change over time by a second-order adaptation principle, and the same applies to the speed factors for different persons of the connection weight adaptation based on the homophily adaptation principle. In Fig. 3 the architecture of the (second-order) multilevel reified network is shown. The middle plane shows how the connection weights have been reified (firstorder reification), and the network relations of the reification states Xi,Y to the states Xi and Y in the base network (lower plane) are shown. These network relations (including their labels, such as combination functions; see below) define the firstorder adaptation principle based on homophily and its semantics. Note that speed factor reification states Y and combination function reification states Ci,Y for the base network states Y have been left out of the middle plane here as for this example focusing on the homophily adaptation principle they are considered constant.

TX1,Y

HX1,Y

TX2,Y

HX2,Y

X1,Y X2,Y X1

Y X2

Fig. 3 Multilevel Reified Conceptual Modeling of Homophily in Social Networks with reified tipping point states and connection weight adaptation speed factors at the second reification level (third, upper plane)

On top of this first-order reified network another reification level has been added (the upper plane), in order to get a second-order reified network. Here reification states TXi,Y are added for the similarity tipping point characteristic of the homophily adaptation principle, and reification states HXi,Y for the speed factor characteristic of the homophily adaptation principle. Also for these reification states (upward and downward) connections have been added that (together with the relevant labels; see below) define the second-order adaptation principle based on them. After having defined the basic architecture, next the labels for the new network are defined.

Base level The speed factors and connection weights in this new network are kept simple: all of them are set at 1. What remains is to define the combination functions; for the base level, this is done by c*Y(V1, …, Vk, W1, …, Wk) = cY(W1V1, …, WkVk) = [ cw1,Y(t) bc1(W1V1, …, WkVk) + … + cwm,Y(t) bcm(W1V1, …, WkVk) ]/ [cw1,Y(t) + … + cwm,Y(t)] where Vi stands for Xi(t), and Wi stands for Xi,Y(t); see also (Treur, 2018b). Here the coefficients cwi,Y(t) are assumed constant. For example, if cY(..) is chosen as the advanced logistic sum function alogistic,(…), which is the third in the row bc1(..), bc2(..)…, then cw1,Y = 0, cw2,Y = 0, and cw3,Y = 1, and this obtains: c*Y(V1, …, Vk, W1, …, Wk) = bc3(W1V1, …, WkVk) = alogistic,(W1V1, …, WkVk) = [1/(1+e–σ(W1V1+ … + Wk Vk -)) – 1/(1+eσ)] (1+e–σ) First reification level Next, consider the combination function defining the homophily adaptation principle at the first reification level (the middle plane); see [7] and [11], Ch 11, Section 11.7: c*Xi,Y(S, V1, V2, T, W) = S (W + Xi,Y W (1-W) (T - |V1 -V2|)) + (1-S) W where S is used for the Xi,Y speed factor reification HXi,Y(t), V1 for Xi(t), V2 for Y(t), T for Xi,Y homophily tipping point reification TXi,Y(t), W for connection weight reification Xi,Y(t), and Xi,Y is a homophily modulation factor. This combination function (together with connection weights and speed factor 1) defines the following difference equation for Xi,Y (see Section 2, Table 1): Xi,Y(t+t) = Xi,Y(t) + [HXi,Y(t) (Xi,Y +  Xi,Y (1-Xi,Y) (TXi,Y(t) - | Xi(t) - Y(t)|)) + (1- HXi,Y(t)) Xi,Y - Xi,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 TXi,Y(t) and decrease will take place when this difference is more. Second reification level Finally, consider how the tipping points should be adapted to circumstances. Here the idea is that the tipping point of a person will become higher if the person lacks strong connections (the person becomes less strict) and will become lower if the person already has strong connections (the person becomes more strict). This is modeled using an average norm weight  for connections. This can be considered to relate to the amount of time or energy available for social contacts. If the connections of a person are on average stronger than , downward regulation takes place: the tipping point will become lower and when the connections of this person are on average weaker than  upward regulation takes place: the tipping point will become higher. This is described by the following combination function for the second reification level (the upper plane): c*TY,Xi(W, T) = T + T Y,Xi T(1-T) (T Y,Xi - (W1 + .. + Wk)/k)

where T is used homophily tipping point reification value TY,Xi(t) for Y,Xi and Wj for connection weight reification value Y,Xj(t); T Y,Xi is a modulation factor, and TY,Xi is a norm for Y for average connection weight for Y,X1 to Y,Xk. This combination function (together with connection weights and speed factor 1) defines the following difference equation for TY,Xi(t) (see Section 2, Table 2): TY,Xi(t+t) = TY,Xi(t) + [TY,Xi(t) + TY,Xi TY,Xi(t) (1- TY,Xi(t)) (TY,Xi - (Y,X1(t) +..+ Y,Xk(t))/k) - TY,Xi(t)]t Note that as a slightly different variant the division by k can 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*TX ,Y(W, T) = T + TX ,Y T(1-T) (TX ,Y - (W1 + .. + Wk)/k) i

i

i

where T is used for Xi,Y homophily tipping point reification value TXi,Y(t) and Wj for connection weight reification value Xj,Y(t) TX ,Y is a modulation factor for TXi,Y i

TX ,Y is a norm for average (incoming) connection weights for Y i 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*HY,Xi(W, S) = S +  S (1-S) (H Y,Xi - (W1 + .. + Wk)/k) where S is used for Y,Xi speed factor reification value HY,Xi(t), Wj for connection

weight reification value Y,Xi(t), HY,Xi is a modulation factor for HY,Xi, and HY,Xi is a norm for average of (outgoing) connection weights for Y. This combination function defines the following differential equation for HY,Xi: HY,Xi(t+t) = HY,Xi(t) + [HY,Xi(t) + HY,Xi HY,Xi(t)(1 - HY,Xi(t))(HY,Xi - (Y,X1(t) +.. + Y,Xk(t))/k) - HY,Xi(t)]t Also here an opposite variant is possible: c*HXi,Y(W, S) = S + HXi,Y S (1-S) (HXi,Y - (W1 + .. + Wk)/k) where S is used for Xi,Y speed factor reification value HXi,Y(t), Wj for connection weight reification value Xj,Y(t), HXi,Y is a modulation factor for HXi,Y, and HXi,Y is a norm for average of (incoming) connection weights for Y

5 Simulation Scenarios The scenarios concern 10 persons X1 to X10. For the first two scenarios only the outgoing connections of X1 have been modelled in an adaptive manner, the other connection weights were kept constant. For all simulations t = 1 was used, and the focus in al three scenarios was on the homophily adaptation with constant connection weight speed factor HXj,Xi = Xj,Xi = 1. Moreover, in Scenarios 1 and 2 the focus is only on the adaptive connections from X1, and the other connections were kept constant. Scenarios 1 and 2 can be found in the Appendix 1. In Table 2 the main parameter values for Scenario 3 can be found. Scenario 3: All connections adaptive For the third scenario all connections were adaptive with main parameters shown in Table 2 and initial connection weight values shown in Table 3. Note that the norm for average connection weight is 0.4 this time. Table 2 Scenario 3: Main parameter values Base level Contagion alogistic steepness Xi for Xi

0.8

Contagion alogistic threshold Xi for Xi

0.15

Speed factor Xi for base state Xi

0.5

First reification level Homophily modulation 1 factor Xj,Xi for Xj,Xi Connection weight speed factor Xj,Xi for Xj,Xi

1

Second reification level Tipping point speed factor TXj,Xi for TXj,Xi 0.5 Tipping point modulation factor TXj,Xi for TXj,Xi 0.4 Tipping point connection norm TXj,Xi for TXj,Xi

0.4

Table 3 Scenario 3: Initial connection weights connections X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

X1 0.5 0.3 0.6 0.2 0.6 0.2 0.6 0.6 0.6

X2 0.5 0.6 0.4 0.5 0.6 0.8 0.5

X3 0.3 0.6 0.6 0.7 0.6 0.6

0.7

X4 0.1 0.3 0.7 0.7 0.5 0.7 0.4 0.7 0.7

X5 0.2 0.4 0.7 0.4

X6 0.6 0.7 0.4 0.6 0.4

X7 0.5 0.7 0.4 0.7 0.7

0.6 0.4 0.4

0.7 0.6

X8 0.2 0.9 0.8 0.4 0.7 0.7

0.5 0.7

0.6

X9 0.3 0.5 0.6 0.9 0.5 0.4

X10 0.4 0.8 0.9 0.4 0.7 0.5 0.6

0.8

In Figs 4 to 7 the simulation outcomes are shown. As can be seen in Fig. 7 eventually all connection weights converge to 0 or 1. Fig. 4 shows in particular the values of the connection weights from X1, and their average, and Fig. 5 shows the corresponding tipping points. 1

http://www.few.vu.nl/~treur/AppendixSimulationScenarios.pdf

  X1,X2  X1,X3  X1,X4  X1,X5  X1,X6  X1,X7  X1,X8  X1,X9  X1,X10 average  X1,Xi

Fig. 4 Scenario 3: Adaptive weights of outgoing connections from X1 over time, with the thick pink line showing the average weight for X1 X ,X  1 2 X ,X  1 3  X1,X4 X ,X  1 5 X ,X  1 6 X ,X  1 7 X ,X  1 8  X1,X9  X1,X10

Fig. 5 Scenario 3: Adaptive tipping points TX1,Xj over time X1,Xi average X2,Xi average X3,Xi average X4,Xi average X5,Xi average X6,Xi average X7,Xi average X8,Xi average X9,Xi average X10,Xi average Overall average

Fig. 6 Scenario 3: Average connection weights for each of X1 to X10 and of all connections over time

Note that Fig. 6 shows that in the emerging process eventually the average connection weights per person stick in some seemingly mysterious 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 TXi,Xj for average connection weight for each person was 0.4). Also in

other simulations this discrete set of multiples of 0.111111 emerges. In Section 6 it will be analysed where these values come from.

Fig. 7 Scenario 3: All connection weights are 0 or 1 at time 1750

In Fig. 7 it is shown that all connection weights converge to 0 or 1. This will also be analysed in Section 6. For the tipping points, for all outgoing connections of X1 they converge to 0 (see also Fig. 6), and for all outgoing connections of the other persons they converge to 1.

6 Analysis of Equilibrium Values In this section it is analysed what are the possible values to which certain states in the second-oreder reified network may converge. This is done by an analysis of equilibrium states. A state Y has a stationary point at t if dY(t)/dt = 0. The network is in equilibrium a t if every state Y of the model has a stationary point at t. Considering the differential equation for a temporal-causal network model (and assuming a nonzero speed factor) a more specific criterion can be found: Criterion for a stationary point in a temporal-causal network Let Y be a state with speed factor Y > 0 and X1, ..., Xk the states with outgoing connections to state Y. Then Y has a stationary point at t if and only if cY(X1,YX1(t), …, Xk,YXk(t)) = Y(t) Note that this can be applied to the states at all levels in the second-order reified network, in particular to the first and second reification level. First, in an equilibrium state, the above criterion applied to tipping point reification states at the second reification level is as follows: c*TXi,Xj(W, TXi,Xj(t)) = TXi,Xj(t) This provides the folowing equation TXi,Xj(t) + TXi,Xj TXi,Xj(t) (1- TXi,Xj(t)) (TXi,Xj - (W1 + .. + Wk)/k) = TXi,Xj(t) which can be rewritten as follows TXi,Xj TXi,Xj(t) (1- TXi,Xj(t)) (TXi,Xj - (Xi,X1(t) + .. + Xi,Xk(t))/k) = 0 Assuming TXi,Xj nonzero, this equation has three solutions:

TXi,Xj(t) = 0

or

TXi,Xj (t) = 1 or (Xi,X1(t)+ .. + Xi,Xk(t))/k = TXi,Xj

Similarly, the criterion can be applied to connection weights at the first reification level: c*Xi,Xj(HXi,Xj (t), Xi(t), Y(t), TXi,Xj(t), Xi,Xj(t)) = Xi,Xj(t) This provides the following equation HXi,Xj(t) (Xi,Xj(t) + Xi,Xj Xi,Xj(t) (1-Xi, Xj(t)) (TXi,Xj(t) - | Xi(t) - Y(t)|)) + (1- HXi,Xj(t)) Xi,Xj(t) = Xi,Xj(t) which can be rewritten as follows HXi,Xj(t) Xi,Xj Xi,Xj(t) (1-Xi, Xj(t)) (TXi,Xj(t) - | Xi(t) - Y(t)|) + Xi,Xj(t) = Xi,Xj(t) HXi,Xj(t) Xi,Xj Xi,Xj(t) (1-Xi, Xj(t)) (TXi,Xj(t) - | Xi(t) - Y(t)|) = 0 Assuming Xi,Xj nonzero, this equation has four solutions: HXi,Xj(t) = 0 or Xi,Xj(t)= 0

or

Xi,Xj(t)= 1

or |Xi(t) – Xj(t)| = TXi,Xj(t)

Combined with the three solutions for TXi,Xj(t) the matrix in Table 4 can be found. HXi,Xj(t) = 0

Xi,Xj= 0

Xi,Xj= 1

|Xi – Xj| = TXi,Xj(t)

TXi,Xj(t) = 0

HXi,Xj(t) = 0 TXi,Xj(t) = 0

Xi,Xj= 0 TXi,Xj(t) = 0

Xi,Xj= 1 TXi,Xj(t) = 0

Xi = Xj TXi,Xj(t) = 0

TXi,Xj(t) = 1

HXi,Xj(t) = 0 TXi,Xj(t) = 1

Xi,Xj= 0 TXi,Xj(t) = 1

Xi,Xj= 1 TXi,Xj(t) = 1

Xi = 0 and Xj = 1 or Xi = 1 and Xj = 0 TXi,Xj(t) = 1

HXi,Xj(t) = 0 m Xi,Xm/k = TXi,Xj m Xi,Xm /k = TXi,Xj

XXXXXXX

XXXXXXX

|Xi – Xj| = TXi,Xj(t) m Xi,Xm /k = TXi,Xj

Table 4 Overview of the different solutions of the equilibrium equations for Xi,Xj and TXi,Xj.

In cases that not |Xi – Xj| = TXi,Xj(t), and HXi,Xj(t) is nonzero (which was the case for the simulations displayed in Section 5), all Xi,Xj are 0 or 1, so with k = 9 the average (Xi,X1 + .. +  Xi,Xk)/k is a multiple of 1/9, which in the simulation case the norm TXi,Xj = 0.4 is not. Therefore the cases (Xi,X1(t) + .. + Xi,Xk(t))/k = TXi,Xj cannot actually occur, so then TXi,Xj(t) = 0 or TXi,Xj(t) = 1 and Xi,Xj(t) = 0 or Xi,Xj(t) = 1 are the only solutions, which is also shown in the simulations (e.g., see Fig. 12), and indeed all averages are multiples of 1/9 = 0.111111, as observed above (see Fig. 11). This explains the specific discrete set of numbers 0.111111, 0.222222, 0.333333, … observed in the simulations. Note that although in general the reified speed factor HXi,Xj(t) may be assumed nonzero, but there are also specific processes in which it converges to 0, for example, the temperature in simulated annealing.

7 Discussion The construction of network reification can provide advantages similar to those found for reification in modeling and programming languages in other areas of AI and Computer Science; e.g., [2, 4, 5, 6, 9, 10, 14]. A reified network including an explicit representation of the network structure enables to model dynamics of the original network by dynamics within the reified network. In this way an adaptive network can be represented by a non-adaptive network. In [13] it is shown how network reification provides a unified manner of modelling adaptation principles, and allows comparison of such principles across different domains. In the current paper it was shown how a multilevel reified network can be obtained by reifying an already reified network. It was shown how first-order adaptation principles which are explicitly represented at the first reification level can be made adaptive themselves by adding explicit representations of their characteristics at the second reification level, and specifying a second-order adaptation principle at that level. Thus a network architecture is obtained for adaptive adaptation principles: (second-order) adaptation principles applied to (first-order) adaptation principles. This was illustrated in particular for a first-order adaptation principle based on homophily from Social Science [7] represented at the first reification level, and a second-order adaptation principle describing change of characteristics of this first-order adaptation principle. This second order adaptation was applied to the characteristics similarity tipping point and speed factor for the firstorder adaptation principle based on homophily. Network reification will increase complexity, but for one reification step this will at most be quadratic in the number of nodes N; see [13]. If this 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 illustration in the current paper, the parameter TXi,Xj for 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 n n reification levels are added, then it is in the order of N(2 ), 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 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. Note that in an nth order reified network still there will be network structures introduced in the last step from n-1 to n that have no reification within the nth order reified network. From a theoretical perspective it can also be considered to repeat the construction infinitely many times, for all natural numbers n; this can be called order reification, where  is the ordinal for the natural numbers. Then an infinite network is obtained, which is theoretically well-defined as a mathematical structure. All network structures in this -order reified network are reified within the network

itself, so it is closed under reification. But it may not be clear whether such an -order construction has a useful application in practice, or be used to explore theoretical research questions. This may be a subject for future research.

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