A multi-adaptive network model for human Hebbian learning, synchronization and social bonding based on adaptive homophily

This paper present a multi-adaptive network model integrating multiple adaptation mechanisms, specifically focusing on five types of such adaptation mechanisms. Two of them address first-order adaptation by learning of responding on others and first-order adaptation by bonding with others based on homophily. Three other adaptation mechanisms addressed are second-order adaptation of the speed of both Hebbian learning and bonding by homophily, and second-order adaptation of the homophily tipping point. The paper provides a comprehensive explanation of these concepts and their role in controlled adaptation within the diverse contextual scenarios of the paper.


Introduction
The phenomenon of human attunement and social bonding has been a curious topic for researchers from various scientific disciplines for decades.From an early age, humans exhibit a remarkable ability to attune their actions, thoughts, and emotions with others, forming strong social bonds that shape our interactions and relationships.This innate capacity for synchronization and social bonding plays a vital role in our social development, cooperation, and overall well-being.An interesting phenomenon regarding human interaction in daily life activities is our ability to effectively coordinate our actions with each other, especially given that it must occur in a dynamic environment, as nothing around us is constant (Van der Steen and Keller, 2013).
To model these various dynamics of human interaction in real-world scenarios, it is essential to consider both cognitive processes in the brain and interpersonal relationships within social networks.The adopted network-oriented modeling approach in the paper, serves as a conceptual tool for understanding the structure and dynamics of complex adaptive processes (Treur, 2016(Treur, , 2020)).The presented approach integrates in a multilevel manner two fundamental principles: Hebbian learning for progressive changes in connections between mental states (modeling plasticity in the brain) and the bonding by homophily principle for adaptive interpersonal connectedness in modeling adaptive social networks (Hebb, 1949;McPherson, Smith-Lovin, Cook, 2001).For both fundamental principles a variant for first-order adaptation and one for second-order adaptation is addressed.
The paper examines how homophily-based adaptation operates in different multi-agent interaction scenarios using a social network model with multiple agents.The objective is on the one hand to understand how the strength of connections between individuals changes based on their similarities, but on the other hand how this can depend on circumstances, in particular on how many or how strong relationships someone already has.
The approach followed is that when individuals are less dissimilar than a tipping point τ, their connection strengthens and otherwise it becomes weaker as applied in (Sharpanskykh and Treur, 2014), see also (Accetto, Treur, Villa, 2018;Blankendaal, Parinussa, Treur, 2016;Kozyreva, Pechina, Treur, 2018;Van den Beukel, Goos, Treur, 2019).To address this, in the current paper an adaptive tipping point for the homophily is used, which is a form of second-order adaptation over the first-order adaptation described as bonding by homophily.By making the homophily tipping point adaptive, it ensures that it fluctuates in response to changes in how many or how strong relations already exist for the person.Two approaches are explored to model these adaptive tipping points: a simple linear tipping point function that focuses on the average weight of the existing connections of a person (indicating how strong are the connections), and a logistic tipping point function that emphasizes the total sum of the connection weights (indicating how many are they).

Background knowledge
In this section we introduce the main concepts used in this paper, both about the human processes addressed and about the modelling approach followed.First the two first-order adaptation principles are briefly explained.

First-order adaptation principles
Two first-order adaptation principles are addressed:

Hebbian learning adaptation principle
Hebbian learning is a type of synaptic plasticity that occurs when there is consistent and correlated activation between two neurons (Choe, 2014).This learning principle, first proposed by Donald Hebb in 1949, suggests that if a presynaptic neuron repeatedly activates a postsynaptic neuron while both neurons are active, the connection between them becomes stronger (Hebb, 1949;Shatz, 1992).In simpler form, Hebbian learning describes how neurons strengthen their communication pathway when they fire together frequently.An important element is the causality involved in the simultaneous firing.It is the repeated participation (consistency) of one neuron in the firing activity (causality) of the other that leads to strengthened connectivity (Keysers and Gazzola, 2014).In other words, the focus is on the sustained and causal engagement of one neuron with the firing of another.

Homophily adaptation principle
In regards to interpersonal relations, a key feature that has an influence on the level of adaptation is the extent of similarity between the individuals in question (McPherson et al.,2001).People who are more 'alike' tend to develop stronger connections, rather than the ones who are less 'alike' (Treur, 2020, Ch 6 and 13).Such behavior can be seen, for example, in children's interaction.Researchers in the field of social science have observed that school children exhibit a tendency to form friendships and engage in play groups more frequently when they share similarities in demographic characteristics, reinforcing their position within the social space (McPherson et al.,2001).This type of adaptation can be explained using the homophily principle.According to this principle, the more alike the states of two connected individuals are, the stronger their connection becomes.

Second-order adaptation principles
In order to apply the above mentioned adaptation principles, it is important to understand the overall structure of the entire model.Adaptive network models can be depicted in a form of a multi-level construction, containing different orders of adaptations on different levels.The base mental network model is depicted at the lower level, with connections between different mental states.As default, these base states have static parameters for weights of connections between each other, which is not a realistic set up if the goal is to mimic a real-world

Table 1
Base level states and their explanation.
Y. Mukeriia et al. scenario.Thus, in order to make them adaptive, a 'first-order adaptation level' is introduced, modeling first-order connection adaptation principles such as Hebbian learning and bonding by homophily, briefly explained in the Section 2.1 above.Now, these first-order adaptation principles themselves also have parameters by default, and these parameters can in reality also be dynamic, in this case by second-order adaptation.In this paper, second-order adaptation is applied in three ways: to the adaptation speed both of the Hebbian learning and of the bonding by homophily, and to the tipping point of the homophily adaptation.Such second-order adaptation mechanisms are sometimes described as metaplasticity or 'plasticity of the plasticity' (Abraham and Bear, 1996;Robinson, Harper, & McAlpine, 2016;Sjöström, Rancz, Roth, & Hausser, 2008).

The self-modeling modeling approach used
The modeling approach used is network-oriented and addresses adaptation of networks by self-modeling networks (Treur, 2016(Treur, , 2020)).

Network models
Within a network model, like in neural network models at each time point t, the impact of state X on state Y is depicted through their connections, as where ω X,Y is the weight of the connection from X to Y and X(t) the activation value of X at time t.
There could be one or multiple connections impacting a single state, and in order to aggregate them, combination functions c Y (..) are used This corresponds to activation functions in neural network models.
Lastly, the speed factor η Y is what determines the timing with which the aggegrated impact aggimpact Y (t) is exerted on state Y. Thus, the full difference and differential equations are the following (Treur, 2016):

Table 2
First-order self-model states and their explanation.

Table 3
Second-order self-model states and their explanation.
The three types of network characteristics ω X,Y , c Y (..), η Y defining a temporal-causal network in principle are constant, in which case they model a nonadaptive network.However, the realistic case is that they also change over time and therefore the network is adaptive.Selfmodelling networks are an easy way to address adaptivity by adding states (called self-model states) to the network that represent these adaptive network characteristics.For example, a self-model state W X,Y can be added to represent a connection weight ω X,Y , or a self-model state H Y can be used to represent a speed factor η Y ; such types of self-model states are also called W-states and H-states, respectively.In such a case in equations ( 2), ( 3) and (4), for the adaptive network characteristics ω and η the values of their corresponding self-model W-states and H-states are used: (5) This can also be done in a higher-order manner, for example, a second-order self-model state H W Y (also called an H W -state) can be used to represent the adaptation speed of W X,Y ; see (Treur, 2020), Ch 4 for more details.This can be used to model metaplasticity, e.g., (Abraham & Bear, 1996;Sjöström et al., 2008).

Combination functions and their parameters.
Next, we will introduce the combination functions used in the paper, and the individual parts of the paper will then delve into the application of the functions that are relevant to their models (Treur, 2019a,b).The identity function id(.) is the combination function that is usually used for the states that have an impacting link from only one state.The numerical representation of the function is The scaled maximum function smax λ (…) is used to ensure a smooth transition from one value to another.In this paper, it is only applied to a self-modeling state for a combination function.Numerically, it is represented as where λ is the scaling factor.The advanced logistic sum function alogistic σ,τ (…), is usually used for the states that have impacting connections from multiple states, and can be expressed as es X,a → bs X,a 1 0.9 0.9 0.9 Inter-Agent Expressive Link ps X,a → es X,a 0.95 0.9 1 0.4 where σ is a steepness parameter and τ the excitability threshold parameter.
The Hebbian function hebb μ (…), is used for adaptation based on Hebbian learning and is expressed for self-model state W X,Y as follows: where μ in [0, 1] is the persistence factor.The simple linear homophily function slhomo α,τ (…), is applied to the homophily adaptation states W X,Y .This function has two parameters: amplification factor α and the tipping point value τ, and can be expressed as The step-once function steponce α,β (…) is used to model changing context factors, and is defined by two parameters: the start point α and the end point β with-in the simulation time t, during which the state needs to be active.This function is applied in cases when a state is not needed throughout the whole simulation.In mathematical terms, the function yields a value of 1 if α ≤ t ≤ β, else 0. The step-modulo function stepmod ρ,δ (…) is used to model behavior with constant intervals, using two parameters: the repeated time duration ρ and the tipping point δ.The function can be expressed as a formula which yields 0 if mod(t, ρ) < δ, else 1.
The simple linear tipping point function sltip ν,α (…), is used for the homophily adaptive tipping point self-model TP-states.It can be expressed as Table 5 Role matrix ms specifying the both the adaptive and nonadaptive speed factors.where α is the modulation factor for the tipping point TP W Xi,Y and ν is the norm for the average incoming connection weights from X i (Treur, 2020), Ch 6 and 13.The logistic tipping point function alogtip ν,α (…), is a varied form of the sltip ν,α (…) function, also used for the homophily adaptive tipping point self-model TP-states.The difference lies in the way the norm ν is approximated.In this function, the norm ν is approximated using the function ν -alogistic σ,τ (W 1 , …,W k ).Hence, the full function is The parameters for the alogistic σ,τ (…) function within the alogtip ν,α (…) function, are set at 5 for the steepness σ and 1.5 for the excitability threshold τ, as they are default values for a sum of W's that is in the range of [0, 3].

Design of the multi-adaptive network model
The designed complex multi-adaptive network model is shown in Fig. 1 and explained in Table 1 (base level), Table 2 (first-order selfmodel level), and Table 3 (second-order self-model level).

Base level processes
The base level comprises mental network models for four distinct individuals, and the social interactions begins upon activation of the world state ws s , which serves as the triggering signal.When the agents sense the beginning of the social interaction ss s , they receive a sensory representation of the action srs a .At the same time, agents initiate their mental models and start interacting with each other individually.Agent A detects interactions from each individual agent via sensor states ss e_B , ss e_C , ss e_D , generates sensory representation states srs e_B , srs e_C , srs e_D for them via states and forms beliefs bs e_B , bs e_C , bs e_D about them.Via a preparation state ps a , executes the (inter)action es a , and forms a belief state bs a regarding the level of its own action.The same procedure is followed by the other three agents.All base level states use combination function id (when they only have one incoming connection) or alogistic (otherwise), see equations ( 8) and (10), respectivily.

First-order self-model level processes
The first-order self-model level (also called first reification level) models the first-order adaptation processes for the base level processes; see Table 2. Adaptation takes place in two ways: individual adaptivity by Hebbian learning and social adaptivity through bonding by homophily.To model this, for each agent or pair of agents self-model W-states X 58 to X 69 for Hebbian learning adaptation and self-model W-states X 70 to X 81 for bonding by homophily adaptation are included.For the Hebbian learning W-states, the combination function hebb is used and for the bonding by homophily W-states the combination function slhom, see equations ( 11) and ( 12), respectively.
The Hebbian learning states gather information from the bs e and ps a states, indicated by blue arrows, while the homophily states receive input from the bs e and bs a states.Note that by using belief states as input for the homophily W-states, a form of subjective homophily is modeled in contrast to the objective homophily based on execution states that usually is applied in literature such as (Accetto et al., 2018;Blankendaal et al., 2016;Kozyreva et al, 2018;Sharpanskykh and Treur, 2014;Van den Beukel et al.,2019).
The impact of the learning states on their related base level states is indicated by red downward arrows: toward ps a for Hebbian learning and toward ss e for bonding by homophily.

Second-order self-model level processes
The second-order self-model level (second reification level) models adaptivity of the adaptivity of the base level; see Table 3.It makes the first-order adaptation level itself adaptive as well: second-order adaptation.This is specified by self-model states for network characteristics used for the first-order self-model level.In particular this level contains a context state and the second-order self-model H W -states for the speed factors for all first-order adaptation W-states, as well as the second-order              10).It expresses the second-order adaptation principle that the speed of adaptation is higher when there is more stimulus exposure (Robinson et al., 2016).
On the other hand, the adaptation principle for the tipping point TP W -states can be described in terms of the intensity of the existing connections of the individual, modeled by the adaptive W-states of the individual.Unlike for the adaptive speed states, in the case of a tipping point self-model state, the base level states are not involved.It receives information only from the W-states and from itself, by the means of a persistence link.Lastly, the context state represents how far someone prefers to have bonding, as a personality characteristic.Its link to the adaptive tipping point states will amplify or reduce the tolerance of the agent when adapting to others.Depending on the addressed scenario, the TP W -states for the tipping points use combination function sltip or alogtip, see equations ( 13) and ( 14), respectively.

Scenario A: Exclusion of an agent
The first scenario explored is concerning a social interaction where initi- ally one agent's, in our case agent D's, connection with other agents is too weak, causing an eventual loss of connections.As a result, the agent is separated from the others.This section covers the processes through which such separation effect is achieved.

Connectivity characteristics
As previously mentioned, various connection weights contribute to interactions.Among them are two types of links: within-agent and interagent.Within-agent links encompass prediction link, perception link, belief of perceived stimulus link, and belief of own execution link.In the scenario separating agent D, agents A, B, and C possess strong prediction, perception, and belief of perceived stimulus links towards each other, while having very weak such links towards agent D. Specifically, in the model, the links between agents A, B, and C fall within the range of [0.7, 0.85], whereas for agent D, they fall within the range of [0.3, 0.4].Hence, the links from all agents to agent D are intentionally weakened, posing a challenge for agent D to perceive any incoming information from them.Moreover, the expressive link from D to other agents is also weakened, further hindering the adaptation of other agents towards D. The aim of this scenario is to make agent D a less favorable candidate for connection.Detailed numerical information can be found in Table 4, while the complete connection weight matrix is presented in Table 8.

Aggregation characteristics
The aggregation of connections leading to any given state is shaped by the combination functions and their parameters, serving as the decisive factors.The combination functions used for this scenario include the identity function, alogistic function, simple linear homophily function, Hebbian function, steponce function, and the simple linear tipping point function.Within the combination functions, their parameters are determining factors for the way the causal impacts on a given state are aggregated.For all of the agents, for the ss s , srs s , ss e , es a , bs e , and bs a states the identity function is used, while the srs e and ps a  states are modeled using the alogistic function, with the same steepness σ and excitability threshold τ parameters; the steepness is 20 for all, and the threshold is 0.6 for srs e and 0.7 for ps a .The world state ws is modeled using the stepmod function, where the repeated time duration ρ of the interaction is set at 50, and the transition point δ which serves as a breaking moment between interactions is set at 20.These parameters result in 30 units of time of interaction and 20 units of time of break.
The Hebbian learning W-states are modeled using their own combination function hebb, and its only parameter, the amplification factor μ, is set to values in range of [0.94, 0.995].The values are all relatively close to 1, yet are not all the same to ensure that differences in learning would be visible.The homophily bonding W-states use the simple logistic homophily function slhomo, where the amplification parameter α is set at 0.2 for all of the homophily W-states.The tipping point parameter τ of the homophily function is adaptive and thus the tipping points follow the TP W -states.When the tipping point is increased, more similarity is perceived and as a consequence more connections are strengthened by the homophily adaptation, so the average connection weight and total sum of connection weights will become higher compared to the norm ν, and vice versa.The speed adaptation H W -states are modeled using the alogistic combination function with the steepness σ and excitability threshold τ being 1.5 and 1 for both the Hebbian and homophily speed adaptation H W -states.
Lastly, the tipping point adaptation TP W -states are modeled using two combination functions, simple linear tipping point function and the alogistic tipping point function.The simple linear tipping point function       focuses on the average weight of connections in comparison to the norm ν and has parameters 0.6 and 0.4 for the modulation factor α and the norm ν, respectively.In how far the agents achieve being close to the precise value for their norm ν, can be found by graphing the average trend of homophily learning for each agent, and then taking the mean of each.As can be seen from Fig. 2, the norm lies at around 0.4, for all four agents.
The alogtip tipping point function, on the other hand, has the same parameter values (0.6 and 0.4) for the modulation factor α and the norm ν, respectively.The norm ν for this function, was found using the alogistic function, where the function parameters (5 for the steepness σ and 1.5 for the excitability threshold τ).The step-size of the simulation was 0.1.All of the values for the function parameters for both functions can be found in Tables 8 to 14.

Timing characteristics
Each state is set at a default speed factor of 10, which defines the speed at which the state changes for the given aggregated impact.In the same manner, for the learning W-states X 58 to X 81 the adaptive speed factors are indicated (in red in Table 5) from the second-order self-model level H W -states X 82 to X 105 , whose own speed factors are set at 1.3.The speeds of the tipping point adaptation TP W -states are nonadaptive and have a range of numbers between 0.05 and 1, to explore the naturally  occurring varying tipping point outcomes for the agents.Due to the identical timing characteristic shared by both scenarios A and B, it is not necessary to include a dedicated section about the speed factor in Scenario B. This omission is deliberate to avoid redundant repetition in the subsequent section.

Scenario B: Group project dynamics
This section introduces a second scenario centered around the dynamic interactions of four students collaborating on a group project.Initially, individuals A and B share a stronger bond, while C and D also possess a closer connection.Each of these four individuals exhibits unique personalities and communication styles, which significantly impact their level of expressiveness and openness to receiving stimuli.
The main aim of this scenario is to explore the extent of adaptation throughout the simulation, as influenced by different ways and levels of willingness to respond to stimuli, which are represented by the varying tipping point.

Connectivity characteristics
Due to the scenario only differing in the values of the connection weights, and function parameters, role matrix mb for base connectivity remains identical to the one for Scenario 1.The difference lies in the weights of these connections.A clear understanding of the different weights can be found in Table 6.Since agents A and B have a closer bond, they have stronger prediction, perception sensitivity, and belief of perceived link towards each other, while a bit weaker towards the other  7.

Aggregation characteristics
In contrast to the previous scenario, where the combination function parameters (the modulation factor α and the norm ν) remained consistent for both functions (sltip and alogtip) and across all agents, this scenario introduces a variation of the modulation factor α among agents and functions.This distinction aims to further explore the various outcomes that can arise from the varying tipping points.However, it is important to note that the norm ν should remain unchanged, as it is either predetermined based on the parameters of the alogistic function or manually determined.For the sltip function, the modulation factor α is set at 0.8 for agents A and D, and 0.5 for agents B and C. Similarly, for the alogtip function, the modulation factor α is set at 0.7 for agents A and D, and 0.4 for agents B and C. For results as gathered in Fig. 3, to ensure uniformity and simplicity, a value of 0.4 is assigned as the norm ν for all agents.All of the values for the function parameters for both functions can be found in the included tables for role matrices.

Simulation results for Scenario A: Separation of an agent
This section showcases the simulation results in graphical form, specifically focusing on Scenario A described earlier.The section is divided into two parts: one presents the results obtained using the sltip function for the tipping point of homophily, while the other focuses on the alogtip function.While the primary emphasis is on the homophily adaptation, the Hebbian learning graphs will not be extensively analyzed here, but can be found in the appendices for reference.As the two tipping point functions model different adaptation principles (average connection vs total sum of connections), the objective is not to compare the effects of the two functions, but rather to highlight the distinct outputs they produce.Therefore, the results of each function will be presented separately, providing a comprehensive understanding of their individual impact.
In this scenario, specific adjustments were made to the connection weight links of the base level states, aiming to discourage Agent D from connecting to others.By gradually weakening selected links, the strength of homophily adaptation between all agents and agent D was significantly reduced.The impact of these weight alterations is visualized in Figs.4-11, where the individual states of each agent exhibit similar patterns across the two functions.However, variations arise in the intensity and frequency of these states, attributable to the distinct characteristics of the sltip and alogtip functions.The introduction of weak weights for prediction, perception sensitivity, belief of perceived stimulus, and expressive links with agent D results in the agent displaying weak sensory, sensory representation, and belief states from the other agents, as indicated by near-zero or zero curves.Thus, in the base level graphs for agent D, the sensory states drop fairly quickly from the start of the simulation as the agent struggles to attain information from others.
The weakened performance of the base level states poses challenges for achieving effective homophily adaptation between agents, seen in Figs. 12, 13.For the homophily adaptation to have a high performance, the agent is expected to have strong inter-agent links with the agent they are adapting to, and strong expressive link from the agent it wants to receive stimuli from.It is important to note that in the graph, the label "Homophily AB" signifies A's influence on B, or in simpler terms, B adapting to A. When using the sltip function, the strongest adaptation occurred in cases where C was adapting to A, C adapting to D, and A adapting to B. This can be attributed to the stronger within-agent link weights and more prominent expressive link of agent C. Conversely, the weakest adaptation was observed in cases where D was adapting to C, D adapting to B, D adapting to A, and A adapting to B. When using the alogistic function, the strongest adaptation was observed in cases where C was adapting to A, C adapting to D, and C adapting to A. On the other hand, the weakest adaptation occurred in cases where D was adapting to C, D adapting to A, D adapting to B, and B adapting to D. In both functions, agent D displayed zero adaptation with all other agents, highlighting how variation in connection weights can lead to the exclusion of an individual.Although agent D could share its information, it was unable to receive any meaningful input due to the weak connections from other agents, preventing it from adapting.
This shows that separation of a person during an initial phase can lead to long-term isolation due to the two adaptation mechanisms considered here.
The TP W -states corresponding to the tipping point and the speed of adaptations are also represented in a graphical form and are essential in confirming that they are functioning as intended.These graphs are used to validate that the expected behavior is observed, ensuring the reliability of the adaptation graphs' results.The graphs are included in the appendix for readers to reference and gain a better understanding of the expected behavior and effectiveness of the adaptation process.

Simulation results for Scenario B: Group project dynamics
The aim of this simulation was to investigate the dynamics of a project group comprising four individuals with distinct connection links.Upon analyzing the base level states graphs in Figs.14-21, it is observed that agent A demonstrates a weak sensitivity towards stimuli from agent B, despite the expectation of a strong connection between them.This weak sensitivity can be attributed to agent B's limited expressiveness.Conversely, although the links with agents C and D were intentionally weakened, agent A still obtains information from agent C due to its strong expressive links.On the other hand, Agent B demonstrates a high level of connection strength with Agent A, attributed to their strong connection links and Agent A's high expressiveness.Additionally, Agent B exhibits moderate connection strengths with the other two agents in the group.Interestingly, agent C, who is expected to have a stronger bond with agent D, demonstrates a weak sensory state in response to stimuli from agent D. This can be attributed to the weak expressive link of agent D, which hampers stimulus detection.As for agent D, its sensory state initially rises rapidly but then quickly drops in response to agent C's stimuli, possibly due to the weak connection links established from agent D's weak expressive link.Consequently, as both agents are more interconnected and one of them has limited expressiveness, their ability to share knowledge gradually weakens over time, as evident in the graph for agent D.

Table 9
Scenario A and B: role matrices with combination functions with sltip.

Table 10
Scenario A: role matrices with combination function parameters with sltip.

Table 11
Scenario A and B: role matrices with combination functions for alogtip.

Table 12
Scenario A: role matrices with combination function parameters for alogtip.

Table 13
Scenario B: role matrices with combination function weight for sltip.

Table 14
Scenario B: role matrices with combination function weight for alogtip.Thus, these results yield that strong connection weights of links between agents rarely cause strong adaptation, regardless of how strong the expressive link is, which at first sight might feel as counter intuitive.However, an explanation is that the internal mental processes are decisive for bonding as well.Without their adaptation, bonding will not be easy.

Evaluation Scenario B: Group project dynamics
Searching for the cause of such unexpected adaptation, the amplification factor function parameter for the tipping point was the first factor that was presumed to have been the cause.Agents A and D, with high amplification factors, exhibited greater variation in tolerance levels.This could explain why Agent A only displayed strong adaptation with Agent C, who had strong expressiveness, while Agent D did not exhibit strong adaptations to any other agent.In order to assess such hypothesis, the alpha parameter for the tipping point of each agent was set to 0.6 for both the sltip and alogtip functions.Remarkably, this adjustment produced equivalent adaptation performance compared to when different alpha values were used.The sole distinction observed was in the wavelength of the tipping point.
Upon further investigation, it was discovered that the link to the belief of own execution for agents A and C was the factor that played the crucial role in the observed adaptation.Initially lowered to 0.7 and 0.6 in scenario B, these links were subsequently adjusted to a value of 1 for evaluation purposes.As anticipated, this adjustment resulted in the desired outcome, indicating that agents with strong within-agent links and expressive links exhibited the most favorable performance (Figs. 24,25).

Discussion
The model presented in this paper explores two scenarios to investigate the dynamics of homophily adaptation by manipulating connection weights and function parameters.Taking inspiration from (Treur, 2020, Ch 6) the model in this paper expands upon it.In addition to incorporating the homophily adaptation and the adaptive tipping point functions, the model contains adaptive speed and Hebbian learning, while exploring both the sltip function and the alogtip function.This extension enhances the capabilities of the model from (Treur, 2020, Ch 6), enabling a more comprehensive analysis of the dynamics of homophily adaptation.
In the first scenario, which involved agent exclusion, the findings revealed successful adaptation among agents who possessed strong within-agent links and expressive links.This supports the understanding that robust connections between agents facilitate effective homophily adaptation.However, the second scenario introduced additional variations, particularly in the amplification factor of the tipping point, within agent links and the expressive links between agents.This scenario initially challenged the previously established notion.Contrary to expectations, originally the combination of strong connection links and expressive links did not lead to strong adaptation outcomes in the context of homophily.Initially, it was presumed that the amplification factor was responsible for the unexpected adaptation observed among certain agents.However, upon evaluation, it became evident that this was not the underlying cause.Surprisingly, it was the belief of own execution link that effected the intensity of adaptation.A slightly weakened belief in an agent's own execution hindered the adaptation of the other agent towards it.Consequently, it can be inferred that the most robust adaptation occurs when an agent possesses a strong within-agent link towards another agent with strong expressive links.
These findings highlight the complex nature of bonding by homophily and its ability to manifest in diverse ways, even when certain link configurations appear advantageous.The adaptability of homophily suggests the existence of underlying mechanisms that go beyond simple connectivity patterns, emphasizing the importance of considering multiple factors in understanding and predicting the dynamics of social interactions.
Other work om bonding by homophily can be found in (Accetto et al., 2018; Blankendaal et al.,2016;Kozyreva et al, 2018;Sharpanskykh and Treur, 2014;Van den Beukel et al.,2019).Here, the second reference combines bonding by homophily with another principle for bonding: the more becomes more principle.The third one considers multicriteria similarity for the homophily.All of these four references do not address the second-order adaptivity of the homophily tipping point and do not consider subjective homophily as is the focus in the current paper.Moreover, the last three do not integrate bonding by homophily with hebbian learning.Other related research also addresses bonding in social interaction but not in relation to homophily but to synchronisation (or mimicry) between persons, e.g., empirically focused research such as (Behrens, Snijdewint, Moulder, Prochazkova, Sjak-Shie, Boker, Kret, 2020;Koole and Tschacher, 2016;Kret and Akyüz, 2022;Prochazkova, Sjak-Shie, Behrens, Lindh, Kret, 2022) and analysis and modeling research such as (Hendrikse et al., 2023a;Hendrikse et al., 2024;Hendrikse, Treur, Koole, 2023b;Hendrikse, Treur, Wilderjans, Dikker, Koole, 2023c).This area has some similarity to the topic of the current paper, but does not address the approach based on homophily addressed here in relation to literature such as (McPherson et al.,2001).An important difference is that synchronisation is more complex as it is based on temporal patterns (time series) for activation levels of states that become attuned whereas homophily is based on states with activation levels that become attuned at single time points.A limitation of the current work is that it only considers bonding by homophily ans does not integrate other adaptation principles such as more becomes more as addressed in (Blankendaal et al, 2018;Van den Beukel, Goos, Treur, 2019).This can be work for future research.Another limitation is that homophily was based on single time point comparison of activation levels for similarity.The work of Hendrikse et al. (2023aHendrikse et al. ( ,b,c, 2024) ) mentioned above points to a (more complex) way to how such similarity can alternatively be based on temporal patterns of activation levels (time series) and synchrony of them.
In future studies using the presented model, an interesting aspect to explore is adjusting the unique values of the link between the context state and the tipping point.This link represents a personal characteristic that reflects an individual's preference for bonding.In this paper, the value for this link was set to 1 for all agents, ensuring they had the same preference.However, varying this value could create more differences among the agents, allowing a better understanding of how individual preferences influence bonding dynamics in the model.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Y. Mukeriia et al. self-model TP W -states for the adaptive tipping points for the adaptation W-states for bonding by homophily.The self-model H W -states for adaptive adaptation speed are unique to each pair of interactions and receive information from the same base-level states as the learning Wstates, along with the learning W-states themselves.The H W -states use combination function alogistic, see equation (

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.Mukeriia et al.

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.Mukeriia et al.

Table 4
Connection weights for Scenario 1.

Table 6
Connection weights for Scenario 2.

Table 7
Role matrix mcw for both adaptive and nonadaptive connection weights at the base level.

Table 8
Scenario A: role matrices with connections, connection weights, and initial values.