The collaborative evolution of trust, information flow, and social cooperation: a study on network stability based on dynamic game models

Interpersonal communication involves the exchange of information between individuals, and the quality and depth of this exchange can provide insights into the strength of communication and the nature of the relationship. The establishment of interpersonal interaction can be analyzed through the framework of complex networks, where each individual in society is represented as an independent node. These nodes form connections through continuous communication with other nodes, creating paths within the complex network. This section introduces a simulation model that simulates the formation of interpersonal networks by increasing the number of nodes participating in communication. It studies the characteristics of nodes and networks during the network formation process. Each node interacts with other nodes in each phase, and successful information transmission is regarded as successful cooperation. With each successful cooperation, both partners receive a positive score, while points are deducted in the event of interaction failure (i.e., failure in information transmission). This paper primarily focuses on the information interaction behaviors and characteristics during the formation of human social relationship networks, emphasizing the relationships established between individuals following successful information interactions.

The existing social relationship network models have limited fitting to real-life social relationship networks, which has aroused the interest of many researchers⁶³. We used ER’s random graph model as the starting point for information system research in this experiment, and the simulation results of other network models are one of the possible extension directions of this study. The network information system designed in this article is to establish a trust foundation for a node within the scope of information dissemination. This system includes two types of scoring: X and Y. X algorithm: If there is already a history of cooperation between interactive nodes, the nodes can evaluate each other’s behavior on their own and determine whether its interacting node is worth investing in cooperation costs again (the higher the x, the more successful the cooperation between both parties). (Between nodes with interaction history, both parties develop a mutual understanding of each other’s behavioral patterns and cooperative intentions. If the behavior of the other party during period t continues to align with the initial assessment, then my expectations remain rational, and my original judgment regarding the other party is validated. Consequently, successful cooperation in this phase strengthens the trust foundational between the two parties, reflecting the value of time. For a more in-depth discussion of this concept, including instances of betrayal, please refer to²⁰). Y is the evaluation of nodes without interaction history after observing the cooperative behavior of other nodes, simulating the social evaluation possessed by individuals, that is, their cooperation history represents the possibility of successful cooperation with them (higher Y values indicate the lowest risk of investment in cooperation costs). (In a cooperative network, nodes that successfully collaborate receive positive scores from their social environment (For strangers we never know, our social judgment of them comes from external factors added to them, such as educational experience and the continuity of marital relationships, which are important criteria for us to judge whether they can maintain a long-term and deep relationship). When a node achieves successful cooperation in phase t, its external evaluations remain unchanged (same logic as we develop X), indicating that there is no reason for questioning its cooperative behavior. Therefore, we propose that if a node exhibits no activity within the entire social environment, its social evaluation is solely influenced by its temporal presence in the network, specifically the duration of its existence). We set the basic model as following:

G is a series of games and related relations in interactive cyberspace. The node set in this space represents each person in the cyberspace, denoted by V= (v1, v2…, vn), and edge set E denotes the relationships for each node. Z is the score system of nodes. The scoring system contains two dimensions.

The formula (2) and (3) represents a time difference between the trust foundation of the previous period (t-1) and the trust foundation of the current period (t). The variable λ represents the time value of an individual in certain social structure. In the absence of behavior, a node’s reputation in the network accumulates over time according to λ, similar to how elderly individuals are respected in society. The variable t signifies different periods of network evolution. The trust basis in the next period is updated by the time value compared to the current period.

For any given node i, there is a probability q for it to interact with its neighboring node j, with a corresponding probability (:{q}_{1}) of unsuccessful interaction. Successful interactions result in benefits α for both parties, while unsuccessful interactions lead to a decrease in evaluation by β. For bystander node h, node i has probability (:{q}^{{prime:}}) of interaction and unsuccessful information dissemination probability (:{q}_{1}). Successful interactions yield benefits γ, while unsuccessful interactions result in a decrease in evaluation by δ. The consistent behavioral preferences of nodes can lead to similar mistakes in different social scenarios, hence the probability of unsuccessful information exchange is the same. The current score matrix of node i comprises two parts:

Table 1 illustrates the score structure of node i within the network. This article employs the Monte Carlo simulation method to construct the network. Table 1 offers a comprehensive overview of the scoring rules governing the nodes during the interaction process. In the first step, our research node (node a) interacts with its neighbor (node b), aiming to achieve successful information transfer. In the second step, the newly added node (node c) interacts with the target node (node a), while the previous period’s interactive object (node b) remains in a ‘wait and see’ state. In the third step, contingent upon the success of the prior interaction, the ‘observation’ node (node b) engages with the newly added node (node c) from the previous period. Simultaneously, node a interacts with another newly added node (node d) in this period’s network, while still actively engaging with nodes b and c according to Eq. (2) and our scoring rule. Simultaneously, nodes b and c observe the interaction. Each time the interaction is successful, the participating nodes update their (:{X}_{ij}^{t}) according to Eq. (2). Concurrently, other nodes in the network that do not engage in the interaction adjust their evaluation ((:{Y}_{i}^{t})) of the successful interaction node in accordance with Eq. (3). For every successful interaction, the participating node updates its score based on the aforementioned matrix.

Information dissemination and node heterogeneity in the process of network formation

This article employs a multi-agent simulation model to investigate the process of network formation. The number and sequence of nodes that interact directly have been previously introduced, along with a designated number of bystander nodes that do not participate in the interaction but still possess a certain probability of engaging with other nodes. Presented here are the experimental results derived from a configuration of 209 bystander nodes, selected as a representative and reader-friendly outcome from over 1,000 simulations. The network formation process is illustrated in Fig. 1.

Depicts the relationship between network information synchronization for direct interacting nodes and bystander nodes, and the time needed for the network to achieve a stable state as the number of direct interacting nodes increases. The horizontal axis indicates the number of iterations, while the vertical axis shows the status of network information synchronization.The number of stable periods for network structure and the number of periods required for information synchronization between two types of nodes (interacting nodes and bystander nodes).

Figure 1 illustrates that as the number of nodes participating in the interaction increases, high-level nodes gradually emerge within the entire network. The identification of the heterogeneity of these nodes will be discussed in detail in Part Four. These high-level nodes possess the largest number of link paths within the network and play a decisive role in its formation. To investigate the state of information flow during this process, we analyzed both the information flow rate and the information propagation state of the entire network, as depicted in Fig. 2.

The bar chart in Fig. 2 illustrates the number of periods required for the network to achieve a stable state and its final stable value, signifying the completion of network construction. The line chart depicts the heterogeneity in information synchronization between nodes that interact directly and those that do so occasionally. Directly interacting nodes require fewer periods for information synchronization compared to bystander nodes, as interaction is essential for acquiring information. Bystander nodes experience a delay in forming cognition as they await the transmission of messages from interacting nodes. This observation elucidates why bystander nodes can still complete the information transmission process without engaging in direct interaction. Information can be synchronized at the overall network level, indicating that information exchange occurs not only among nodes directly involved in interactions but also across all nodes within the simulation. This finding is particularly intriguing as it suggests that, in real life, even in the absence of deliberate participation in information exchange, individuals are, to some extent, involved in various types of information flow. The information reception behavior of nodes can be both conscious and unconscious. Here we have obtained the ‘Image Scoring’ in Nowak’s work⁶⁴, where the existence of indirect reciprocity necessarily involves the transmission of information through bystanders, thereby providing social feedback to collaborators. As members of this complex society, we are interconnected within various social networks, positioning us as nodes in the processes of information production, transmission, and re-manufacturing.

Based on the conclusions drawn from the above simulation, we posit that the primary factor influencing network formation is not merely the transmission of information, but rather additional determinants. Not all random nodes possess the capability to interact in a manner that results in the establishment of a network with a specific structure. The formation of a network necessitates the repeated flow of information between nodes, thereby fostering relatively stable relationships through continuous interactions. Throughout this process, nodes may encounter various challenges, including behavioral errors and information mismatches. It is only through persistent error correction and interaction that the connections between nodes can achieve stability.To investigate the information flow mechanisms of nodes, it is essential to examine their behaviors in receiving and sending information more closely. The X scoring system we have established reflects the final outcome of a node’s success or failure in cooperative interactive behavior. A high X score signifies that the frequency of successful interactions significantly surpasses that of interaction failures, as illustrated in the following figure (Fig. 3).

Illustrates the score levels of both directly participating interactive nodes and bystander nodes throughout the iterative process of network formation with varying numbers of directly participating nodes (The number of observer nodes is randomly generated during the simulation process). The horizontal axis denotes different iteration periods, while the vertical axis indicates the score level. Node score status during network information synchronization process.

In a three-node network, where only three nodes engage in direct interaction, the scores of participating nodes are similar to those of non-participating nodes, indicating minimal heterogeneity. However, as the number of nodes involved in direct interaction increases, we observe significant differences in the efficiency of cooperative success among nodes with varying network statuses. The simulation results demonstrate that the success efficiency of nodes directly participating in the interaction is notably high, with scoring efficiency rising rapidly. These nodes emerge as key players within the overall social network organization. In a four-node network, nodes classified as either direct interaction or bystander nodes exhibit distinct rates of information exchange, while a five-node network shows fluctuations in scores for these node types. When the number of nodes participating in direct interaction exceeds five, the complex network begins to exhibit score fluctuations. These fluctuations occur not only among nodes directly interacting but also among those engaged in indirect interaction. Even when we define the initial form of network formation, these fluctuations persist. After repeated calculations of network dynamics, we find that this phenomenon arises due to the complexity of information transmission within networks beginning from the five-node configuration, necessitating a certain iterative period to synchronize network information. In networks comprising six and seven nodes, the scores of directly interacting nodes exhibit a rapid increase, leading to a significant enhancement in their success rates. A careful analysis of the nodes involved in these interactions reveals a distinct hierarchy among them. The node score depicted in Fig. 3 represents the average score of these nodes. Within the directly interacting nodes, the scores of higher-level nodes are markedly greater than those of lower-level nodes, while the scores of low-level nodes are even inferior to those of nodes that do not participate directly in the interaction. Bystander nodes experience fluctuations in their scores; however, these variations are not easily discernible due to their overall low score levels. A similar trend is observed in networks with eight and nine nodes, where the highest score attainable by heterogeneous nodes increases as the number of nodes engaged in direct interaction rises. The repeated fluctuations in scores reveal the true insight of cooperation. Node interactions cannot form stable network relationships as mentioned above. Only when behavioral deviations are corrected and trust relationships are re-established through continuous interactions can stable connections be formed during the repeated process of deviation. Only then can a network path be considered established. The formation of network pathways will further promote stable cooperation between nodes, as the cost of forgiving erroneous behavior is much lower than that of rebuilding new node pathways. However, this is not absolute. When the disappearance of nodes in the network continues to occur in our simulation, we realize that the forgiveness of node behavior is not endless. Further research on this issue requires a deepening of the model, which represents another potential research direction of this article.

This pattern remains consistent across all simulations conducted in our study. Furthermore, the exchange of information among nodes is pivotal in determining the overall efficiency of a network. The node score status depicted in Fig. 3 represents the outcomes of the first nine simulations conducted. It is important to note that the simulation process is ongoing. In our simulations, the number of designed nodes typically exceeds 200; diagrams featuring more than 300 nodes become overly complex, making the connections exceedingly difficult to display. After nine iterations, the simulation proceeds until the network cluster coefficient, illustrated in Fig. 2, attains a stable state, indicating that the network has reached a steady state.

Node behavior selection and the influence of social trust foundation

The construction of interpersonal relationships is predicated on a foundational level of trust²⁰. The evolution of this trust is contingent upon the history of prior interactions. During these interactions, individuals may inevitably exhibit behavioral deviations, which refer to the extent to which their behaviors diverge from the expected norms of both parties involved. When such deviations occur, either party may opt to discontinue the interaction, as these behaviors can adversely affect the likelihood of successful cooperation and, consequently, undermine the initial investment in cooperation. However, in real-world scenarios, it is quite common for individuals to forgive the mistakes of others. This article introduces a metric, termed the ‘y-value,’ which primarily aims to observe the behavioral signals of interactive nodes as perceived by bystanders. It assigns scores to these nodes based on the observed signals, ultimately leading to an evaluation of the interactive nodes by bystanders. In essence, the y-value represents the trust foundation ascribed to behaving nodes by the entire network organization. If a node exhibits no behavior within the network, its y-value is derived from the time value accumulated during its presence in the network. The existence of such a node indicates that it occupies a functional role within the network, and its value is determined by the number of iterative periods of its existence.

The node with the highest Y value is likely to achieve the most successful cooperation. In the context of unfamiliar nodes, a higher Y value indicates a greater probability of successful collaboration, thereby minimizing potential cooperation costs. Our simulation reveals a co-integration relationship between the scores of X and Y, suggesting that nodes with elevated levels within the network typically possess exceptionally high Y values. When the score of a cooperative entity is significantly high, deviations from the rational expectations of interacting nodes may occur; however, the historical high score of this node contributes to its ‘social reputation,’ indicating a history of excellent behavior. Such deviations are often attributable to transient errors or mistakes made by its interactive counterparts. Consequently, errors committed by high-scoring nodes tend to be more readily forgiven by other nodes within the network. This forgiveness mechanism is reflected not only in the Y value but also in the X value. Forgiveness is justified by specific reasons, which may stem from familial ties or from the exemplary behavioral history of the node in question. In our simulation, we observed that the value of Y exhibits significant nonlinear characteristics.

X and Y together constitute the network information system. The presence of Y enhances the significance of successful cooperation between nodes, making it far more critical than merely completing the information flow. Given the interaction involves two types of information, the cost of unsuccessful cooperation can lead to a decline in network ratings (The credible threat mentioned above). Consequently, each node exercises extreme caution when selecting its behavior during interactions. As illustrated in Fig. 2, prior to reaching a stable state, the number of nodes actively participating in the network and the degree of network aggregation in the initial phase surpass the level of aggregation in its stable state. This indicates that during the early stages of network formation, there is a distinct period of ‘over trust,’ which we refer to as the ‘honeymoon period.’ In this initial phase, nodes exhibit cautious behavioral choices (Prosocial behavior beyond its cooperation preference), resulting in their Y values being inflated compared to their actual behavioral strategies. However, as iterations continue, nodes will ultimately employ their genuine behavioral strategies in cooperation. This may lead to errors in selecting behavioral strategies, causing fluctuations in the behavior information of nodes and increasing the complexity of the network information. The number of iteration periods required for network formation significantly exceeds the number of periods necessary for network information synchronization. The establishment of the final connection path within the network results from the continuous mistakes, forgiveness, and re-connections of nodes as informed by dynamic game theory. Ultimately, this process leads to the formation of a stable interaction relationship. The emergence of such a relationship implies that even if a network node deviates from expected behavior, the errors made by that node can be accommodated by the established stable path. When examining the prisoner’s dilemma, it is evident that the underlying motivation for repeated games is also rooted in repeated interactions, which dissuades rational opponents from opting to betray during a specific game. However, as previously mentioned, the capacity to forgive the other party’s transgressions is not limitless. In simulations, it is often observed that cooperative nodes may ‘die’ in the network in subsequent iterations due to the deduction of all their scores. Behavioral deviations can adversely affect an individual’s fitness within the social structure.

The presence of the y value provides a positive incentive for nodes that have already established connections within the network to promote cooperative behavior. The cooperative actions of a particular node yield significantly greater rewards than the actions themselves might suggest. Moreover, this behavior conveys information to other nodes in the network, indicating that the cooperative node is capable of fulfilling its commitments without resorting to deceit, thereby establishing its trustworthiness. Within the context of the information system discussed in this article, the ‘honeymoon period’ of nodes during the initial stages of network formation is evident; specifically, nodes participating in interactions for the first time tend to exhibit a preference for cooperation when selecting their behaviors.