2.1 Computational model

The model utilised for this paper expands upon previous work by Yang et al [17], in order to explore how the presence of public information as an observable history of previous decisions can affect cooperation and network well-being. In the model utilised by Yang et al [17], newcomers joining a network of a fixed size select a role-model from individuals already present in the network based on their level of fitness. A newcomer will then make use of both public and private information to determine which individuals it should form connections with. Any connected individuals will engage in a round of a donation game.

We utilise a special case of the prisoner’s dilemma game [6] to investigate the conflict that can emerge between cooperators and defectors, called a donation game. Here, an individual may engage in cooperative behaviour to produce benefits for each of its neighbours, which will come at some personal cost. An individual engaging in defection will produce no benefits incurring no costs whilst still taking any benefits produced by others. The payoff matrix utilises parameter b, which determines the value of benefits produced from a cooperative individual. Parameter c is used to determine the cost that is incurred when an individual engages in cooperation. Throughout simulations, we assume that b is set to 10 and c is set to 8, which will adhere to the condition of b > c > 0. The payoff P_i of individuals is determined by the sum of all payoff obtained by all interactions that individuals have with their neighbours. For example, assuming a cooperator has K cooperative neighbours and J defective neighbours, they will receive the payoff described in Eq (1). For defectors, assuming they have K cooperative neighbours and J defective neighbours, will instead receive a payoff described in Eq (2). (1) (2)

Parameter δ is used to determine the level of selection strength that is present in networks and will influence the likelihood that nodes are selected as role-models based on their payoff P. This parameter is used to calculate the fitness of a node i as described in Eq (3). (3)

At each step during a simulation, a newcomer is incorporated into the network (Fig 1). Firstly, an existing node is randomly selected to act as the newcomers role-model (Fig 1a). This selection is influenced the fitness of nodes, obtained as in Eq (3), with a higher value meaning a node is more likely to be chosen as a role-model over other nodes. Next, the newcomer will determine if it will adopt the strategy of its role-model (Fig 1b). Parameter μ is used to determine the likelihood that a mutation will occur. Following [17], μ is set to 0.0001 to allow for comparison against previous work. If a mutation occurs, the newcomer will instead adopt the opposing strategy of its role-model. The role of mutation within networks is to avoid networks only consisting of either cooperators or defectors, and allows for the possibility of a previously absent strategy to invade a network. Following [17], public and private information are then used to establish which connections the newcomer will form. The potential connections an individual may form are restricted to the chosen role-model and any individuals that are connected to the role-model, and is done using the decision-making process described in Section 2.4 (Fig 1c).

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Fig 1. An illustration of the process that is undertaken as a newcomer joins the network at each step during the simulation.

Firstly, a newcomer will select an existing node to act as its role-model, which is influenced by node fitness and δ = 0.001 (a). Following this, the newcomer will adopt a strategy, by either copying the role-model or mutating by adopting the opposing strategy (b). Lastly, the newcomer will then proceed to choose which of the role-models neighbours to form connections with, which is done by making use of available public and private information (c). To model private information, two Gaussian distributions are used to represent an individuals ability to identify either cooperators or defectors respectively (d). Specificity and sensitivity represents an individuals ability to correctly reject a connection with a defector or correctly accept a connection with a cooperator respectively.

https://doi.org/10.1371/journal.pone.0275909.g001

2.2 Private information

Following [17], we utilise two fixed Gaussian distributions (See Fig 1d) to model a newcomers ability to determine the intentions of a potential neighbour as private information. Both distributions are created with a variance of σ² = 0.5 and both peaks are set 1 apart from each other, with a μ of -0.5 for φ_c and a μ of 0.5 for φ_d. The use of these two overlapping distributions φ_c and φ_d are used in order to emulate an individuals imperfect ability to differentiate between potential cooperators and defectors respectively. For each node x (either the role-model or any nodes connected to the role model) where a connection can be formed, a value is sampled from either φ_c if the node x is a cooperator or φ_d if node x is a defector. Should the decision threshold τ exceed the sampled value, the private information will indicate that a connection should be made towards node x. Otherwise, the private information will indicate that a connection should not be made towards x. Increasing τ throughout simulations increases the likelihood that private information will indicate that a connection should be made. Whilst this will mean that newcomers are open to form more connections, this also means that more connections with defectors are increasingly likely (see Fig 1d).

2.3 Public information

2.3.1 Connection average.

To understand how public information based on previous decisions can influence structured populations. we review a type of public information, referred to as connection average (see Fig 2), that has been studied in previous works [17, 21], which does not consider previous decisions but only the current connections of a potential neighbour. In this case, public information will be obtained by examining the number of connections a node x possesses and if that count exceeds the average connection count across the network.

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Fig 2. Example scenario of connection average method being utilised.

Here, the newcomer is determining if it should form a connection with node x (a). The connection average across the network and the current connection count of x is obtained (b). As the current connection count for node x does not exceed the network average, public information will indicate that a connection should not be made (c).

https://doi.org/10.1371/journal.pone.0275909.g002

Given the node x, public information will give an indication whether or not a connection should be formed with x, based on the following conditional statements:

• If node x has connection count that exceeds connection average, public information will indicate that a connection should be formed with x
• If node x has connection count that does not exceed connection average, public information will indicate that a connection should not be formed with x

2.3.2 Global history.

We expand upon previous work [17] by considering a kind of public information based on the sequence of decisions undertaken by previous newcomers which we generally refer to as ‘global history’ (as it represents the global history of decisions taken by previous newcomers). Here, we introduce global history into the model by adding to the model a list H = {h₁, h₂…h_n} of recorded previous undertaken decisions by newcomers (i,e., H represents then the ‘global history’). After a newcomer has joined a network, a 3-tuple entry h_i = {e₁, e₂, e₃} is appended to H, recording the accepted and rejected connections during decision making (see Fig 3b). For each entry h_i, e₁ is used to record the ID of the newcomer that this entry in H relates to. e₂ and e₃ records the individuals the newcomer accepted (a) and rejected(r) connections with respectively. To restrict the amount of information available to newcomers, we implement parameter S to define the number of entries that can be present in H. When a newcomers actions are logged, and when the length of H exceeds S, the oldest entry is removed from H. To ensure that information present in H is accurate and to account for the Moran process, any accepted or rejected information relating to a node x is removed from H if that node x is removed by a newcomer. We define AcceptTotal_x as the total number of accepted connections made with a node x present on the network that are currently recorded in H. We define RejectTotal_x as the total number of connections that are rejected towards a node x present on the network that are currently recorded in H. We define fAcceptIndexes^H(x) as a list of indexes that correspond to the position of entries in H where a connection to the specified node x was accepted. We define fRejectIndexes^H(x) as a list of indexes that correspond to the position of entries in H where a connection to the specified node x was rejected.

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Fig 3. Process of appending new information to the global history list H.

As newcomers N determine which connections they form with other nodes x, the connections they accepted and rejected are noted (a). Following a newcomer N joining the network, a tuple h_i comprised of the newcomer ID e₁ and all accepted e₂ and rejected e₃ connection decisions are appended into H (b). The oldest log is also removed from H, keeping H from exceeding a size of S (c).

https://doi.org/10.1371/journal.pone.0275909.g003

The first of the methods using the global history is called ‘Bikchandani Method’, based on previous works carried out by Bikchandani et al [18]. When utilising this method, a newcomer will examine the difference d_x between the total previously accepted AcceptTotal_x and rejected RejectTotal_x connections for node x(see Fig 4). For this method, should the difference exceed either 1 or -1, the indication from private information will be overwritten to match the given indication from public information to either accept or reject a connection.

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Fig 4. Example scenario of ‘Bikchandani’ method being utilised.

Here, the newcomer is determining if it should form a connection with node x (a). The total number of accepted and rejected connections for node x are obtained (b). With these values, the difference d_x is calculated and the final decision of this evaluation is based on the conditional statements that have been defined for this method (c).

https://doi.org/10.1371/journal.pone.0275909.g004

Given the node x, public information will give an indication whether or not a connection should be formed with x, based on the following conditional statements:

• If d_x > 1, then public information and private information will indicate connection should be formed with node x, the previous indication from private information is overwritten
• If d_x = 1, then public information will indicate a connection should be formed with node x if private information also indicates likewise, otherwise, public information will indicate connection should be formed with node x with probability 0.5.
• If d_x = 0, then public information will indicate connection should not be formed with node x
• If d_x = −1, then public information will indicate a connection should not be formed with node x if private information also indicates likewise, otherwise, public information will indicate connection should not be formed with node x with probability 0.5
• If d_x < −1, then public information and private information will indicate connection should not be formed with node x, the previous indication from private information is overwritten

The second method to use global history implemented into the model we refer to as ‘Probabilistic Aggregation’. The intuition behind this method is that public information will be more likely to indicate a connection should be formed where a node x has more previously accepted connections towards it than rejections (see Fig 5). The probability P_x calculated for this method is described in (4), where AcceptTotal_x and RejectTotal_x are defined in Section 2.4. P_x will determine the likelihood that public information will indicate a connection should be formed based on the information currently stored in H for a node x. When evaluating a prospective neighbour x, if less than two entries of both accepted and rejected connections are available, a newcomer will instead utilise a coin-toss to determine if public information will indicate a connection should be made with node x.

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Fig 5. Example scenario of probabilistic aggregation method being utilised.

Here, the newcomer is determining if it should form a connection with node x (a). The total number of accepted and rejected connections for node x are obtained (b). These values are then used to calculate P_x, which determines the likelihood of public information indicating a connection should be made (c).

https://doi.org/10.1371/journal.pone.0275909.g005

Given the node x, public information will give an indication whether or not a connection should be formed with x, based on the following conditional statements:

• If AcceptTotal_x+ RejectTotal_x > 1, then public information will indicate a connection should be formed with node x with probability P_x
• If AcceptTotal_x+ RejectTotal_x ≤ 1, then public information will indicate a connection should be formed with node x with probability 0.5

(4)

The last method to utilise global history is a variation of the previously described method we refer to as ‘Diminishing Probabilistic Aggregation’. Here the age of information is taken into account, where the latest information available to individuals has a greater relevance and indication of current trends rather than older information [22] (see Fig 6). Lists fAcceptIndexes^H and fRejectIndexes^H, as described in Section 2.4, are utilised as part of (5) and (6) to calculate the values for AcceptWeighted_x and RejectWeighted_x that denote the sum of accepted and rejected connections towards node x, taking into consideration their position within H and its current size S. With these additional parameters, P_x is calculated as described in (7) and is utilised to determine the likelihood of public information indicating a connection should be formed with a node x (see Fig 6). As with the previous method, if the total weighted sum of accepted and rejected connections available in H does not exceed one, a newcomer will instead utilise a coin-toss to determine if public information will indicate a connection should be made.

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Fig 6. Example scenario of diminishing probabilistic aggregation method being utilised.

Here, the newcomer is determining if it should form a connection with node x (a). First the indexes of accepted and rejected connections within H are obtained as the two defined lists (b). Next, the weights of both accepted and rejected connections are calculated by utilising the obtained index lists and the current value of S (c). Lastly, these values are then used to calculate P_x, which determines the likelihood of public information indicating a connection should be made (d).

https://doi.org/10.1371/journal.pone.0275909.g006

Given the node x, public information will give an indication whether or not a connection should be formed with x, based on the following conditional statements:

• If AcceptWeighted_x+ RejectWeighted_x > 1, then public information will indicate a connection should be formed with node x with Probability P_x
• If AcceptWeighted+ RejectWeighted_x ≤ 1, then public information will indicate a connection should be formed with node x with Probability 0.5

(5) (6) (7)

2.4 Information based decision-making

The indications provided by private and public information are combined to determine if a newcomer will form a connection with a node x, which is either the role-model or one of its neighbours (Fig 1c). The indications on whether or not to establish a connection are as described in Sections 2.2 and 2.3. The indications provided by public and private information are weighted to establish the likelihood of a connection being created if only one indicates a connection with a node x should be made. Parameter p will determine the likelihood that a newcomer will form a connection when only public information indicates a connection should be made. Parameter q governs a similar likelihood for when only private information indicates a connection should be made. Parameters p and q are modified throughout simulations to assign more weight to one source of information over the other.

Given the node x, the chosen role-model or a neighbour of the chosen role-mode and the indications from public and private information, the newcomer decides whether to connect with a node x in the following way:

• If both public and private information indicate a connection should be made, form connection with node x
• If both public and private information indicate a connection should not be made, reject connection with node x
• If public information indicates to make a connection but private information does not, a connection with node x will be created with probability p
• If private information indicates to make a connection but public information does not, a connection with node x will be created with probability q

Once a newcomer has determined which connections it will form (see Fig 1c), the network is then updated. Following this, an individual is randomly selected to be replaced by the newcomer, keeping the population sized fixed as in the Moran process [6]. Parameter N sets the number of nodes present in the network. Parameter E sets the initial number of connections that are present in the network. Throughout simulations, we assume that the starting conditions of networks will utilise N = 100 and E = 200 with the structure being generated as a random graph. When calculating fitness, we assume δ = 0.001.

2.5 Network metrics

We study the dynamics of the model using several metrics. Cooperation is used to determine the average number of cooperators that are present on the network during a simulation. Fitness is used to determine the average level of fitness of individuals present during within a simulation. This value is normalised based on the maximum possible average fitness. Connections is used to determine the average number of connections individuals had during simulations, which is also normalised based on the maximum number of connections an individual can possess (N−1). As simulations progress, the choices made by newcomers are evaluated, which includes if they either correctly accept a connection with a cooperator or correctly reject a connection with a defector. If a newcomer correctly forms a connection with a cooperator, this will constitute a True Positive (TP); if a newcomer correctly rejects a connection with a defector, this will constitute a True Negative (TN); if a newcomer incorrectly forms a connection with a defector, this will constitute a False Positive (FP); if a newcomer incorrectly rejects a connection with a cooperator, this will constitute a False Negative (FN). To evaluate individual ability to differentiate between cooperators and defectors, specificity and sensitivity metrics are utilised. Specificity is considered as Eq (8) and sensitivity is considered as Eq (9). Specificity will determine how well newcomers are able to identify defectors and sensitivity indicates similarly for cooperators. As this model utilises randomly initialised networks in conjunction with the Moran process [6], these metrics will be logged during simulations following the first 10⁴ steps. (8) (9)

Another metric that is recorded during simulations is the number of transitions that occurred. Within the scope of this model, a transition is considered to have occurred when a previously absent strategy emerges as a consequence of mutation and spreads across the network to the point that it is the only remaining strategy. Should this value be large, this will indicate that a network configuration is somewhat unstable.

With both private and public information implemented into the model, a conflict between differing indications can arise, which can lead to erroneous decisions to be taken by individuals [17]. As described above, parameters p and q are utilised to determine the likelihood of a connection being formed when only public or private information indicates a connection should be formed, respectively. This potentially means that a conflict between public and private information may result in a newcomer making an incorrect decision when forming connections, either incorrectly rejecting a connection with a cooperator or forming a connection with a defector. As newcomers continue to join the network during simulations, these incorrect decisions can potentially propagate to other newcomers as they form connections based on their choice of role-model (see Fig 8). During simulations, the spread of these erroneous decisions are monitored as information cascades [17]. Here, we observe the emergence of two different kinds of information cascades. Illustrations of these cascades can be examined in Figs 7 and 8. The first kind of cascade that is monitored is the P cascade. Within the model, a P cascade is considered to have occurred when a newcomer rejects a connection with a potential cooperative neighbour when only private information indicates the connection should have been made and public information indicates a connection should not be made. The other kind of cascade that is monitored is the N cascade. An N cascade is considered to have occurred when a newcomer forms a connection with a defector when only private information indicates that the connection should have been rejected and public information indicates a connection should be made. For both types of cascades, the number of cascades and their length are recorded, which is calculated by determining the number of nodes that are present as part of the cascade (Figs 7 and 8). An increase in these information cascades will indicate that more newcomers are making incorrect decision when forming connections due to a conflict between public and private information.

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Fig 7. Truth table for determining which combination of choices made by newcomers are considered as contributing to cascades.

The first column indicates whether the choice to connect (C) or not connect (NC) is correct. The second and third columns show the indications from public and private information. The forth column is the actual choice made by the newcomer. The fifth column indicates if a cascade has occurred, either P or N cascade, or if none has occurred with 0.

https://doi.org/10.1371/journal.pone.0275909.g007

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Fig 8. An illustration of the two types of cascades that are monitored and and how they can potentially propagate as newcomers join the network.

A P cascade is considered to have occurred when a node does not form a connection with another cooperator based on misleading public information. A N cascade is considered to have occurred when a node incorrectly forms a connection with a defector based on misleading public information.

https://doi.org/10.1371/journal.pone.0275909.g008

3.1 Connection average

For the first series of simulations, newcomers only utilise the current connection average of a node x to serve as public information, as described in Section 2.3. As S does not need to be adjusted for these simulations, each network configuration will only consist of one series of simulations each.

The first set of these simulations utilises private information as the primary indication for if a connection should be formed (p = 0.25, q = 0.75). As can be observed in Fig 9a, the average level of cooperation gradually increases as τ is increased, which aligns with what has been observed in previous works [17, 21]. However, further increasing τ past a value of 1 eventually results in a collapse of average cooperation (Fig 9a). This spike in cooperation aligns with what can be observed for the level of fitness (Fig 9b) with significant increase followed by a sharp decline as τ continues to increase. Also aligning with these observations (Fig 9a and 9b) is the number of connections formed between individuals (Fig 9c) with a sharp increase in overall connections for greater values of τ. Something to note is the sharp increase in the number of transitions (Fig 9d) which occurs as a sharp decline in cooperation occurs (Fig 9a). With public information also present within these networks, the occurrence of information cascades are also recorded during simulations. There is a spike in N cascades for simulations with τ ≈ 0 and are generally short in length (Fig 9e and 9f). P cascades occur significantly more than N cascades throughout simulations (Fig 9g) and are generally quite long in length for lower values of τ (Fig 9h).

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Fig 9. Overall cooperation that occurs in networks gradually increases as τ is increased in networks.

However, a collapse in cooperation eventually occurs, along with a sharp increase in network transitions. The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising connection average as the method for public information where private information is prioritised. τ represents the decision threshold. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g009

For the next series of simulations, nodes are configured to prioritise public information over private information (p = 0.75, q = 0.25). With these configurations, some network metrics are largely similar to simulations where connection average was utilised (Fig 9) with some slight decreases. The average level of cooperation present in networks is largely similar, but the greatest point in cooperation is noticeably lower than simulations prioritising private information (Fig 10a). The level of observed fitness is also noted to be somewhat lower (Fig 10b) than simulations prioritising private information (Fig 9b), following similar observations noted in previous works [17]. With public information prioritised, the number of connections formed between individuals is also impacted, with overall less connections being formed (Fig 10c). Although the number of transitions still increases for higher values of τ (Fig 10d), this peak is somewhat more subdued than simulations prioritising private information (Fig 9d). Prioritising public information also leads to some differences in the occurrences of information cascades (Fig 10). When compared to simulations prioritising private information (Fig 9), the number of N cascades that occur generally increase, particularly at higher values of τ (Fig 10e). The average lengths of these N cascades are also significantly greater (Fig 10f) than previously recorded occurrences (Fig 9b). For P cascades, there is an overall decrease in occurrences, with a peak in P cascades occurring for τ ≈ 0 (Fig 10g). There is also a marked increase in average length for P cascades at lower values of τ (Fig 10h).

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Fig 10. Prioritising public information leads to an overall decrease in cooperation occurring in networks.

The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising connection average as the method for public information where public information is prioritised. τ represents the decision threshold. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g010

3.2 Bikchandani method

We investigate the effects of the different methods that use global history in the decision-making process and how they can affect cooperation within networks. For this series of simulations, newcomers utilise the ‘Bikchandani’ method as it is described in Section 2.3.2. As previously described (Section 3), S is initially be set as 10 and increased in increments of 10, up to a maximum of 50. We remind that S represents that maximum number of entries that can be stored within H, i.e. the amount of public information available to newcomers. As with simulations utilising connection-average, the values of p and q are altered to adjust the balance of public and private information throughout simulations (see Section 2.4).

The first series of simulations utilising the ‘Bikchandani’ method is carried out in networks where private information is prioritised over public information (p = 0.25, q = 0.75). Examining data from these simulations shows that both the value of S and how information in H is utilised by individuals can affect the level of cooperation occurring in networks (Fig 11a). Firstly, increasing the value of S in general leads to an increase in cooperation present in networks (Fig 11a). Although, when compared to simulations utilising connection average (Fig 9a), there is a general decrease in the level of cooperation, especially for lower values of S (Fig 11a). A similar trend can also be observed when examining the average fitness that occurs within these networks (Fig 11b). Increasing S leads to an increase in fitness, with the eventual spike aligning just before the eventual collapse in cooperation for higher values of τ (Fig 11a). Something to note here is that there comes a point where increasing S can be detrimental to fitness. Networks set as S = 40 are able to attain a higher point of fitness than networks where S = 50 (Fig 11b). One of the most significant changes with this method is the change in the number of connections formed between individuals (Fig 11c). Overall, the average number of connections is significantly lower when compared to connection average simulations (Fig 9c), which may suggest that individuals are less open to forming connections with others when historical information is available to them rather than limiting their observations of others to a specific point in time and just considers well-connected individuals (see Section 2.3.1). Similarly, lower values of S further decreases the number of connections formed, suggesting that individuals are also less likely to form connections when little information is available (Fig 11c). Lastly, similar to observations regarding fitness (Fig 11b), setting S = 50 slightly impacts the number of connections formed when compared to simulations set as S = 40, which aligns with observed fitness. Utilising this method also has some effect on the number of transitions that occurred (Fig 11d). At higher values of τ, increasing S leads to some decreases for the number of transitions occurring in networks.

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Fig 11. Networks utilising global history are able to generate more cooperation and connections amongst individuals as more information is made available by increasing S, which comes at a cost of increased information cascades.

The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising the ‘Bikhchandani’ method for public information where private information is prioritised. τ represents the decision threshold. S represents the number of entries present in global history H. Shading represents level of standard error calculated for each dataset. S represents the number of entries present in global history H. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g011

When compared against simulations utilising connection average (Fig 9d), the number of transitions overall decreased when this method is utilised, which may suggest that network stability may be more viable as individuals are enabled to make more nuanced decisions when observing others in networks. One of the more prominent areas where the effects of this information method could be observed is in the occurrences of information cascades (Fig 11), mainly N cascades. When the ‘Bikchandani’ method is utilised by networks, the occurring spike in N cascades occurrences shift towards higher values of τ (Fig 11e), rather than centered towards 0 in networks utilising the connection average method (Fig 9e). As S is increased, the occurrences in N cascades significantly increased, with a difference in average length also noted (Fig 11f). These observations suggest that newcomers are more likely to form connections with defectors when pubic information indicates a connection should be made, which aligns with how this method functions as the indications of private information can be overwritten by the indication of public information (see Section 2.3.2). Depending on the value of S, networks can either exceed or not exceed the number of N cascades occurrences observed during connection average simulations (Fig 9e). Some notable changes in P cascades are also noted here. As information availability increases, there is a slight increase in P cascades at τ ≈ 0.5 (Fig 11g) and no significant difference in average length (Fig 11h). However, when compared to connection average simulations (Fig 9c and 9d), significantly less P cascades occurred overall when newcomers utilised the Bikchandani method when evaluating potential neighbours.

For the next series of simulations, networks are configured to prioritise public information over private information (p = 0.75, q = 0.25) when newcomers determine which potential neighbours they should form connections with. In previous works [17, 21] and simulations utilising connection average (Fig 10), prioritising public information leads to some changes in network metrics, although they still largely resemble trends that occur in networks that prioritise private information (Fig 9). However, in networks where the ‘Bikchandani’ method is utilised, the result is stagnation, which we consider here as no significant increases or decreases in cooperation and connections, across all networks that utilise different τ values (Fig 12).

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Fig 12. Cooperation in networks stagnate when public information is prioritised whilst utilising the ‘Bikhchandani’ method in networks regardless of the value of S.

The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising the ‘Bikhchandani’ method for public information where public information is prioritised. τ represents the decision threshold. S represents the number of entries present in global history H. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g012

Cooperation in these networks (Fig 12a) results in an average level of ≈0.5 and networks are able to avoid a sharp collapse in cooperation regardless of the current value of τ. The trade off to this is that no networks are able to generate higher levels of cooperation. This observation aligns with the observed levels of fitness in these networks (Fig 11b). Under these conditions, there is no significant difference in fitness, networks with higher values of S are only able to attain a slightly higher level of fitness. The number of transitions that occur also stagnate (Fig 12d), suggesting that both strategies had a difficult time propagating throughout networks under these conditions. Examining observed connections may offer some reasoning behind these observations (Fig 12c). Whilst networks with greater information availability results in slightly more connections, all networks under these configurations result in very few connections being made between individuals, especially when compared against simulations utilising connection average (Fig 10c). Connection average also gradually increases as S is increased, suggesting that newcomers become more open to forming connections with more information via S, although they are still severely limited when prioritising public information. Given the rules defined for the ‘Bikchandani’ method in section 2.3.2, these observations suggest that when prioritising public information, newcomers are far less likely to form connections due to a entries in H encouraging the formation of more connections. This possibly resulted in more rejected connections and this resulting information may have further dissuaded later newcomers from forming connections.

In simulations utilising connection average (Fig 10), prioritising public information leads to some increases in the number of information cascades that occur. However, when the ‘Bikchandani’ method is utilised under these configurations, a significant decrease in N information cascades occurs (Fig 12e). Increasing S still results in a gradual increase in the number of N cascades that occur within networks (Fig 12e) with their average length also gradually increasing (Fig 12f). P cascades, in general, also increase in occurrence as S is increased (Fig 12g). Although here, P cascade occurrence remains high for higher values of τ, unlike simulations prioritising private information where P cascades eventually drop off (Fig 11g). Here, it is also possible to observe some notable changes in P cascade length, with networks with higher values of S resulting in somewhat shorter P cascades (Fig 11h).

3.3 Probabilistic aggregation method

Following on from simulations utilising the ‘Bikchandani’ method, public information is updated to utilise the probabilistic aggregation method, as described in section 2.3.2. Like previous simulations, the amount of available information via S is gradually increased in increments of 10, starting at 10 up to a maximum of 50.

For the first series of simulations utilising probabilistic aggregation, networks prioritise private information over public information (p = 0.25, q = 0.75). Similar to ‘Bikchandani’ simulations (Fig 11a), the level of cooperation that occurs within networks gradually increases as S is increased, with networks generating slightly more cooperation (Fig 13a). Although, the difference between the highest and lowest maximum observed level of cooperation is much smaller than in the case of ‘Bikchandani’ simulations (Fig 11a). A collapse in cooperation continues to occur once τ exceeds a threshold of ≈1. A similar trend can be observed for the level of fitness in these networks (Fig 13b) with greater values of S leading to a higher level of fitness and are capable of generating a higher level of fitness than ‘Bikchandani’ simulations (Fig 11b). Networks observed with these parameters are also able to result in more connections being formed between individuals (Fig 13c) than ‘Bikchandani’ simulations (Fig 11c). The trade-off to some of these improvements is a slight increase in the number of transitions that occurred during simulations (Fig 13d) after a threshold of τ is crossed at ≈1, which aligns with observations regarding the collapse in cooperation (Fig 13a). Although higher values of S can suppress the number of transitions at certain points (τ ≈ 1 & τ ≈ 2), this does result in less cooperation being generated at the highest values of τ (Fig 13a).

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Fig 13. Networks generate higher levels of cooperation as S is increased.

When public information is based on the probabilistic aggregation method and private information is prioritised, this results in a sharp increase in the number of information cascades that occur within networks. The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising the probabilistic aggregation as the method for public information where private information is prioritised. τ represents the decision threshold. S represents the number of entries present in global history H. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g013

One area where a dramatic difference occurs is with the number of information cascades observed in networks (Fig 13). The number of cascades here not only exceed connection average simulations (Fig 9) but greatly exceed the number of occurrences observed for the ‘Bikchandani’ method (Fig 11). In general, as S is increased, the number of N cascades decreases (Fig 13e). Although more N cascades occur under these parameters, the number of these greatly diminishes as τ is increased, and reaches a similar number of cascades observed in ‘Bikchandani’ simulations (Fig 11e) prior to the increase in cooperation at τ ≈ 1 (Fig 13a). Increasing S also results in slightly shorter N cascades occurring (Fig 13f). The number of P cascades that occur for lower values of τ exceed that of ‘Bikchandani’ simulations (Fig 11g), with similar levels of occurrence observed for higher values of τ (Fig 13g). The average length of P cascades (Fig 11h) also significantly decreases compared to ‘Bikchandani’ simulations (Fig 11h), which may be due to the increase in these cascades occurring. Increasing S also results in slightly longer P cascades at lower values of τ (Fig 11h).

To further evaluate the effects of probabilistic aggregation, networks are configured to prioritise public information over private information (p = 0.75, q = 0.25). Similar to what occurs with the ‘Bikchandani’ method under these conditions (Fig 12), networks here also stagnate (Fig 14). Here, networks result in a middling level of cooperation with no collapse occurring (Fig 14a). When examining values of τ ≈ 1, where previous peaks in cooperation tend to occur when private information is prioritised (Fig 14a), increasing S leads to a minor increase in cooperation. This aligns with observations regarding fitness (Fig 14b), where increased availability of information via S leads to a slight increase in fitness. Networks here also appear to generate slightly higher levels of fitness with additional information when compared to ‘Bikchandani’ simulations (Fig 12b). A minor increase also occurs for connection formation as S is increased (Fig 12c), which aligns with observations regarding cooperation and fitness. The connection count continues to remain very low under these conditions. Similar to ‘Bikchandani’ simulations (Fig 12d), the number of transitions here (Fig 14d) are relatively lower than connection average simulations (Fig 10d) and largely remain at the same level of occurrence regardless of the current value of τ.

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Fig 14. Stagnation of cooperation, fitness and stability occurs in networks utilising probabilistic aggregation when public information is prioritised over private information.

The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising the probabilistic aggregation as the method for public information where public information is prioritised. τ represents the decision threshold. S represents the number of entries present in global history H. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g014

Similar to simulations prioritising private information (Fig 13), the number of cascades observed continues to occur more frequently (Fig 14) than other utilised information types (Figs 10 & 12). As with simulations prioritising private information (Fig 13), N cascades occur most at the lowest value of τ and gradually decrease in occurrence as τ is increased (Fig 14e). Average length also gradually decreases, with networks with greater values of S resulting in slightly longer N cascades (Fig 14f). Unlike ‘Bikchandani’ simulations (Fig 12g), the occurrences of P cascades remain frequent throughout all simulations here regardless of both values of S and τ (Fig 14g). Increasing S slightly decreases the average length of P cascades at higher values of τ (Fig 14h). One noticeable change when compared to ‘Bikchandani’ simulations (Fig 12h) is that the average length of P cascades here is much shorter (Fig 14h), although this is likely due to an increased occurrence of these cascades at lower values of τ.

3.4 Diminishing probabilistic aggregation method

We next observe the effects of the method diminishing probabilistic aggregation, as described in section 2.3.2. As with previous simulations, S is initially set as 10 and is increased in increments of 10 up to a maximum of 50, the weights of information types is also adjusted throughout simulations.

Simulations here begin with networks that prioritise private information over public information (p = 0.25, q = 0.75). Similar to simulations utilising probabilistic aggregation (Fig 13a), the level of cooperation observed in these networks gradually increases as S is increased (Fig 15a). Compared with probabilistic aggregation (Fig 13a), networks attain a higher average of cooperation for higher values of S, whilst networks utilising lower values of S attain a lower average of cooperation (Fig 15a). The level of fitness that occurs here (Fig 15b) follows similar patterns to probabilistic aggregation simulations (Fig 13b) with increasing S leading to an increase in fitness. However, in this case, there is a greater decrease here in fitness for networks that utilise the least amount of information to work with (Fig 15b). Networks continue to garner an increase in connectivity between individuals as S is increased (Fig 15c), similar to previously observed patterns (Fig 13c). A decrease in connections formed can be observed for networks utilising little information (Fig 15c), which aligns with observations regarding the changes in both cooperation and fitness (Fig 15a and 15c). Increasing S in general leads to an increase in transitions occurring (Fig 15d), although observations suggest there comes a point where enough information can start to decrease the number of transitions within these networks, particularly for τ ≈ 0.8. The difference between these observed transitions (Fig 15d) also increases when compared with probabilistic aggregation (Fig 13d).

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Fig 15. Cooperation greatly increases in networks as S is increased when diminishing probabilistic aggregation is utilised by networks.

The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising the Probabilistic Aggregation as the method for public information where private information is prioritised. τ represents the decision threshold. S represents the number of entries present in global history H. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g015

The observations regarding information cascades (Fig 15) largely resemble that of simulations utilising probabilistic aggregation (Fig 13). Compared with connection average simulations (Figs 9 and 11), a significant increase in both N and P cascades occur within these networks, particularly for lower values of τ. Networks that utilise lower values of S appear to generally result in more N cascades occurring with a slight increase in length (Fig 15e and 15f). Findings from these networks also suggest that in some cases, increasing the amount of information available to newcomers can lead to increases in P cascades (Fig 15g), particularly at τ ≈ −1 and τ ≈ 1. Increasing S also results in slightly longer P cascades occurring (Fig 15h).

Further simulations are then carried out where networks prioritise public information (p = 0.75, q = 0.25) whilst utilising diminishing probabilistic aggregation. As with networks utilising other information types (Figs 12a & 14a), networks stagnate with no collapse or significant increases of cooperation occurring (Fig 16a). When compared to probabilistic aggregation simulations (Fig 14a), there is little difference in cooperation between networks that utilise different values of S (Fig 16a). Little difference in the level of fitness can also be observed in these networks, with only a slight increase for networks that utilised greater values of S (Fig 16b). Fitness here still remains very low compared to simulations where private information was prioritised (Fig 15b). The number of connections formed between individuals also decreases slightly (Fig 16c) which may offer some reasoning behind some of the changes observed for cooperation and fitness (Fig 16a and 16b). As with simulations utilising ‘Bikchandani’ and probabilistic aggregation methods under these conditions (Figs 12d & 14d), transitions remain low with no significant difference between networks utilising differing levels of information occurring (Fig 16d).

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Fig 16. Stagnation occurs when public information is prioritised whilst utilising diminishing probabilistic aggregation.

The plots visualise and compare the metrics (see Section 2.5) of cooperation, fitness, average connectivity, number of transitions and the count and average lengths of information cascades that occurred. Data obtained by running the model for 10⁸ steps utilising the probabilistic aggregation as the method for public information where public information is prioritised. τ represents the decision threshold. S represents the number of entries present in global history H. Shading represents level of standard error calculated for each dataset. Data interpolated utilising RStudio [23]. Plots produced via RStudio.

https://doi.org/10.1371/journal.pone.0275909.g016

The occurrences of information cascades here (Fig 16) largely resemble the patterns observed in probabilistic aggregation simulations (Fig 14) with only minor differences. Similar to probabilistic aggregation simulations (Fig 14e), the majority of N cascades occur at lower values of τ (Fig 16e) with no significant difference between networks utilising different values of S. Overall, the occurrences of N cascades appear to have decreased (Fig 16e) when compared to probabilistic aggregation simulations (Fig 14e). The average length of these N cascades remain fairly short (Fig 16f). No significant changes to the occurrences of P cascade (Fig 16g) can be noted when compared against the previous method of information evaluation (Fig 14g), with their average length also remaining fairly short (Fig 16h), although a slight increase in length can be observed for networks utilising more information via S.