2.1 Institution and population of Individuals

Let us consider a setting with two types of agents: Institutions and Individuals. We assume one Institution and a finite population of N Individuals subject to classification. Fig 1a provides a visual representation of the interaction between one Institution and multiple Individuals that could be part of a social network. The Institution aims at accurately classifying Individuals in order to provide a certain service. In practice, this could be applied in different contexts, from a college deciding to admit a student, to a bank deciding to offer a loan based on the customer creditworthiness. We take the case of credit lending. We assume that the Institution builds a classification algorithm in order to assign a score to each client to determine the final decision of granting or not credit. The transparency level of the feedback shared by the Institution with Individuals is modelled in our setting via parameter (see Eq. (7) for more details).

At time t, a generic Individual i is characterized by a (normalized) real feature , directly causing individuals success (i.e., actually repaying a loan), and an observable feature value , which is used by the Institution to classify users. We recall that, in real world cases, x₁ and x₂ do not necessarily match for all Individuals as declared features might be misreported.

2.2 Classification task

Let be the real success threshold determining an Individual’s real success, e.g., capturing the real ability to repay a loan in the credit lending application analysed in this paper. This means that we can state what follows, :

• when , Individual i is successful (e.g., is able to successfully repay a loan),
• when , Individual i is not successful (e.g., is not able to successfully repay a loan).

The algorithm deployed by the institution aims at classifying individuals as Positive or Negative to offer them a service (e.g., credit or loan). Let us denote with the probability of success of Individual i at time t which depends on the real feature x₁(i,t) and the real success threshold as follows

(1)

with capturing the noise between the real feature and the probability of success, and function being the Heaviside step function

(2)

Function is also referred to as the true or target classifier in [12]. Parameter introduces a potential disturbance between the real feature x₁ and the probability of success, thus linked with the possibility of predicting whether someone will to repay a loan. In this sense, represents classification noise and allows us to control for the complexity of the classification task faced by institutions.

Based on the level of classification noise , we have:

(3)

meaning that, for all Individuals i, irrespective of their real feature x₁(i,t), as becomes very large (), the probability of being a positive example (i.e., repaying a loan) gets closer to 50%. This represents a random scenario.

On the contrary, when we observe a split, based on the value of the real feature w.r.t. the real success threshold . In particular:

• i) all individuals i with real feature will successfully repay the loan:

(4)

• ii) all Individuals with will be negative examples, i.e., associated to a null probability of success

(5)

The Institution runs the internal classification model and sets a threshold , where is a parameter learned through methods such as empirical risk minimization. This parameter defines a decision boundary on the observable feature x₂, self-reported by each Individual, which determines a binary classification outcome , given by

(6)

Observe that, while identifies the real ability to repay a loan based on x₁, the threshold determines the classification outcome on the loan application only based on the information self-reported by Individuals x₂.

In practice, Institutions may have different classification thresholds , and decide to set higher or lower ones depending on how strict they wish to be in terms of requirements on the application process. A lower threshold can be translated into potential acceptance of lower quality applications, while a high threshold would imply potential higher standards set to receive a favourable outcome from the classification.

For each Individual, based on the real probability of success, , and inferred success based on , the classification task can have one of four outcomes: False Positive (FP), False Negative (FN), True Positive (TP), True Negative (TN). Fig 1b provides an overview of these outcomes.

2.3 Feedback and adaptation

A generic Individual i infers the classification threshold used by the Institution, based on the feedback provided by the latter, as

(7)

where denotes the estimate of classification threshold by Individual i, x₂(i,t) is the observable feature self-reported by Individual i and represents a value sampled from a normal distribution with mean and standard deviation . Parameter controls for the accuracy of the feedback provided by the Institution to Individuals.

We tune the transparency level of the feedback provided by the Institution through parameter (). In particular, captures the uncertainty associated to the knowledge – for Individuals – of the real decision threshold set by the Institution on the observable feature x₂. Parameter can be interpreted as a measure of noise around the real value of the threshold, capturing information uncertainty as representative of different levels of transparency. In this sense, we also refer to it as level of feedback noise. This enables us to create and assess scenarios ranging from full transparency (i.e., Individuals have full information about the exact position of the Institution decision boundary, ) to obscurity (i.e., individuals can only guess where the decision boundary lies, ). This setting builds on the model proposed in [27,28] and considers a similar conceptual framework to describe the interaction between Individuals and the Institution.

Notice that, intuitively, a high value of means that Individuals cannot do better than randomly guessing the threshold used by the Institution. On the contrary, a low value of implies that all Individuals have the exact information about the real threshold and consider strategic adaptation accordingly.

2.3.1 Adaptation.

Once information is provided, individuals build their estimate of the threshold used by the Institution, in Eq. (7), and act based on their adaptation decisions. Let us indicate the amount of fake information provided at time t by Individual i with

(8)

measuring the difference between the observable feature and the real feature for the same individual. We denote the probability of detection for Individual i at time t with d[f(i,t)], given by

(9)

Since , we introduce parameter to measure detection effectiveness. When detection always fails; when , we assume a linear dependence of the detection probability on the amount of fake information; when , detection never fails and faking is always identified.

Individuals adaptation is not free. To mimic real situations, we assume that individuals must exert an effort to adapt their features. Let us denote with , respectively, the cost of truthfully improving and faking regarding the information self-reported by Individuals to Institutions; denotes the cost for an Individual of being detected when faking. To assess the impacts of the different costs, we formally define the cost functions as follows:

(10)

In practice, the cost of faking, , results from, e.g., the effort of cheating on an exam, committing plagiarism or declaring non-existent financial information on income. In general, without considering detection, it is clear that faking in providing information involves a cost naturally much lower than the cost of honestly improving. Taking into account controls put in place by the Institution, detection plays a key role. In this sense, the cost of being detected, , is imputed when an Individual fakes and the misreported information is detected, e.g., resulting in suspension from college, a fine or an audit. In principle, costs can vary independently. However, parameter helps exploring different reasonable combinations of cost values. By tuning we can interpolate between extreme scenarios in terms of complexity faced by the Institution: i) for , Individuals are unlikely to improve (e.g., assuming faking cheap and low detection costs) and ii) for , Individuals are likely to improve (e.g., assuming faking as expensive as truthful improvement and high detection costs). As main baseline case for the analysis, we consider an intermediate scenario in which , in order to explore the core impacts deriving from transparency and behavioural imitation on complex networks without having extreme cases for the costs.

Let us consider the set of Individuals i who received a negative classification outcome. For each Individual i in this set, let us consider the estimate about the decision threshold in Eq. (7) and the detection mechanism with probability of detection d[f(i,t)] in Eq. (9). Once received the feedback from the Institution, these individuals define at time t + 1 the new pair of features, potentially different from the original one, depending on their adaptation decision and strategy. We assume that individuals decide the vector of features at time t + 1, namely , by maximizing the (expected) utility function u(i,t + 1), given as:

(11)

with indicating the variation in the information regarding the real feature

(12)

Observe that represents the output of the binary classification model deployed by the Institution given in Eq. (6); indicates the estimate on the expected classification done by Individual i; f(i,t + 1) is given in Eq. (8) and d[f(i,t + 1)] in Eq. (9). Parameter indicates the benefit of receiving a positive classification, i.e., , while , denote, respectively, the cost of improving or faking and the cost incurred after being detected given in Eq. (10).

Each term of the expected utility function can be explained as follows. We assume that individuals receive a benefit b if they are classified as positive and are not detected to be faking, ; individuals pay an improvement cost for each unit of improvement in the information regarding their real feature, ; individuals pay a faking cost for each unit of change in their observable feature alone, ; and, finally, individuals pay a detection cost in the case of faking and effective detection, . Note that, if faking does not occur, the last term of the utility function is null, i.e., individuals expect to never pay any detection cost since d[f(i,t)]=0.

2.3.2 Adaptation effort.

We keep track of individuals’ total adaptation effort and evaluate how this measure depends on social networks’ topology. The idea is to evaluate the average variation in the information regarding the real and observable features.

Let us define the average improvement effort, namely , by each user. For a given population of N individuals, let us define the average adaptation effort (improve) over the period (t,t + 1) as

(13)

with

as in Eq. (12), representing the variation in the information regarding the real feature x₁ between time t + 1 and time t for a generic individual i.

Let us recall that all individuals in the population can decide whether to fake (by only modifying their observable feature x₂), or truthfully improve (by modifying their real feature x₁. We can thereby define the average faking effort, namely , as

(14)

with

representing the variation in the information regarding the observable feature x₂ between time t + 1 and time t for a generic Individual i.

2.4 Behavioural imitation over networks

Let us introduce the possibility of imitation and assume that Individuals in the population can decide to imitate the behaviour of others Individuals in the population. Recall that adaptation consists in modifying feature values and can thereby be formalized via an adaptation vector , as

(15)

taking as elements the variation in the information regarding both the real and observable feature, for each Individual i. Let us denote with:

• the vector resulting from utility maximization by Individual i (i.e., best response), based on utility function in Eq. (11), and
• the vector resulting from the imitation of behaviour g of an observed pool of individuals in the population, for a given network topology identified via the type of network net and the specific adjacency matrix . In particular, by introducing complex network effects, we assume that is the set of neighbors (nodes) directly connected to individual i, which may change depending on the network topology . The denser the network, the bigger is the size of . Parameter g indicates whether Individual i imitates on the basis of the average value or maximum value, i.e., – emulating real cases in which individuals use the information collected to imitate an average observed behaviour or an extreme observed behaviour.

The general imitation dynamics is formally introduced by assuming that individuals willing to imitate will adapt their behaviour by setting

(16)

where parameter captures the imitation strength in the adaptation process. Fig 1d reports an illustrative example of the network structure and the imitation dynamics, meaning how each node can decide to act, i.e., based on i) best response, ii) imitation, highlighting as imitation strength parameter. In this setting, limiting the set of imitators and individuals to be imitated to specific groups implies that, in a population, not everyone has the same visibility and likelihood to be influenced by others. This is directly connected to the complex network, its structure and topology. We shall also clarify that our model assumes a one-step imitation process and does not consider long-term dynamics of information diffusion (as in alternative works such as [31]).

2.4.1 Networks and network topologies.

We model social networks where each Individual i can be seen as a node or vertex (Fig 1a). Let us assume G = (V,E) to be an undirected graph with V representing vertices and E edges. Let us denote with N = |V| the number of vertices, which correspond to the nodes for each Individual in our problem and m = |E| the number of edges in the graph. As indicator of the social network density, we consider the average degree of the undirected network. Formally, each undirected network is also characterized by a specific adjacency matrix and associated network topology that we use to define the imitation process in Eq. (16).

We consider different complex network topologies, capturing characteristics of individuals’ social networks. We test a set of alternative topologies to cover a wide range of networks, varying according to average degree, degree distribution and clustering coefficient. Fig 2 reports a visualization of the network structure for each type of topology considered in the paper: a) complete graph, in which all nodes are connected with each other; b) random-regular graph, in which all nodes are connected to the same number of neighbors [32], chosen randomly (i.e., all nodes have the same degree); c) small-world graph, characterized by high clustering coefficient and low average path length [33]; d) scale-free (Barabasi-Albert) graph, where nodes are connected through preferential attachment [34].

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Fig 2. Complex networks.

Illustrative example of the networks considered in this paper. Panel a) complete graphs, where all nodes are connected with each other. Panel b) random-regular graph, where all nodes are connected to the same number of neighbors, chosen randomly (i.e., all nodes have the same degree) [32]. Panel c) small-world graphs, characterized by high clustering coefficient and low average path length [33]; we consider a Watts-Strogatz network with p = 0.01 rewire probability. Panel d) scale-free (Barabasi-Albert) graphs [34] where nodes are connected through preferential attachment leading to heterogeneity in their degree distribution.

https://doi.org/10.1371/journal.pone.0346241.g002

2.5 Experimental setup

We consider a population size of N = 1000 and thereby networks with N = 1000 nodes. At the beginning of each simulation (run) we sample the x₁(i,0) and x₂(i,0) of each Individual i from a probability distribution with support [0,1]. We assume an initial setting without faking, , where x.(i,0) are distributed uniformly. In reality, individuals can misreport initially. However, the assumption at t = 0 is introduced to isolate the direct effect of strategic adaptation in a single step of classification. By assuming a setting where we start from , we prevent False Positives or False Negatives before our strategic adaptation step. We repeat our simulations over 50 runs, starting from random initial conditions; the reported results are an average over these 50 runs. Furthermore, while in general we assign each initial feature independently, in Fig 8 we sort the sampled feature values and node degrees, assuming an assignment where the higher features values are assigned to the most connected nodes (higher degree). Each plot includes a shaded area showing the standard deviation across runs to reflect significance and sensitivity to initial conditions.

We first initialize the setting for agents, then consider the Institution classifying them. The classification problem at stake is a simple one, given the low number of dimensions in our feature space and the existence of a clear rule that determines individuals’ success (see Eqs. (1)-(2)). We perform classification with a linear SVM (Support Vector Machine), following the default configuration of scikit-learn [35].

After the classification of Individuals, we iterate over examples of negative classification outcome (i.e., rejection) so that these users may have a chance to adapt. We perform synchronous adaptation: first, we compute the best response adaptation vector for all agents (see in Eq. (16), Section 2.4). Afterwards, we compute the imitation vector for all agents, based on the best-response vector of their neighbors. Ultimately, these two vectors are combined (given the imitation strength ) following the dynamic in Eq. (16). After adaptation, we re-run the classification algorithm and compute the difference in the confusion matrix obtained (i.e., , , , ). The S1 Appendix provides the pseudo-code underlying the experimental setup [29].

3.1 Imitation and social networks affect the risks and benefits of algorithmic transparency

Transparent feedback may impact strategic adaptation by Individuals. When the Institution provides exact information about the classification threshold used for the internal classification model (i.e., full transparency), there is a higher likelihood that Individuals decide to react by providing a new value of their observable feature x₂(i,t + 1), higher than the real feature (x₁(i,t + 1)). As a result, there is an increased likelihood in potential frauds, misreported information and data distribution shifts: in turn, it might lead to a high number of Individuals wrongly classified as positive. This has been shown in previous papers [28] and is also visible from Fig 3, where we observe that increasing leads to a decrease in . This outcome results however from a setting where imitation is not considered () and all agents act based on best response.

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Fig 3. Network effects without imitation.

The plot reports the difference in the number of False Positives after and before strategic adaptation ( on the y-axis) as a function of feedback noise ( on the x-axis). We consider a population size of N = 1000 and thereby networks with N nodes. The reported results are an average over 50 runs starting from random conditions, with each individual starting with feature values sampled from a uniform distribution. The shaded area represents the standard deviation across independent runs. As reference, here we plot the variation in when imitation is switched off, i.e., dynamic in Eq. (16) with . Results are derived for the case of Best response, i.e., utility maximization. Network topology: Scale-free (Barabasi-Albert) graph, as in Fig 2d. Results are reported for different values of the network degree As expected, network properties, such as degree, have no impact in the observed results. Parameters: b = 1.0, , , , , .

https://doi.org/10.1371/journal.pone.0346241.g003

When imitation plays a role in individuals’ adaptation (), we observe that increasing the level of feedback noise can lead to a rise in the number of False Positives. Fig 4 reveals that for all social network topologies considered, and for high average degrees (), initially increasing feedback noise leads to an increase in . In such settings, transparency results in a relatively low risk of malicious adaptation. These results reveal that the potential risks and benefits of algorithmic transparency are also impacted by individuals’ behaviour imitation patterns and potential social network effects.

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Fig 4. Transparency and accuracy of strategic classification on networks when imitating the average behaviour of neighbors.

These plots report the difference in the number of False Positives after and before strategic adaptation ( on the y-axis) as a function of feedback noise ( on the x-axis). We observe that increasing can increase , translating in higher costs to the institution, meaning that an opaque decision threshold can negatively impact the institutions’ utility. This effect is more pronounced in networks with a higher degree. We consider a population size of N = 1000 and thereby networks with N nodes. The reported results are an average over 50 runs starting from random conditions, with each individual starting with feature values sampled from a uniform distribution. The shaded area represents the standard deviation across independent runs. For each plot, we consider a different network topology, following the examples in Fig 2 and as depicted in the small inset networks. For each plot, each curve is associated to a different value of average degree . Parameters: b = 1.0, , , , . The imitation dynamic is defined in Eq. (16). For the complex network, results are based on an imitation strength , assuming that Individuals imitate based on the average, i.e. . Results are reported for different values of the network degree .

https://doi.org/10.1371/journal.pone.0346241.g004

By assessing alternative metrics to , we can observe that the variation in False Positives derives, essentially, from a transfer between True Negatives and False Positives (see Fig 5). The variation of False Negatives is negligible and the variation in True Positives has a slight increase only for the level of feedback noise above a certain value. This is explained by considering that in our simulation we assume an initial setting without faking, . As a consequence, at time t = 0 we have zero False Positives and zero False Negatives. After adaptation, Individuals initially classified as True Negative are switching to either False Positives or True Positives.

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Fig 5. Transparency, Accuracy and Complex networks: metrics overview.

The four plots report, respectively, from left to right, the difference in the number of True Positives, False Negatives, True Negatives and False Positives, after and before strategic adaptation (, , , on y-axis) as function of the level of feedback noise ( on x-axis). We can observe that the changes in False Positives observed in the previous figures (and also panel d) is mainly due to a change in the number of True Negatives (panel c). This means that the effects of noise () and network topology observed are mainly due to strategic adaptation by the users. Results are reported for different values of the network degree Parameters: b = 1.0, , , , . The imitation dynamic is defined in Eq. (16). For the complex network, results are based on an imitation strength , assuming that Individuals imitate based on the average, i.e. g = avg. Observe that associated to this case is Panel d) in Fig 4 (same as the right plot in this figure). This refers to a scale-free (Barabasi-Albert) network topology (Fig 2d).

https://doi.org/10.1371/journal.pone.0346241.g005

Fig 6 provides an intuition for the impact of feedback noise on networks, when imitating based on average behaviour. Let us consider Individuals’ adaptation for different levels of feedback noise , when social networks are defined by a scale-free network (Barabasi-Albert, N = 200 and ). Red circles represent True Negatives, after adaptation; yellow circles represent False Positives and blue circles represent True Positives. Let us observe the plot on the left, for : feedback noise and are low, according to Fig 4. This happens because imitating the average adaptation of a large pool of individuals in the population induces negative examples with a low value for the real feature (x₁) to undershoot the decision threshold even when increasing x₂, thereby remaining classified as negative. The plot in the middle reports the case of an intermediate value of the feedback noise and is high: noisy information about the Institution’s decision threshold induces Individuals to adapt by overshooting the decision threshold. In this scenario, imitating average behaviour leads thereby to sufficient adaptation for negative examples to be classified as False Positives. Finally, the plot on the right captures the case of , thus high feedback noise and low : as uncertainty about the decision threshold increases, this can induce some nodes to over-adapt and become True Positives.

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Fig 6. Intuition for the effect of noise on networks.

Here we zoom-in on individuals’ adaptation for different levels of noise, when social networks are defined by a scale-free network (Barabasi-Albert, N = 200 and ). Red circles represent True Negatives, after adaptation; yellow circles represent False Positives and blue circles represent True Positives. The leftmost plot corresponds to a setting where noise () is low and is also low, according to Fig 4. This occurs as imitating the average adaptation of a large pool of individuals in the population induces negative examples with a low real feature (x₁) to undershoot the decision threshold even if increasing x₂, thereby remaining classified as negative. The centre plot corresponds to a setting where noise () is medium () and is also high; this occurs as noisy information about the institution’s decision rule induces individuals to adapt by overshooting the decision threshold; imitating average behaviour leads thereby to sufficient adaptation for negative examples to be classified as False Positives. Finally, the rightmost plot corresponds to a setting where noise () is high () and is low; this happens as increased uncertainty about the decision threshold can induce some nodes to over-adapt and become True Positives.

https://doi.org/10.1371/journal.pone.0346241.g006

3.2 When imitating maximum behaviour, sparse networks reduce the risks of strategic classification

Next, we analyse how the previous results depend on individuals’ adaptation mode. In our model we assume that the social learning mechanism can be driven by the average or the max g = max feature observed in other individuals’. When g = max, we consider the role of imitating the most extreme behaviour. In Fig 7 we observe that, across social network topologies and network degrees, imitating maximum behaviour results in a negative correlation between feedback noise and . We also observe that the risks of strategic classification in transparent settings are reduced in sparser social networks, i.e., with lower average degree.

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Fig 7. Strategic classification when imitating the most extreme behaviour of neighbors.

These plots report the difference in the number of False Positives after and before strategic adaptation ( on the y-axis) as a function of feedback noise ( on the x-axis). We consider a population size of N = 1000 and thereby networks with N nodes. The reported results are an average over 50 runs starting from random conditions, with each individual starting with feature values sampled from a uniform distribution. The shaded area represents the standard deviation across independent runs. For each plot, we consider a different network topology, following the examples in Fig 2 and as depicted in the small inset networks. For each plot, each curve is associated to a different value of average degree . Parameters: b = 1.0, , , , . The imitation dynamic is defined in Eq. (16). For the complex network, results are based on an imitation strength , assuming that Individuals imitate based on the most extreme behaviour, i.e. . Results are reported for different values of the network degree In this setting, feedback noise leads to a decrease in , implying that uncertainty about the decision threshold reduces users’ gaming behaviour. We note however that also here networks play a role: the number of is reduced in networks with lower average degree . For each plot, we consider a different network topology, following the examples in Fig 2 and as depicted in the small inset networks. Basically, this is the same configuration as in Fig 4 apart from the imitation behaviour, i.e. here.

https://doi.org/10.1371/journal.pone.0346241.g007

3.3 Feature-degree assortment exacerbates the effects of social networks in strategic classification

We have assumed that individuals’ features are distributed randomly in the networks. In reality, individuals’ network position could be correlated with their socioeconomic status [36]. In our subsequent experiments, we assume that features are correlated with network degree. We distribute features on networks assuming that the most connected node has the highest (initial) value of x₁(i,0) and x₂(i,0). Fig 8 presents the results in this setting. Our previous conclusions are also verified in this (more realistic) case.

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Fig 8. Results when features are sorted according to degree.

These plots report the difference in the number of False Positives after and before strategic adaptation ( on the y-axis) as a function of feedback noise ( on the x-axis). Same parameters and configuration as in Fig 4 for the plots at the top (Imitate average, g = avg); same parameters and configuration as in Fig 7 for the plots at the bottom (Imitate maximum, g = max). In reality, individuals’ network position could be correlated with their socioeconomic status [36]. Here we assume that features are correlated with network degree. We distribute features on networks assuming that the most connected node has the highest (initial) value of x₁(i,0) and x₂(i,0). We confirm the effect of noise () and network topology remains when there are degree-feature correlations in the networks. This can occur when highly connected nodes (i.e., with higher social capital) are also the individuals with more financial resources to re-pay a loan. We can observe that our previous results, discussed in Figs 4 and 7 are confirmed.

https://doi.org/10.1371/journal.pone.0346241.g008

We note, however, that in this configuration, differences in the degree distribution of networks impact to a larger extent the observed results. The negative effects of feedback noise are more pronounced in scale-free networks, when users imitate average behaviour and networks have a higher average degree (Section 2.4.1). Fig 9 provides an analysis to assess the impacts of both network topology and degree-features correlation. When considering the topology of a scale-free network (Barabasi-Albert N = 200, ), we observe that nodes in highly connected positions have higher feature values (x₁ and x₂). Thus, they receive a favourable outcome by being classified as positive. As a result, based on our assumptions, they do not imitate and are not imitated by other nodes in the network: this effect drastically reduces the number of links over which imitation can occur. By interpreting this result, we can say that in practice, this has a similar effect as reducing the networks’ average degree which, as observed in Fig 4, also leads to an increase in in a setting with low feedback noise (). Considering Random regular networks (right plot), where all nodes have the same degree, prevents feature-degree correlations, thereby reducing the effect of removing highly connected nodes from the pool of negative examples.

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Fig 9. Intuition for the effect of network topology and degree-feature correlations.

In a scale-free network (Barabasi-Albert N = 200, ), nodes in highly connected positions have higher feature values (x₁ and x₂), thereby being classified as positive. As a result, they do not imitate and are not imitated by other nodes, which drastically reduces the number of links over which imitation can occur. In practice, this has a similar effect as reducing the networks’ average degree which, as observed in Fig 4, also leads to an increase in in a setting of low noise (). Considering Random regular networks (right plot), where all nodes have the same degree, prevents feature-degree correlations, thereby reducing the effect of removing highly connected nodes from the pool of negative examples.

https://doi.org/10.1371/journal.pone.0346241.g009

3.4 Network degree affects individuals’ adaptation effort

Here we dive deeper into individuals adaptation process. Fig 10 reports the extent to which individuals adapt as function of the level of feedback noise (), highlighting the dynamics of the measures introduced. In particular, Fig 10 (bottom) reports the behaviour of the average adaptation effort to improve, i.e., in Eq. (13) and Fig 10 (top) reports the behaviour of the average adaptation effort to fake, i.e., in Eq. (14).

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Fig 10. Adaptation effort.

This figure reports the average adaptation effort of users ( on y-axis) for different levels of the feedback noise ( on x-axis). We distinguish between improvement effort (bottom panels, , defined in Eq. (13)) and faking effort (top panels, , defined in Eq. (14)). We also consider adaptation based on imitation of average behaviour, i.e. g = avg (left panels) or imitation of the maximum behaviour, i.e. g = max (right panels). For each plot, we consider the same network topology, following one of the examples in Fig 2b, namely, a scale-free Barabasi-Albert network where nodes have on average connections. We can observe that users’ change their features to a large extent in networks with a smaller degree , when imitating the average behaviour. The opposite trend is observed when imitating the maximum behaviour. Moreover, we note that adaption decreases as noise increases and, overall, imitating maximum behaviour leads to a higher average adaptation. We consider a population size of N = 1000 and thereby networks with N nodes. The reported results are an average over 50 runs starting from random conditions, with each individual starting with feature values sampled from a uniform distribution. The shaded area represents the standard deviation across independent runs. For each plot, each curve is associated to a different value of average degree . Parameters: b = 1.0, , , , . The imitation dynamic is defined in Eq. (16). For the complex network, results are based on an imitation strength . Results are reported for different values of the network degree We use the same parameters as in: Fig 4b for Panel a) Imitate average, Fig 7b for Panel b) Imitate maximum.

https://doi.org/10.1371/journal.pone.0346241.g010

We observe that both and are decreasing w.r.t. the level of feedback noise . We also observe that the magnitude of adaptation is higher when considering imitation behaviour based on maximum g = max, in general, when compared with average g = avg.

Despite these similarities between honest () and malicious () adaptation, we observe important differences between these two adaptation modes. We note that has a more limited range of variation compared to . Moreover, is systematically higher than for all levels of feedback noise. This implies that, while imitation and social network effects could possibly influence both improving and faking, their impact is more pronounced in the case of faking. In this sense, as a result, faking behaviours during adaptation create an additional source of risk for institutions, since the negative effects could be further amplified. From a risk perspective, it is crucial to raise awareness about the potential burdens and risk scenarios, especially for the case of faking by imitating the maximum in a social network with high degree.

3.5 Effective detection mechanisms mitigate the risks of strategic classification on networks

First-party fraud includes intentionally misrepresenting intentions of financial situations and represents one of the main sources of fraud in digital transactions. With the present paper – besides allowing to study the role of networks in mitigating the risk of strategic adaptation – we account for the possibility that more effective faking detection methods are in place to mitigate this risk. It is crucial to ensure robustness of algorithmic design w.r.t. strategic manipulation and build strong detection mechanisms. In Fig 11 we observe that increasing detection effectiveness () reduces the overall increase in the number of False Positives . We consider simulations for different levels of detection effectiveness . We observe that our previous conclusions are verified when detection effectiveness is relatively low (). For higher levels of detection effectiveness, the number of false positives decreases, overall (). In fact, as detection effectiveness increases, its impact dominates the ones deriving from networks, probably by creating more incentive to truthfully improve, regardless of the networks structure. This highlights the importance of having strong detection mechanism in place both for institutions and society, as this could contribute to mitigate the direct risks deriving from faking behaviours not based on real creditworthiness.

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Fig 11. Strategic classification on complex networks under different levels of detection effectiveness.

These plots report the difference in the number of False Positives after and before strategic adaptation ( on the y-axis) as a function of feedback noise ( on the x-axis). We consider a population size of N = 1000 and thereby networks with N nodes. The reported results are an average over 50 runs starting from random conditions, with each individual starting with feature values sampled from a uniform distribution. The shaded area represents the standard deviation across independent runs. The plots assume a scale-free Barabasi-Albert network [34] as in Fig 2d. For the rest, we use the same configuration as Fig 4. For each plot, each curve is associated to a different value of average degree . Parameters: b = 1.0, , , . The imitation dynamic is defined in Eq. (16). For the complex network, results are based on an imitation strength , assuming that Individuals imitate based on the average, i.e. g = avg. Results are reported for different values of the network degree For each plot, we consider a specific value of the detection effectiveness (only variation w.r.t. Fig 4), left and right plot. We consider different detection probability factors (). For higher levels of detection, the number of false positives decreases, overall. Observe that Fig 4d depicts the same results as Fig 11b.

https://doi.org/10.1371/journal.pone.0346241.g011