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Structural power and the evolution of collective fairness in social networks

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ABSTRACT

Mug: From work contracts and group buying platforms to political coalitions and international climate and economical summits, often individuals assemble in groups that must collectively reach decisions that may favor each part unequally. Here we quantify to which extent our network ties promote the evolution of collective fairness in group interactions, modeled by means of Multiplayer Ultimatum Games (). We show that a single topological feature of social networks—which we call structural power—has a profound impact on the tendency of individuals to take decisions that favor each part equally. Increased fair outcomes are attained whenever structural power is high, such that the networks that tie individuals allow them to meet the same partners in different groups, thus providing the opportunity to strongly influence each other. On the other hand, the absence of such close peer-influence relationships dismisses any positive effect created by the network. Interestingly, we show that increasing the structural power of a network leads to the appearance of well-defined modules—as found in human social networks that often exhibit community structure—providing an interaction environment that maximizes collective fairness.

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Impact of structural power on fair collective action.We interpolate between a regular triangle-free ring (high SP, r = 0, panel c) and a homogeneous random graph (r = 1, low SP, panel d) by rewiring a fraction r of all edges in the network while keeping the degree distribution unchanged. Our starting topology (r = 0) differs from the conventional regular rings (illustrated, for comparison, in panel b) as, by construction, it avoids the creation of triangles, leading to a CC = 0. Panel a) shows how different global network properties change as we change r (note that in this case networks have <k> = 6, corresponding to group size N = 7) and, importantly, how they correlate with properties emerging from playing the MUG on these networks: besides the average values of offer, <p>, and acceptance threshold, <q>, we also depict the dependence of CC, APL and SP. Whereas the value of CC remains negligible for all r, (growing from 0 at r = 0 to 0.003 at r = 1) the dependence of <p> and <q> is fully correlated with that of SP and with none of the other variables plotted. Other parameters (see Methods): M = 0.5, Z = 1000, <k> = 6, μ = 0.001, ε = 0.05 and β = 10.
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pone.0175687.g003: Impact of structural power on fair collective action.We interpolate between a regular triangle-free ring (high SP, r = 0, panel c) and a homogeneous random graph (r = 1, low SP, panel d) by rewiring a fraction r of all edges in the network while keeping the degree distribution unchanged. Our starting topology (r = 0) differs from the conventional regular rings (illustrated, for comparison, in panel b) as, by construction, it avoids the creation of triangles, leading to a CC = 0. Panel a) shows how different global network properties change as we change r (note that in this case networks have <k> = 6, corresponding to group size N = 7) and, importantly, how they correlate with properties emerging from playing the MUG on these networks: besides the average values of offer, <p>, and acceptance threshold, <q>, we also depict the dependence of CC, APL and SP. Whereas the value of CC remains negligible for all r, (growing from 0 at r = 0 to 0.003 at r = 1) the dependence of <p> and <q> is fully correlated with that of SP and with none of the other variables plotted. Other parameters (see Methods): M = 0.5, Z = 1000, <k> = 6, μ = 0.001, ε = 0.05 and β = 10.

Mentions: Despite the fact that both classes of networks exhibit the same Degree Distribution (DD), they have quite different Clustering Coefficients (CC) and also Average Path Lengths (APL) [36, 37]. The regular ring networks warrant a high CC which, in turn, ensures that individuals appear repeatedly in the interaction groups of others. The prevalence of a given individual in the interaction groups of another may be understood as a power relation [15, 38, 39], that is, as a measure of the influence that an individual A has in the goals (here, fitness) of another individual B. This influence is enhanced by the fraction of interaction groups of B in which A appears (see Methods). To further characterize this property, we define an explicit quantity, that we call the Structural Power (SP). At the individual level, the structural power of an individual A over another individual B is given by the fraction of all groups in which B participates that also include A. This quantity, conveniently normalized between 0 and 1, is further extended to define the (average) SP of a node in a network, as well as the (average) SP of an entire network. Full details are provided in Methods. It is important to point out, however, that SP and CC convey different properties of a network: For instance, whereas CC only accounts for the triangular motifs present in a network, the computation of SP also reflects existing square motifs. To isolate the effect of SP from CC—and also from APL and DD—we calculate the average proposals <p> and average acceptance threshold <q> emerging when MUG is played in a class of networks in which CC always remains close to 0, but SP is not negligible (see Fig 3 and Methods).


Structural power and the evolution of collective fairness in social networks
Impact of structural power on fair collective action.We interpolate between a regular triangle-free ring (high SP, r = 0, panel c) and a homogeneous random graph (r = 1, low SP, panel d) by rewiring a fraction r of all edges in the network while keeping the degree distribution unchanged. Our starting topology (r = 0) differs from the conventional regular rings (illustrated, for comparison, in panel b) as, by construction, it avoids the creation of triangles, leading to a CC = 0. Panel a) shows how different global network properties change as we change r (note that in this case networks have <k> = 6, corresponding to group size N = 7) and, importantly, how they correlate with properties emerging from playing the MUG on these networks: besides the average values of offer, <p>, and acceptance threshold, <q>, we also depict the dependence of CC, APL and SP. Whereas the value of CC remains negligible for all r, (growing from 0 at r = 0 to 0.003 at r = 1) the dependence of <p> and <q> is fully correlated with that of SP and with none of the other variables plotted. Other parameters (see Methods): M = 0.5, Z = 1000, <k> = 6, μ = 0.001, ε = 0.05 and β = 10.
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getmorefigures.php?uid=PMC5391959&req=5

pone.0175687.g003: Impact of structural power on fair collective action.We interpolate between a regular triangle-free ring (high SP, r = 0, panel c) and a homogeneous random graph (r = 1, low SP, panel d) by rewiring a fraction r of all edges in the network while keeping the degree distribution unchanged. Our starting topology (r = 0) differs from the conventional regular rings (illustrated, for comparison, in panel b) as, by construction, it avoids the creation of triangles, leading to a CC = 0. Panel a) shows how different global network properties change as we change r (note that in this case networks have <k> = 6, corresponding to group size N = 7) and, importantly, how they correlate with properties emerging from playing the MUG on these networks: besides the average values of offer, <p>, and acceptance threshold, <q>, we also depict the dependence of CC, APL and SP. Whereas the value of CC remains negligible for all r, (growing from 0 at r = 0 to 0.003 at r = 1) the dependence of <p> and <q> is fully correlated with that of SP and with none of the other variables plotted. Other parameters (see Methods): M = 0.5, Z = 1000, <k> = 6, μ = 0.001, ε = 0.05 and β = 10.
Mentions: Despite the fact that both classes of networks exhibit the same Degree Distribution (DD), they have quite different Clustering Coefficients (CC) and also Average Path Lengths (APL) [36, 37]. The regular ring networks warrant a high CC which, in turn, ensures that individuals appear repeatedly in the interaction groups of others. The prevalence of a given individual in the interaction groups of another may be understood as a power relation [15, 38, 39], that is, as a measure of the influence that an individual A has in the goals (here, fitness) of another individual B. This influence is enhanced by the fraction of interaction groups of B in which A appears (see Methods). To further characterize this property, we define an explicit quantity, that we call the Structural Power (SP). At the individual level, the structural power of an individual A over another individual B is given by the fraction of all groups in which B participates that also include A. This quantity, conveniently normalized between 0 and 1, is further extended to define the (average) SP of a node in a network, as well as the (average) SP of an entire network. Full details are provided in Methods. It is important to point out, however, that SP and CC convey different properties of a network: For instance, whereas CC only accounts for the triangular motifs present in a network, the computation of SP also reflects existing square motifs. To isolate the effect of SP from CC—and also from APL and DD—we calculate the average proposals <p> and average acceptance threshold <q> emerging when MUG is played in a class of networks in which CC always remains close to 0, but SP is not negligible (see Fig 3 and Methods).

View Article: PubMed Central - PubMed

ABSTRACT

Mug: From work contracts and group buying platforms to political coalitions and international climate and economical summits, often individuals assemble in groups that must collectively reach decisions that may favor each part unequally. Here we quantify to which extent our network ties promote the evolution of collective fairness in group interactions, modeled by means of Multiplayer Ultimatum Games (). We show that a single topological feature of social networks&mdash;which we call structural power&mdash;has a profound impact on the tendency of individuals to take decisions that favor each part equally. Increased fair outcomes are attained whenever structural power is high, such that the networks that tie individuals allow them to meet the same partners in different groups, thus providing the opportunity to strongly influence each other. On the other hand, the absence of such close peer-influence relationships dismisses any positive effect created by the network. Interestingly, we show that increasing the structural power of a network leads to the appearance of well-defined modules&mdash;as found in human social networks that often exhibit community structure&mdash;providing an interaction environment that maximizes collective fairness.

No MeSH data available.


Related in: MedlinePlus