Trust based attachment

Abstract

In social systems subject to indirect reciprocity, a positive reputation is key for increasing one's likelihood of future positive interactions [1-13]. The flow of gossip can amplify the impact of a person's actions on their reputation depending on how widely it spreads across the social network, which leads to a percolation problem [14]. To quantify this notion, we calculate the expected number of individuals, the "audience", who find out about a particular interaction. For a potential donor, a larger audience constitutes higher reputational stakes, and thus a higher incentive, to perform "good" actions in line with current social norms [7, 15]. For a receiver, a larger audience therefore increases the trust that the partner will be cooperative. This idea can be used for an algorithm that generates social networks, which we call trust based attachment (TBA). TBA produces graphs that share crucial quantitative properties with real-world networks, such as high clustering, small-world behavior, and powerlaw degree distributions [16-21]. We also show that TBA can be approximated by simple friend-of-friend routines based on triadic closure, which are known to be highly effective at generating realistic social network structures [19, 22-25]. Therefore, our work provides a new justification for triadic closure in social contexts based on notions of trust, gossip, and social information spread. These factors are thus identified as potential significant influences on how humans form social ties.

Fig 1. Calculating the size of the audience of a good deed.

a, Individual i (blue square) is connected to individual j (orange circle) on a social network. b, If i performs a cooperative act toward j, who among the neighborhood of i (dark nodes and edges) will find out about it? c, We assume that gossip originates from the recipient, j, and flows with probability p to other individuals as long as they know the donor, i.d, Depending on the structure of the network, individuals have certain probabilities to learn about the cooperative act (shown in green). Summing over these probabilities gives the expected number of individuals n_ij, which is the size of the audience for an action from i to j.

Fig 2. Properties of n_ij on local structures and real-world networks.

Values of n_ij are shown as the colors of the respective arrows. a, the value of n_ij varies between recipients (orange circles) of the same actor (blue square). The coloring is as in Fig 1 description. If the recipient, j, is central to the neighborhood of the actor, i, and connected to several others, then n_ij is large (yellow arrow). For a recipient, who is only peripherally connected, n_ij is lower (red arrow). b, Jthe value of n_ij can differ from n_ji. In both cases, the overall networks are identical and the actor and recipient have a single mutual neighbor in common (highlighted in green). In the left case, the mutual neighbor is the actor’s only additional neighbor, and n_ij is low. In the right case, the actor has several other neighbors who can learn about the interaction, and n_ij is higher. c, the value of n_ij is large between nodes that are part of the same highly connected community (magenta and orange arrows). Two such communities may only be weakly connected to each other, and thus n_ij is low for inter-community interactions. By adding some new links (thick green edges), higher incentives to cooperate (red arrow) can be established. Enhancing the interconnectivity between communities builds trust. Parameters: p = 0.4. For clarity we here omit coloring actors and recipients separately. d, Values for n_ij are shown for 1000 random edges on various real social networks [59, 60, 64] (see S1 File for details). Highly connected nodes have values of n_ij near the maximum possible k − 1 (see “Size of the expected audience” in S1 File; black dashed diagonal). The total number of nodes N and the mean degree $\overline{k}$ are given for each network. Parameters: p = 0.2.

Fig 3. Trust based attachment.

The attachment process (a-c) begins by a new individual (blue square) being introduced to a social group by a random individual (green) of that group. The new individual j selects additional friends k proportional to the trustworthiness of that link—that is the value n_kj—if it were formed. The broken lines, in color, indicate the values of n_kj on the edges that could be formed. Every time the new individual actually forms a link, the values of n_kj for future potential friends change. The new individual in this case selects two additional friends for a total of three connections (green). d, comparing graphs generated by TBA (red circles) and by friend-of-friend attachment (blue triangles). The TBA graphs display high and constant (as N → ∞) clustering. The dashed line indicates the limiting value. The networks also show small world behavior, i.e. logarithmically growing mean shortest path length (the dashed line indicates logN behavior), as well as a power law degree distribution (the dashed line has a slope of −2). Trust based attachment generates some of the same structural features as exhibited by real-world social networks, even in the simplified local form of friend-of-friend attachment. Error bars in both plots are significantly smaller than the size of the symbols. Parameters: p = 0.1, k = 6, N ∈ [15, 2000]. For other values of p, see Extended Data Fig 6 in S1 File.

Fig 4. Illustrating n_ij on a Facebook and a TBA network.

A subset of a Facebook social network [60, 64] is shown at the top, and a network generated by TBA with the same number of nodes (N) and average degree (k) is shown below. Edges are colored by n_ij and nodes are colored by the value of n_ij averaged over all neighbors j, which we denote as n_i*. While n_ij is specific to the relationship from i to j, n_i* can be seen as a measure of “average trustworthiness” of the individual i. JWhile these networks are quite small, they allow a clearer view into the local structure. The networks have strongly connected central hubs and high clustering. The highest values of n_ij appear with strongly embedded nodes in the center, while the lowest values appear with the isolated peripheral nodes. Parameters: p = 0.2. See Extended Data Figs 7 and 8 in S1 File for more details.