Structural Group Unfairness: Measurement and Mitigation by means of the Effective Resistance

TL;DR

This work denotes the social capital disparity among different groups in a network as structural group unfairness, and proposes to mitigate it by means of a budgeted edge augmentation heuristic that systematically increases the social capital of the most disadvantaged group.

Abstract

Social networks contribute to the distribution of social capital, defined as the relationships, norms of trust and reciprocity within a community or society that facilitate cooperation and collective action. Therefore, better positioned members in a social network benefit from faster access to diverse information and higher influence on information dissemination. A variety of methods have been proposed in the literature to measure social capital at an individual level. However, there is a lack of methods to quantify social capital at a group level, which is particularly important when the groups are defined on the grounds of protected attributes. To fill this gap, we propose to measure the social capital of a group of nodes by means of the effective resistance and emphasize the importance of considering the entire network topology. Grounded in spectral graph theory, we introduce three effective resistance-based measures of group social capital, namely group isolation, group diameter and group control, where the groups are defined according to the value of a protected attribute. We denote the social capital disparity among different groups in a network as structural group unfairness, and propose to mitigate it by means of a budgeted edge augmentation heuristic that systematically increases the social capital of the most disadvantaged group. In experiments on real-world networks, we uncover significant levels of structural group unfairness when using gender as the protected attribute, with females being the most disadvantaged group in comparison to males. We also illustrate how our proposed edge augmentation approach is able to not only effectively mitigate the structural group unfairness but also increase the social capital of all groups in the network.

Figures (12)

• Figure 1: Illustration of the three proposed group social capital metrics on the same graph. The color of the nodes corresponds to $\mathop{\mathrm{\mathsf{R_{tot}}}}\nolimits(u)$, $\mathop{\mathrm{\mathcal{R}_{\text{diam}}}}\nolimits(u)$ and $\mathop{\mathrm{\mathsf{B_R}}}\nolimits(u)$, respectively. The nodes are grouped according to three different values of the protected attribute $S$ indicated as green, blue and red.
• Figure 2: Illustration of the impact of adding one edge on the information flow of $G$. Note how all the effective resistances between the star node and the rest of nodes in the network ($R_{\bigstar,v}$) decrease in $G'$ even if there is no change in the geodesic distance between them.
• Figure 3: Pareto front of the structural group unfairness (X-axis) vs the sum of the group's isolation of all the groups (Y-axis) using $\mathop{\mathrm{\mathsf{R_{tot}}}}\nolimits$ and $\mathop{\mathrm{\mathcal{R}_{\text{diam}}}}\nolimits$ (denoted by X-axis' label). Best results correspond to the bottom left corner of the graph.
• Figure 4: Evolution of the group metrics (top-row) and structural group unfairness metrics (bottom-row) as the number of added edges increases on the Google+ dataset with a total budget of 5,000 new edges.
• Figure 5: Original degree distribution per dataset and gender. We also include the average degree per group and the composition of the top 5% of nodes.

Theorems & Definitions (4)

• Theorem B.1
• proof
• Theorem B.2
• proof