  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. Social capital exists in the
relations among individuals, such that 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. Furthermore, state-of-the-art approaches fail to model the role of
long-range interactions between nodes in the network and their contributions to
social capital. To fill this gap, we propose to measure the social capital of a
group of nodes by means of their information flow and emphasize the importance
of considering the whole 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. 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 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.
