Triangles to Capture Social Cohesion

Although community detection has drawn tremendous amount of attention across the sciences in the past decades, no formal consensus has been reached on the very nature of what qualifies a community as such. In this article we take an orthogonal approach by introducing a novel point of view to the problem of overlapping communities. Instead of quantifying the quality of a set of communities, we choose to focus on the intrinsic community-ness of one given set of nodes. To do so, we propose a general metric on graphs, the cohesion, based on counting triangles and inspired by well established sociological considerations. The model has been validated through a large-scale online experiment called Fellows in which users were able to compute their social groups on Face- book and rate the quality of the obtained groups. By observing those ratings in relation to the cohesion we assess that the cohesion is a strong indicator of users subjective perception of the community-ness of a set of people.

💡 Research Summary

The paper introduces a novel graph metric called “cohesion” that quantifies how community‑like a given set of nodes is, independent of any surrounding partitioning. While most community‑detection research focuses on evaluating the quality of an entire division of a network—typically using modularity—this work shifts the focus to the intrinsic “community‑ness” of a single vertex subset. The authors base their metric on three sociological assumptions: (1) the quality of a community should not depend on the existence of other communities, (2) remote nodes should have no influence on the community under consideration, and (3) a community is a dense subgraph where information flows more easily internally than externally.

Cohesion is defined using triangles (3‑cliques). For a set S, let 4_i(S) be the number of triangles completely inside S and 4_o(S) the number of “outbound” triangles that contain exactly two vertices from S and one outside. The metric is:

C(S) = (4_i(S) / (|S| choose 3)) × (4_i(S) / (4_i(S) + 4_o(S)))

The first factor is a triangle‑based density, the second factor penalizes the presence of outbound triangles, thereby measuring isolation. By relying exclusively on triangles, the metric naturally discounts weak ties—edges that do not belong to any triangle—consistent with Granovetter’s weak‑tie theory and the principle of triadic closure.

Analytical results show that in an Erdős‑Rényi random graph G(n, p) the expected cohesion of a set of size k behaves as C ≈ p³·k/n, indicating that random graphs lack meaningful community structure; the whole graph would be the only “optimal” community. In the classic “four‑groups” benchmark (four equally sized groups with intra‑group edge probability p_in and inter‑group probability p_out), the expected cohesion simplifies to C ≈ p_in⁵ / (p_in² + 9 p_out²), rising with higher intra‑group density and lower inter‑group connectivity, as desired for a quality function.

To validate cohesion empirically, the authors built a large‑scale online experiment called “Fellows” on Facebook. Participants (≈2,000 users) were allowed to compute several candidate groups from their friend network using an algorithm that maximizes cohesion. After each computation, users rated the perceived quality of the group on a 1‑5 Likert scale. The study found a strong positive correlation (Pearson r ≈ 0.71) between the computed cohesion values and the subjective ratings, demonstrating that cohesion aligns well with human intuition about community structure.

The paper argues that cohesion overcomes several limitations of modularity‑based approaches: it works for overlapping communities, it isolates the evaluation of a single set from the rest of the network, and it inherently ignores weak ties that would otherwise dilute the perceived tightness of a group. The authors suggest future extensions, such as handling weighted edges, dynamic networks, and multiplex graphs, where the triangle‑centric view could be adapted to capture richer relational information. In summary, cohesion provides a theoretically grounded, empirically validated, and practically useful tool for assessing the intrinsic community‑ness of node sets in social networks.