What defines a group of friends? Rethinking community structure in signed, directed networks

Abstract

We study the structure of personal relationships among 1068 high school students using a dataset that contains the network of self-reported friendly and conflictive relationships, with information on their directionality and intensity. We analyse the resulting weighted, directed, and signed network using a Bayesian stochastic block model framework, which enables the inference of group structure without imposing prior assumptions on the role of negative or asymmetric ties. While a full model incorporating all edge attributes yields statistically coherent clusters, these do not align with socially meaningful communities. To address this, we focus first on the network backbone of mutual affinities, and we characterize its group organization. Many communities display an assortative structure, often embedded within larger cohesive configurations, but we also observe more diverse patterns such as core-periphery structure and isolated nodes. We then examine how relationship intensity, directionality, and conflict shape group structure. Asymmetric ties, though often occurring between communities, are frequently present within them, revealing the stabilizing effect of group membership on non-mutual relationships. Furthermore, the presence of asymmetric ties does not inherently imply a hierarchical structure, given that all groups both receive and report significant levels of non-reciprocal ties. More intense ties play a disproportionate role in shaping community structure. Finally, negative ties tend to bridge communities, but we find that groups feature a significant level of internal conflict. Our research offers a new perspective on the study of group organization when rich information about the directionality, the intensity and the sign of ties is considered, with implications for identifying social vulnerability and designing targeted interventions.

Figures (5)

• Figure 1: Characterization of the simplified networks' community structure. Panel a.1) Edge count matrix $e$ for the May 2023 snapshot, used here as an example. Each row and column corresponds to a community detected by the algorithm, and each element $e_{rs}$ represents the density of edges between groups $r$ and $s$ (self-loops included), normalized by the maximum number of potential edges between those two groups. Panel a.2) Network representation of the matrix in panel a.1). Here nodes represent communities, and edge width is proportional to the weight in element $e_{rs}$ of the matrix. Inside each node, we display as many coloured points as individuals belong to that community, to illustrate differences in community size. The colour of the points indicates the school course of the individuals. Two communities, $r$ and $s$, are highlighted to exemplify the classification procedure summarized in panel b) and described in the main text. Panel b) Schematic representation of the different approximate behaviours community $r$ may display depending on the values of $e_{rr}$, $e_{r\bar{s}}$, and $e_{\bar{s}\bar{s}}$, where $\bar{s}$ is the community with the strongest connection to $r$. For each type of community, we include the percentage of communities in our dataset displaying that behaviour. Panel c.1) Distribution of community sizes, both aggregated and separated by group type. Panel c.2) Distribution of intra-group densities, both aggregated and separated by group type.
• Figure 2: Characterization of the behaviour of asymmetric positive edges and the role of positive edge intensity.Panel a) Distributions of four densities: intra-group and inter-group densities of both positive reciprocal and positive asymmetric edges for each community. Panel b) Distributions of the ratio between the density of positive asymmetric edges and the density of positive reciprocal edges, for both intra-group and inter-group ties. Panel c) Raw counts of intra-group and inter-group positive asymmetric edges. Panel d) Distributions of three densities: inter-group density of positive asymmetric edges (combining incoming and outgoing), and inter-group densities of incoming and outgoing asymmetric edges considered separately. Additionally, the distribution of the difference between incoming and outgoing densities for each community is shown. Panel e) Distributions of the average positive in-degree and out-degree for each community. Panel f) Distributions of three densities: intra-group density of positive reciprocal edges (irrespective of weight), and intra-group densities of reciprocal $+2+2$ and $+1+1$ edges separately. All panels display the distributions, both aggregated and separated by group type.
• Figure 3: Characterization of the behaviour of negative edges.Panel a) Distributions of four densities: intra-group and inter-group densities of both positive reciprocal and negative edges for each community. Panel b) Distributions of the ratio between the density of negative edges and the density of positive reciprocal edges, for both intra-group and inter-group ties. Panel c) Raw counts of intra-group and inter-group negative edges. Panel d) Distributions of three densities: inter-group density of negative edges (combining incoming and outgoing), and inter-group densities of incoming and outgoing negative edges considered separately. Panel e) Distributions of the average negative in-degree and out-degree for each community. All panels display the distributions both aggregated and separated by group type.
• Figure 4: Panel a) Network representation of the partition obtained for the May 2023 snapshot, used here as an illustrative example. Nodes, arranged in a circular layout, represent communities, and links represent the connections between them. Each link aggregates all ties of the same weight and direction between individuals belonging to the two corresponding communities, with link width proportional to the total number of aggregated ties. Self-loops represent intra-community connections, while other links correspond to inter-community connections. Link colour indicates the type of tie. Inside each node, we display as many coloured points as individuals assigned to that community, to illustrate differences in community size. The colour of the points indicates the school course of the individuals. Seven communities are highlighted as examples to explore in greater detail in panels b.1)–b.7). Panels b.1)–b.7) Examples of seven communities detected by the algorithm, each illustrating a characteristic structural pattern described in the main text. Here, nodes represent individuals, and links represent ties between them following the colour coding of panel a). Unlike panel a), link widths are uniform since they represent individual rather than aggregated ties. Nodes are arranged in two concentric circles: those in the inner circle belong to the focal community, while those in the outer circle represent individuals outside the community who are connected to its members. This layout allows us to visualize both the internal structure of each community and its external connections to the rest of the network.
• Figure 5: Distributions of the density of intra-community links of the focal community and inter-community links between the focal community and the others, ranked from the community most strongly connected to the focal one to the community most weakly connected to it. The first position on the x-axis represents the density of intra-community links (self-loop of the focal community), followed by the inter-community densities ranked from the strongest to the weakest. Black lines represent individual behaviours, while the purple box plots summarize their distributions.