Inferring signed social networks from contact patterns

TL;DR

This work tackles the problem of inferring signed social networks from indirect contact data while distinguishing cases where no interaction is due to lack of opportunity versus active avoidance. It introduces a Bayesian generative model that ties a latent signed network $\mathbf{A}$, interaction- opportunity groups $\mathbf{g}$, and edge-specific probabilities together with a baseline rate $q$ to observed interaction counts, and it uses a coordinate-wise Metropolis-Hastings MCMC sampler to recover $\mathbf{A}$ with quantifiable uncertainty. Synthetic experiments show superior detection of negative edges compared with baselines, and a real-world application to French high school data demonstrates the method yields a signed network structure consistent with friendship surveys, validated by posterior predictive checks. The approach enables robust, uncertainty-aware reconstruction of signed networks from proximity data with potential applicability to diverse domains beyond schooling.

Abstract

Social networks are typically inferred from indirect observations, such as proximity data; yet, most methods cannot distinguish between absent relationships and actual negative ties, as both can result in few or no interactions. We address the challenge of inferring signed networks from contact patterns while accounting for whether lack of interactions reflect a lack of opportunity as opposed to active avoidance. We develop a Bayesian framework with MCMC inference that models interaction groups to separate chance from choice when no interactions are observed. Validation on synthetic data demonstrates superior performance compared to natural baselines, particularly in detecting negative edges. We apply our method to French high school contact data to reveal a structure consistent with friendship surveys and demonstrate the model's adequacy through posterior predictive checks.

Figures (5)

• Figure 1: Data generation and network reconstruction process. (Top) The latent signed network is the target of inference. (Middle) We observe recorded contact patterns created with the binomial model of Eq. \ref{['eq:obs_desc']}. In this example, individuals $a$, $b$, and $d$ end up in close proximity (i.e., in the same group) and can interact with high probability, while individual $c$ is by itself. This leads to, for example, 43 recorded interactions between individuals $a$ and $d$, a comparatively high number since they have a positive relationship, and four interactions between individuals $a$ and $b$, since they have a negative relationship. Individual $c$ has few interactions with others, as it is not in the same group. (Bottom) Our inference algorithm ascribes a probability to every interaction from the recorded contact patterns alone. For instance, we find that the interaction between $a$ and $d$ is very likely positive (probability 1.0), while $a$ and $b$ very likely have a negative relationship (probability 0.86). We are less certain about the interaction between $c$ and the other individuals because the model treats observations between groups as noisy.
• Figure 2: Reconstruction accuracy for our MCMC algorithm and two baselines on (left) positive and (right) negative edge classification. We generated synthetic signed networks with 64 nodes and randomly assigned them to groups. We varied the number of groups in the network partition and recorded the fraction of edges within groups (internal edge fraction), which varied from $\sim0.15$ to $1.0$. For each network and partition, we generated an observation matrix $\bm{X}$. AUC scores are shown for one-versus-rest classification as a function of the internal edge fraction.
• Figure 3: Signed network reconstruction from contact data among high school students. (a) Mean number of interactions of students during day averaged over the five days of the study. In-class interactions are shown in orange, while cross-class interactions are shown in blue. The vertical lines represent the start and end of the breaks, which are estimated from the rise of cross-class interactions. (b) Dichotomized matrix, in which pairs of students who interacted at all during the breaks are shown with a dark square. Classes are delimited with blue lines and denoted by focus, e.g., "PC" for the "Physics-Chemistry" class. (c) Inference results, in which $\hat{A}_{ij}$ is the posterior mean of the sampled interaction signs, computed from 64 samples. (d) Entropy of the estimated distribution over signs for each pair of students. Pairs with high entropy, corresponding to low certainty, are shown in black. The pairs with the lowest entropy consist of students in the same class. (e) Using questionnaire data, we identified 262 reciprocated and 144 unreciprocated friendships. This histogram shows the average of the sample means across each of the five networks, one for each day, that we inferred. A control group, comprised of 300 pairs of nodes selected uniformly at random, is shown in blue.
• Figure 4: Posterior predictive analysis. The latent signed network (top) is never observed, but it determines the observations (right). Our method can be used to estimate a network using these observations alone (bottom). In posterior predictive analysis, we then generate a new set of synthetic observations using the generative model (left), and compare them to the actual observations (middle arrow). If they are dissimilar, then the model is a poor fit and needs to be revised.
• Figure 5: (a) Example of the posterior predictive for one observation period. The diagonal line indicates equal values of the discrepancies. This test yields a Bayesian $p$-value of $0.25$, corresponding to the fraction of nodes above the dotted line. (b) Histogram of Bayesian p-values for the high school dataset. The dotted line represents the (arbitrary) $0.05$ significance level.

Theorems & Definitions (2)

• Theorem 1
• proof