Social systems often exhibit multilevel structures, such as multiple classrooms, firms, or groups. These structures are assumed to be independent but share a common structure. Within each group, social actors are interconnected through various social relationships, such as friendship and bullying networks. Analyzing repeated observations of these multiplex relations across different groups allows for more generalizable results and addresses group-level research questions.

We propose a hierarchical extension to the multiplex p2 model, a mixed-effects model for cross-sectional binary multiplex network data. This multilevel multiplex p2 model allows for dependency modeling at the actor, dyad, and group levels. At the actor level, ties sent and/or received by the same actors are dependent across different relational dimensions as fixed and random effects. At the dyad level, ties between two actors are dependent across different relational dimensions as fixed effects. At the group level, multiplex relationships in each group share common structures across groups, meaning the fixed effects parameters in each multiplex network are correlated between groups. The model also incorporates group-level covariates, such as class size, in addition to actor and dyad-level covariates.

Compared to fitting each multiplex network separately and aggregating by meta-analysis, our approach estimates parameters more efficiently through pooling. The degree of pooling, which controls how much information is shared between groups, is determined by the data and a prior on the amount of pooling. We demonstrate the utility of the multilevel multiplex p2 model through an original study on gossip, as perceived by gossip senders and targets, and their differing perspectives. The study is based on data from 34 Hungarian elementary school classes.

Network data consists of dyadic observations. In contrast to monadic data, which observes aspects on an individual level, network data have two indices at a minimum. Furthermore, the two sets of indices are self-referent, i.e. they are sets of labels pointing to the same objects. For example, consider a classroom of 10 students. Each individual student receives a unique label, say i\in\left\{1,2,\ldots,10\right\}. Observing ages of the students could be represented as a monadic data variable, A_i. However, the dyadic `Friendship’ relation between all pairs of students is represented in the dyadic variable F_{ij}, where i,j\in\left\{1,2,\ldots,10\right\}. This immediately shows the inherent dependence of observations, as each student is involved in 18 observations (9 sending and 9 receiving), for example, the observations in the subset {F_{1j},\ F_{i1}\ \forall\ i,j\ \lnot\ 1} all are dependent on the friendship behavior of student with the label "1”. Classical statistical tests will lead to underestimation of parameter variance under such conditions. A remedy against such dependence is to use permutation tests, that relabel the network, hence generating a random data set with equal dependence, under the null-hypotheses. This approach is very succesful, but subject to stringent conditions. For example, multicollinearity and skewed distributions require specific adaptations of the permutation approach. Another less studied issue is due to heteroskedasticity which violates the basic assumption of permutation tests, namely the exchangeability assumption. Depending on its source heteroskedasticty can be dealt with through HC-consistent estimators, but also through specific permutaion schemes. A more persistent issue lays in the fact that some non-linear models, such as the logit model inherently induce heteroskedasticity. In this paper we study how different estimators (Maximum Likelihood, GMM) exacerbate the problem of heteroskedasticity with network data. Furthermore, we present a permutation test for the presence of heteroskedasticity, which may prevent overuse of remedies.

Consider for example the determinants of the choice to buy an electric vehicle (EV) or to use modern contraceptives (MC). Your choice of EV will be influenced by your concern for the environment, economic sensibilities, etc, but also the opinions about these things of your friends. Your choice of using MC will depend on whether you think it causes conflict, whether you think others approve, and whether it is important to you that others use MC but, in a small village, the opinion about these things of the people you spend your free time with will also be important. We propose a multivariate autologistic attribute model (MALAAM) for studying the social influence on multiple binary outcomes simultaneously for cross-sectional data. This MALAAM has a regular graphical model for contingency tables as a special case. Furthermore, setting some interactions to zero yields a product ALAAM, where independent ALAAMs are estimated jointly for multiple outcomes. We demonstrate a Bayesian inference procedure for obtaining the posteriors of the model parameters and a Bayesian model selection approach using DIC. Applying this modelling framework to data on MC use in a Kenyan village we find that there are different types of social influence on different outcomes and that the social dependencies confound associations that would be inferred using a regular graphical model.

Understanding structural balance in signed graphs is a central challenge in network science, with applications in social networks, international relations, and organizational structures. One emerging approach to quantifying balance behavior is through balance correlation, a measure that captures the extent to which triadic relations follow balance theory principles. However, existing statistical tests for balance correlation rely on the expected degree distribution, which imposes strong assumptions about the underlying probability distributions. These assumptions can lead to inefficiencies in generating random graphs and, consequently, a loss of statistical power.

Our study introduces a new Fixed Degree Test for assessing balance correlations in signed and multigraphs. Unlike the expected degree distribution test, which generates random networks under more restrictive conditions, our approach preserves the observed degree marginals while allowing for more flexible network structures. Through extensive simulations, we demonstrate that the Fixed Degree Test improves the power of balance correlation significance tests, ensuring more reliable detection of balance-driven behavior in real-world networks.

Our results indicate that the expected degree test, while widely used, may over-constrain network structures, leading to misleading conclusions about balance prevalence. In contrast, the Fixed Degree Test provides a more accurate baseline, making it particularly useful for studying balance in networks with heterogeneous tie distributions. Beyond balance correlation, we will also explore how this test generalizes to other network statistics, offering a versatile framework for analyzing signed and multigraph structures.

By refining the statistical toolkit for signed network analysis, our work contributes to a more robust and flexible approach to studying balance theory in complex networks. We invite discussion on its applications across disciplines and its potential integration into broader statistical models for network dynamics.