Background <p>Many networks contain node and edge data in the form of node-specific covariates and edge weights, respectively. Established methods for investigating graph structure often rely on dichotimizing edge data. This simplification motivates the development of techniques for multivariate analysis. In the context of biological networks, such as in gene co-expression analysis, flexible tools and models are needed to interrogate data for enrichment and differential interactions. We present Multigraph Estimation Models (mGEM) as flexible alternatives to correlation-based methods for analyzing co-occurrence and co-expression data.</p> Methods <p>The mGEM approach interprets non-negative integer-valued edge weights as multiple edges in a multigraph and models the expected number of edges between nodes through a propensity-based relation. Overdispersion and node-specific attributes are treated under a unified parametric framework by changing distributional assumptions and reparameterization, respectively. By providing a background model for comparison, we are able to compute residuals that, in turn, are used to rank associations between nodes using mutual ranks and identify dependent components.</p> Conclusions <p>We consider several simulated and real data examples including a classic nematode neural network, a manually-added co-authorship network derived from Google Scholar, and <i>Saccharomyces cerevisiae</i> co-expression data from RNA-seq experiments. These examples illustrate how our models can be used in exploratory data analysis to generate potentially interesting hypotheses. For example, our co-authorship analysis extracts known collaborations between universities and reveals connections between fields of interest. In analyzing co-expression data, mGEM posits several gene modules that share similarities with correlation-based analyses yet exhibit greater diversity in terms of co-expression propensities.</p>