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When droplets are tightly packed in a 2D microchannel, coalescence of a pair of droplets can trigger an avalanche of coalescence events that propagate through the entire emulsion. This propagation is found to be stochastic, i.e., every coalescence event does not necessarily trigger another. To study how the local probabilistic propagation affects the dynamics of the avalanche, as a whole, a stochastic agent-based model is used. Taking as input, (i) how the droplets are packed (configuration) and (ii) a measure of local probabilistic propagation (experimentally derived; function of fluid and other system parameters), the model predicts the average size distribution of avalanches. In this article, we investigate how droplet configuration affects the avalanche dynamics. We find the mean size of these avalanches to depend non-trivially on how droplets are packed together. Large variations in the avalanche dynamics are observed when droplet packing are different, even when the other system properties (number of droplets, fluid properties, channel geometry, etc.) are kept constant. Bidisperse emulsions show less variation in the dynamics and they are surprisingly more stable than monodisperse emulsions. To get a systems-level understanding of how a given droplet-configuration either facilitates or impedes the propagation of an avalanche, we employ a graph-theoretic analysis, where emulsions are expressed as graphs. We find that the properties of the underlying graph, namely the mean degree and the algebraic connectivity, are well correlated with the observed avalanche dynamics. We exploit this dependence to derive a data-based model that predicts the mean avalanche sizes from the properties of the graph. [ABSTRACT FROM AUTHOR]