A biclique-centric exponential random graph model for common-knowledge networks

Common knowledge (CK) is a social phenomenon where individuals in a group share the same information and each knows that the others share it as well, recursively. This mutual awareness enables coordinated action and gives rise to contagion dynamics that differ from standard models such as threshold and susceptible-infectious-recovered processes. In Facebook networks, bicliques are key graph substructures for generating CK because all nodes in a biclique can share information through their walls. Motivated by the need to systematically study biclique structure and CK-based contagion, we develop a biclique-centric exponential random graph model (BC-ERGM). By incorporating a qualitative biclique term, the model is designed to generate new networks with biclique-related features similar to those of an original network, providing a framework for studying CK-based contagion on the generated networks. We also introduce a structural reduction approach to make model estimation more computationally feasible for large networks. We evaluate the framework using Erdős–Rényi (ER) graphs across multiple network sizes and average-degree settings and a stochastic block model (SBM) with pronounced community structure. In the ER experiments, BC-ERGM generally reproduces the structural properties and CK-based contagion dynamics of the original networks, while structural reduction lowers computational cost. In the SBM experiment, the edge-only, edge-plus-biclique, and edge-plus-2-star models reproduce network density similarly, but BC-ERGM more closely reproduces the biclique count, biclique-size distribution, and node-participation distribution.