Network biology aims to understand cell behavior through the analysis of underlying complex biomolecular networks. Inference of condition-specific interaction networks from epigenomic data enables the characterization of the structural plasticity that regulatory networks can acquire in different tissues of the same organism. From this perspective, uncovering specific patterns of variation by comparing network structure among tissues could provide insights into systems-level mechanisms underlying cell behavior. Following this idea, here we propose an empirical framework to analyze mammalian tissue-specific networks, focusing on characterizing and contrasting their structure and behavior in response to perturbations. We structurally represent the state of the cell/tissue by condition specific transcription factor networks generated using DNase-seq chromatin accessibility data, and we profile their systems behavior in terms of the structural robustness against random and directed perturbations. Using this framework, we unveil the structural heterogeneity existing among tissues at different levels of differentiation. We uncover a novel and conserved systems property of regulatory networks underlying embryonic stem cells (ESCs): in contrast to terminally differentiated tissues, the promiscuous regulatory connectivity of ESCs produces a globally homogeneous network resulting in increased structural robustness. We show that this property is associated with a more permissive, less restrictive chromatin accesibility state in ESCs. Possible biological consequences of this property are discussed.