Evaluation of security robustness against information leakage in Gaussian polytree graphical models

Extensive works have been undertaken to develop efficient statistical inference algorithms based on graphical models. However, there still lacks sufficient understanding about how topological properties affect certain information related metrics for certain graphs. In this paper, we are particularly interested in finding out how topological properties of rooted polytrees for Gaussian random variables determine its security robustness, which is measured by our proposed max-min information (MaMI) metric. MaMI is defined as the maximin value of the conditional mutual information between any two random variables (nodes) in a given DAG, conditioned on the value of a third random variable, which is at full disposal of an eavesdropper, under a constraint of a given fixed joint entropy. We show some general topological properties which the desired max-min solutions satisfy. Under such properties, we prove the superior max-min feature of the linear topology for a simple but non-trivial case. The results not only help us understand the security strength of different rooted polytree type DAGs, which is critical when we evaluate the information leakage issues for various jointly Gaussian distributed measurements in networks, but also provide us another algebraic and analysis perspective in grasping some fundamental properties of such DAGs.