Topological and Algebraic Properties for Classifying Unrooted Gaussian Trees under Privacy Constraints

In this paper, our objective is to find out how topological and algebraic properties of unrooted Gaussian tree models determine their security robustness, which is measured by our proposed max-min information (MaMI) metric. Such metric quantifies the amount of common randomness extractable through public discussion between two legitimate nodes under an eavesdropper attack. We show some general topological properties that the desired max-min solutions shall satisfy. Under such properties, we develop conditions under which comparable trees are put together to form partially ordered sets (posets). Each poset contains the most favorable structure as the poset leader, and the least favorable structure. Then, we compute the Tutte-like polynomial for each tree in a poset in order to assign a polynomial to any tree in a poset. Moreover, we propose a novel method, based on restricted integer partitions, to effectively enumerate all poset leaders. The results not only help us understand the security strength of different Gaussian trees, which is critical when we evaluate the information leakage issues for various jointly Gaussian distributed measurements in networks, but also provide us both an algebraic and a topological perspective in grasping some fundamental properties of such models.