  This paper explores the information-theoretic limitations of graph property
testing in zero-field Ising models. Instead of learning the entire graph
structure, sometimes testing a basic graph property such as connectivity, cycle
presence or maximum clique size is a more relevant and attainable objective.
Since property testing is more fundamental than graph recovery, any necessary
conditions for property testing imply corresponding conditions for graph
recovery, while custom property tests can be statistically and/or
computationally more efficient than graph recovery based algorithms.
Understanding the statistical complexity of property testing requires the
distinction of ferromagnetic (i.e., positive interactions only) and general
Ising models. Using combinatorial constructs such as graph packing and strong
monotonicity, we characterize how target properties affect the corresponding
minimax upper and lower bounds within the realm of ferromagnets. On the other
hand, by studying the detection of an antiferromagnetic (i.e., negative
interactions only) Curie-Weiss model buried in Rademacher noise, we show that
property testing is strictly more challenging over general Ising models. In
terms of methodological development, we propose two types of correlation based
tests: computationally efficient screening for ferromagnets, and score type
tests for general models, including a fast cycle presence test. Our correlation
screening tests match the information-theoretic bounds for property testing in
ferromagnets.
