  The epidemic threshold of a social system is the ratio of infection and
recovery rate above which a disease spreading in it becomes an epidemic. In the
absence of pharmaceutical interventions (i.e. vaccines), the only way to
control a given disease is to move this threshold by non-pharmaceutical
interventions like social distancing, past the epidemic threshold corresponding
to the disease, thereby tipping the system from epidemic into a non-epidemic
regime. Modeling the disease as a spreading process on a social graph, social
distancing can be modeled by removing some of the graphs links. It has been
conjectured that the largest eigenvalue of the adjacency matrix of the
resulting graph corresponds to the systems epidemic threshold. Here we use a
Markov chain Monte Carlo (MCMC) method to study those link removals that do
well at reducing the largest eigenvalue of the adjacency matrix. The MCMC
method generates samples from the relative canonical network ensemble with a
defined expectation value of $\lambda_{max}$. We call this the
"well-controlling network ensemble" (WCNE) and compare its structure to
randomly thinned networks with the same link density. We observe that networks
in the WCNE tend to be more homogeneous in the degree distribution and use this
insight to define two ad-hoc removal strategies, which also substantially
reduce the largest eigenvalue. A targeted removal of 80\% of links can be as
effective as a random removal of 90\%, leaving individuals with twice as many
contacts.
