  Modern threats have emerged from the prevalence of social networks. Hostile
actors, such as extremist groups or foreign governments, utilize these networks
to run propaganda campaigns with different aims. For extremists, these
campaigns are designed for recruiting new members or inciting violence. For
foreign governments, the aim may be to create instability in rival nations.
Proper social network counter-measures are needed to combat these threats. Here
we present one important counter-measure: penetrating social networks. This
means making target users connect with or follow agents deployed in the social
network. Once such connections are established with the targets, the agents can
influence them by sharing content which counters the influence campaign. In
this work we study how to penetrate a social network, which we call the
follow-back problem. The goal here is to find a policy that maximizes the
number of targets that follow the agent.
  We conduct an empirical study to understand what behavioral and network
features affect the probability of a target following an agent. We find that
the degree of the target and the size of the mutual neighborhood of the agent
and target in the network affect this probability. Based on our empirical
findings, we then propose a model for targets following an agent. Using this
model, we solve the follow-back problem exactly on directed acyclic graphs and
derive a closed form expression for the expected number of follows an agent
receives under the optimal policy. We then formulate the follow-back problem on
an arbitrary graph as an integer program. To evaluate our integer program based
policies, we conduct simulations on real social network topologies in Twitter.
We find that our polices result in more effective network penetration, with
significant increases in the expected number of targets that follow the agent.
