  Becker, Greaves-Tunnell, Kontorovich, Miller, Ravikumar, and Shen determined
the long term evolution of virus propagation behavior on a hub-and-spoke graph
of one central node and $n$ neighbors, with edges only from the neighbors to
the hub (a $2$-level starlike graph), under a variant of the discrete-time SIS
(Suspectible Infected Suspectible) model. The behavior of this model is
governed by the interactions between the infection and cure probabilities,
along with the number $n$ of $2$-level nodes. They proved that for any $n$,
there is a critical threshold relating these rates, below which the virus dies
out, and above which the probabilistic dynamical system converges to a
non-trivial steady state (the probability of infection for each category of
node stabilizes). For $a$, the probability at any time step that an infected
node is not cured, and $b$, the probability at any time step that an infected
node infects its neighbors, the threshold for the virus to die out is $b \leq
(1-a)/\sqrt{n}$. We extend this analysis to $k$-level starlike graphs for $k
\geq 3$ (each $(k-1)$-level node has exactly $n_k$ neighbors, and the only
edges added are from the $k$-level nodes) for infection rates above and below
the critical threshold of $(1-a)/\sqrt{n_1+n_2+\dots+n_{k-1}}$. We do this by
first analyzing the dynamics of nodes on each level of a $3$-level starlike
graph, then show that the dynamics of the nodes of a $k$-level starlike graph
are similar, enabling us to reduce our analysis to just $3$ levels, using the
same methodology as the $3$-level case.
