  Contagion, broadly construed, refers to anything that can spread infectiously
from peer to peer. Examples include communicable diseases, rumors,
misinformation, ideas, innovations, bank failures, and electrical blackouts.
Sometimes, as in the 1918 Spanish flu epidemic, a contagion mutates at some
point as it spreads through a network. Here, using a simple
susceptible-infected (SI) model of contagion, we explore the downstream impact
of a single mutation event. Assuming that this mutation occurs at a random node
in the contact network, we calculate the distribution of the number of
"descendants," $d$, downstream from the initial "Patient Zero" mutant. We find
that the tail of the distribution decays as $d^{-2}$ for complete graphs,
random graphs, small-world networks, networks with block-like structure, and
other infinite-dimensional networks. This prediction agrees with the observed
statistics of memes propagating and mutating on Facebook, and is expected to
hold for other effectively infinite-dimensional networks, such as the global
human contact network. In a wider context, our approach suggests a possible
starting point for a mesoscopic theory of contagion. Such a theory would focus
on the paths traced by a spreading contagion, thereby furnishing an
intermediate level of description between that of individual nodes and the
total infected population. We anticipate that contagion pathways will hold
valuable lessons, given their role as the conduits through which single
mutations, innovations, or failures can sweep through a network as a whole.
