  We study how the behavior of viral spreading processes is influenced by local
structural properties of the network over which they propagate. For a wide
variety of spreading processes, the largest eigenvalue of the adjacency matrix
of the network plays a key role on their global dynamical behavior. For many
real-world large-scale networks, it is unfeasible to exactly retrieve the
complete network structure to compute its largest eigenvalue. Instead, one
usually have access to myopic, egocentric views of the network structure, also
called egonets. In this paper, we propose a mathematical framework, based on
algebraic graph theory and convex optimization, to study how local structural
properties of the network constrain the interval of possible values in which
the largest eigenvalue must lie. Based on this framework, we present a
computationally efficient approach to find this interval from a collection of
egonets. Our numerical simulations show that, for several social and
communication networks, local structural properties of the network strongly
constrain the location of the largest eigenvalue and the resulting spreading
dynamics. From a practical point of view, our results can be used to dictate
immunization strategies to tame the spreading of a virus, or to design network
topologies that facilitate the spreading of information virally.
