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We study how the behavior of viral spreading processes is influenced by localstructural properties of the network over which they propagate. For a widevariety of spreading processes, the largest eigenvalue of the adjacency matrixof the network plays a key role on their global dynamical behavior. For manyreal-world large-scale networks, it is unfeasible to exactly retrieve thecomplete network structure to compute its largest eigenvalue. Instead, oneusually have access to myopic, egocentric views of the network structure, alsocalled egonets. In this paper, we propose a mathematical framework, based onalgebraic graph theory and convex optimization, to study how local structuralproperties of the network constrain the interval of possible values in whichthe largest eigenvalue must lie. Based on this framework, we present acomputationally efficient approach to find this interval from a collection ofegonets. Our numerical simulations show that, for several social andcommunication networks, local structural properties of the network stronglyconstrain the location of the largest eigenvalue and the resulting spreadingdynamics. From a practical point of view, our results can be used to dictateimmunization strategies to tame the spreading of a virus, or to design networktopologies that facilitate the spreading of information virally.

Structural Analysis of Viral Spreading Processes in Social and Communication Networks Using Egonets