  We study the interplay between a dynamic process and the structure of the
network on which it is defined. Specifically, we examine the impact of this
interaction on the quality-measure of network clusters and node centrality.
This enables us to effectively identify network communities and important nodes
participating in the dynamics. As the first step towards this objective, we
introduce an umbrella framework for defining and characterizing an ensemble of
dynamic processes on a network. This framework generalizes the traditional
Laplacian framework to continuous-time biased random walks and also allows us
to model some epidemic processes over a network. For each dynamic process in
our framework, we can define a function that measures the quality of every
subset of nodes as a potential cluster (or community) with respect to this
process on a given network. This subset-quality function generalizes the
traditional conductance measure for graph partitioning. We partially justify
our choice of the quality function by showing that the classic Cheeger's
inequality, which relates the conductance of the best cluster in a network with
a spectral quantity of its Laplacian matrix, can be extended from the
Laplacian-conductance setting to this more general setting.
