@article {Yan2016peerj, title = {Capturing the interplay of dynamics and networks through parameterizations of Laplacian operators.}, journal = {PeerJ Computer Science}, volume = {2}, year = {2016}, pages = {e57+}, abstract = {We study the interplay between a dynamical process and the structure of the network on which it unfolds using the parameterized Laplacian framework. This framework allows for defining and characterizing an ensemble of dynamical processes on a network beyond what the traditional Laplacian is capable of modeling. This, in turn, allows for studying the impact of the interaction between dynamics and network topology on the quality-measure of network clusters and centrality, in order to effectively identify important vertices and communities in the network. Specifically, for each dynamical process in this framework, we define a centrality measure that captures a vertex{\textquoteright}s participation in the dynamical process on a given network and also define a function that measures the quality of every subset of vertices as a potential cluster (or community) with respect to this process. We show that the 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{\textquoteright}s inequality, which relates the conductance of the best cluster in a network with a spectral quantity of its Laplacian matrix, can be extended to the parameterized Laplacian. The parameterized Laplacian framework brings under the same umbrella a surprising variety of dynamical processes and allows us to systematically compare the different perspectives they create on network structure.}, keywords = {social-dynamics, social-networks}, doi = {https://doi.org/10.7717/peerj-cs.57}, url = {https://peerj.com/articles/cs-57/}, author = {Yan, Xiaoran and Teng, Shanghua and Ghosh, Rumi and Lerman, Kristina} }