  Graphs and networks are common ways of depicting biological information. In
biology, many different biological processes are represented by graphs, such as
regulatory networks, metabolic pathways and protein--protein interaction
networks. This kind of a priori use of graphs is a useful supplement to the
standard numerical data such as microarray gene expression data. In this paper
we consider the problem of regression analysis and variable selection when the
covariates are linked on a graph. We study a graph-constrained regularization
procedure and its theoretical properties for regression analysis to take into
account the neighborhood information of the variables measured on a graph. This
procedure involves a smoothness penalty on the coefficients that is defined as
a quadratic form of the Laplacian matrix associated with the graph. We
establish estimation and model selection consistency results and provide
estimation bounds for both fixed and diverging numbers of parameters in
regression models. We demonstrate by simulations and a real data set that the
proposed procedure can lead to better variable selection and prediction than
existing methods that ignore the graph information associated with the
covariates.
