Statistical Inference in a Directed Network Model with Covariates

Networks are often characterized by node heterogeneity for which nodes exhibit different degrees of interaction and link homophily for which nodes sharing common features tend to associate with each other. In this paper, we propose a new directed network model to capture the former via node-specific parametrization and the latter by incorporating covariates. In particular, this model quantifies the extent of heterogeneity in terms of outgoingness and incomingness of each node by different parameters, thus allowing the number of heterogeneity parameters to be twice the number of nodes. We study the maximum likelihood estimation of the model and establish the uniform consistency and asymptotic normality of the resulting estimators. Numerical studies demonstrate our theoretical findings and a data analysis confirms the usefulness of our model.Comment: 29 pages. minor revisio

Mapping Molecular Orientation with Phase Sensitive Vibrationally Resonant Sum-Frequency Generation Microscopy

We demonstrate a phase sensitive, vibrationally resonant sum-frequency generation (PSVR-SFG) microscope that combines high resolution, fast image acquisition speed, chemical selectivity, and phase sensitivity. Using the PSVR-SFG microscope, we generate amplitude and phase images of the second-order susceptibility of collagen I fibers in rat tail tendon tissue on resonance with the methylene vibrations of the protein. We find that the phase of the second-order susceptibility shows dependence on the effective polarity of the fibril bundles, revealing fibrous collagen domains of opposite orientations within the tissue. The presence of collagen microdomains in tendon tissue may have implications for the interpretation of the mechanical properties of the tissue. [Image: see text