Many problems of practical interest can be represented by graphs. In practice, each edge of a graph is associated with some scalar eight denoting the similarity or the relevance between two vertices. However these edge weights might not directly reflect the relative ̀̀importance'' of edges in maintaining the structure of graphs. The study of edge ranking of graphs is to determine the graph edges by the relative importance under various criteria and is therefore essential for our fundamental understanding of graphs. In this thesis, we study various methods to determine the relative importance of the edges under several graphs models. Then we will use the edge ranking to examine two interrelated problems -- graph sparsification and graph partitioning. First, we study the edge ranking problem in the (usual) graph. We use PageRank vectors to define the edge ranking and give an improved algorithm for computing approximate (personalized) PageRank vectors on a graph of n vertices with tight error bounds which can be as small as O(1/n̂p) for any fixed positive integer p. The improved PageRank algorithm is crucial for computing the quantitative ranking of edges in a given graph. Our graph sparsification algorithm samples edges of a given graph with probabilities proportional to the edge ranking defined by PageRank. It can be used as a preprocess for graph partitioning. The combination of the graph sparsification and the partitioning algorithms using PageRank vectors leads an improved partitioning algorithm. Next, we consider the edge ranking and sparsification in connection graphs. Connection graphs arise in dealing with high-dimensional data sets in which each edge is associated with both a scalar edge weight and a d- dimensional linear transformation. We generalized the PageRank and the effective resistance in the usual graph to their vectorized versions in connection graphs. They can be used as basic tools for organizing and analyzing complex data sets. For example, the generalized PageRank and effective resistance can be utilized to derive and modify diffusion distances for vector diffusion maps in data and image processing. Furthermore, the edge ranking of connection graphs determined by the vectorized PageRank and effective resistance are an essential part of sparsification algorithms which simplify and preserve the global structure of connection graphs. Finally, we further explore the use of the vectorized version of PageRank vector in measuring the (local) consistency or inconsistency of a small area around a vertex in the connection graph. Since a PageRank vector can also be expressed as a geometric sum of random walks with different number of steps, we use the PageRank to define the (local) inconsistency coefficient for measuring the the portion of "probability'' vanished in the local random walk process. Furthermore, we develop an algorithm by using vectorized version of PageRank for finding the local consistent area around a vertex

Ferroelectrics and related materials (e.g., non-traditional ferroelectrics such as relaxors) have long been used in a range of applications, but with the advent of new ways of modeling, synthesizing, and characterizing these materials, continued access to astonishing breakthroughs in our fundamental understanding come each year. While we still rely on these materials in a range of applications, we continue to re-write what is possible to be done with them. In turn, assumptions that have underpinned the use and design of certain materials are progressively being revisited. This perspective aims to provide an overview of the field of ferroelectric/relaxor/polar-oxide thin films in recent years, with an emphasis on emergent structure and function enabled by advanced synthesis, processing, and computational modeling.

Relaxor ferroelectrics have been intensely studied for decades based on their unique electromechanical responses which arise from local structural heterogeneity involving polar nanoregions or domains. Here, we report first studies of the ultrafast dynamics and reconfigurability of the polarization in freestanding films of the prototypical relaxor 0.68PbMg_1/3Nb_2/3O₃-0.32PbTiO₃ (PMN-0.32PT) by probing its atomic-scale response via femtosecond-resolution, electron-scattering approaches. By combining these structural measurements with dynamic phase-field simulations, we show that femtosecond light pulses drive a change in both the magnitude and direction of the polarization vector within polar nanodomains on few-picosecond time scales. This study defines new opportunities for dynamic reconfigurable control of the polarization in nanoscale relaxor ferroelectrics.

The rapid development of computing applications demands novel low-energy consumption devices for information processing. Among various candidates, magnetoelectric heterostructures hold promise for meeting the required voltage and power goals. Here, a route to low-voltage control of magnetism in 30 nm Fe0.5Rh0.5/100 nm 0.68PbMg1/3Nb2/3O3-0.32PbTiO3 (PMN-PT) heterostructures is demonstrated wherein the magnetoelectric coupling is achieved via strain-induced changes in the Fe0.5Rh0.5 mediated by voltages applied to the PMN-PT. We describe approaches to achieve high-quality, epitaxial growth of Fe0.5Rh0.5 on the PMN-PT films and, a methodology to probe and quantify magnetoelectric coupling in small thin-film devices via studies of the anomalous Hall effect. By comparing the spin-flop field change induced by temperature and external voltage, the magnetoelectric coupling coefficient is estimated to reach ≈7 × 10−8 s m−1 at 325 K while applying a −0.75 V bias.