Local graph clustering methods aim to find a cluster of nodes by exploring a small region of the graph. These methods are attractive because they enable targeted clustering around a given seed node and are faster than traditional global graph clustering methods because their runtime does not depend on the size of the input graph. However, current local graph partitioning methods are not designed to account for the higher-order structures crucial to the network, nor can they effectively handle directed networks. Here we introduce a new class of local graph clustering methods that address these issues by incorporating higher-order network information captured by small subgraphs, also called network motifs. We develop the Motif-based Approximate Personalized PageRank (MAPPR) algorithm that finds clusters containing a seed node with minimal motif conductance, a generalization of the conductance metric for network motifs. We generalize existing theory to prove the fast running time (independent of the size of the graph) and obtain theoretical guarantees on the cluster quality (in terms of motif conductance). We also develop a theory of node neighborhoods for finding sets that have small motif conductance, and apply these results to the case of finding good seed nodes to use as input to the MAPPR algorithm. Experimental validation on community detection tasks in both synthetic and real-world networks, shows that our new framework MAPPR outperforms the current edge-based personalized PageRank methodology.

A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucial to understanding complex networks, and the clustering behavior with respect to such higher-order network structures is not well understood. Here we introduce higher-order clustering coefficients that measure the closure probability of higher-order network cliques and provide a more comprehensive view of how the edges of complex networks cluster. Our higher-order clustering coefficients are a natural generalization of the traditional clustering coefficient. We derive several properties about higher-order clustering coefficients and analyze them under common random graph models. Finally, we use higher-order clustering coefficients to gain new insights into the structure of real-world networks from several domains.

The phenomenon of edge clustering in real-world networks is a fundamental property underlying many ideas and techniques in network science. Clustering is typically quantified by the clustering coefficient, which measures the fraction of pairs of neighbors of a given center node that are connected. However, many common explanations of edge clustering attribute the triadic closure to a "head" node instead of the center node of a length-2 path-for example, "a friend of my friend is also my friend. " While such explanations are common in network analysis, there is no measurement for edge clustering that can be attributed to the head node. Here we develop local closure coefficients as a metric quantifying head-node-based edge clustering. We define the local closure coefficient as the fraction of length-2 paths emanating from the head node that induce a triangle. This subtle difference in definition leads to remarkably different properties from traditional clustering coefficients. We analyze correlations with node degree, connect the closure coefficient to community detection, and show that closure coefficients as a feature can improve link prediction.

With the rapidly growing demand for large-scale online education and the advent of big data, numerous research works have been performed to enhance learning quality in e-learning environments. Among these studies, adaptive learning has become an increasingly important issue. The traditional classification approaches analyze only the surface characteristics of students but fail to classify students accurately in terms of deep learning features. Meanwhile, these approaches are unable to analyze these high-dimensional learning behaviors in massive amounts of data. Hence, we propose a learning style classification approach based on the deep belief network (DBN) for large-scale online education to identify students’ learning styles and classify them. The first step is to build a learning style model and identify indicators of learning style based on the experiences of experts; then, relate the indicators to the different learning styles. We improve the DBN model and identify a student’s learning style by analyzing each individual’s learning style features using the improved DBN. Finally, we verify the DBN result by conducting practical experiments on an actual educational dataset. The various learning styles are determined by soliciting questionnaires from students based on the ILS theory by Felder and Soloman (1996) and the Readiness for Education At a Distance Indicator. Then, we utilized those data to train our DBNLS model. The experimental results indicate that the proposed DBNLS method has better accuracy than do the traditional approaches.

Recent work studying triadic closure in undirected graphs has drawn attention to the distinction between measures that focus on the “center” node of a wedge (i.e., length-2 path) versus measures that focus on the “initiator,” a distinction with considerable consequences. Existing measures in directed graphs, meanwhile, have all been center-focused. In this work, we propose a family of eight directed closure coefficients that measure the frequency of triadic closure in directed graphs from the perspective of the node initiating closure. The eight coefficients correspond to different labeled wedges, where the initiator and center nodes are labeled, and we observe dramatic empirical variation in these coefficients on real-world networks, even in cases when the induced directed triangles are isomorphic. To understand this phenomenon, we examine the theoretical behavior of our closure coefficients under a directed configuration model. Our analysis illustrates an underlying connection between the closure coefficients and moments of the joint in- and out-degree distributions of the network, offering an explanation of the observed asymmetries. We also use our directed closure coefficients as predictors in two machine learning tasks. We find interpretable models with AUC scores above 0.92 in class-balanced binary prediction, substantially outperforming models that use traditional center-focused measures.