  In this paper, we introduce the notion of motif closure and describe
higher-order ranking and link prediction methods based on the notion of closing
higher-order network motifs. The methods are fast and efficient for real-time
ranking and link prediction-based applications such as web search, online
advertising, and recommendation. In such applications, real-time performance is
critical. The proposed methods do not require any explicit training data, nor
do they derive an embedding from the graph data, or perform any explicit
learning. Existing methods with the above desired properties are all based on
closing triangles (common neighbors, Jaccard similarity, and the ilk). In this
work, we investigate higher-order network motifs and develop techniques based
on the notion of closing higher-order motifs that move beyond closing simple
triangles. All methods described in this work are fast with a runtime that is
sublinear in the number of nodes. The experimental results indicate the
importance of closing higher-order motifs for ranking and link prediction
applications. Finally, the proposed notion of higher-order motif closure can
serve as a basis for studying and developing better ranking and link prediction
methods.
