  Learning low-dimensional embeddings of knowledge graphs is a powerful
approach used to predict unobserved or missing edges between entities. However,
an open challenge in this area is developing techniques that can go beyond
simple edge prediction and handle more complex logical queries, which might
involve multiple unobserved edges, entities, and variables. For instance, given
an incomplete biological knowledge graph, we might want to predict "em what
drugs are likely to target proteins involved with both diseases X and Y?" -- a
query that requires reasoning about all possible proteins that {\em might}
interact with diseases X and Y. Here we introduce a framework to efficiently
make predictions about conjunctive logical queries -- a flexible but tractable
subset of first-order logic -- on incomplete knowledge graphs. In our approach,
we embed graph nodes in a low-dimensional space and represent logical operators
as learned geometric operations (e.g., translation, rotation) in this embedding
space. By performing logical operations within a low-dimensional embedding
space, our approach achieves a time complexity that is linear in the number of
query variables, compared to the exponential complexity required by a naive
enumeration-based approach. We demonstrate the utility of this framework in two
application studies on real-world datasets with millions of relations:
predicting logical relationships in a network of drug-gene-disease interactions
and in a graph-based representation of social interactions derived from a
popular web forum.
