  The study of complex networks is a significant development in modern science,
and has enriched the social sciences, biology, physics, and computer science.
Models and algorithms for such networks are pervasive in our society, and
impact human behavior via social networks, search engines, and recommender
systems to name a few. A widely used algorithmic technique for modeling such
complex networks is to construct a low-dimensional Euclidean embedding of the
vertices of the network, where proximity of vertices is interpreted as the
likelihood of an edge. Contrary to the common view, we argue that such graph
embeddings do not}capture salient properties of complex networks. The two
properties we focus on are low degree and large clustering coefficients, which
have been widely established to be empirically true for real-world networks. We
mathematically prove that any embedding (that uses dot products to measure
similarity) that can successfully create these two properties must have rank
nearly linear in the number of vertices. Among other implications, this
establishes that popular embedding techniques such as Singular Value
Decomposition and node2vec fail to capture significant structural aspects of
real-world complex networks. Furthermore, we empirically study a number of
different embedding techniques based on dot product, and show that they all
fail to capture the triangle structure.
