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.

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三角丰富的复杂网络的低秩表示的不可能性

复杂网络的研究是现代科学的重大发展，丰富了社会科学、生物学、物理学和计算机科学。此类网络的模型和算法在我们的社会中无处不在，并通过社交网络、搜索引擎和推荐系统等影响人类行为。一种广泛用于对此类复杂网络进行建模的算法技术是构建网络顶点的低维欧几里得嵌入，其中顶点的接近度被解释为边缘的可能性。与普遍观点相反，我们认为这种图嵌入不能}捕获复杂网络的显着特性。我们关注的两个属性是低度和大的聚类系数，这已被广泛证实在现实世界网络的经验上是正确的。我们在数学上证明了任何可以成功创建这两个属性的嵌入（使用点积来衡量相似性）必须在顶点数量上具有几乎线性的等级。除其他含义外，这表明流行的嵌入技术（如奇异值分解和 node2vec）无法捕捉现实世界复杂网络的重要结构方面。此外，我们根据经验研究了许多基于点积的不同嵌入技术，并表明它们都无法捕获三角形结构。这表明流行的嵌入技术，如奇异值分解和 node2vec 无法捕捉现实世界复杂网络的重要结构方面。此外，我们根据经验研究了许多基于点积的不同嵌入技术，并表明它们都无法捕获三角形结构。这表明流行的嵌入技术，如奇异值分解和 node2vec 无法捕捉现实世界复杂网络的重要结构方面。此外，我们根据经验研究了许多基于点积的不同嵌入技术，并表明它们都无法捕获三角形结构。