Graph representation learning (also called graph embeddings) is a popular technique for incorporating network structure into machine learning models. Unsupervised graph embedding methods aim to capture graph structure by learning a low-dimensional vector representation (the embedding) for each node. Despite the widespread use of these embeddings for a variety of downstream transductive machine learning tasks, there is little principled analysis of the effectiveness of this approach for common tasks. In this work, we provide an empirical and theoretical analysis for the performance of a class of embeddings on the common task of pairwise community labeling. This is a binary variant of the classic community detection problem, which seeks to build a classifier to determine whether a pair of vertices participate in a community. In line with our goal of foundational understanding, we focus on a popular class of unsupervised embedding techniques that learn low rank factorizations of a vertex proximity matrix (this class includes methods like GraRep, Deep-Walk, node2vec, NetMF). We perform detailed empirical analysis for community labeling over a variety of real and synthetic graphs with ground truth. In all cases we studied, the models trained from embedding features perform poorly on community labeling. In constrast, a simple logistic model with classic graph structural features handily outperforms the embedding models. For a more principled understanding, we provide a theoretical analysis for the (in)effectiveness of these embeddings in capturing the community structure. We formally prove that popular low-dimensional factorization methods either cannot produce community structure, or can only produce “unstable” communities. These communities are inherently unstable under small perturbations. This theoretical result suggests that even though “good” factorizations exist, they are unlikely to be found by computational methods.

Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and$n$be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, one removes a small fraction of edges of the graph controlled by a proximity parameter to get connected components of size independent of$n$. An important tool for sublinear algorithms and property testing for such classes is the partition oracle, introduced by the seminal work of Hassidim-Kelner-Nguyen-Onak (FOCS 2009). A partition oracle is a local procedure that gives consistent access to a hyperfinite decomposition, without any preprocessing. Given a query vertex v, the partition oracle outputs the component containing v in time independent of n. All the answers are consistent with a single hyperfinite decomposition. The partition oracle of Hassidim et al. runs in time exponential in the proximity parameter per query. They pose the open problem of whether partition oracles which run in time polynomial in reciprocal of proximity parameter can be built. Levi-Ron (ICALP 2013) give a refinement of the previous approach, to get a partition oracle that runs in quasipolynomial time per query. In this paper, we resolve this open problem and give polynomial time partition oracles (in reciprocal of proximity parameter) for bounded degree graphs in any minor-closed family. Unlike the previous line of work based on combinatorial methods, we employ techniques from spectral graph theory. We build on a recent spectral graph theoretical toolkit for minor-closed graph families, introduced by the authors to develop efficient property testers. A consequence of our result is an efficient property tester for any monotone and additive with running time property of minor-closed families (such as bipartite planar graphs). Our result also gives query efficient algorithms for additive approximations for problems such as maximum matching, minimum vertex cover, maximum independent set, and minimum dominating set for these graph families.

Let G be a graph with n vertices and maximum degree d. Fix some minor-closed property P (such as planarity). We say that G is ε-far from P if one has to remove ε dn edges to make it have P. The problem of property testing P was introduced in the seminal work of Benjamini-Schramm-Shapira (STOC 2008) that gave a tester with query complexity triply exponential in ε−1. Levi-Ron (TALG 2015) have given the best tester to date, with a quasipolynomial (in ε−1) query complexity. It is an open problem to get property testers whose query complexity is (dε−1), even for planarity.

Let $G$ be a graph with $n$ vertices and maximum degree $d$. Fix some minor-closed property $\mathcal{P}$ (such as planarity). We say that $G$ is $\varepsilon$-far from $\mathcal{P}$ if one has to remove $\varepsilon dn$ edges to make it have $\mathcal{P}$. The problem of property testing $\mathcal{P}$ was introduced in the seminal work of Benjamini, Schramm, and Shapira (Symposium on the Theory of Computing 2008) that gave a tester with query complexity triply exponential in $\varepsilon^{-1}$. Levi and Ron [ ACM Trans. Algorithms, 11 (2005), 24 2015] have given the best tester to date, with a quasi-polynomial (in $\varepsilon^{-1}$) query complexity. It remained an open problem to show whether there is a property tester whose query complexity is $\mathrm{poly}(d\varepsilon^{-1})$, even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity $d\cdot \mathrm{poly}(\varepsilon^{-1})$. The previous line of work on (independent of $n$, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (Foundations of Computer Science 2018) analyzing random walk algorithms that find forbidden minors.

Let G be an undirected, bounded degree graph with n vertices. Fix a finite graph H, and suppose one must remove ε n edges from G to make it H-minor free (for some small constant ε > 0). We give an n1/2+o(1)-time randomized procedure that, with high probability, finds an H-minor in such a graph. As an application, suppose one must remove ε n edges from a bounded degree graph G to make it planar. This result implies an algorithm, with the same running time, that produces a K3,3or K5minor in G. No prior sublinear time bound was known for this problem. By the graph minor theorem, we get an analogous result for any minor-closed property. Up to no(1)factors, this resolves a conjecture of Benjamini-Schramm-Shapira (STOC 2008) on the existence of one-sided property testers for minor-closed properties. Furthermore, our algorithm is nearly optimal, by an Ω(√n) lower bound of Czumaj et al (RSA 2014). Prior to this work, the only graphs H for which non-trivial one-sided property testers were known for H-minor freeness are the following: H being a forest or a cycle (Czumaj et al, RSA 2014), K2,k, (k× 2)-grid, and the k-circus (Fichtenberger et al, Arxiv 2017).

We introduce HyperHeadTail, a streaming algorithm for estimating the degree distribution of a graph from a stream of edges using very little storage space. Real world graph streams, such as those generated by network traffic or other communication networks, tend to contain repeated elements as well as a temporal nature. Our algorithm handles these situations by extending the HeadTail algorithm of Simpson, Seshadhri, and McGregor [20]. We provide an implementation of HyperHeadTail and demonstrate its utility on both synthetic and real-world data sets. We show that HyperHeadTail offers similar performance to HeadTail, while also providing additional functionality for tracking dynamic graphs that previous algorithms cannot efficiently achieve. We show that with a space usage on the order of 8% of the number of vertices in a graph, we were able to achieve a Relative Hausdorff distance of .27.