We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Prediction tasks over nodes and edges in networks require careful effort in engineering features used by learning algorithms. Recent research in the broader field of representation learning has led to significant progress in automating prediction by learning the features themselves. However, present feature learning approaches are not expressive enough to capture the diversity of connectivity patterns observed in networks. Here we propose node2vec, an algorithmic framework for learning continuous feature representations for nodes in networks. In node2vec, we learn a mapping of nodes to a low-dimensional space of features that maximizes the likelihood of preserving network neighborhoods of nodes. We define a flexible notion of a node's network neighborhood and design a biased random walk procedure, which efficiently explores diverse neighborhoods. Our algorithm generalizes prior work which is based on rigid notions of network neighborhoods, and we argue that the added flexibility in exploring neighborhoods is the key to learning richer representations. We demonstrate the efficacy of node2vec over existing state-of-the-art techniques on multi-label classification and link prediction in several real-world networks from diverse domains. Taken together, our work represents a new way for efficiently learning state-of-the-art task-independent representations in complex networks.

/pdf/node2vec-scalable-feature-learning-for-networks-yhhqueundw.pdf

node2vec: Scalable Feature Learning for Networks

Matrix Factorization Techniques for Recommender Systems

We explore the effect of dimensionality on the nearest neighbor problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 10-15 dimensions. These results should not be interpreted to mean that high-dimensional indexing is never meaningful; we illustrate this point by identifying some high-dimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate high-dimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple linear scan, and are evaluated over workloads for which nearest neighbor is not meaningful. Often, even the reported experiments, when analyzed carefully, show that linear scan would outperform the techniques being proposed on the workloads studied in high (10-15) dimensionality!.

When is nearest neighbor meaningful

Recent analyses of self-supervised learning (SSL) ﬁnd the following data-centric properties to be critical for learning good representations: invariance to task-irrelevant semantics, separability of classes in some latent space, and recoverability of labels from augmented samples. However, given their discrete, non-Euclidean nature, graph datasets and graph SSL methods are unlikely to satisfy these properties. This raises the question: how do graph SSL methods, such as contrastive learning (CL), work well? To systematically probe this question, we perform a generalization analysis for CL when using generic graph augmentations (GGAs), with a focus on data-centric properties. Our analysis yields formal insights into the limitations of GGAs and the necessity of task-relevant augmentations. As we empirically show, GGAs do not induce task-relevant invariances on common benchmark datasets, leading to only marginal gains over naive, untrained baselines. Our theory motivates a synthetic data generation process that enables control over task-relevant information and boasts pre-deﬁned optimal augmentations. This ﬂexible benchmark helps us identify yet unrecognized limitations in advanced augmentation techniques (e.g., automated methods). Overall, our work rigorously contextualizes, both empirically and theoretically, the effects of data-centric properties on augmentation strategies and learning paradigms for graph SSL.

Analyzing Data-Centric Properties for Contrastive Learning on Graphs

While most network embedding techniques model the proximity between nodes in a network, recently there has been significant interest in structural embeddings that are based on node equivalences, a notion rooted in sociology: equivalences or positions are collections of nodes that have similar roles--i.e., similar functions, ties or interactions with nodes in other positions--irrespective of their distance or reachability in the network. Unlike the proximity-based methods that are rigorously evaluated in the literature, the evaluation of structural embeddings is less mature. It relies on small synthetic or real networks with labels that are not perfectly defined, and its connection to sociological equivalences has hitherto been vague and tenuous. With new node embedding methods being developed at a breakneck pace, proper evaluation and systematic characterization of existing approaches will be essential to progress. To fill in this gap, we set out to understand what types of equivalences structural embeddings capture. We are the first to contribute rigorous intrinsic and extrinsic evaluation methodology for structural embeddings, along with carefully-designed, diverse datasets of varying sizes. We observe a number of different evaluation variables that can lead to different results (e.g., choice of similarity measure, classifier, label definitions). We find that degree distributions within nodes' local neighborhoods can lead to simple yet effective baselines in their own right and guide the future development of structural embedding. We hope that our findings can influence the design of further node embedding methods and also pave the way for more comprehensive and fair evaluation of structural embedding methods.

Towards Understanding and Evaluating Structural Node Embeddings

Network alignment, or the task of finding meaningful node correspondences between nodes in different graphs, is an important graph mining task with many scientific and industrial applications. An important principle for network alignment is matched neighborhood consistency (MNC): nodes that are close in one graph should be matched to nodes that are close in the other graph. We theoretically demonstrate a close relationship between MNC and alignment accuracy. As many existing network alignment methods struggle to preserve topological consistency in difficult scenarios, we show how to refine their solutions by improving their MNC. Our refinement method, RefiNA, is straightforward to implement, admits scalable sparse approximation, and can be paired post hoc with any network alignment method. Extensive experiments show that RefiNA increases the accuracy of diverse unsupervised network alignment methods by up to 90%, making them robust enough to align graphs that are 5x more topologically different than were considered in prior work.

Refining Network Alignment to Improve Matched Neighborhood Consistency

We introduce a general framework for leveraging graph stream data for temporal prediction-based applications. Our proposed framework includes novel methods for learning an appropriate graph time-series representation, modeling and weighting the temporal dependencies, and generalizing existing embedding methods for such data. While previous work on dynamic modeling and embedding has focused on representing a stream of timestamped edges using a time-series of graphs based on a specific time-scale (e.g., 1 month), we propose the notion of an $\epsilon$-graph time-series that uses a fixed number of edges for each graph, and show its superiority over the time-scale representation used in previous work. In addition, we propose a number of new temporal models based on the notion of temporal reachability graphs and weighted temporal summary graphs. These temporal models are then used to generalize existing base (static) embedding methods by enabling them to incorporate and appropriately model temporal dependencies in the data. From the 6 temporal network models investigated (for each of the 7 base embedding methods), we find that the top-3 temporal models are always those that leverage the new $\epsilon$-graph time-series representation. Furthermore, the dynamic embedding methods from the framework almost always achieve better predictive performance than existing state-of-the-art dynamic node embedding methods that are developed specifically for such temporal prediction tasks. Finally, the findings of this work are useful for designing better dynamic embedding methods.

From Static to Dynamic Node Embeddings.

Viruses and bacteria are critical components of the human microbiome and play important roles in health and disease Most previous work has relied on studying microbes and viruses independently, thereby reducing them to two separate communities Such approaches are unable to capture how these microbial communities interact, such as through processes that maintain community stability or allow phage-host populations to co-evolve We developed and implemented a network-based analytical approach to describe phage-bacteria network diversity throughout the human body We accomplished this by building a machine learning algorithm to predict which phages could infect which bacteria in a given microbiome This algorithm was applied to paired viral and bacterial metagenomic sequence sets from three previously published human cohorts We organized the predicted interactions into networks that allowed us to evaluate phage-bacteria connectedness across the human body We found that gut and skin network structures were person-specific and not conserved among cohabitating family members High-fat diets and obesity were associated with less connected networks Network structure differed between skin sites, with those exposed to the external environment being less connected and more prone to instability This study quantified and contrasted the diversity of virome-microbiome networks across the human body and illustrated how environmental factors may influence phage-bacteria interactive dynamics This work provides a baseline for future studies to better understand system perturbations, such as disease states, through ecological networks

http://www.biorxiv.org/content/early/2017/09/01/144642.full.pdf

Danai Koutra

Papers

Analyzing Data-Centric Properties for Contrastive Learning on Graphs

Towards Understanding and Evaluating Structural Node Embeddings

Refining Network Alignment to Improve Matched Neighborhood Consistency

From Static to Dynamic Node Embeddings.

Biogeography & Environmental Conditions Shape Phage & Bacteria Interaction Networks Across The Human Microbiome