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Questions and Answers
What is a primary use of embeddings in graph-based machine learning?
What is a primary use of embeddings in graph-based machine learning?
Which of the following tasks can be performed using graph embeddings?
Which of the following tasks can be performed using graph embeddings?
What does the adjacency matrix of a graph represent?
What does the adjacency matrix of a graph represent?
In which case would you likely use anomalous node detection in graph analysis?
In which case would you likely use anomalous node detection in graph analysis?
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Which of the following statements about graph G is accurate?
Which of the following statements about graph G is accurate?
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What is the primary purpose of using negative sampling in this context?
What is the primary purpose of using negative sampling in this context?
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When sampling negative nodes, what characteristic should ideally be prioritized?
When sampling negative nodes, what characteristic should ideally be prioritized?
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What is the effect of increasing the number of negative samples, k?
What is the effect of increasing the number of negative samples, k?
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What function is used to distinguish the target node v from sampled nodes in the background?
What function is used to distinguish the target node v from sampled nodes in the background?
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In the objective function ℒ, what is being minimized?
In the objective function ℒ, what is being minimized?
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What method is suggested for optimizing the objective function?
What method is suggested for optimizing the objective function?
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What probability distribution is used to sample the negative nodes?
What probability distribution is used to sample the negative nodes?
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What is the limitation of using any node for negative sampling?
What is the limitation of using any node for negative sampling?
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What does the parameter $p$ represent in the context of biased random walks?
What does the parameter $p$ represent in the context of biased random walks?
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In the biased random walk model, what does the term '1/𝑞' signify?
In the biased random walk model, what does the term '1/𝑞' signify?
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What aspect of the neighbors of a node is highlighted when considering the bias in a random walk?
What aspect of the neighbors of a node is highlighted when considering the bias in a random walk?
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What does the term 'walk away' parameter refer to in the context of biased random walks?
What does the term 'walk away' parameter refer to in the context of biased random walks?
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What impact does remembering the previous edge have on the biased random walk process?
What impact does remembering the previous edge have on the biased random walk process?
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What does the BFS strategy provide in the context of neighborhood definition?
What does the BFS strategy provide in the context of neighborhood definition?
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Which parameter in a biased fixed-length random walk indicates returning to the previous node?
Which parameter in a biased fixed-length random walk indicates returning to the previous node?
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What is the main difference in perspective between BFS and DFS in neighborhood definition?
What is the main difference in perspective between BFS and DFS in neighborhood definition?
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Which statement about feature engineering is correct?
Which statement about feature engineering is correct?
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What defines a random walk that is biased towards moving outwards?
What defines a random walk that is biased towards moving outwards?
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Which of the following best describes the local microscopic view provided by BFS?
Which of the following best describes the local microscopic view provided by BFS?
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What is a characteristic feature of a DFS search strategy?
What is a characteristic feature of a DFS search strategy?
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Why is hand-engineering domain-specific features considered tedious?
Why is hand-engineering domain-specific features considered tedious?
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What is the primary function of the encoder in the discussed framework?
What is the primary function of the encoder in the discussed framework?
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Which of the following statements about the similarity function is true?
Which of the following statements about the similarity function is true?
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What does the notation $ENC \ v = 𝐳𝑣$ signify in the encoding process?
What does the notation $ENC \ v = 𝐳𝑣$ signify in the encoding process?
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In the simplest encoding approach, how is the node embedding vector represented?
In the simplest encoding approach, how is the node embedding vector represented?
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What happens in the embedding lookup process?
What happens in the embedding lookup process?
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What role does the decoder play in the encoder + decoder framework?
What role does the decoder play in the encoder + decoder framework?
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Which methods for node embedding are mentioned as examples in the text?
Which methods for node embedding are mentioned as examples in the text?
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What is the aim of optimizing the parameters of the encoder?
What is the aim of optimizing the parameters of the encoder?
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What defines the dimensionality of the embeddings in the simplest encoding approach?
What defines the dimensionality of the embeddings in the simplest encoding approach?
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In the context of the discussed framework, what primarily constitutes a node's embedding?
In the context of the discussed framework, what primarily constitutes a node's embedding?
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What does the introduction of a 'virtual node' in graph embedding aim to achieve?
What does the introduction of a 'virtual node' in graph embedding aim to achieve?
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Which characteristic defines anonymous walks in graph analysis?
Which characteristic defines anonymous walks in graph analysis?
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In the example provided, how many anonymous walks of length 3 can be formed?
In the example provided, how many anonymous walks of length 3 can be formed?
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What is the output representation of a graph using anonymous walks of length 3?
What is the output representation of a graph using anonymous walks of length 3?
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How does sampling anonymous walks contribute to graph representation?
How does sampling anonymous walks contribute to graph representation?
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What is a key outcome of performing a random walk on a graph?
What is a key outcome of performing a random walk on a graph?
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What characteristic of anonymous walks results in their exponential growth?
What characteristic of anonymous walks results in their exponential growth?
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What is the primary focus of the technique proposed by Li et al., 2016?
What is the primary focus of the technique proposed by Li et al., 2016?
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Study Notes
CS224W: Node Embeddings
- CS224W is a machine learning course focusing on graphs at Stanford University.
- The course covers node embeddings, a technique for representing nodes in a graph in a low-dimensional vector space.
- Traditional machine learning for graphs requires feature engineering for each individual graph.
- Graph representation learning removes the need to manually engineer features for each graph.
- The goal of graph representation learning is to efficiently learn task-independent features for machine learning tasks on graphs.
- Node embeddings aim to map nodes to vectors, where a node's similarity in the embedding space corresponds to similarity in the network.
- Embeddings are potentially useful for various downstream tasks, including node classification, link prediction, graph classification, anomalous node detection, and clustering.
- Zachary's Karate Club network is an example used to illustrate 2D node embeddings.
Recap: Traditional ML for Graphs
- Traditional machine learning for graphs involves extracting node, link, and graph-level features from an input graph.
- A machine learning model (e.g., SVM, neural network) is then trained to map these extracted features to graph-related labels.
- Feature engineering is a critical step in this approach.
Graph Representation Learning
- Graph representation learning automates feature extraction by directly learning features from the graph structure.
- It alleviates the need for manual feature engineering.
- Learning the features automatically from the graph means better scalability.
Why Embedding?
- Graph embeddings aim to map nodes into an embedding space, making nodes with similar relationships close in the space.
- Similarity in embeddings indicates similarity in the original graph structure.
Setup
- A graph G is represented with a vertex set V and an adjacency matrix A.
- The adjacency matrix is assumed to be binary.
- No additional node features are considered for simplicity.
Embedding Nodes
- The goal is to create node embeddings that capture the similarity structure of the original graph.
- The similarity in the embedding space should reflect the relationships in the graph.
Learning Node Embeddings
- The encoder transforms nodes into embeddings.
- A node similarity function defines how similarities in the graph map to the embedding space.
- The decoder maps embeddings to similarity scores.
- The encoder parameters are optimized such that the encoded similarity reflects the original graph similarity.
“Shallow” Encoding
- The simplest encoding approach where the encoder directly provides an embedding for each node from a learned embedding matrix.
- The embedding matrix contains a low-dimensional vector (embedding) for every node in the graph
Framework Summary
- The framework involves a shallow encoder with an embedding lookup, where the parameters to optimize are the embeddings.
- The decoder is based on node similarity, aiming to maximize the dot product between similar nodes' embeddings.
How to Define Node Similarity?
- The method that defines node similarity is key to the node embedding technique.
- Similarity can be determined based on links, shared neighbors, or structural roles.
Note on Node Embeddings
- Node embeddings are learned using an unsupervised/self-supervised approach.
- Node labels and features are not used in the learning process.
- The goal is to preserve network structure by capturing coordinates (embedding) of the nodes.
- Embeddings are task-independent, usable for various tasks.
Random Walk Approaches for Node Embeddings
- The focus here is on using random walks to learn node embeddings.
- The approach aims to estimate the probability of visiting a node during a random walk.
- Non-linear functions, like softmax, are utilized to normalize predicted probabilities of visits on random walks.
Random-Walk Embeddings
- The goal is represented through similarity in the embedding space.
- The strategy involves estimating probabilities of each node being visited from a start node over a graph walk.
- Optimizing embeddings, via a maximum likelihood approach, aims to accurately encode random walk statistics.
Why Random Walks?
- Random walks provide a flexible way to define node similarity.
- They capture the local and higher-order structural information from the graph better than just node neighbors or links.
- Random walks are computationally efficient, considering only co-occurring node pairs during training.
Unsupervised Feature Learning
- The intuition is to embed nodes in a low dimensional space that preserves their relationships in the network.
Feature Learning as Optimization
- The goal is to learn a function that maps each node to its embedding (low-dimensional representation) in the graph.
Random Walk Optimization
- Embeddings are optimized by examining random walks over nodes in the graph, finding walks that co-occur on the graph, and optimizing for the likelihood of those co-occurrences.
Random Walk Optimization
- Optimizing involves calculating probabilities from random walk co-occurrences and determining an equivalent function.
- These calculations initially face high computational complexity that necessitates approximation via negative sampling.
Negative Sampling
- Negative sampling is used for efficient computation.
- Unlike normalizing over all nodes, it normalizes over a subset of sampled negative nodes for each node in the graph, thus reducing computation complexity.
Stochastic Gradient Descent
- Gradient descent is used to minimize the objective function.
- Stochastic gradient descent is used to calculate the gradient for each training example.
Random Walks: Summary
- The summary describes the steps to obtain and find useful random walks on graphs for node embedding.
How Should We Randomly Walk?
- The summary mentions different strategies for random walks.
- DeepWalk from Perozzi et al. (2013) is presented as a simple strategy.
- The current algorithm has limitations that need to be addressed or generalized.
Overview of Node2Vec
- node2vec embeds nodes to be close when they have similar neighborhoods in the graph.
- The method is independent of the downstream task.
- Flexible neighborhoods result in richer node embeddings.
- It uses a biased random walk to generate neighborhoods around each node, which addresses the limitations of unconstrained random walks..
node2vec: Biased Walks
-The technique uses biased random walks to generate node neighborhoods.
- Biased random walks trade off between local and global exploration in the graph (e.g., BFS vs. DFS).
Biased Random Walks
- Biased random walks, such as those implemented in node2vec, explore network neighborhoods by considering the second-order relationships between nodes.
- The walk is modified to allow for a bias towards either node neighborhood, based on a given parameter
- Parameters (p, q) model probabilities associated with either returning to the previous node or a different node. They control the bias towards local and global neighborhoods.
node2vec Algorithm
- The node2vec algorithm involves computing random-walk probabilities.
- Simulating random walks for a determined length, initialized from each node in the graph.
- Optimizing the node2vec objective using stochastic gradient descent yields node embeddings.
- The three steps are individually parallelizable for reduced runtime.
Other Random Walk Ideas
- Other techniques using biased random walks, alternative optimization schemes, or network preprocessing methods were discovered.
Summary So Far
- A core concept of graph embedding is embedding nodes so distances reflect original graph similarity
- Different approaches to node similarity were illustrated.
- Naive similarity methods, neighborhood overlap, and random walk approaches were highlighted.
Summary So Far
- No single method performs across all situations; the choice depends on the particular application's requirements.
- Random walk approaches are often efficient.
Embedding Entire Graphs
- The goal is to embed entire subgraphs or graphs.
- Embeddings are obtained for individual graphs.
- Various types of graph embeddings are proposed.
Approach 1
- A simple approach to graph embedding is summing or averaging the node embeddings in a subgraph.
Approach 2
- The method introduces a virtual node to represent the entire (sub)graph for standard graph embeddings.
Approach 3: Anonymous Walk Embeddings
- Anonymous walk embeddings are agnostic to the identity of the nodes visited.
- A random walk gives the index of the first time we visit a node.
- Walks are represented by probability distributions.
Sampling Anonymous Walks
- Generate a set of random anonymous walks to represent the graph.
- The number of walks needed to achieve a desired level of error is discussed.
Learn Walk Embeddings
- The core concept is learning graph embeddings for a whole graph together with all anonymous walk embeddings.
- An objective function is formulated for learning the embedding parameters, considering co-occurrence of walks in a context window.
- Random walks enable a more nuanced understanding of graph relationships.
Learn Walk Embeddings
- Learning walk embeddings involves obtaining graph embeddings by optimizing a function based on the likelihood of random walks co-occurring during training.
Summary
- Three distinct strategies were presented for obtaining graph embeddings.
Preview: Hierarchical Embeddings
- More advanced embedding methods are available.
- Clustering nodes hierarchically and summing/averaging node embeddings within each cluster is a method.
How to Use Embeddings
- Embeddings can be used for various applications including clustering, node and graph classification.
- Methods to measure distance between embeddings are also discussed; for example, these are concatenation, Hadamard product, summation or average, and distance measure.
Today's Summary
- Graph representation learning is reviewed, a technique for learning node and graph embeddings used for downstream tasks that do not require feature engineering.
- Encoder and decoder components of the framework, including node similarity measures, are discussed.
- Different types of graph embedding are presented, including DeepWalk and node2vec.
- Extensions include graph embedding, anonymous walk embeddings, and techniques for creating hierarchical embeddings are discussed.
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Description
Test your knowledge on graph embeddings and their applications in machine learning. This quiz covers topics such as adjacency matrices, anomalous node detection, and negative sampling techniques. Perfect for those looking to deepen their understanding of graph-based algorithms.
