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Graph Edit Distance (GED) is a measure used to quantify the difference or similarity between two graphs. It is defined as the minimum cost required to transform one graph into another through a series of allowable edit operations. These operations typically include: 1. **Node Insertion**: Adding a new node to one graph. 2. **Node Deletion**: Removing a node from one graph. 3. **Edge Insertion**: Adding a new edge between two nodes in one graph.

Graph embedding is a technique used to represent the nodes, edges, or entire graphs in a continuous vector space. The main idea behind graph embedding is to map discrete graph structures into a lower-dimensional space such that the semantic information and relationships within the graph are preserved as much as possible. This representation can then be used for various machine learning tasks, such as classification, clustering, or visualization. ### Key Concepts: 1. **Nodes and Edges**: In a graph, nodes represent entities (e.

A Graph Neural Network (GNN) is a type of neural network specifically designed to work with data represented as graphs. Graphs are mathematical structures consisting of nodes (or vertices) connected by edges, which can represent various types of relationships between entities. Common applications for GNNs include social networks, molecular chemistry, recommendation systems, and knowledge graphs. ### Key Features of Graph Neural Networks: 1. **Graph Structure**: Unlike traditional neural networks that operate on grid-like data (e.g.

Graph reduction is a concept that originates from computer science and mathematics, particularly in the fields of graph theory and functional programming. It involves simplifying or transforming a graph into a simpler or reduced form while preserving certain properties or relationships among its components. Here are some key aspects of graph reduction: 1. **Graph Theory Context**: In graph theory, graph reduction may involve removing certain nodes or edges from a graph to simplify its structure, often with the goal of making algorithms that operate on the graph more efficient.

HCS stands for Hierarchical Clustering using Single-linkage. It is a type of hierarchical clustering algorithm that builds a hierarchy of clusters by progressively merging or splitting existing clusters based on some distance metric. Here’s a brief overview of how HCS operates: 1. **Distance Matrix**: The algorithm starts by calculating the pairwise distances between all data points, usually using a metric like Euclidean distance or Manhattan distance. This forms a distance matrix.

Hall-type theorems for hypergraphs are generalizations of Hall's Marriage Theorem, which originally deals with bipartite graphs. Hall's theorem states that a perfect matching exists in a bipartite graph if and only if for every subset of vertices in one part, the number of neighbors in the other part is at least as large as the size of the subset.

Hierarchical clustering of networks is a method used to group nodes in a network into clusters based on their similarities and relationships. It is particularly useful in the analysis of complex networks, such as social networks, biological networks, and communication networks, where the goal is to uncover underlying structures or patterns within the data.

Initial attractiveness refers to the immediate appeal or allure that a person, object, or idea holds for an individual upon first encounter. In the context of interpersonal relationships, it often pertains to the physical appearance or charisma of a person that can create an instant attraction. This can be influenced by various factors, including physical traits, body language, grooming, and even social signals such as confidence and warmth.

Iterative compression is a technique used primarily in computer science and optimization, particularly for solving hard problems like those in NP-hard categories. The method involves breaking down a problem into smaller parts while iteratively refining a solution until an optimal or satisfactory solution is found. ### Key Concepts: 1. **Compression**: The idea is akin to compressing the problem space—removing unnecessary components or simplifying aspects of the problem to make it more manageable.

Iterative Deepening Depth-First Search (IDDFS) is a search algorithm that combines the space-efficiency of Depth-First Search (DFS) with the completeness of Breadth-First Search (BFS). It is particularly useful in scenarios where the search space is very large, and the depth of the solution is unknown.

PET-MRI, or Positron Emission Tomography-Magnetic Resonance Imaging, is a hybrid imaging technique that combines the functional imaging capabilities of PET with the detailed anatomical imaging of MRI. This technology aims to provide comprehensive insights into both the physiological and structural aspects of tissues and organs. ### Key Components: 1. **Positron Emission Tomography (PET)**: - PET utilizes radioactive tracers (often fluorodeoxyglucose, or FDG) that emit positrons.

Jump Point Search (JPS) is an optimization technique used in pathfinding algorithms, particularly in grid-based environments. It significantly enhances the efficiency of A* (A-star) pathfinding by reducing the number of nodes that need to be evaluated and explored. ### How Jump Point Search Works: 1. **Concept of Jump Points**: - In a typical grid layout, movement is often restricted to adjacent cells (up, down, left, right).

The Junction Tree Algorithm is a method used in probabilistic graphical models, notably in Bayesian networks and Markov networks, to perform exact inference. The algorithm is designed to compute the marginal probabilities of a subset of variables given some evidence. It operates by transforming a graphical model into a junction tree, which is a specific type of data structure that facilitates efficient computation. ### Key Concepts 1. **Graphical Models**: These are representations of the structure of probability distributions over a set of random variables.

Karger's algorithm is a randomized algorithm used to find a minimum cut in a connected undirected graph. The minimum cut of a graph is a partition of its vertices into two disjoint subsets such that the number of edges between the subsets is minimized. This is a fundamental problem in graph theory and has applications in network design, image segmentation, and clustering. ### Overview of Karger's Algorithm: 1. **Random Edge Selection**: The algorithm works by randomly selecting edges and contracting them.

The Kleitman-Wang algorithms refer to a class of algorithms used primarily in combinatorial optimization and graph theory. These algorithms are particularly known for their application in finding maximum independent sets in certain types of graphs. The most notable contribution by David Kleitman and Fan R. Wang was the development of an efficient algorithm to find large independent sets in specific kinds of graphs, particularly bipartite graphs or specific sparse graphs. Their work often explores the relationships between graph structures and combinatorial properties.

Knowledge graph embedding is a technique used to represent entities and relationships within a knowledge graph in a continuous vector space. A knowledge graph is a structured representation of knowledge where entities (such as people, places, or concepts) are represented as nodes and relationships between them are represented as edges. The primary goal of knowledge graph embedding is to capture the semantics of this information in a way that can be effectively utilized for various machine learning and natural language processing tasks.

Lexicographic breadth-first search (Lex-BFS) is a specific order of traversal used in graph theory, particularly for directed and undirected graphs. It operates similar to a standard breadth-first search (BFS), but incorporates a lexicographic ordering to determine the order in which nodes are explored. ### Key Concepts: 1. **BFS Overview**: In a standard BFS, nodes are explored level by level, starting from a given source node.

Link prediction is a task in network science and machine learning that aims to predict the likelihood of a connection or relationship forming between two nodes in a graph, based on the existing structure of the network and the features of the nodes. This problem is particularly relevant in various domains, including social networks, biological networks, recommendation systems, and information retrieval. ### Applications of Link Prediction 1. **Social Networks**: Predicting new friendships or connections between users based on their mutual acquaintances and interactions.

The Longest Path Problem is a common problem in graph theory and computational optimization, where the goal is to find the longest simple path in a given graph. A simple path is defined as a path that does not visit any vertex more than once. ### Key Characteristics of the Problem: 1. **Graph Types**: The problem can be applied to both directed and undirected graphs.

METIS can refer to different things depending on the context. Here are a few of the more common meanings: 1. **Mythological Reference**: In Greek mythology, Metis is a Titaness and the first wife of Zeus. She is associated with wisdom and cunning. According to myth, she was the mother of Athena, the goddess of wisdom and warfare.