Concept Taxonomies

Our next step is to take the list of concepts and add a classification code.

Prompt

That was very good.  Next, let's Create a taxonomy to classify
these concepts. For the list of concepts, can you suggest a
taxonomy of up to ten categories on how to classify each concept?

For example:

1. Foundational Concept (graph model)
2. Types of Graphs
3. Graph Theory
4. Basic Graph Algorithms
5. Advanced Graph Algorithms
6. Graph Machine Learning

What would your list of these categories be?
Give some examples of concepts within each category.

Response

Certainly! Based on your list of concepts for the Graph Algorithms course, here is a taxonomy of ten categories to classify them, along with examples of concepts within each category:

1. Foundational Concepts and Definitions

Description: Basic definitions and elements that form the foundation of graph theory.

Examples:

• Graph Definition
• Nodes
• Edges
• Directed Graph (Digraph)
• Undirected Graph
• Weighted Graphs
• Unweighted Graphs
• Self-loop
• Adjacent Nodes
• Incident Edges
• Degree of a Node
• In-degree
• Out-degree
• Graph Density
• Path
• Walk
• Cycle
• Subgraph
• Graph Isomorphism
• Multigraph

2. Types and Properties of Graphs

Description: Various classifications of graphs and their inherent properties.

Examples:

• Cyclic Graphs
• Acyclic Graphs (DAG)
• Connected Graph
• Disconnected Graph
• Bipartite Graph
• Complete Graph
• Planar Graphs
• Graph Embedding
• Face of a Graph (Planar)
• Graph Minor Theorem
• Graph Duality
• Kuratowski’s Theorem
• Planarity Testing
• Percolation Theory
• Small-World Networks
• Scale-Free Networks

3. Graph Representations and Data Structures

Description: Methods and data structures used to represent graphs in computational contexts.

Examples:

• Graph Adjacency Matrix
• Graph Adjacency List
• Graph Incidence Matrix
• Graph Laplacian
• Priority Queue
• Disjoint Set Data Structure
• Union-Find Algorithm
• Graph Embedding
• Node Embeddings

4. Basic Graph Algorithms

Description: Fundamental algorithms for traversing and exploring graphs.

Examples:

• Graph Traversal
• Depth-First Search (DFS)
• Breadth-First Search (BFS)
• Stack (DFS)
• Queue (BFS)
• Backtracking
• Recursion (DFS)
• Topological Sorting
• Cycle Detection (MST)
• Strongly Connected Components (SCC)
• Tarjan’s Algorithm (SCC)
• Kosaraju’s Algorithm (SCC)
• Articulation Points
• Bridge Finding Algorithm
• Cut-Vertices
• Bridges (Edge Cut)

5. Shortest Path and Spanning Tree Algorithms

Description: Algorithms for finding the shortest paths and constructing spanning trees.

Examples:

• Shortest Path Problem
• Dijkstra’s Algorithm
• Bellman-Ford Algorithm
• A* Search Algorithm
• Floyd-Warshall Algorithm
• All-Pairs Shortest Path (APSP)
• Single-Source Shortest Path (SSSP)
• Negative Weight Cycles
• Minimum Spanning Tree (MST)
• Kruskal’s Algorithm
• Prim’s Algorithm

6. Network Flow and Matching Algorithms

Description: Algorithms related to network flows and matching in graphs.

Examples:

• Network Flow
• Max Flow Problem
• Ford-Fulkerson Algorithm
• Edmonds-Karp Algorithm
• Dinic’s Algorithm
• Push-Relabel Algorithm
• Capacity Scaling Algorithm
• Min-Cut Max-Flow Theorem
• Circulation Problem
• Multi-Commodity Flow
• Graph Matching
• Bipartite Matching
• Hopcroft-Karp Algorithm
• Stable Matching Problem
• Gale-Shapley Algorithm

7. Advanced Graph Algorithms and Problems

Description: Complex algorithms and computationally intensive graph problems.

Examples:

• Eulerian Path
• Eulerian Circuit
• Hamiltonian Path
• Hamiltonian Cycle
• Traveling Salesman Problem (TSP)
• Approximation Algorithms (TSP)
• Heuristics (TSP)
• Graph Coloring
• Chromatic Number
• Greedy Coloring Algorithm
• Clique
• Maximum Clique Problem
• Independent Set
• Maximal Independent Set
• Dominating Set
• Vertex Cover
• Minimum Vertex Cover

8. Graph Measures and Centrality Metrics

Description: Quantitative measures to assess properties of graphs and importance of nodes.

Examples:

• Centrality Measures
• Degree Centrality
• Betweenness Centrality
• Closeness Centrality
• Eigenvector Centrality
• Katz Centrality
• Graph Diameter
• Graph Radius
• Eccentricity
• Graph Girth

9. Spectral Graph Theory and Clustering

Description: Study of graph properties through eigenvalues and eigenvectors; methods for grouping similar nodes.

Examples:

• Spectral Graph Theory
• Graph Laplacian
• Cheeger’s Inequality
• Expander Graphs
• Graph Partitioning
• Graph Clustering
• k-Cuts Problem
• Modularity (Clustering)
• Spectral Clustering
• Girvan-Newman Algorithm
• Louvain Method (Clustering)
• Random Walk
• Markov Chains (Graphs)
• Pagerank Algorithm
• HITS Algorithm

10. Graph Machine Learning and Network Models

Description: Application of machine learning techniques to graphs and study of network structures.

Examples:

• Graph Neural Networks (GNN)
• DeepWalk Algorithm
• Node2Vec Algorithm
• GraphSAGE
• GCN (Graph Convolutional Network)
• GAT (Graph Attention Network)
• Graph Recurrent Neural Networks
• Link Prediction
• Graph Anomaly Detection
• Knowledge Graphs
• RDF vs LPG Models
• Probabilistic Graph Models
• Belief Propagation
• Label Propagation Algorithm
• Small-World Networks
• Scale-Free Networks
• Graph Randomization
• Erdős–Rényi Model
• Barabási–Albert Model
• Watts-Strogatz Model
• Percolation Theory

Notes:

• Foundational Concepts and Definitions serve as the building blocks for understanding more complex topics in graph theory and algorithms.
• Types and Properties of Graphs help in recognizing and categorizing different kinds of graphs, which is crucial for selecting appropriate algorithms.
• Graph Representations and Data Structures are essential for implementing graph algorithms efficiently.
• Basic Graph Algorithms provide fundamental techniques for exploring and analyzing graphs.
• Shortest Path and Spanning Tree Algorithms are critical for optimization problems in networks.
• Network Flow and Matching Algorithms address problems related to resource allocation and pairing in graphs.
• Advanced Graph Algorithms and Problems cover computationally challenging topics often encountered in research and advanced applications.
• Graph Measures and Centrality Metrics allow for the analysis of node importance and network connectivity.
• Spectral Graph Theory and Clustering utilize mathematical properties to understand graph structures and identify communities.
• Graph Machine Learning and Network Models represent cutting-edge areas where machine learning intersects with graph theory, enabling advanced data analysis and applications.

This taxonomy provides a structured way to approach the teaching and learning of graph algorithms, ensuring that foundational concepts are established before progressing to more advanced topics. It also highlights the interconnections between different areas, facilitating a comprehensive understanding of the subject.

Add TaxonomyIDs to CSV

Perfect.  We will call this our taxonomy and the number will be the ConceptID.  Now for each row in the CSV file above that enumerate the concepts and their dependencies, please add the primary taxonomy ID as a new column to the CSV file.  Return the CSV information as a single file that can be downloaded.

ChatGPT o1-preview Thought for 43 seconds

Here is the updated CSV file with the primary taxonomy ID added as a new column:

ConceptID,ConceptName,Dependencies,TaxonomyID
1,Graph Definition,,1
2,Nodes,1,1
3,Edges,1,1
4,Directed Graph (Digraph),1|3,1
5,Undirected Graph,1|3,1
6,Weighted Graphs,1|3,1
7,Unweighted Graphs,1|3,1
8,Cyclic Graphs,1|16,2
9,Acyclic Graphs (DAG),1|16,2
10,Graph Density,1|2|3,1
11,Degree of a Node,1|2|3,1
12,In-degree,4|11,1
13,Out-degree,4|11,1
14,Path,1|2|3,1
15,Walk,1|2|3,1
16,Cycle,14,1
17,Connected Graph,14,2
18,Disconnected Graph,17,2
19,Subgraph,1|2|3,1
20,Bipartite Graph,1|2|3,2
21,Complete Graph,1|2|3,2
22,Multigraph,1|3,1
23,Graph Isomorphism,1|2|3|19,1
24,Self-loop,3,1
25,Adjacent Nodes,2|3,1
26,Incident Edges,2|3|11,1
27,Graph Adjacency Matrix,2|3|25,3
28,Graph Adjacency List,2|3|25,3
29,Graph Incidence Matrix,2|3|26,3
30,Graph Traversal,1|2|3|14,4
31,Depth-First Search (DFS),30|33|36,4
32,Breadth-First Search (BFS),30|34,4
33,Stack (DFS),,3
34,Queue (BFS),,3
35,Backtracking,31|36,4
36,Recursion (DFS),,4
37,Shortest Path Problem,14,5
38,Dijkstra’s Algorithm,37|6|151,5
39,Bellman-Ford Algorithm,37|6,5
40,A* Search Algorithm,37|6|58,5
41,Floyd-Warshall Algorithm,42|6,5
42,All-Pairs Shortest Path (APSP),37,5
43,Single-Source Shortest Path (SSSP),37,5
44,Negative Weight Cycles,6|16,5
45,Minimum Spanning Tree (MST),3|6|17,5
46,Kruskal’s Algorithm,45|48,5
47,Prim’s Algorithm,45,5
48,Union-Find Algorithm,152,3
49,Cycle Detection (MST),16|45,4
50,Cut-Vertices,31|17,4
51,Bridges (Edge Cut),31|17,4
52,Eulerian Path,1|11|14,7
53,Eulerian Circuit,52,7
54,Hamiltonian Path,14|16,7
55,Hamiltonian Cycle,54,7
56,Traveling Salesman Problem (TSP),55|6,7
57,Approximation Algorithms (TSP),56|58,7
58,Heuristics (TSP),,7
59,Graph Coloring,2|3,7
60,Chromatic Number,59,7
61,Greedy Coloring Algorithm,59,7
62,Clique,2|3|21,7
63,Maximum Clique Problem,62,7
64,Independent Set,2|3|62,7
65,Maximal Independent Set,64,7
66,Dominating Set,2|3|64,7
67,Vertex Cover,2|3|64,7
68,Minimum Vertex Cover,67,7
69,Planar Graphs,1,2
70,Graph Embedding,1|69,2
71,Kuratowski’s Theorem,69,2
72,Graph Minor Theorem,1|69,2
73,Graph Duality,69,2
74,Planarity Testing,69|71,2
75,Face of a Graph (Planar),69|70,2
76,Topological Sorting,9,4
77,DAG Applications,9|76,4
78,Strongly Connected Components (SCC),4|14|31,4
79,Tarjan’s Algorithm (SCC),78|31,4
80,Kosaraju’s Algorithm (SCC),78|31,4
81,Articulation Points,31|17,4
82,Bridge Finding Algorithm,51|31,4
83,Graph Matching,2|3|20,6
84,Bipartite Matching,83|20,6
85,Hopcroft-Karp Algorithm,84,6
86,Stable Matching Problem,83,6
87,Gale-Shapley Algorithm,86,6
88,Network Flow,1|3|6,6
89,Max Flow Problem,88,6
90,Ford-Fulkerson Algorithm,89,6
91,Edmonds-Karp Algorithm,90,6
92,Min-Cut Max-Flow Theorem,89|98,6
93,Push-Relabel Algorithm,89,6
94,Dinic’s Algorithm,89,6
95,Capacity Scaling Algorithm,89,6
96,Circulation Problem,88,6
97,Multi-Commodity Flow,88,6
98,Cut-Set,3|14,4
99,Cut-Set Theorem,98,4
100,Flow Decomposition,88|14,6
101,Flow Networks,88,6
102,Graph Partitioning,1|98|103,9
103,Graph Clustering,102,9
104,k-Cuts Problem,98|102,9
105,Modularity (Clustering),103,9
106,Spectral Clustering,103|119|120,9
107,Girvan-Newman Algorithm,103|115,9
108,Louvain Method (Clustering),103,9
109,Random Walk,1|14,9
110,Markov Chains (Graphs),109|1,9
111,Pagerank Algorithm,110|109,9
112,HITS Algorithm,110|109,9
113,Centrality Measures,1|2|3|14,8
114,Degree Centrality,113|11,8
115,Betweenness Centrality,113|14,8
116,Closeness Centrality,113|14,8
117,Eigenvector Centrality,113|27|120,8
118,Katz Centrality,117|113,8
119,Graph Laplacian,27|11,9
120,Spectral Graph Theory,119|27,9
121,Cheeger’s Inequality,120,9
122,Expander Graphs,120|121,9
123,Graph Diameter,14|2|3,8
124,Graph Radius,123|125,8
125,Eccentricity,14|2,8
126,Graph Girth,16|1,8
127,Percolation Theory,1|130,10
128,Small-World Networks,130|133,10
129,Scale-Free Networks,130|132,10
130,Graph Randomization,1,10
131,Erdős–Rényi Model,130,10
132,Barabási–Albert Model,130,10
133,Watts-Strogatz Model,130,10
134,Community Detection,103|102,9
135,Block Models,134,9
136,Stochastic Block Model,135|130,9
137,Label Propagation Algorithm,134|1,10
138,Belief Propagation,1|153,10
139,Graph Neural Networks (GNN),154|1,10
140,Node Embeddings,1|155,10
141,DeepWalk Algorithm,140|109,10
142,Node2Vec Algorithm,140|109,10
143,GraphSAGE,140|139,10
144,GCN (Graph Convolutional Network),139|119,10
145,GAT (Graph Attention Network),139|156,10
146,Graph Recurrent Neural Networks,139|157,10
147,Link Prediction,1|140|155,10
148,Graph Anomaly Detection,1|155|140,10
149,Knowledge Graphs,1,10
150,RDF vs LPG Models,149,10
151,Priority Queue,,3
152,Disjoint Set Data Structure,,3
153,Probabilistic Graph Models,1|158,10
154,Neural Networks,155,10
155,Machine Learning,159,10
156,Attention Mechanism,154,10
157,Recurrent Neural Networks,154,10
158,Probability Theory,,1
159,Data Science,,1

Instructions for Downloading:

You can download the CSV content by copying it into a text editor and saving it with a .csv extension.

Taxonomy Categories:

Here is a reminder of the taxonomy categories and their corresponding IDs:

1. Foundational Concepts and Definitions
2. Types and Properties of Graphs
3. Graph Representations and Data Structures
4. Basic Graph Algorithms
5. Shortest Path and Spanning Tree Algorithms
6. Network Flow and Matching Algorithms
7. Advanced Graph Algorithms and Problems
8. Graph Measures and Centrality Metrics
9. Spectral Graph Theory and Clustering
10. Graph Machine Learning and Network Models

Notes:

• The TaxonomyID column indicates the primary category each concept belongs to, based on the taxonomy.
• All concepts have been assigned to one of the ten taxonomy categories.
• Dependencies remain the same as in the previous CSV file.

Legend Prompt

Convert this taxonomy list into an HTML table with the the following format:

<table>
            <tr>
                <td>Foundational Concepts and Definitions</td>
                <td style="background-color: red; color: white;">Red</td>
            </tr>
            <tr>
                <td>Level 1</td>
                <td style="background-color: orange;">Orange</td>
            </tr>

Response

<table>
    <tr>
        <td>Foundational Concepts and Definitions</td>
        <td style="background-color: red; color: white;">Red</td>
    </tr>
    <tr>
        <td>Types and Properties of Graphs</td>
        <td style="background-color: orange;">Orange</td>
    </tr>
    <tr>
        <td>Graph Representations and Data Structures</td>
        <td style="background-color: yellow;">Yellow</td>
    </tr>
    <tr>
        <td>Basic Graph Algorithms</td>
        <td style="background-color: green; color: white;">Green</td>
    </tr>
    <tr>
        <td>Shortest Path and Spanning Tree Algorithms</td>
        <td style="background-color: blue; color: white;">Blue</td>
    </tr>
    <tr>
        <td>Network Flow and Matching Algorithms</td>
        <td style="background-color: purple; color: white;">Purple</td>
    </tr>
    <tr>
        <td>Advanced Graph Algorithms and Problems</td>
        <td style="background-color: pink; color: black;">Pink</td>
    </tr>
    <tr>
        <td>Graph Measures and Centrality Metrics</td>
        <td style="background-color: lightgray;">Light Gray</td>
    </tr>
    <tr>
        <td>Spectral Graph Theory and Clustering</td>
        <td style="background-color: brown; color: white;">Brown</td>
    </tr>
    <tr>
        <td>Graph Machine Learning and Network Models</td>
        <td style="background-color: teal; color: white;">Teal</td>
    </tr>
</table>