network graph

business analytics

slug: topic-map-business-analytics-network-graph

Vocabulary:

• Graph: Mathematical structure of vertices and edges
• Vertex/Node: Fundamental unit in a graph
• Edge/Link/Arc: Connection between two vertices
• Directed graph/Digraph: Graph with directional edges
• Undirected graph: Graph with bidirectional edges
• Weighted graph: Graph with values assigned to edges or vertices
• Degree: Number of edges connected to a vertex
• In-degree: Number of incoming edges (directed graphs)
• Out-degree: Number of outgoing edges (directed graphs)
• Path: Sequence of vertices connected by edges
• Cycle: Path that starts and ends at the same vertex
• Connected graph: Graph where path exists between all vertex pairs
• Component: Maximal connected subgraph
• Subgraph: Graph formed from subset of vertices and edges
• Adjacency: Relationship between connected vertices
• Neighbor: Vertex directly connected to another vertex
• Walk: Sequence of vertices with repeated vertices allowed
• Trail: Walk with no repeated edges
• Clique: Complete subgraph where all vertices are connected
• Independent set: Set of vertices with no edges between them
• Tree: Connected acyclic graph
• Forest: Collection of disjoint trees
• Spanning tree: Subgraph that includes all vertices
• Bipartite graph: Graph with vertices divided into two disjoint sets
• Planar graph: Graph that can be drawn without edge crossings
• Complete graph: Graph where every vertex connects to every other
• Multigraph: Graph allowing multiple edges between vertices
• Simple graph: Graph with no loops or multiple edges
• Hypergraph: Generalization where edges connect multiple vertices
• Diameter: Maximum shortest path between any two vertices
• Radius: Minimum eccentricity among all vertices
• Eccentricity: Maximum distance from a vertex to all others
• Centrality: Measure of vertex importance in network
• Betweenness: Number of shortest paths passing through a vertex
• Closeness: Average distance from a vertex to all others
• Eigenvector centrality: Importance based on neighbor importance
• PageRank: Algorithm measuring vertex importance
• Clustering coefficient: Measure of local clustering around vertex
• Modularity: Measure of community structure strength
• Assortativity: Tendency of similar vertices to connect
• Transitivity: Probability of forming triangles
• Homophily: Tendency of similar nodes to connect
• Bridge: Edge whose removal increases components
• Cut vertex/Articulation point: Vertex whose removal disconnects graph
• Minimum cut: Smallest set of edges to disconnect graph
• Maximum flow: Largest flow through network
• Hamiltonian path: Path visiting each vertex exactly once
• Eulerian path: Path using each edge exactly once
• Isomorphism: Structural equivalence between graphs
• Automorphism: Isomorphism from graph to itself
• Chromatic number: Minimum colors needed for vertex coloring
• Matching: Set of edges without common vertices
• Network motif: Recurring significant subgraph pattern
• Hub: Highly connected vertex
• Authority: Vertex receiving many connections
• Sink: Vertex with no outgoing edges
• Source: Vertex with no incoming edges
• Adjacency matrix: Matrix representation of graph connections
• Incidence matrix: Matrix showing vertex-edge relationships
• Laplacian matrix: Matrix encoding graph structure
• Graph kernel: Similarity measure between graphs
• Random walk: Stochastic path through graph
• Percolation: Study of connectivity under random removal
• Small-world property: Short paths between most vertex pairs
• Scale-free network: Network with power-law degree distribution
• Assortative mixing: Pattern of connections by vertex attributes
• Graph embedding: Mapping graph to vector space
• Topology: Study of graph structural properties
• Graph symmetry: Invariance under vertex permutations

Concepts:

• Graph representation and mathematical formalization
• Structural properties and invariants
• Connectivity and reachability analysis
• Network topology characterization
• Centrality and importance measures
• Community detection and clustering
• Graph traversal strategies
• Optimization on graphs
• Random graph models
• Network evolution and dynamics
• Information diffusion in networks
• Percolation theory and phase transitions
• Spectral graph theory
• Algebraic graph theory
• Topological properties of networks
• Graph isomorphism and equivalence
• Network resilience and robustness
• Scale-free and small-world phenomena
• Hierarchical network structures
• Temporal and dynamic networks
• Multilayer and multiplex networks
• Spatial networks and geographic graphs
• Bipartite network projections
• Graph symmetry and automorphism groups
• Matching theory and perfect matchings
• Graph coloring and chromatic properties
• Planarity and graph embeddings
• Network entropy and information theory
• Cascading failures in networks
• Epidemic spreading on networks
• Game theory on networks
• Network controllability
• Synchronization in networks
• Network inference from data
• Graph limits and convergence
• Extremal graph theory
• Random geometric graphs
• Preferential attachment models
• Core-periphery structure
• Network motifs and subgraph patterns

Procedures:

• Graph construction and representation:
  • Define vertex and edge sets
  • Choose representation (adjacency matrix, list, etc.)
  • Initialize data structures
  • Add/remove vertices and edges
• Graph traversal algorithms:
  • Breadth-First Search (BFS)
  • Depth-First Search (DFS)
  • Implement iterative or recursive approaches
  • Track visited vertices
• Shortest path computation:
  • Dijkstra's algorithm (weighted graphs)
  • Bellman-Ford algorithm (negative weights)
  • Floyd-Warshall algorithm (all pairs)
  • A* search algorithm (heuristic)
• Minimum spanning tree:
  • Kruskal's algorithm (edge-based)
  • Prim's algorithm (vertex-based)
  • Sort edges by weight
  • Use union-find data structure
• Network flow analysis:
  • Ford-Fulkerson algorithm
  • Edmonds-Karp algorithm
  • Push-relabel algorithm
  • Find augmenting paths
• Centrality calculation:
  • Compute degree centrality
  • Calculate betweenness centrality
  • Calculate closeness centrality
  • Calculate eigenvector centrality/PageRank
• Community detection:
  • Modularity optimization (Louvain method)
  • Girvan-Newman algorithm
  • Label propagation
  • Spectral clustering
• Graph coloring:
  • Greedy coloring algorithm
  • Backtracking approaches
  • Check chromatic number bounds
• Cycle detection:
  • DFS-based cycle detection
  • Union-find for undirected graphs
  • Identify strongly connected components
• Connectivity analysis:
  • Find connected components
  • Identify bridges and articulation points
  • Test biconnectivity
  • Compute network diameter
• Matching algorithms:
  • Hungarian algorithm
  • Blossom algorithm
  • Hopcroft-Karp algorithm
• Topological sorting:
  • Kahn's algorithm
  • DFS-based approach
  • Detect cyclic dependencies
• Graph generation:
  • Erdős-Rényi random graphs
  • Barabási-Albert preferential attachment
  • Watts-Strogatz small-world model
  • Configuration model
• Network analysis pipeline:
  • Load and preprocess network data
  • Compute basic statistics
  • Calculate centrality measures
  • Detect communities
  • Visualize network structure
  • Generate reports and insights
• Subgraph mining:
  • Enumerate motifs
  • Find frequent patterns
  • Test statistical significance
• Graph comparison:
  • Compute graph edit distance
  • Test isomorphism
  • Calculate graph kernels
• Network visualization:
  • Apply force-directed layout algorithms
  • Use hierarchical layouts
  • Apply dimensionality reduction
  • Color/size nodes by properties

Topics:

• Fundamental graph theory concepts
• Graph algorithms and complexity
• Network topology and structure
• Centrality and importance measures
• Community structure and clustering
• Random graph theory
• Scale-free networks
• Small-world networks
• Network robustness and resilience
• Epidemic spreading models
• Information diffusion and cascades
• Network dynamics and evolution
• Temporal networks
• Multilayer networks
• Spatial networks
• Bipartite networks and projections
• Directed acyclic graphs (DAGs)
• Trees and hierarchical structures
• Social network analysis
• Biological network analysis
• Brain network analysis (connectomics)
• Transportation networks
• Communication networks
• Power grid networks
• Financial networks
• Citation networks
• Knowledge graphs
• Semantic networks
• Protein interaction networks
• Gene regulatory networks
• Metabolic networks
• Ecological networks (food webs)
• Internet and web graphs
• Network visualization techniques
• Graph drawing algorithms
• Spectral graph theory
• Algebraic graph theory
• Probabilistic graphical models
• Graph neural networks
• Network embedding methods
• Graph kernels and similarity
• Network inference and reconstruction
• Link prediction
• Missing data in networks
• Network sampling methods
• Graph compression
• Network motifs and patterns
• Network controllability
• Synchronization phenomena
• Percolation theory
• Network games and strategic behavior

Categories:

• Pure Mathematics
• Applied Mathematics
• Computer Science - Algorithms
• Data Science
• Network Science
• Computational Biology
• Social Sciences
• Physics - Statistical Mechanics
• Operations Research
• Information Theory
• Machine Learning
• Systems Biology
• Computational Neuroscience
• Transportation Engineering
• Telecommunications
• Epidemiology

Themes:

• Structure and function in complex systems
• Emergence of global properties from local interactions
• Universality in network organization
• Trade-offs between efficiency and robustness
• Self-organization in networks
• Information flow and propagation
• Centralization vs decentralization
• Modularity and hierarchical organization
• Network resilience under attack or failure
• Dynamics on networks vs dynamics of networks
• Multi-scale network organization
• Network topology shapes function
• Optimization principles in network formation
• Statistical mechanics of networks
• Phase transitions in network processes
• Universality classes in network science
• Network inference from incomplete data
• Causality in networked systems
• Control and manipulation of networks

Trends:

• Graph neural networks and deep learning on graphs
• Temporal and dynamic network analysis
• Hypergraph and higher-order network analysis
• Network digital twins
• Quantum networks and quantum graph algorithms
• Causal inference in networks
• Fairness and bias in network algorithms
• Privacy-preserving network analysis
• Federated learning on graphs
• Explainable AI for graph models
• Graph transformers and attention mechanisms
• Self-supervised learning on graphs
• Geometric deep learning
• Network analysis at scale (big graph analytics)
• Real-time streaming graph analytics
• Heterogeneous information networks
• Knowledge graph embedding and reasoning
• Graph generation using GANs and diffusion models
• Differentiable graph algorithms
• Integration of symbolic and neural approaches
• Multi-modal network analysis
• COVID-19 contact tracing networks
• Climate and sustainability networks
• Misinformation spread analysis
• Decentralized networks and blockchain graphs
• Neuro-symbolic graph reasoning
• Foundation models for graphs
• Active learning on graphs
• Few-shot learning for graph tasks
• Graph-based drug discovery and molecular design

Use_cases:

• Social network analysis and influence detection
• Recommendation systems based on user-item graphs
• Fraud detection in financial transaction networks
• Disease outbreak tracking and epidemic modeling
• Protein-protein interaction analysis
• Drug target identification
• Gene regulatory network inference
• Brain connectivity analysis (fMRI, EEG)
• Route optimization in transportation networks
• Traffic flow prediction
• Supply chain optimization
• Power grid stability analysis
• Telecommunication network design
• Internet routing protocols
• Web page ranking (search engines)
• Citation analysis and research impact
• Knowledge graph construction and reasoning
• Natural language processing (dependency parsing)
• Computer vision (scene graphs)
• Molecular structure analysis and drug design
• Recommendation of movies, products, content
• Community detection in online platforms
• Fake news and bot detection
• Cybersecurity threat detection
• Malware analysis through call graphs
• Compiler optimization using control flow graphs
• Program analysis and bug detection
• Chip design and circuit optimization
• Ecological food web analysis
• Climate system modeling
• Urban planning and smart city design
• Logistics and delivery optimization
• Airline route network optimization
• Collaboration network analysis
• Criminal network detection
• Market basket analysis
• Customer segmentation through purchase graphs
• Sensor network optimization
• Robot path planning
• Game AI and decision trees
• Workflow optimization
• Project dependency management
• Organizational structure analysis
• Infrastructure vulnerability assessment
• Pandemic preparedness planning
• Content moderation at scale
• Linked data and semantic web
• Blockchain transaction analysis