11.1 Introduction to graph theory and network analysis

Graph theory is the study of relationships between objects, represented as vertices and edges. It's a powerful tool for modeling complex systems, from social networks to transportation routes, providing insights into their structure and behavior.

In this section, we'll explore fundamental concepts of graph theory, its applications, and ways to represent graphs. We'll also dive into key graph properties and measures that help analyze network characteristics and identify important nodes.

Fundamental Concepts of Graph Theory

Graph Components and Structure

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• Graphs consist of vertices (nodes) and edges connecting pairs of vertices
• Vertices represent entities or data points in a network
• Edges represent relationships or connections between entities
• Directed edges indicate one-way relationships (arrows)
• Undirected edges indicate two-way relationships (lines)
• Weighted graphs assign numerical values to edges (strength or cost of connections)

Graph Classifications

• Simple graphs have no self-loops or multiple edges
• Multigraphs allow multiple edges between vertices
• Pseudographs permit self-loops (edges connecting a vertex to itself)
• Complete graphs connect all vertices to each other
• Bipartite graphs divide vertices into two disjoint sets
• Trees are connected graphs without cycles

Applications of Graph Theory

Social and Recommendation Systems

• Model relationships between individuals, organizations, or online entities
• Analyze user preferences and item similarities for personalized suggestions
• Facilitate targeted marketing and community detection
• Identify influential users and information spread patterns

Transportation and Biological Networks

• Optimize route planning and network flow analysis in logistics
• Model protein-protein interaction networks
• Represent gene regulatory networks
• Analyze metabolic pathways and ecological food webs

Advanced Graph Applications

• Graph Neural Networks (GNNs) perform node classification and link prediction
• Knowledge graphs represent complex relationships between entities
• Support information retrieval and reasoning in various domains
• Enable fraud detection in financial networks
• Optimize computer network topologies

Representing Graphs

Matrix-based Representations

• Adjacency matrices store graph information in 2D arrays
  • Rows and columns represent vertices
  • Entries indicate presence or weight of edges
• Incidence matrices use rows for vertices and columns for edges
  • Entries show vertex-edge relationships
• Laplacian matrices combine degree and adjacency information
  • Used in spectral graph theory and clustering algorithms

List-based and Specialized Representations

• Adjacency lists use collections of lists for vertex neighbors
• Edge lists store pairs (or triples for weighted graphs) representing edges
• Graph database management systems (GDBMS) employ specialized structures
  • Enable efficient storage and querying of graph data
• Standardized file formats include GraphML, GML, and DOT language
  • Facilitate data exchange and visualization

Graph Properties

Connectivity and Structural Measures

• Vertex connectivity measures minimum number of vertices to disconnect graph
• Edge connectivity represents minimum number of edges to disconnect graph
• Connected components identify subgraphs where all vertices are reachable
• Graph density calculates ratio of existing edges to maximum possible edges
• Clustering coefficient quantifies tendency of vertices to form tight groups
• Diameter determines maximum shortest path length between any two vertices
• Average path length indicates overall efficiency of information flow

Centrality and Degree Measures

• Degree distribution describes probability distribution of vertex degrees
  • Often follows power-law distributions in real-world networks (scale-free networks)
• Degree centrality counts number of connections a vertex has
• Betweenness centrality measures frequency of a vertex in shortest paths
• Closeness centrality calculates average distance to all other vertices
• Eigenvector centrality assesses influence based on neighbors' importance
• PageRank algorithm (variation of eigenvector centrality) ranks web pages