The Mathematics of Networks
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is not a plot of data, but a collection of nodes and connections.
Components of a Graph
A graph G consists of two sets:
Vertices (V): A set of nodes or points.
Edges (E): A set of connections between pairs of vertices.
Types of Graphs
Undirected Graph: Edges have no orientation. The connection between vertex A and B is the same as from B to A.
Directed Graph (Digraph): Edges have a direction, represented by arrows. The connection from A to B is distinct from the connection from B to A.
Unweighted Graph: Edges simply represent a connection.
Weighted Graph: Each edge is assigned a numerical value, or 'weight', which can represent cost, distance, or capacity.
Graph Traversal
A graph traversal is the process of visiting (checking and/or updating) each vertex in a graph.
Breadth-First Search (BFS): An algorithm for traversing a graph. It starts at a chosen 'root' vertex and explores all of the neighbor vertices at the present depth prior to moving on to the vertices at the next depth level. It explores the graph layer by layer.
Depth-First Search (DFS): An algorithm that explores as far as possible along each branch before backtracking.
Applications
Graph theory is an extremely useful tool for modeling networks and relationships in the real world.
Computer Science: Computer networks, social networks (Facebook, Twitter), the structure of the World Wide Web.
Logistics and Operations: Finding the shortest path for a delivery route (e.g., Google Maps), optimizing airline networks.
Biology: Modeling protein-protein interaction networks, gene regulatory networks.
Chemistry: Representing molecules, where atoms are vertices and bonds are edges.