Graphs: The Web of Connections

[!NOTE] This module explores the core principles of Graphs: The Web of Connections, deriving solutions from first principles and hardware constraints to build world-class, production-ready expertise.

1. The Concept: Everything is Connected

In previous modules, data was either linear (Arrays, Linked Lists) or hierarchical (Trees). But the real world is messy. Roads loop back on themselves, friends have mutual friends, and the internet is a chaotic web of links.

To model this, we need Graphs.

A Graph G = (V, E) consists of:

  • Vertices (V): Also called Nodes. These are the entities (e.g., Cities, Users, Routers).
  • Edges (E): The connections between them (e.g., Roads, Friendships, Cables).

Why Graphs Matter?

Why do we need a special structure? Because relationships matter as much as the data itself.

  1. Social Networks (Facebook/LinkedIn):
    • Nodes: People.
    • Edges: Friendships.
    • Problem: “People you may know” (Triangle closing).
  2. Navigation (Google Maps/Uber):
    • Nodes: Intersections.
    • Edges: Roads (Weighted by traffic/distance).
    • Problem: Shortest Path (A* or Dijkstra).
  3. The Internet (PageRank):
    • Nodes: Web pages.
    • Edges: Hyperlinks.
    • Problem: Which page is most important?

2. Interactive: The Cyberpunk Network Builder

Build your own network. Click to add nodes. Drag from one node to another to create edges.

Nodes: 0 | Edges: 0
Click to Add Node • Drag to Connect • Right Click to Remove

3. Types of Graphs

  1. Undirected Graph: Connections are two-way (like a handshake). If A is connected to B, B is connected to A.
  2. Directed Graph (Digraph): Connections are one-way (like a Twitter follow). A → B doesn’t imply B → A.
  3. Weighted Graph: Edges have a cost or value (distance, time, bandwidth cost).

4. Module Roadmap

  1. Introduction to Graphs
  2. Representations: Adjacency Matrix vs List.
  3. Traversals: BFS (Waves) and DFS (Mazes).
  4. Shortest Path: Dijkstra’s Algorithm.
  5. Spanning Trees: Connecting everything efficiently.
  6. Real World: How Google and Uber use graphs.

Module Chapters