Building Interactive Graphs for the Web

Graph Fundamentals: A Beginner’s GuideA graph is a mathematical structure used to model pairwise relationships between objects. At its core, a graph consists of two sets: a set of vertices (also called nodes) and a set of edges that connect pairs of vertices. Graphs are widely used across computer science, mathematics, engineering, biology, social sciences, and many applied fields because they provide a flexible way to represent relationships, networks, and connections.

1. Basic terminology

• Vertex (Node): An individual object or point in the graph.
• Edge: A connection between two vertices.
• Adjacent/Neighbor: Two vertices connected directly by an edge are adjacent or neighbors.
• Degree: The number of edges incident to a vertex. In directed graphs, we distinguish in-degree and out-degree.
• Path: A sequence of edges that connects a sequence of distinct vertices.
• Cycle: A path that starts and ends at the same vertex without repeating edges (and typically without repeating other vertices).
• Connected graph: In an undirected graph, there is a path between every pair of vertices.
• Component: A maximal connected subgraph.
• Subgraph: A graph formed from a subset of the vertices and edges of a larger graph.

2. Types of graphs

• Undirected vs. directed graphs:

  • In an undirected graph, edges have no orientation — the connection is bidirectional.
  • In a directed graph (digraph), each edge has a direction from one vertex to another.

• Weighted vs. unweighted:

  • Weighted graphs assign a numerical value (weight) to edges, representing cost, distance, capacity, etc.
  • Unweighted graphs treat all edges equally.

• Simple vs. multigraph:

  • A simple graph has at most one edge between any pair of vertices and no loops (edges connecting a vertex to itself).
  • A multigraph may have multiple parallel edges and loops.

• Bipartite graph:

  • Vertices can be divided into two disjoint sets such that all edges connect a vertex from one set to a vertex in the other set.

• Trees and forests:

  • A tree is a connected acyclic undirected graph. It has exactly n-1 edges for n vertices.
  • A forest is a disjoint union of trees.

• Special graphs (complete graph, path graph, cycle graph, etc.):

  • Complete graph (K_n): Every pair of distinct vertices is connected by an edge.
  • Path graph (P_n): Vertices arranged in a single line (chain).
  • Cycle graph (C_n): Vertices form a closed loop.

3. Representing graphs in code

Two common ways to represent graphs in programs:

• Adjacency matrix:

  • A 2D array where entry (i, j) indicates the presence (and possibly weight) of an edge between vertices i and j.
  • Fast to check edge existence (O(1)), but uses O(n^2) space.

• Adjacency list:

  • For each vertex, store a list (or vector) of its neighbors.
  • Space-efficient for sparse graphs: O(n + m) where m is the number of edges.
  • Iterating neighbors is efficient; checking whether a specific edge exists may require searching the list.

Example (Python adjacency list):

graph = {     'A': ['B', 'C'],     'B': ['A', 'D'],     'C': ['A', 'D'],     'D': ['B', 'C'] } 

4. Core graph algorithms

• Traversal:

  • Breadth-first search (BFS): Explores vertices by increasing distance from the start; useful for shortest path in unweighted graphs and layer-order processing.
  • Depth-first search (DFS): Explores as deep as possible along each branch before backtracking; useful for cycle detection, topological sorting, and connectivity.

• Shortest paths:

  • Dijkstra’s algorithm: Finds shortest paths from a source in graphs with non-negative edge weights.
  • Bellman–Ford: Handles negative weights and detects negative cycles.
  • A* search: Uses heuristics to speed up pathfinding (common in games and maps).

• Minimum spanning tree (MST):

  • Kruskal’s and Prim’s algorithms compute a tree connecting all vertices with minimal total edge weight.

• Connectivity and components:

  • Union-Find (disjoint-set) structure: Efficiently track components for dynamic connectivity.
  • Kosaraju’s or Tarjan’s algorithms: Find strongly connected components in directed graphs.

• Topological sort:

  • Orders vertices of a directed acyclic graph (DAG) so that for every directed edge u → v, u comes before v.

5. Practical applications

• Computer networks: Routers and switches form network graphs; routing protocols use graph algorithms.
• Social networks: Users are vertices; friendships/follows are edges; used for community detection, recommendation, influence analysis.
• Biology: Protein interaction networks, gene regulatory networks.
• Transportation: Road, rail, and airline networks for route planning and optimization.
• Recommendation systems: Graphs model relationships between users and items.
• Knowledge graphs: Represent entities and relationships for search and reasoning.

6. Modeling tips and common pitfalls

• Choose representation based on density: adjacency matrix for dense graphs; adjacency list for sparse.
• Be careful with zero-based vs. one-based indexing in implementations.
• Watch for integer overflow when summing many weights.
• When using BFS/DFS recursively, consider recursion depth limits for large graphs.
• For weighted shortest-path problems, ensure edge weights meet algorithm assumptions (e.g., non-negative for Dijkstra).

7. Simple examples

• Finding connected components (conceptually): run DFS/BFS from any unvisited node, mark all reachable nodes as one component, repeat.
• Detecting cycles in undirected graph: if during DFS you find a visited neighbor that is not the parent, a cycle exists.
• Shortest path on unweighted graph: BFS from source gives minimum number of edges to each vertex.

8. Further learning resources (topics to explore next)

• Advanced algorithms: network flow (Ford–Fulkerson, Edmonds–Karp), matching (Hopcroft–Karp), and graph isomorphism.
• Graph databases: Neo4j, JanusGraph for storing and querying large graphs.
• Graph machine learning: node embeddings (DeepWalk, Node2Vec), Graph Neural Networks (GNNs).
• Visualization tools: Gephi, Cytoscape, D3.js for interactive graph visualizations.

Graphs turn complex relationships into structures you can analyze and compute on. Start by practicing with small examples and implementing BFS/DFS and Dijkstra; those foundations unlock many real-world problems.