Graph Degree Distribution Calculator

What is Graph Degree Distribution?

The degree distribution of a graph is a fundamental concept in network science that describes the probability of a randomly chosen vertex (or node) in the network having a certain degree. In simpler terms, it tells us how many connections each node typically has and how these connections are spread across the entire network. Understanding this distribution helps us characterize the structure and behavior of complex systems, from social networks to biological systems.

Types of Degree Distributions: Classifying Network Structures

Different types of networks exhibit distinct degree distributions, which reveal their underlying formation mechanisms and functional properties. Analyzing these patterns helps classify and understand various real-world systems.

• Scale-Free Networks:
  • Power Law Distribution: Characterized by a few highly connected nodes (hubs) and many nodes with only a few connections. The probability P(k) of a node having degree k follows a power law, meaning P(k) ~ k⁻γ. This distribution is often observed in real-world networks like the internet, social media, and biological networks.
  • Hub Formation: The presence of these "hubs" makes scale-free networks robust to random failures but vulnerable to targeted attacks on their highly connected nodes.
  • Preferential Attachment: A common mechanism for their formation, where new nodes are more likely to connect to existing nodes that already have many connections (the rich get richer).
  • Network Growth: These networks typically grow over time, with new nodes joining and attaching to existing ones based on their popularity.
• Random Networks:
  • Poisson Distribution: In a random network, the degrees of nodes tend to follow a Poisson distribution, meaning most nodes have a degree close to the average degree, and extreme degrees are rare.
  • Erdős-Rényi Model: The classic model for random graphs, where each pair of nodes is connected with a fixed probability 'p', independently of other pairs.
  • Random Connectivity: Connections are formed randomly, without any specific preference or underlying structure.
  • Homogeneous Structure: These networks tend to be more uniform, without the distinct hubs found in scale-free networks. They are often used as a baseline for comparison with real-world networks.
• Small-World Networks:
  • High Clustering: Nodes in a small-world network tend to form tightly knit groups (clusters), meaning if node A is connected to B and C, B and C are also likely to be connected to each other.
  • Short Path Lengths: Despite high clustering, the average path length between any two nodes in the network is surprisingly short, often referred to as "six degrees of separation."
  • Watts-Strogatz Model: A well-known model that generates small-world networks by starting with a regular lattice and randomly rewiring a small fraction of its edges.
  • Social Networks: Many real-world social networks exhibit small-world properties, allowing information to spread quickly across seemingly distant individuals.

Network Properties: Beyond Degree Distribution

While degree distribution is crucial, other network properties provide a more complete picture of a graph's structure and function. These metrics help us understand how information flows, how robust a network is, and how communities are formed.