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Scale Free Network

A scale-free network is a type of complex network that exhibits a specific degree distribution pattern. In a scale-free network, the distribution of node degrees (the number of connections or links each node has) follows a power-law distribution, which means that there are a few nodes with a very high degree of connectivity (hubs), while most nodes have relatively few connections.

Hubs: Scale-free networks have a small number of highly connected nodes, called hubs, which have a much higher degree than the average node in the network. These hubs play a critical role in the network's structure and resilience.

Power-law degree distribution: The degree distribution of nodes in a scale-free network follows a power-law distribution, which means that the probability of a node having a certain degree is inversely proportional to that degree raised to a power. Mathematically, it can be represented as P(k) ~ k^(-γ), where P(k) is the probability of a node having degree k, and γ is a parameter typically between 2 and 3.

Robustness: Scale-free networks are often robust to random failures but vulnerable to targeted attacks on hubs. Removing a few hubs can disrupt the entire network, making it less resilient than networks with a more uniform degree distribution.

Real-world examples: Scale-free networks can be found in various real-world systems, such as social networks, the World Wide Web, biological networks (e.g., protein-protein interaction networks), and transportation networks. These networks often exhibit the "small world" property, where most nodes can be reached from every other node in a relatively small number of steps.


Scale-free networks have been extensively studied in the field of network science, and they have important implications for understanding the structure and dynamics of complex systems. They have also been used to model and analyze various real-world phenomena and systems.


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