Betweennesscentrale

Betweenness-centrality is a measure used in network analysis to identify the most important nodes in a graph. It quantifies the number of times a node acts as a bridge along the shortest path between two other nodes. In other words, it calculates how often a node lies on the shortest path between any two other nodes in the network. Nodes with high betweenness-centrality scores are often considered critical because they play a vital role in connecting different parts of the network. This measure is widely used in various fields such as sociology, biology, and computer science to understand the structure and function of complex networks.

The concept of betweenness-centrality was first introduced by Linton C. Freeman in 1977. It is defined mathematically as follows: for a node v, the betweenness-centrality B(v) is given by the sum of the fraction of all-pairs shortest paths that pass through v. Specifically, B(v) = Σ (σst(v)/σst) for all pairs of nodes s and t, where σst is the total number of shortest paths from s to t, and σst(v) is the number of those paths that pass through v. This formula ensures that nodes that are not on any shortest path between two nodes do not contribute to the betweenness-centrality score.

Betweenness-centrality can be normalized to account for the size of the network. The normalized betweenness-centrality is given by B(v) = (2/((n-1)(n-2))) Σ (σst(v)/σst), where n is the number of nodes in the network. This normalization helps in comparing betweenness-centrality scores across networks of different sizes.

In practice, betweenness-centrality can be computed using various algorithms, such as Brandes' algorithm, which efficiently calculates betweenness-centrality for all nodes in the network. This measure is particularly useful for identifying key players or hubs in social networks, critical infrastructure nodes in transportation networks, and essential genes in biological networks. By understanding the betweenness-centrality of nodes, researchers and practitioners can gain insights into the structure and dynamics of complex systems.