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There are many network properties that you can extract for a given node. Two I have encountered that sound conceptually very similar to me and that I have difficulty distinguishing are betweenness centrality and efficiency.

Definitions:

Betweenness: Node betweenness centrality is the fraction of all shortest paths in the network that contain a given node. Nodes with high values of betweenness centrality participate in a large number of shortest paths... measures the extent to which a vertex lies on paths between other vertices. Vertices with high betweenness may have considerable influence within a network by virtue of their control over information passing between others.

Local Efficiency: The global efficiency is the average inverse shortest path length in the network, and is inversely related to the characteristic path length. The local efficiency is the global efficiency computed on the neighborhood of the node, and is related to the clustering coefficient... quantifies the exchange of information across the whole network where information is concurrently exchanged.

To me, these sound quite similar, at least by their conceptual descriptions. Specifically, descriptions of both describe them as governing or controlling the exchange of information. But perhaps this is a time when the mathematical notation and understanding is necessary to understand the distinction.

Can anyone explain how these two network properties differ?

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Yes, if you look into the definitions, there will be obvious differences. I won’t do this here since there is no commonly agreed upon definition of local efficiency to begin with and what you write is already contradictory. First the local efficiency is compared with the clustering coefficient, which is an inherently local metrics as it does not care about what happens in distant regions of the network. But then you say it quantifies the entire network.

Either way, there are some differences that are inevitable:

  1. Betweenness very roughly measures to what extent a node is a bottleneck. It takes into account what happens in distant parts of the network (i.e., remote from the node in question).

  2. A purely local version of efficiency per definition of local does not take into account what happens in distant parts of the network.

  3. A more global node-wise efficiency measure usually takes into account how close other nodes are to the node in question. It takes into account what happens in distant parts of the network.

It is clear that that if anything 1 and 3 could be identical, but now consider the following example: Your nodes are cities in a country and your edges are roads. Now, suppose you have city that lies on the isthmus that connects say one fifths of the country to the rest, which is well connected. That city will have a very high betweenness since all connections going between the two parts of the country have to go over this city. However, it will not have a very high local efficiency as per Point 3; cities in the middle of the major part will score better here.

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