The underlying idea (and main assumption) is that the more distant two nodes are in the network, the less efficient their communication will be.
The concept of efficiency can be applied to both local and global scales in a network.
On a global scale, efficiency quantifies the exchange of information across the whole network where information is concurrently exchanged.
The local efficiency quantifies a network's resistance to failure on a small scale.
characterizes how well information is exchanged by its neighbors when it is removed.
Distances can be measured in different ways, depending on the type of networks.
The most natural distance for unweighted networks is the length of a shortest path between a nodes
The shortest path distance can also be generalised to weighted networks, see the weighted shortest path distance, but in this case
and the average communication efficiency needs to be properly normalised in order to be comparable among different networks.
dividing it by the efficiency of an idealised version of the network
In the unweighted case every edge has unitary weight,
When the edges are weighted, a sufficient condition (for having a proper normalisation, i.e.
) on the distances in the ideal network, called this time
A common choice is to take them as the geographical or physical distances in spatial networks[2] or as the maximum cost over all links, e.g.
[4] However, in [3] the authors highlight the issues of these choices when dealing with real world networks, which are characterised by heterogeneous structure and flows.
makes the global measure very sensitive to outliers in the distribution of weights and tends to under-estimate the actual efficiency of a network.
The authors also propose a normalising procedure, i.e. a way for building
using all and only the information contained in the edge weights (and no other meta-data such as geographical distances), which is statistically robust and physically grounded.
The global efficiency of a network is a measure comparable to
measures efficiency in a system where only one packet of information is being moved through the network,
measures the efficiency of parallel communication, that is when all the nodes are exchanging packets of information with each other concurrently.
A local average of pairwise communication efficiencies can be used as an alternative to the clustering coefficient of a network.
Efficiency can also be used to determine cost-effective structures in weighted and unweighted networks.
Furthermore, global efficiency is easier to use numerically than its counterpart, path length.
[5] For these reasons the concept of efficiency has been used across the many diverse applications of network science.
It is used to help determine how cost-efficient a particular network construction is, as well as how fault tolerant it is.
Studies of such networks reveal that they tend to have high global efficiency, implying good use of resources, but low local efficiency.
Efficiency is used in neuroscience to discuss information transfer across neural networks, where the physical space and resource constraints are a major factor.
[5] Efficiency has also been used in the study of ant colony tunnel systems, which are usually composed of large rooms as well as many sprawling tunnels.
[7] This application to ant colonies is not too surprising because the large structure of a colony must serve as a transportation network for various resources, most namely food.