In network science, the efficiency of a network is a measure of how efficiently it exchanges information and it is also called communication efficiency. 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. That is the local efficiency of a node ${\displaystyle i}$ characterizes how well information is exchanged by its neighbors when it is removed.

The definition of communication efficiency assumes that the efficiency is inversely proportional to the distance, so in mathematical terms

where ${\displaystyle \epsilon _{ij}}$ is the pairwise efficiency of nodes ${\displaystyle i,j\in V}$ in network ${\displaystyle G=(V,E)}$ and ${\displaystyle d_{ij}}$ is their distance.

The average communication efficiency of the network ${\displaystyle G}$ is then defined as the average over the pairwise efficiencies:

where ${\displaystyle N=|V|}$ denotes the number of nodes in the network.

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 ${\displaystyle i}$ and ${\displaystyle j}$, i.e. a shortest path between ${\displaystyle i,j}$ is a path with minimum number of edges and the number of edges is its length. Observe that if ${\displaystyle i=j}$ then ${\displaystyle d_{ij}=0}$—and that is why the sum above is over ${\displaystyle i\neq j}$— while if there is no path connecting ${\displaystyle i}$ and ${\displaystyle j}$, ${\displaystyle d_{ij}=\infty }$ and their pairwise efficiency is zero. Being ${\displaystyle d_{ij}}$ a count, for ${\displaystyle i\neq j}$ ${\displaystyle d_{ij}\geq 1}$ and so ${\displaystyle E(G)}$ is bounded between 0 and 1, i.e. it is a normalised descriptor.

The shortest path distance can also be generalised to weighted networks, see the weighted shortest path distance, but in this case ${\displaystyle d_{ij}^{W}\in [0,+\infty ]}$ and the average communication efficiency needs to be properly normalised in order to be comparable among different networks.

In the authors proposed to normalise ${\displaystyle E(G)}$ dividing it by the efficiency of an idealised version of the network ${\displaystyle G}$: