# Efficiency (network science)

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In network science, the efficiency of a network is a measure of how efficiently it exchanges information. [1] 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.

## Definition

The average efficiency of a network ${\displaystyle G}$ is defined as:[1]

${\displaystyle E(G)={\frac {1}{n(n-1)}}\sum _{i\neq j\in G}{\frac {1}{d(i,j)}}}$

where ${\displaystyle n}$ denotes the total nodes in a network and ${\displaystyle d(i,j)}$ denotes the length of the shortest path between a node ${\displaystyle i}$ and another node ${\displaystyle j}$.

As an alternative to the average path length ${\displaystyle L}$ of a network, the global efficiency of a network is defined as:

${\displaystyle E_{glob}(G)={\frac {E(G)}{E(G^{ideal})}}}$

where ${\displaystyle G^{ideal}}$ is the "ideal" graph on ${\displaystyle n}$ nodes wherein all possible edges are present. The global efficiency of network is a measure comparable to ${\displaystyle 1/L}$, rather than just the average path length itself. The key distinction is that ${\displaystyle 1/L}$ measures efficiency in a system where only one packet of information is being moved through the network and ${\displaystyle E_{glob}(G)}$ measures the efficiency where all the nodes are exchanging packets of information with each other.

As an alternative to the clustering coefficient of a network, the local efficiency of a network is defined as:

${\displaystyle E_{loc}(G)={\frac {1}{n}}\sum _{i\in G}E(G_{i})}$

where ${\displaystyle G_{i}}$ is the local subgraph consisting only of a node ${\displaystyle i}$'s immediate neighbors, but not the node ${\displaystyle i}$ itself.

## Applications

Broadly speaking, the efficiency of a network can be used to quantify small world behavior in networks. Efficiency can also be used to determine cost-effective structures in weighted and unweighted networks. [2] Comparing the two measures of efficiency in a network to a random network of the same size to see how economically a network is constructed. Furthermore, global efficiency is easier to use numerically than its counterpart, path length. [3]

For these reasons the concept of efficiency has been used across the many diverse applications of network science.[2] [4] Efficiency is useful in analysis of man-made networks such as transportation networks and communications networks. 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. This is because, for example, a subway network is not closed, and passengers can be re-routed, by buses for example, even if a particular line in the network is down.[1]

Beyond human constructed networks, efficiency is a useful metric when talking about physical biological networks. In any facet of biology, the scarcity of resource plays a key role, and biological networks are no exception. Efficiency is used in neuroscience to discuss information transfer across neural networks, where the physical space and resource constraints are a major factor.[3] 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.[5] 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.[4]

## References

1. ^ a b c Latora, Vito; Marchiori, Massimo (17 October 2001). "Efficient Behavior of Small-World Networks". Phys. Rev. Lett. 87. arXiv:cond-mat/0101396. Bibcode:2001PhRvL..87s8701L. doi:10.1103/PhysRevLett.87.198701.
2. ^ a b Latora, Vito; Marchiori, Massimo (March 2003). "Economic small-world behavior in weighted networks". The European Physical Journal B. 32 (2): 249–263. arXiv:cond-mat/0204089. Bibcode:2003EPJB...32..249L. doi:10.1140/epjb/e2003-00095-5.
3. ^ a b Bullmore, Ed; Sporns, Olaf (March 2009). "Complex brain networks graph theoretical analysis of structural and functional systems". Nature Reviews Neuroscience. 10: 186–198. doi:10.1038/nrn2575. PMID 19190637.
4. ^ a b Bocaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U. (February 2006). "Complex networks: Structure and dynamics". Physics Reports. 424 (4–5): 175–308. Bibcode:2006PhR...424..175B. doi:10.1016/j.physrep.2005.10.009.
5. ^ Buhl, J.; Gautrais, J.; Solé, R.V.; Kuntz, P.; Valverde, S.; Deneubourg, J.L.; Theraulaz, G. (November 2002). "Efficiency and robustness in ant networks of galleries". The European Physical Journal B. 42 (1): 123–129. Bibcode:2004EPJB...42..123B. doi:10.1140/epjb/e2004-00364-9.