> [!definition]+ Definition. ([[closeness centrality]]) > Fix a node $i$ in a [[network]] $G$ with $|G|=n$. The **closeness centrality** $C_{i}$ of node $i$ is defined to be the *inverse of* the mean [[geodesic distance in networks|shortest distance]] from $i$ to every node in the [[network]]: $\text{ If }\ell_{i}:=\frac{1}{n}\sum_{j=1}^{n} d_{ij}, \text{ then }C_{i}:= \frac{1}{\ell_{i}}.$ > This [[centrality]] takes high values for nodes that are separated from others by only a short distance on average. ^Definition > [!justification]+ Motivation. > The $\ell_{i}$ quantity takes low values for nodes that are separated from others by only a short distance on average. It is plausible that such nodes might have more direct influence on others or better access. In a social network, for instance, a person with lower mean distance to others might find that their opinions spread through the community more quickly than other people’s. (Newman p.170) ^Motivation > [!basicexample]+ > ![[CleanShot 2023-09-19 at 21.24.44.jpg]] ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch03