Measures of network topological features

```
network_core(object, membership = NULL)
network_factions(object, membership = NULL)
network_modularity(object, membership = NULL, resolution = 1)
network_smallworld(object, method = c("omega", "sigma", "SWI"), times = 100)
network_scalefree(object)
network_balance(object)
```

`{signnet}`

by David Schoch

- object
An object of a migraph-consistent class:

matrix (adjacency or incidence) from

`{base}`

Redgelist, a data frame from

`{base}`

R or tibble from`{tibble}`

igraph, from the

`{igraph}`

packagenetwork, from the

`{network}`

packagetbl_graph, from the

`{tidygraph}`

package

- membership
A vector of partition membership.

- resolution
A proportion indicating the resolution scale. By default 1.

- method
There are three small-world measures implemented:

"sigma" is the original equation from Watts and Strogatz (1998), $$\frac{\frac{C}{C_r}}{\frac{L}{L_r}}$$, where \(C\) and \(L\) are the observed clustering coefficient and path length, respectively, and \(C_r\) and \(L_r\) are the averages obtained from random networks of the same dimensions and density. A \(\sigma > 1\) is considered to be small-world, but this measure is highly sensitive to network size.

"omega" (the default) is an update from Telesford et al. (2011), $$\frac{L_r}{L} - \frac{C}{C_l}$$, where \(C_l\) is the clustering coefficient for a lattice graph with the same dimensions. \(\omega\) ranges between 0 and 1, where 1 is as close to a small-world as possible.

"SWI" is an alternative proposed by Neal (2017), $$\frac{L - L_l}{L_r - L_l} \times \frac{C - C_r}{C_l - C_r}$$, where \(L_l\) is the average path length for a lattice graph with the same dimensions. \(SWI\) also ranges between 0 and 1 with the same interpretation, but where there may not be a network for which \(SWI = 1\).

- times
Integer of number of simulations.

`network_core()`

: Returns correlation between a given network and a core-periphery model with the same dimensions.`network_factions()`

: Returns correlation between a given network and a component model with the same dimensions.`network_modularity()`

: Returns modularity based on nodes' membership in pre-defined clusters.`network_smallworld()`

: Returns small-world metrics for one- and two-mode networks. Small-world networks can be highly clustered and yet have short path lengths.`network_scalefree()`

: Returns the exponent of the fitted power-law distribution. Usually an exponent between 2 and 3 indicates a power-law distribution.`network_balance()`

: Returns the structural balance index on the proportion of balanced triangles, ranging between`0`

if all triangles are imbalanced and`1`

if all triangles are balanced.

Modularity measures the difference between the number of ties within each community from the number of ties expected within each community in a random graph with the same degrees, and ranges between -1 and +1. Modularity scores of +1 mean that ties only appear within communities, while -1 would mean that ties only appear between communities. A score of 0 would mean that ties are half within and half between communities, as one would expect in a random graph.

Modularity faces a difficult problem known as the resolution limit (Fortunato and Barthélemy 2007). This problem appears when optimising modularity, particularly with large networks or depending on the degree of interconnectedness, can miss small clusters that 'hide' inside larger clusters. In the extreme case, this can be where they are only connected to the rest of the network through a single tie.

Borgatti, Stephen P., and Martin G. Everett. 2000.
“Models of Core/Periphery Structures.”
*Social Networks* 21(4):375–95.
doi:10.1016/S0378-8733(99)00019-2

Murata, Tsuyoshi. 2010. Modularity for Bipartite Networks.
In: Memon, N., Xu, J., Hicks, D., Chen, H. (eds)
*Data Mining for Social Network Data. Annals of Information Systems*, Vol 12.
Springer, Boston, MA.
doi:10.1007/978-1-4419-6287-4_7

Watts, Duncan J., and Steven H. Strogatz. 1998.
“Collective Dynamics of ‘Small-World’ Networks.”
*Nature* 393(6684):440–42.
doi:10.1038/30918
.

Telesford QK, Joyce KE, Hayasaka S, Burdette JH, Laurienti PJ. 2011.
"The ubiquity of small-world networks".
*Brain Connectivity* 1(5): 367–75.
doi:10.1089/brain.2011.0038
.

Neal Zachary P. 2017.
"How small is it? Comparing indices of small worldliness".
*Network Science*. 5 (1): 30–44.
doi:10.1017/nws.2017.5
.

`network_transitivity()`

and `network_equivalency()`

for how clustering is calculated

Other measures:
`centralisation`

,
`centrality`

,
`closure`

,
`cohesion()`

,
`diversity`

,
`holes`

,
`tie_centrality`

```
network_core(ison_adolescents)
#> [1] 0.323
network_core(ison_southern_women)
#> [1] 0.343
network_factions(ison_adolescents)
#> [1] 0.174
network_factions(ison_southern_women)
#> [1] 0.485
network_modularity(ison_adolescents,
node_kernighanlin(ison_adolescents))
#> [1] -0.205
network_modularity(ison_southern_women,
node_kernighanlin(ison_southern_women))
#> [1] -0.458
network_smallworld(ison_brandes)
#> [1] 0.825
network_smallworld(ison_southern_women)
#> [1] -1.04
network_scalefree(ison_adolescents)
#> [1] 3.69
network_scalefree(generate_scalefree(50, 1.5))
#> [1] 2.82
network_scalefree(create_lattice(100))
#> Note: Kolgomorov-Smirnov test that data could have been drawn from a power-law distribution rejected.
#> [1] 12.1
network_balance(ison_marvel_relationships)
#> [1] 0.668
```