smallworldness: Compute the small-worldness index.

In qgraph: Graph Plotting Methods, Psychometric Data Visualization and Graphical Model Estimation

Compute the small-worldness index.

Description

Compute the small-worldness index (Humphries & Gurney, 2008) relying on the global transitity of the network (Newman, 2003) and on its average shortest path length.

Usage

smallworldness(x, B = 1000, up = 0.995, lo = 0.005)

Arguments

x	A graph. Can be a qgraph object object, an igraph object, an adjacency matrix, a weight matrix and an edgelist, or a weighted edgelist.
B	The number of random networks.
up	The upper quantile.
lo	the lower quantile.

Details

The function computes the transitivity of the target network and the average shortest path length. Then it computes the average of the same indices on B random networks. The small-worldness index is then computed as the transitivity (normalized by the random transitivity) over the average shortest path length (normalized by the random average shortest path length). The lo and up quantiles of the distribution of the random networks are also returned for both the transitivity and the average shortest path length.

A network can be said "smallworld" if its smallworldness is higher than one (a stricter rule is smallworldness>=3; Humphries & Gurney, 2008). To consider a network as "smallworld", it is also suggested to inspect that the network has a transitivity substantially higher than comparable random networks and that its average shortest path length is similar or higher (but not many times higher) than that computed on random networks. Edge weights, signs and directions are ignored in the computation of the indices.

Value

smallworldness	the "small-worldness" index proposed by Humphries & Gurney (2008)
trans_target	the global transitivity of the target network (Newman, 2003)
averagelength_target	the average shortest path length in the target network
trans_rnd_M	the average transitivity in the B random networks
trans_rnd_lo	the lo quantile of the transitivity in the B random networks
trans_rnd_up	the up quantile of the transitivity in the B random networks
averagelength_rnd_M	the average shortest path length in the B random networks
averagelength_rnd_lo	the lo quantile of the shortest path length in the B random networks
averagelength_rnd_up	the up quantile of the shortest path length in the B random networks

Note

If a directed network is given as input, an edge between every two nodes i and j is considered present if there is an arrow either from i to j or from j to i or both.

Author(s)

Giulio Costantini ([e-mail]), Sacha Epskamp ([e-mail])

References

Costantini, G., Epskamp, S., Borsboom, D., Perugini, M., Mottus, R., Waldorp, L., Cramer, A. O. J., State of the aRt personality research: A tutorial on network analysis of personality data in R. Manuscript submitted for publication.

Humphries, M. D., & Gurney, K. (2008). Network "small-world-ness": a quantitative method for determining canonical network equivalence. PLoS One, 3(4), e0002051.

Newman, M. E. J. (2003). The structure and function of complex networks*. SIAM Review, 45(3), 167–256.

Examples

set.seed(1)
# a regular lattice. Even if the small-worldness is higher than three, the average path length is 
# much higher than that of random networks.
regnet<-igraph::watts.strogatz.game(dim=1, size=1000, nei=10, p=0, loops=FALSE, multiple=FALSE)
smallworldness(regnet, B=10)

## Not run: 
# a small-world network: the transitivity is much higher than random, the average path length is 
# close to that of random networks
swnet<-igraph::watts.strogatz.game(dim=1, size=1000, nei=10, p=.1, loops=FALSE, multiple=FALSE)
smallworldness(swnet, B=10)

# a pseudorandom network: both the average path length and the transitivity are similar to random 
# networks.
rndnet<-igraph::watts.strogatz.game(dim=1, size=1000, nei=10, p=1, loops=FALSE, multiple=FALSE)
smallworldness(rndnet, B=10)

## End(Not run)

qgraph documentation built on Nov. 3, 2023, 5:07 p.m.