Topology: Generate an adjacency matrix.
In GiulioCostantini/TOMproject: Simulation of Topological Overlap

Description Usage Arguments Details Value Author(s) Examples

Generate an adjacency matrix and a partial correlation matrix encoding the same conditional independencies. It allows to specify the desired number of nodes, the number of edges and the topology.

Topology(N, m, topology = c("ER", "BA", "WS"), exact.m = TRUE, force.connected = TRUE, negpro = 0, not = NULL, pw_ba = 1, p_ws = 0.1, minpcor = 0.2, maxpcor = 0.9, maxiter = c(50, 1000, 50), verbose = TRUE)

`N`	Integer. Number of nodes
`m`	Integer. Number of edges.
`topology`	Character. The topology on which the network is based. One of "ER" = Erdos-Renyi random network. "BA" = Barabasi & Albert scale-free network. "WS" = Watts & Strogatz random networks.
`exact.m`	Logical. Specify whether the network should have exactly `m` edges (`TRUE`) or at least `m` edges. If `TRUE` the computation can be very slow in some cases.
`force.connected`	Logical. Specify whether the network should be connected.
`negpro`	Integer in [0,1]. The desired proprtion of negative edges. Suggested value is 0 for the simulation, otherwise positive-definiteness problems are more likely to occur.
`not`	List of matrices. A list of adjacency matrices to prevent extracting networks isomorphic to networks extraced at earlier stages.
`pw_ba`	Numeric, specify the power of the preferential attachment. Only relevant if `topology = "BA"`.
`p_ws`	Numeric in [0,1]. Probability of random rewiring. Only relevant if `topology = "WS"`
`minpcor`	Numeric. Minimum allowed value for partial correlation.
`maxpcor`	Numeric. Maximum allowed value for partial correlation.
`maxiter`	Vector of three integers, specifying the maximum allowed iterations for three stages of computation. `maxiter[1]`. Maximum number of attempts to search for a network that is not isomorphic to other networks in "not" and that results also into a positive definite partial correlation matrix. This is the most global level of iteration. `maxiter[2]`. Maximum number of attempts to find a connected network, when `topology = "ER"` or `topology = "WS"` are specified. `maxiter[3]`. Maximum number of attemtps to find a positive definite correlation matrix in `ggm2pcm`. I reccommend a relatively low value for this because in case of repeated failure (maxiter[3] failures), a new network is drawn (up to maxiter[1] times) and the procedure is repeated: insisting too much on a topological configuration that does not appear to work may not be the best solution.
`verbose`	Logical. Do you want the function to tell you how it is going and where it is struggling?

This function creates a network with the desired topology, allowing to regulate directly the number of edges (which is not always possible using igraph functions). The number of edges is ensured for topology = "ER" or topology = "WS" by specifiying the nearest number of edge larger than m and then removing edges randomly. If force.connected = TRUE, then whenever an edge is removed, a check is done to preven removing a bridge, therefore disconnecting the network (see bridges). The function also tests for graph isomorphisms: if the obtained graph is isomorphic to a graph in the "not" list, the function keeps searching until a non-isomorphic graph is found (or until maxiter[1] calls are done).

A list of two elements

`ggm`	Adjacency matrix
`pcm`	Partial correlation matrix

Giulio Costantini

# this should make it struggle a bit, but then come up with a positive definite matrix
net <- Topology(N=40, m=60, topology="WS", exact.m=TRUE, force.connected=TRUE, negpro=0, p_ws=.2, minpcor=.1, maxpcor=.5)
par(mfrow = c(1,2))
lay <- qgraph(net$ggm)
qgraph(net$pcm, layout = lay$layout)