WGCNA (version 1.10-2)

conformityDecomposition: Conformity vector(s) and factorizability measure(s) of a network

Description

The function calculates the conformity vector CF and factorizability measure Factorizability (number between 0 and 1) for the input adjacency matrix adj. If a module (class) assignment vector Cl is provided then the function automatically carries out the module- and conformity based decomposition of the adjacency matrix. In this case, it outputs a list of (intramodular) conformity vectors, an estimate of the intermodular adjacency matrix, the overall Factorizability (number between 0 and 1), and a vector of intramodular factorizabilities.

Usage

conformityDecomposition(adj, Cl = NULL)

Arguments

adj

a symmetric numeric matrix (or data frame) whose entries lie between 0 and 1.

Cl

a vector (or factor variable) of length equal to the number of rows of adj. The variable assigns each network node (row of adj) to a a module. The entries of Cl could be integers or character strings.

Value

• A list with the following components
• Conformity
  If Cl=NULL then it is a numeric vector of length equal to the number of nodes. Otherwise code{Conformity} is a list whose components provide one (intramodular) conformity vector for each module (class level) specified in Cl. In this case, the list components correspond to the entries of Cl.level.
• IntermodularAdjacency
  A symmetric data frame whose rows and columns correspond to the number of modules specified in Cl. Interpretation: it measures the similarity (adjacency) between the modules. In this case, the rows (and columns) of IntermodularAdjacency correspond to the entries of Cl.level.
• Factorizability
  is a number between 0 and 1. If Cl=NULL then it equals 1, if (and only if) adj is exactly factorizable. If Cl is a vector, then it measures how well the module- and CF based decomposition approximates adj.
• IntramodularFactorizability
  is a numeric vector of length equal to the number of modules specified by Cl. Its entries report the factorizability measure for each module. The components correspond to the entries of Cl.level.
• Cl.level
  is a vector of character strings which correspond to the factor levels of the module assignment Cl. Incidentally, the function automatically turns Cl into a factor variable. The components of Conformity and IntramodularFactorizability correspond to the entries of Cl.level.

Details

This function calculates the conformity vector for a general, possibly non-factorizable network by minimizing a quadratic loss function. Warning: the algorithm may only converge to a local optimum and it may not converge at all. The conformity and factorizability for an adjacency matrix is defined in (Dong and Horvath 2007, Horvath and Dong 2008) but we briefly describe it in the following. A network is called exactly factorizable if the pairwise connection strength (adjacency) between 2 network nodes can be factored into node specific contributions, named node 'conformity', i.e. if adj[i,j]=CF[i]*CF[j] where CF is the conformity vector. The conformity CF turns out to be highly related to the network connectivity (aka degree). If adj is not exactly factorizable, then the function conformityDecomposition calculates a conformity vector of the exactly factorizable network that best approximates adj. The factorizability measure Factorizability is a number between 0 and 1. The higher Factorizability, the more factorizable is adj. One can easily show that the conformity vector is not unique if adj contains only 2 nodes. However, for more than 2 nodes the conformity is uniquely defined when dealing with an exactly factorizable weighted network whose entries adj[i,j] are larger than 0. In this case, one can get explicit formulas for the conformity (Dong and Horvath 2007). If the module assignment vector Cl is provided then the function calculates the module- and CF-based approximation of adj (explained in chapter 2 in Horvath 2011). Currently, the function only implements a heuristic algorithm for arriving at approximate solutions of the module- and CF-based approximation of adj (chapter 2 of Horvath 2011). Another, more accurate Majorization Minorization (MM) algorithm for the decomposition is implemented in the function propensityDecomposition by Ranola et al (2011).

References

Dong J, Horvath S (2007) Understanding Network Concepts in Modules. BMC Systems Biology 2007, June 1:24 Horvath S, Dong J (2008) Geometric Interpretation of Gene Co-Expression Network Analysis. PloS Computational Biology. 4(8): e1000117. PMID: 18704157 Horvath S (2011) Weighted Network Analysis. Applications in Genomics and Systems Biology. Springer Book. ISBN: 978-1-4419-8818-8 Ranola JMO, Langfelder P, Song L, Horvath S, Lange K (2011) An MM algorithm for the module- and propensity based decomposition of a network. Currently a draft.

See Also

conformityBasedNetworkConcepts,propensityDecomposition

Examples

# the examples assume that number of nodes is divisible by 6
n=12
# here is a perfectly factorizable matrix
A=matrix(1,nrow=n,ncol=n)
# this provides the conformity vector and factorizability measure
conformityDecomposition(adj=A)
# now assume we have a class assignment
Cl=rep(c(1,2),c(n/2,n/2))
conformityDecomposition(adj=A,Cl=Cl)
# here is a block diagonal matrix
blockdiag.A=A
blockdiag.A[1:(n/3),(n/3+1):n]=0
blockdiag.A[(n/3+1):n , 1:(n/3)]=0
block.Cl=rep(c(1,2),c(n/3,2*n/3))
conformityDecomposition(adj= blockdiag.A,Cl=block.Cl)