Conformity and module based decomposition of a network adjacency matrix.

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Description

The function calculates the conformity based approximation A.CF of an adjacency matrix and a factorizability measure codeFactorizability. If a module assignment Cl is provided, it also estimates a corresponding intermodular adjacency matrix. In this case, function automatically carries out the module- and conformity based decomposition of the adjacency matrix described in chapter 2 of (Horvath 2011).

Usage

1

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 module. The entries of Cl could be integers or character strings.

Details

We distinguish two situation depending on whether or not Cl equals NULL. 1) Let us start out assuming that Cl = NULL. In this case, the function calculates the conformity vector for a general, possibly non-factorizable network adj by minimizing a quadratic (sums of squares) loss function. 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]=Conformity[i]*Conformity[j]. The conformity 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. Warning: the algorithm may only converge to a local optimum and it may not converge at all. Also see the notes below.

2) Let us now assume that Cl is not NULL, i.e. it specifies the module assignment of each node. Then the function calculates a module- and CF-based approximation of adj (explained in chapter 2 in Horvath 2011). In this case, the function calculates a conformity vector Conformity and a matrix IntermodularAdjacency such that adj[i,j] is approximately equal to Conformity[i]*Conformity[j]*IntermodularAdjacency[module.index[i],module.index[j]] where module.index[i] is the row of the matrix IntermodularAdjacency that corresponds to the module assigned to node i. To estimate Conformity and a matrix IntermodularAdjacency, the function attempts to minimize a quadratic loss function (sums of squares). Currently, the function only implements a heuristic algorithm for optimizing the objective function (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).

Value

A.CF

a symmetric matrix that approximates the input matrix adj. Roughly speaking, the i,j-the element of the matrix equals Conformity[i]*Conformity[j]*IntermodularAdjacency[module.index[i],module.index[j]] where module.index[i] is the row of the matrix IntermodularAdjacency that corresponds to the module assigned to node i.

Conformity

a numeric vector whose entries correspond to the rows of codeadj. If Cl=NULL then Conformity[i] is the conformity. If Cl is not NULL then Conformity[i] is the intramodular conformity with respect to the module that node i belongs to.

IntermodularAdjacency

a symmetric matrix (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.

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.

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.

listConformity

Note

Regarding the situation when Cl=NULL. 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).

Author(s)

Steve Horvath

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

Examples

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# assume the number of nodes can be divided by 2 and by 3
n=6
# 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)

# another block diagonal matrix
blockdiag.A=A
blockdiag.A[1:(n/3),(n/3+1):n]=0.3
blockdiag.A[(n/3+1):n , 1:(n/3)]=0.3
block.Cl=rep(c(1,2),c(n/3,2*n/3))
conformityDecomposition(adj= blockdiag.A,Cl=block.Cl)

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