Conformity and module based decomposition of a network adjacency matrix.
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).
a symmetric numeric matrix (or data frame) whose entries lie between 0 and 1.
a vector (or factor variable) of length equal to the number of rows of
We distinguish two situation depending on whether or not
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
module.index[i] is the row of the matrix
IntermodularAdjacency that corresponds to the module assigned to node i.
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).
a symmetric matrix that approximates the input matrix
a numeric vector whose entries correspond to the rows of codeadj. If
a symmetric matrix (data frame) whose rows and columns correspond to the number of modules specified in
is a number between 0 and 1. If
is a vector of character strings which correspond to the factor levels of the module assignment
is a numeric vector of length equal to the number of modules specified by
Regarding the situation when
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).
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
# 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)
Want to suggest features or report bugs for rdrr.io? Use the GitHub issue tracker.