conformityBasedNetworkConcepts: Calculation of conformity-based network concepts.

Description

This function computes 3 types of network concepts (also known as network indices or statistics) based on an adjacency matrix and optionally a node significance measure.

Usage

conformityBasedNetworkConcepts(adj, GS = NULL)

Arguments

adj

adjacency matrix. A symmetric matrix with components between 0 and 1.

optional node significance measure. A vector with length equal the dimension of adj.

Value

A list with the following components:
Factorizabilitynumber between 0 and 1 giving the factorizability of the matrix. The closer to 1 the higher the evidence of factorizability, that is, A-I is close to outer(CF,CF)-diag(CF^2).
fundamentalNCsfundamental network concepts, that is network concepts calculated directly from the given adjacency matrix adj. A list with components ScaledConnectivity (giving the scaled connectivity of each node), Connectivity (connectivity of each node), ClusterCoef (the clustering coefficient of each node), MAR (maximum adjacency ratio of each node), Density (the mean density of the network), Centralization (the centralization of the network), Heterogeneity (the heterogeneity of the network). If the input node significance GS is specified, the following additional components are included: NetworkSignificance (network significance, the mean node significance), and HubNodeSignificance (hub node significance given by the linear regression of node significance on connectivity).
conformityBasedNCsnetwork concepts based on an approximate adjacency matrix given by the outer product of the conformity vector but with unit diagonal. A list with components Conformity (the conformity vector) and Connectivity.CF, ClusterCoef.CF, MAR.CF, Density.CF, Centralization.CF, Heterogeneity.CF giving the conformity-based analogs of the above network concepts.
approximateConformityBasedNCsnetwork concepts based on an approximate adjacency matrix given by the outer product of the conformity vector. A list with components Conformity (the conformity vector) and Connectivity.CF.App, ClusterCoef.CF.App, MAR.CF.App, Density.CF.App, Centralization.CF.App, Heterogeneity.CF.App giving the conformity-based analogs of the above network concepts.

Details

This function computes 3 types of network concepts (also known as network indices or statistics) based on an adjacency matrix and optionally a node significance measure. Specifically, it computes I) fundamental network concepts, II) conformity based network concepts, and III) approximate conformity based network concepts. These network concepts are defined for any symmetric adjacency matrix (weighted and unweighted). The network concepts are described in Dong and Horvath (2007) and Horvath and Dong (2008). In the following, we use the term gene and node interchangeably since these methods were originally developed for gene networks. In the following, we briefly describe the 3 types of network concepts:

Type I: fundamental network concepts are defined as a function of the off-diagonal elements of an adjacency matrix A and/or a node significance measure GS. Type II: conformity-based network concepts are functions of the off-diagonal elements of the conformity based adjacency matrix A.CF=CF*t(CF) and/or the node significance measure. These network concepts are defined for any network for which a conformity vector can be defined. Details: For any adjacency matrix A, the conformity vector CF is calculated by requiring that A[i,j] is approximately equal to CF[i]*CF[j]. Using the conformity one can define the matrix A.CF=CF*t(CF) which is the outer product of the conformity vector with itself. In general, A.CF is not an adjacency matrix since its diagonal elements are different from 1. If the off-diagonal elements of A.CF are similar to those of A according to the Frobenius matrix norm, then A is approximately factorizable. To measure the factorizability of a network, one can calculate the Factorizability, which is a number between 0 and 1 (Dong and Horvath 2007). The conformity is defined using a monotonic, iterative algorithm that maximizes the factorizability measure. Type III: approximate conformity based network concepts are functions of all elements of the conformity based adjacency matrix A.CF (including the diagonal) and/or the node significance measure GS. These network concepts are very useful for deriving relationships between network concepts in networks that are approximately factorizable.

References