In ECGen/coNet: Model and analyze co-occurrence based networks

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conetto

A toolbox for co-occurrence based network modeling

Network models are a useful representation of systems [@Woodward2010]. Because it is not always possible to directly observe relationships (i.e. ecological interactions) spatial or temporal patterns of co-occurrence can be used to render models of interaction networks [@Araujo2011]. conetto adapts a method previously developed for ecological networks [@Araujo2011] with the application of Bayesian probability to generate network models and provides several functions useful for comparative analysis of sets of co-occurrence based networks.

Installation

Currently, the package is still in beta but it can be installed via the devtools package.

install.packages("devtools")
devtools::install_github("ECGen/conetto")

if (!("conetto" %in% installed.packages()[,1])){
  devtools::install_github("ECGen/conetto", ref = "dev")
}
library(conetto)

How to use the package

The basic usage of the conetto package is to generate network models. To do this, we need a matrix of co-occurrence values, which will be passed into the coNet function:

A <- c(1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1)
B <- c(1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1)
C <- c(1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1)
D <- c(0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0)
M <- data.frame(A, B, C, D)
coNet(M)

The coNet function returns of square matrix with the same number of rows and columns equal to the number of columns in the co-occurrence matrix. There are a couple of features that are useful to note here:

First, the matrix is not symmetric, that is, the top and bottom halves are not equal. This is because the dependency of two variables is based on both their independent occurrences and their co-occurrences: $P(A|B) = P(A,B)/P(B)$ and $P(B|A) = P(A,B)/P(A)$.
Some of the values in the network are zero. This is because coNet conducts a test that compares the observed co-occurrences to a theoretical "null" co-occurrence interval. If the observed co-occurrences are within the interval, the conditional probability is set equal to the joint probability. Then, when relativized, the resulting values are zero (i.e. $P(A,B) - P(A,B) = 0$). The threshold for the null interval test can be adjusted using the ci.p argument (DEFAULT = 95).
Also, some values are negative. This is because the network is comprised of the difference between the interval adjusted conditional probabilities and the observed independent probabilities (i.e. $P(A|B) - P(A)$). The "raw", un-relativized, values can be returned via the raw argument.

Network comparisons

The coNet function was developed for use with replicated datasets, where multiple networks have been observed. Two other functions were written to provide a way to analyze these network sets.

One function, meanNet, will calculate the "mean" or average of a set of networks. The result is the cell-wise summation of all matrices divided by the total number of matrices. This function requires that the set of networks all have the same dimensionality and that they are organized into a list.

net.l <- lapply(1:3, function(x) matrix(runif(9), nrow = 3))
net.l
meanNet(net.l)

Another function, distNet, facilitates the calculation of a distance matrix for a set of networks. Two metrics are provided, Euclidean and Bray-Curtis. The resulting matrix is the distance among all networks in the set. Similar to meanNet, distNet requires that all matrices have the same dimensionality. The result is a distance, or dissimilarity in the case of Bray-Curtis, matrix that can now be used for multivariate analyses, such as ordination.

net.d <- distNet(net.l)
net.d

Method: Co-occurrence based network models

The network modeling is based on conditional probability. Ultimately, we are interested in an estimate of how much one variable (A) affects another (B), and visa-versa. conetto approaches this by calculating the difference between the conditional and independent probabilities of species paris using the observed occurrences ($A$ and $B$) and co-occurrences ($A,B$) of the variables of interest from a set of observations ($N$).

Independent probabilities ($P(A) = A/N$ and $P(B) = B/N$)
Joint probabilities $P(A,B) = (A,B) / N$
Conditional probabilities $P(A|B) = P(B|A) P(A) / P(B)$

Since there is uuncertainly in whether or not two species are dependent, conetto uses a interval based test to determine whether or not to use the observed joint probability $O[P(A,B)] = (A,B)/N$ or the expected probability using an analytical null model $E[P(A,B)] = P(A)P(B)$. Via the Chain Rule of probabililty $P(A,B) = P(B|A)P(A)$, such that if $E[P(A,B)]$ is used then $P(A|B) = P(A)P(B) / P(B)$, which reduces to $P(A|B) = P(A)$ and A is conditionally independent from B. We can then subtract the independent probability matrix from the interval test modified conditional probability matrix, which produces a matrix of the dependency among the variables of interest.