Singular value decomposition: a multitool for network ecology

Ecological networks can be represented by their adjacency matrices, i.e. a matrix for which every entry represents a species pair, and takes either a value of 1 if there is an interaction between the two species, or 0 when there is none. Singular Value Decomposition ([SVD) is the factorisation of a matrix $\mathbf{A}$ (where $\mathbf{A}_{m,n} \in\mathbb{B}$, in the context of adjacency matrices) into the form:

$$ \mathbf{U}\cdot\mathbf{\Sigma}\cdot\mathbf{V}^T $$

Where $\mathbf{U}$ is an $m \times m$ orthogonal matrix and $\mathbf{V}$ an $n \times n$ orthogonal matrix. The columns in these matrices are, respectively, the left- and right-singular vectors of $\mathbf{A}$. $\mathbf{\Sigma}$ is a diagonal matrix that contains only non-negative $\sigma$ values. Where $\sigma_{i} = \Sigma{ii}$, and contains the singular values of $\mathbf{A}$. When the values of $\mathbf{\sigma}$ are arranged in descending order, the singular values ($\mathbf{\Sigma}$) are unique. These singular values can broadly be viewed as a measure of how informative that rank of the matrix is.

SVD Entropy as a measure of complexity

We can use SVD and more specifically SVD entropy as a measure of network complexity. This approach looks at the ‘physical complexity’ of networks - which is in contrast to more traditional measures such as connectence and nestedness which look at the ‘behavioural complexity’ of networks.

SVD Entropy and network assembly

An additional concept that I want to explore (if there is enough time) is the relationship between network assembly and constraints on complexity (SVD entropy). This will be building on a small section of the SVD entropy manuscript
relating to network assembly and the constraints on complexity.