Exploring the complexity of ecological networks using SVD entropy

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# Exploring the complexity of ecological networks using SVD entropy
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Tanya Strydom1,2 Gullio V. Dalla Riva3 & Timothée Poisot1,2

 
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1Département de Sciences Biologiques, Université de Montréal; 2Québec Centre for Biodiversity Sciences; 3School of Mathematics and Statistics, University of Canterbury
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# The complexity of communities

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Quantifying the complexity and stability of communities remains a challenge.
]

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Network theory provides a 
 powerful approach 
 and allows us 
 to infer ecological properties 
 and processes

 

 <img src="img/Network.png" width="100%" />

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Complexity and stability are issues that have been studied but has remained a challenge to quantify. The debate is still prevalent in ecology

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# A traditional definition of complexity

Assume that a system is complex because it has many parts *i.e.* measuring structural complexity

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**Nestedness**

Do species assemblages form subsets of larger assemblages?
]

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**Connectance**

What proportion of potential interaction are realised?
]

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Nestedness would look at specialists vs generalists
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# Taking a physical approach to complexity

Instead we can measure the 'physical complexity' *i.e.* the amount of information required to encode the system

<img src="img/Basquait.png" height="350 px" />

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# Estimating complexity using SVD

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# SVD Entropy as complexity

Using the `$\sigma$` values from `$\mathbf{\Sigma}$` and following Pielou's evenness  (Pielou 1975) we can ensure that `$\sigma$` values are less than one:

$$
s_i = \sigma_i/\text{sum}(\sigma)
$$

and estimate entropy following Shannon (1948) as:

$$
J = -\frac{1}{\ln(k)}\sum_{i=1}^k s_i\cdot\ln(s_i)
$$

High values of SVD entropy reflects that all vectors are equally important, *i.e.* we cannot efficiently compress the network, indicating high complexity

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where k is the max rank

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# SVD Entropy as complexity

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# SVD Entropy of Networks

Ecological networks are *extremely* complex

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# Network Structure Captures Network Complexity

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Interstingly although we see this strong relationship its also interesting to note that while the networks span the almost the 'complete' range of possible values for connectance and nesetedness - something we don't see for entropy

Arguably these measures are telling a similar story but what happens when we take a bit more of an ecological approach e.g. resilliance

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# Defining Complexity Matters

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First do the axes and then have the point appear
How do we define resilience in ecological terms -> focus on how we defined resilience

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# Connecting the Dots

1. Not all measures of 
 complexity are created 
 equal

2. Complexity is not 
 always intuitive

unified 
 definition

SVD Entropy

.footnote[
For a more complete breakdown the preprint can be found at [doi.org/fmpt](https://doi.org/fmpt)
]

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