# Ecological and evolutionary dynamics of interconnectedness and modularity

This paper studies how Modularity of ecological systems can emerge from microscopic evolutionary dynamics. The approach aims to distinguish three factors: ecological, evolutionary, and environmental.

## Theory

### Model formulation

The model assumes a high-dimensional space that includes all possible traits. Any phenotype can be a vector $x \in \Omega$, where $\Omega \subseteq \mathbb{R}^k$. The time-varying density function can be written as:

$\frac{\partial}{\partial t} n(x,t) = n(x,t)\mathcal{A}\{n, E(t)\} + \nabla \cdot (g(x,t) \nabla n(x,t))$

where $\mathcal{A}$ is a growth/decay rate that depends on the current density and the environmental condition ($E(t)$). The second term captures the diffusive intergenerational phenotypic changes, where $g(x,t)$ can be interpreted as ‘evolvability’.

The environment also changes

$\frac{\partial}{\partial t} E(t) = \mathcal{F}(E, n)$

Then the individuals with different traits can interact and this interaction is captured by an incremental impact function $\alpha(x, x'; n) = D\mathcal{A}$, where $D$ is Fréchet derivative. If $\mathcal{A}$ is linear, $\alpha$ can be decomposed and $\mathcal{A}$ can be written as:

$\mathcal{A}\{n\} = r(x) + \int_{\Omega} \alpha(x, x')n(x') dx'$

### Definition of observed interconnectedness

“Observed interconnectedness” $H_x\{n\}$ is the connections observed from the perspective of $x$ and considered to follow the following “axioms”.

- It must be positive.
- If all members have identical traits (homogeneous population) it must be zero.
- No interaction, no connectedness.
- It cannot decrease when new individuals are added to the population.
- (additional property) for a single-species-population, the interconnectedness should not decrease when the variance of traits increases.

How about self interaction with the individuals with very similar traits? Somewhat similar to the axioms for defining Entropy?

The proposed form of $H_x\{n\}$ is

$H_{x}\{n\}=\frac{1}{N} \int_{\Omega} n\left(x^{\prime}\right) \underline{h}\left(x, x^{\prime}\right) d x^{\prime}$

where $h(x,x') = d(x,x')\phi(x,x')$ is an observed interconnectedness kernel, where $d(x,x')$ is a measure of distance between two traits and $\phi(x,x')$ is the interaction strength, which is later defined as $\phi(x,x';n)=|\alpha(x,x';n)|$. The other side of the interconnectedness (imposed interconnected) is defined by flipping the kernel.

$H_{x}^{*}\{n\}=\frac{1}{N} \int_{\Omega} n\left(x^{\prime}\right) \bar{h}\left(x^{\prime}, x\right) d x^{\prime}$

Then the total interconnectedness can be written as

$H_{\Omega}\{n\}=\int_{\Omega} H_{x'}\{n\} d x^{\prime} \text { and } H_{\Omega}^{*}\{n\}=\int_{\Omega} H_{x'}^{*}\{n\} d x^{\prime}$

and the mean interconnectedness

$H\{n\}=\frac{1}{N} \int_{\Omega} n(x) H_{x}\{n\} d x=\frac{1}{N} \int_{\Omega} n(x) H_{x}^{*}\{n\} d x$

$H\{n\}=\frac{1}{N^{2}} \iint_{\Omega} n(x) n\left(x^{\prime}\right) h\left(x, x^{\prime}\right) d x^{\prime} d x$

where

$h\left(x, x^{\prime}\right)=\frac{1}{2}\left(\underline{h}\left(x, x^{\prime}\right)+\bar{h}\left(x, x^{\prime}\right)\right)=\frac{1}{2} d\left(x, x^{\prime}\right)\left(\phi\left(x, x^{\prime}\right)+\phi\left(x^{\prime}, x\right)\right)$

### Connection with modularity

The continuous (Graphon) version of the modularity can be defined as

$Q\{n\}=\frac{1}{N^{2}} \iint_{\Omega} n(x) n\left(x^{\prime}\right) q\left(x, x^{\prime}\right) d x^{\prime} d x$

where

$q\left(x, x^{\prime}\right)=h\left(x, x^{\prime}\right)-\frac{h_{m}(x) h_{m}\left(x^{\prime}\right)}{\iint_{\Omega} h\left(x, x^{\prime}\right) d x d x^{\prime}}$

and $h_{m}(x)=\int_{\Omega} h\left(x, x^{\prime}\right) d x^{\prime}$.

Then, the modularity and interconnectedness can be linked using the following formula:

$Q\{n\}=H\{n\}-\frac{\left(H_{\Omega}\{n\}+H_{\Omega}^{*}\{n\}\right)^{2}}{4 H\{1\}}$

### Evolution

The evolution of interconnectedness and modularity is modeled with three components (ecological, evolutionary, and environmental)

$\frac{d H}{d t}=E c\{n\}+E v\{n\}+E n\{n\}$