Modularity: Networks are modular if they contain
highly connected clusters of nodes that are sparsely connected to nodes in
other clusters.
“Intuitively, modular systems seem more adaptable, a
lesson well known to human engineers, because it is easier to rewire a modular
network with functional subunits than an entangled, monolithic network.
However, because this evolvability only provides a selective advantage over the
long term, such selection is at best indirect and may not be strong enough to
explain the level of modularity in the natural world.”
“The leading hypothesis is that modularity mainly emerges
because of rapidly changing environments that have common subproblems, but
different overall problems. Computational simulations demonstrate that in such
environments (called modularly varying
goals: MVG), networks evolve both modularity and evolvability. By contrast,
evolution in unchanging environments produces non-modular networks that are
slower to adapt to new environments.
One theory, which assumes a constantly changing environment and
selection for evolvability, is that modularity arises to enable modifying one
subcomponent without affecting others.
Other theories are that variation
mechanisms, such as gene duplication, create a bias towards the generation of
modular structures and that modularity evolves because of selection to make
phenotypes robust to environmental perturbations.
An alternative hypothesis: Modularity evolves not because it conveys evolvability,
but as a byproduct from selection to reduce connection costs in a network.
Method:
-
- An eight-pixel retina receives visual stimuli.
- Patterns shown on the retina’s left and right halves may each contain an “object” (i.e., a pattern of interest).
- Networks evolve (using an evolutionary algorithm) to answer either:
- L-AND-R: whether an object is present on both the right and left sides, or
- L-OR-R: whether an object is displayed on either side.
- Each network iteratively sees all 256 possible patterns and answers TRUE or FALSE.
- Performance = percentage of correct answers.
- Each treatment is repeated 50 times, with different random seeds. Population size = 1000, and evolution lasts 25,000 generations.
- Treatments – evolutionary fitness based on either:
- Maximizing performance alone (PA)
- Maximizing performance and minimizing connection costs (P&CC)
- cost objective affects selection probabilistically only 25% of the time.
- Modularity evaluated with approximation of modularity metric Q, which first optimally divides networks into modules and then measures the difference between the number of edges within each model and the number expected for random networks with the same number of edges.
Results:
- P&CC treatment produced significantly more modular networks than PA treatment.
- Only P&CC networks achieved perfect performance in the L-AND-R task.
- P&CC networks achieved high modularity and low cost (connection cost), which PA networks were high cost (because this wasn’t selected for) and low modularity.
- When all possible networks were searchers, there was found an inverse relationship between cost and modularity.
- There are some high-performing, low-modularity, high-cost networks, which explains why modularity does not evolve due to performance alone (because of fitness landscape).
- There was also found to be an inverse correlation between modularity and cost, regardless of network performance (Fig. S12).
- Evolvability
- They ran trials until 50 P&CC and 50 PA networks each had a perfectly performing network, and then transferred these networks into the L-OR-R environment, such that the problem is different – it has the same subproblems in a different combination.
- They also ran the reverse condition, evolving first in L-OR-R condition and then switching to L-AND-R environment.
- In both experiments, P&CC networks required fewer generations to adapt to the new environment.
- Modularity thus evolves because sparse connectivity has lower connection costs, but such modularity also aids performance and evolvability because the problem is modular.
- P&CC modularity is also higher than PA even on problems that are non-modular, even though such modularity is lower than on modular problems.
- Non-modular problem: answer whether any four pixels are on (black).
- More problems: one with 5 XOR modules and one with hierarchically nested XOR problems.
- P&CC created separate modules for the decomposed problems in nearly every trial, whereas PA almost never did.
- P&CC performance was also significantly higher.
