Evolving low-cost, high-performing networks that are non-modular reveals that independent of modularity, a connection cost promotes the evolution of hierarchy.

<p><b>(A)</b> Networks from the 16 highest-performing P&CC-NonMod replicates (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004829#pcbi.1004829.s004" target="_blank">S4 Fig</a> shows networks from all 30 trials). The networks are hierarchal, but not highly modular. <b>(B)</b> There is no significant difference in modularity between P&CC-NonMod and PA networks, but P&CC-NonMod networks are significantly more hierarchical <b>(C)</b> and solve significantly more sub-problems <b>(D)</b> than PA networks. <b>(E-G)</b> P&CC-NonMod networks also adapt significantly faster to a new environment than PA networks, suggesting that hierarchy promotes evolvability independently of modularity. <b>(E)</b> The base and target problem for this evolvability experiment. <b>(F)</b> A perfect-performing network evolved for the base problem (left) and a descendant network evolved on the target problem (right). The example networks are those with median hierarchy: <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004829#pcbi.1004829.s011" target="_blank">S11 Fig</a> shows all pairs. <b>(G)</b> P&CC-NonMod networks adapt significantly faster to the new problem.</p

Solving sub-problems is correlated with both hierarchy (left) and modularity (right).

<p>The shape sizes and enclosed numbers indicate the number of networks at that coordinate (an empty shape indicates only one network is present). The Pearson’s correlation coefficient is 0.96 for hierarchy and 0.87 for modularity, indicating strong, linear, positive relationships. Both correlations are significant (<i>p</i> < 0.00001) according to a t-test with a correlation of zero as the null hypothesis.</p

Lower cost networks are more hierarchical and modular.

<p>The hierarchy (left) and modularity (right) of randomly generated (i.e. non-functional) networks is shown for each cost after being normalized per cost value and then smoothed by a Gaussian kernel density estimation function. Colors indicate the probability of a network being generated at that location (heat map). Networks evolved in either the P&CC or PA treatment are overlaid as green circles or blue triangles, respectively. Circle or triangle size and the enclosed number indicate the number of networks at that coordinate (no number means 1). All evolved P&CC networks are in the high-hierarchy, low-cost region. Most evolved PA networks are in the high-cost, low-hierarchy region.</p

P&CC networks adapt significantly faster and solve significantly more sub-problems in new environments.

<p>In these experiments, networks first evolve to solve a problem perfectly in a base environment (left) and are then placed in a target environment (right) where they continue evolving to solve a different problem. The evolvability of PA and P&CC networks is quantified as the number of generations they take to solve the new problem perfectly. A pair of evolved networks is shown for both treatments. The left one shows the network with median hierarchy (here and elsewhere, rounding up) of 30 replicates in the base environment; the right one shows the median hierarchy network of the 30 runs in the target environment started with the network on the left. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004829#pcbi.1004829.s005" target="_blank">S5</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004829#pcbi.1004829.s010" target="_blank">S10</a> Figs show all network pairs.</p

A cost for network connections produces networks that are significantly more hierarchical, modular, high-performing, and likely to functionally decompose a problem.

<p>The algorithms for quantifying hierarchy and modularity are described in Methods. The bars below plots indicate at which generation a significant difference exists between the two treatments. <b>(A)</b> The hierarchical AND-XOR-AND problem (the default for our experiments). The top eight nodes are inputs to the problem and the bottom node is an output. <b>(B)</b> P&CC networks are significantly more hierarchical than PA networks. <i>p</i>-values are from the Mann-Whitney-Wilcoxon rank-sum test, which is the default statistical test throughout the paper unless otherwise stated. <b>(C)</b> P&CC networks are also significantly more modular than PA networks, confirming a previous finding [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004829#pcbi.1004829.ref017" target="_blank">17</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004829#pcbi.1004829.ref038" target="_blank">38</a>]. <b>(D)</b> P&CC networks evolve a solution to the problem significantly faster. <b>(E)</b> Evolved networks from the 16 highest-performing replicates in the PA treatment. The networks are non-hierarchical, non-modular, and do not tend to decompose the problem. Each network panel reports fitness/performance (F), hierarchy (H), and modularity (M). Nodes are colored if they solve one of the logic sub-functions in (<b>A</b>). <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004829#pcbi.1004829.s001" target="_blank">S1 Fig</a> shows networks from all 30 replicates for both treatments. <b>(F)</b> Evolved networks from the 16 highest-performing replicates in the P&CC treatment. The networks are hierarchical, modular, and decompose the problem. <b>(G)</b> A comparison of P&CC and PA networks from the final generation. P&CC networks are significantly more hierarchical, modular, and solve significantly more sub-problems.</p

The main problem (pictured in Fig 2A).

<p>Networks receive 8-bit vectors as inputs. As shown, a successful network could AND adjacent input pairs, XOR the resulting pairs, and AND the result. Performance is a function only of the final output, and thus does not depend on how the network solves the problem; other, non-hierarchical solutions also exist.</p

The main hypothesis.

<p>Evolution with selection for performance only results in non-hierarchical and non-modular networks, which take longer to adapt to new environments. Evolving networks with a connection cost, however, creates hierarchical and functionally modular networks that can solve the overall problem by recursively solving its sub-problems. These networks also adapt to new environments faster.</p

Network modularity and hierarchy can independently vary, and high-performing networks exist with a wide range of modularity and hierarchy scores.

<p>The highest-performing networks evolution discovered (with the MAP-Elites algorithm) for each combination of modularity and hierarchy. A few example networks are shown, along with their fitness (F), hierarchy (H), and modularity (M). The best network from each of the PA and P&CC treatments are also overlaid as blue triangles and green circles, respectively. The size of the circles or triangles and the enclosed number indicate the number of networks at that coordinate (no number means 1).</p