Modular network evolution under selection for robustness to noise

Real networks often exhibit modularity, which is defined as the degree to which a network can be decomposed into several subnetworks. The question of how a modular network arises is still open to discussion. The leading hypothesis is that high modularity evolves under multiple goals, which are decomposable to subproblems, as well as under the evolutionary constraint that selection prefers sparse links in a network. In the present study, we investigate an alternative evolutionary constraint entailing increased robustness to noise. To examine this, we present noise-interfused network models involving an analytically solvable linear system and biologically inspired nonlinear systems. The models demonstrate that it is possible to evolve a modular network under both modularly changing goal orientations and enhancing robustness to noise, thereby reducing sensitivity to noise. By performing theoretical analyses of linear systems, it is shown that the evolutionary constraint enforces the establishment of well-balanced noise sensitivities of multiple noise sources and leads to a modular network underlying a modular structure in goals. Moreover, computer simulations confirm that the presented mechanisms of modular network evolution are robust to variations of nonlinearity in network functions. Our findings suggest a positive role for the presence of noise in network evolution.

Figure 1

Schematic of evolutionary constraints, based on the assumption that robustness to noise leads to the evolution of modular networks. The region in gray represents one of the leading hypotheses based on reduction of link costs. The region with diagonal lines represents the area we are presently investigating wherein the network evolves under the imperative of increasing robustness to noise.

Figure 2

Network model configurations. The squares and the circles indicate input and output nodes, respectively. The recurrently connected line in red represents the links with respect to $a_{ij}$; the line in blue from inputs to outputs represents the links with respect to $b_{ik}$. The $f_i$ take the place of ${}^{\mathcal{L}}{f}_i$, ${}^{\mathcal{T}}{f}_i$, and ${}^{\mathcal{L}}{f}_i$ depending on the function model.

Figure 3

Network function models: (a) linear function model ${}^{\mathcal{L}}f\left(x\right)$, (b) hyperbolic tangent function model ${}^{\mathcal{T}}f\left(x\right)$ with $\beta$ variations, and (c) the Hill function model ${}^{\mathcal{H}}f\left(x\right)$ with $\gamma$ variations.

Figure 4

Block diagram of the network model, indicating the points of noise interfusion. The function $f$ varies according to the linear, hyperbolic tangent, and Hill models. The first-order lag element $1/(s+1)$ is derived by using the Laplace transform of $\dot{\mathbf{x}}=-\mathbf{x}+\left(\text{external}\text{input}\right)$.

Figure 5

Goal setting under MVG, showing the pairs of outputs $\mathbf{P}$ (left) and inputs $\mathbf{U}$ (right) in the present model. White squares indicate the on state; gray squares indicate the off state in required outputs and inputs. The goals involve three subproblems, which are determined according to the set of invariance states with regard to variation of the number of goals.

Figure 6

Evolutionary dynamics in a one-dimensional linear system. (a) Evolutionary pathway of the network with highest fitness. Blue (dark gray) lines represent the pathway under evolution with noise; gray (light gray) lines represent the pathway without noise. Open squares and circles indicate the states at the initial and later generations after $10000$, respectively. The straight line $b=-a+1$ represents the solution set, satisfying $p=-b/(a-1)u$. The ellipse is determined using ${\sigma }_a^2{(a-1)}^2+{\sigma }_b^2{b}^2-({\sigma }_a^2+{\sigma }_c^2)=0$, which is the condition for maximizing robustness to noise. (b) Stochastic density of the output $x$, at 20, 30, 3000, and $10000$ generations, as visualized by a Gaussian distribution.

Figure 7

Results from simulations using different network function models. The evolved networks and evolving metrics were evaluated in (a) and (d) for the linear model, (b) and (e) for the hyperbolic tangent model, and (c) and (f) for the Hill model. (a)–(c) Evolved network with the highest fitness wherein the lattices represent the link weights (with matrices $\mathbf{A}$ on the left-hand side and $\mathbf{B}$ on the right-hand side of each panel). Networks evolved with robustness to noise (the upper panes) comprising higher modular structures than those evolving without noise (the lower panes) for each network function model. (d)–(f) Box plots of the final metrics evaluated for ten trials. Boxes in blue represent metrics distributions evolved with noise and those in gray represent metrics distributions evolved without noise. On each box plot, thick lines represent the median, box length represents the 25th to 75th data percentile, the whisker indicates one and a half of the interquartile range, and the plus symbols represent outliers.

Figure 8

Evolutionary pathway results from simulations using different network function models. The blue (dark grey) lines represent evolving metrics in the case with noise and the gray (light grey) lines represent those in the case without noise. Squares and circles indicate the initial state and the state at $30000$ generations. Evolutionary pathways were recorded for (a) the linear model, (b) the hyperbolic tangent model, and (c) the Hill model. The upper and lower figures illustrate the evolution of robustness to noise $R$ and modularity $Q$ versus the fairness of trade-off between noise sensitivities $C$, respectively.