The ability to innovate depends on a genotype's position in the neutral network. (*a*) A tradeoff between robustness and innovation. The horizontal axis shows mutational robustness *R*_{μ}, the fraction of a network's *w* topological neighbors that share the same equilibrium expression state, *S*_{∞}, with *w*. For each network *w* whose robustness is displayed on the horizontal axis, the vertical axis shows the fraction of networks of genotype distance *D* < 0.1 around *w*, whose equilibrium state is different from S_{∞}. This fraction declines with increasing robustness. *n* = 8, *c* = 0.25, and *d* = 0.5. (*b*) The horizontal axis shows genotype distance *D*_{12} of two networks (*w*_{1} and *w*_{2}) with the same phenotype. The vertical axis shows the mean fraction *f* of unique new phenotypes, as defined in the main text, found in a *k*-neighborhood (see legend for *k*) around these networks. If *f* is close to zero, then all or most of the phenotypes of networks in the two neighborhoods are identical. If *f* is close to one, then almost all phenotypes in the two neighborhoods are different. Standard deviations around each data point are no greater than 8 × 10^{−3}. (*c*) Like *b*, except for a sample of 2,210 network pairs (*w*_{1} and *w*_{2}) chosen at random from the neutral network and with mutational robustness *R*_{μ} in the interval (0.45, 0.60). *n* = 8, *c* = 0.25, *d* = 0.5, and *k* = 3. As opposed to the strong and positive statistical association between genotype distance and *f* for networks at small *D*_{12}, this association is considerably weaker at larger distances. Notice the large fraction of unique new phenotypes for almost all network pairs shown (mean *f* = 0.73). (*d*) Histogram of *f* for 1-neighbors (blue), 2-neighbors (red), and 3-neighbors (green) of 2,210 randomly chosen network pairs with *R*_{μ} in the interval (0.45, 0.60). Data are shown for *n* = 8, *c* = 0.25, and *d* = 0.5. For one-mutant neighbors, the robustness *R*_{μ} of a network *w* is the fraction of a network's neighbors that has the same gene expression pattern S_{∞}. For *k*-neighbors with *k* > 1, we define *R*_{μ} as the fraction of all networks that differ from *w* by no more than *k* regulatory interactions and that have the same gene expression pattern S_{∞}.

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