The stabilization of rankings and convergence towards the final ranking.

<p>The colored lines (top) track the weekly rankings of teams, which show large fluctuations in the early stages that attenuate as the seasons progress. The fluctuations are quantified by the numbers of line crossings (middle) that show an exponential decrease. This is also reflected in the Spearman Ranking Correlation of the weekly rankings with the final ones (bottom), which reaches 0.9 when fewer than 50% of the games have been played with the exception of NFL in 2013, where 64.7% of the games had to be played.</p

The weekly prediction accuracies of our method and four other methods compared.

<p>Predictions were made based on cumulative data. In the case of NFL (top panels) our method shows a noticeably higher prediction accuracy in comparison with others, while for the EPL (bottom panels) the methods exhibit smaller differences, mainly due to the significantly higher connectance .</p

The final regular season ratings and rankings of the teams for two professional sports, NFL (top) and EPL (bottom).

<p>The errors were estimated using the jackknife method <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113685#pone.0113685-Efron1" target="_blank">[18]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0113685#pone.0113685-Newman2" target="_blank">[19]</a>. In NFL 2013 the top-ranked team (Seattle Seahwaks) enjoyed an exceptionally regular season and won the Super Bowl championship in a dominant fashion.</p

Basic concept of network centralities and our ranking method.

<p>(a) Three basic network centralities. The <i>Degree</i> is the number of node's neighbors; the shaded node in the most central. The <i>Eigenvalue Centrality</i> considers the quality of a connection, so that being connected to a central node raises one's centrality in turn; the larger shaded node is more central than the smaller shaded node, although their degrees are equal. The <i>Betweenness</i> quantifies the node's role in acting as an intermediary between nodes by measuring how often it sits on the geodesic (shortest) paths between two nodes; the shaded node, even though its degree is low, is the most central. (b) In PageRank, with probability a random walker follows a randomly chosen outgoing link (solid lines) to travel to another node, and with probability makes a random jump to any node in the network (red dotted lines). The nodes are ranked by their stationary occupation probability. (c) A competition network is a directed network with weighted directional edges, where the weights can represent the number of wins or the points scored by one node against another. Our ranking method is based on random walk where the edge weights define a gradient between nodes. We use the stationary occupation probability as the measure of node's strength or weakness, depending on the defined directionality of the gradient. (d) A high degree can unfairly favor and penalize a node, necessitating a degree-neutralizing procedure.</p

The schedule networks for (a) the National Football League (NFL) in 2013 and (b) the English Premier League (EPL) of 2012–2013.

<p>The NFL consists of 32 teams divided equally into American Football Conference and National Football Conference, further divided into four divisions (shaped differently) corresponding to regions in the country. The EPL consists of 20 teams, forming a complete network or a round-robin.</p

Schematic depiction of the GP map properties of redundancy, phenotype bias and neutral correlations.

<p>Phenotypes are represented by colours, genotypes as nodes and mutations as edges. A) Each colour appears multiple times with uniform redundancy. B) Some colours appear more often than others, demonstrating a phenotype bias. C) A rearrangement of the colours from the middle plot illustrates positive neutral correlations where the same colours are more likely to appear near each other than would be expected by random chance arrangement. The black box surrounding the six orange genotypes depicts a single component (a set of genotypes connected by neutral point mutations, also called a neutral network) of the orange phenotype. Such positive neutral correlations enhance the probability that such neutral networks occur.</p

Biological GP maps have much larger and fewer neutral components than their random counterparts due to neutral correlations.

<p>A) The logarithm of the largest neutral component for a given phenotype is plotted as a function of frequency for random null models (with <i>K</i> = 4, <i>L</i> = 12) and three biological GP maps, RNA12, <i>S</i>_2,8 and HP24. The vertical dotted line denotes the giant component threshold <i>δ</i> ≈ 1/36, defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e020" target="_blank">Eq (5)</a>, for the schematic random model with <i>K</i> = 4, <i>L</i> = 12. The vertical dashed line denotes the single component threshold <i>λ</i> ≈ 0.37, defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e022" target="_blank">Eq (6)</a>, for the schematic random model. The biological GP maps show much larger connected components below these thresholds, due to the presence of positive neutral correlations. B) The logarithm of the total number of neutral components against frequency is plotted for the same models. The theoretical thresholds <i>δ</i> and <i>λ</i> work well for random model but again the number of components in the biophysical models differ greatly from the random model expectation due to the presence of correlations. In both plots, error bars represent a single standard deviation from the 100 independent realisations of the random null model used to derive the neutral component statistics.</p

Greater mutational robustness indicates the presence of neutral correlations.

<p>A) The phenotype robustness <i>ρ</i>_<i>p</i> is plotted as a function of frequency <i>f</i>_<i>p</i> for all phenotypes in the RNA secondary structure models: RNA12, RNA 15, RNA20, the Polyomino models for protein quaternary structure: <i>S</i>_2,8 <i>S</i>_3,8 and the HP protein folding model HP24. Each model has an associated random model with the same frequencies, but we only show one example, with <i>K</i> = 4 and <i>L</i> = 12 and a set of phenotypes chosen with a broad range of frequencies to best illustrate the relationship (red points). All random models closely follow the expected theoretical curve <i>ρ</i>_<i>p</i> = <i>f</i>_<i>p</i> (grey line). The biophysical models exhibit a much larger robustness than the random models, which indicates the presence of positive neutral correlations. The red dotted line is <i>δ</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e020" target="_blank">Eq (5)</a>) for <i>K</i> = 4, <i>L</i> = 12. If (<i>ρ</i> > <i>δ</i>) then large neutral networks are expected, which is much more likely for the biophysical models than for the random model. B) The average <i>n</i>-robustness 〈<i>ρ</i>^(<i>n</i>)〉, defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e012" target="_blank">Eq 3</a>, for each of the three biological GP maps, along with the expected values 〈<i>ρ</i>^(<i>n</i>)〉 = 1/<i>N</i>_<i>P</i> for the associated random null models (flat coloured horizontal lines) is plotted against <i>n</i>. Across all three GP maps, we see a typical decay in robustness towards the random null model expectation with increasing mutational distance. From this decay a neutral correlation length can be defined which is shorter for the HP model than for the other two models. Error bars for HP24 are the standard error on the mean of the average <i>n</i>-robustness.</p

Enhanced robustness in Maynard Smith’s 4-letter word game.

<p>The single mutation path WORD → WORE → GORE → GONE → GENE is shown in red. All valid words within a one letter mutation of “WORD” and “GENE” are also depicted. According to the Merriam-Webster Official Scrabble Players Dictionary 2014, only 4,175 of the 456,976 possible 4-letter words are valid English words (at least for Scrabble). Since each word has 100 neighbours, for a random model, the expected number of valid words within a one letter mutation is < 1. Nevertheless, due to positive neutral correlations, the probability that a <i>valid</i> word has another valid word as a 1-mutant neighbour is more than ten times greater, as illustrated above for “WORD” and “GENE”. As pointed out by Maynard Smith, in a biological system, such correlations (in his case between “meaningful sequences”) can facilitate evolutionary dynamics [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.ref001" target="_blank">1</a>].</p

Phenotype mutation probabilities scale with global frequency.

<p>We present results for the three GP maps: A) RNA20, B) <i>S</i>_3,8 and C) HP5x5. We plot the relationship between <i>ϕ</i>_<i>qp</i> (circles) and <i>f</i>_<i>q</i> for the largest non-deleterious phenotype <i>p</i> in <i>S</i>_3,8 and HP5x5, and for the second largest in RNA20 (not the largest due to computational expense). We see in each case a strong positive correlation (<i>p</i>-value ≪ 0.05 in all cases), very similar to the expectation for the null model (not shown here, but for which the correlation is exact to within statistical fluctuations, see ref. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.ref014" target="_blank">14</a>] and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.s001" target="_blank">S1 Text</a>). Spearman rank correlation coefficients are shown in the top-left of each plot. Differences from <i>ϕ</i>_<i>qp</i> = <i>f</i>_<i>q</i> are relatively small compared to the overall range of variation, except for sets of phenotypes that are not connected at all, which typically arise due to biophysical constraints. These are shown as downward triangles along the lower horizontal dotted line which represents <i>ϕ</i>_<i>qp</i> = 0. For each plot, the upward triangle indicates <i>ϕ</i>_<i>pp</i> = <i>ρ</i>_<i>p</i>, the phenotype robustness, which is always over-represented (<i>ρ</i>_<i>p</i> ≫ <i>f</i>_<i>p</i>) due to neutral correlations.</p