Patent records and the associated collaboration networks.

<p>(a) Patent records contain collaborations at both the inventor and company levels. (b) By drawing a undirected link between nodes <i>i</i> and <i>j</i> if they filed a patent application at least once, we can construct the collaboration networks of inventors (or companies). The total times of collaborations between nodes <i>i</i> and <i>j</i> over the whole patent record is defined to be the weight of the link (<i>i</i>, <i>j</i>) (shown in black). The total number of collaborators of nodes <i>i</i> over the whole patent record is defined to be the weight of the node <i>i</i> (shown in pink). (c) The inventors (or companies) listed in each patent record forms a clique. For each patent, we calculated its repeat collaboration number (<i>R</i>_l) of its inventors (or companies) by averaging the accumulated number of collaborations among all the inventor (or company) pairs in the team (shown in black). The productivity of node <i>i</i> in a patent is defined to be the accumulated number of patents that node <i>i</i> has contributed. We calculated the team productivity (<i>R</i>_n) by averaging the productivity of all its nodes (shown in pink).</p

Patent records and collaboration networks used in this paper.

<p>The collaboration networks at the inventor and company levels were constructed from the Japan and U.S. patent records of several decades, with number of patents denoted by <i>N</i>_P. For each collaboration network we show the number of nodes (<i>N</i>), edges (<i>L</i>), mean degree (⟨<i>k</i>⟩ = 2<i>L</i>/<i>N</i>), relative size of the largest connected component (<i>s</i>_lc = <i>N</i>_lc/<i>N</i>, where <i>N</i>_lc is the number of nodes in the largest connected component), fraction of isolated nodes (<i>n</i>₀), clustering coefficient (<i>C</i>) and degree correlation (<i>r</i>). In graph theory, the connected components of a graph <i>G</i> are the set of largest subgraphs of <i>G</i> that are each connected (i.e., any two vertices or nodes in a connected component are connected to each other by paths), which can be easily computed using either breadth-first search or depth-first search. The largest connected component (often referred to as the giant component) is the connected component of the largest size (number of nodes). The clustering coefficient <i>C</i> of a graph is a measure of the degree to which the nodes in the graph tend to cluster together (i.e., form triangles) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121973#pone.0121973.ref023" target="_blank">23</a>]. <i>C</i> can be calculated as (3 × number of triangles)/(number of connected triples). The degree correlation <i>r</i> of a network is given by the Pearson correlation coefficient of degrees between pairs of linked nodes [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121973#pone.0121973.ref024" target="_blank">24</a>].</p

Track records of individual teams with at least three patent records.

<p>Different colors represent different individual teams. (a, b) Inventors. (c, d) Companies.</p

Effect of repeat collaboration on innovation performance.

<p>(a, b) Inventors. (c, d) Companies. (a, c) Probability distributions of repeat collaboration number of teams. (b, d) Patent impact as a function of repeat collaboration number.</p

Effect of team size on innovation performance.

<p>(a, b, c, g, h) Inventors. (d, e, f, i, j) Companies. (a, d) The average impacts of patents filed by solos and teams. (b, c, e, f) The impact distribution of patents filed by solos (<i>m</i> = 1) and teams (<i>m</i> ≥ 2). (g, i) The team size distribution. (h, j) The patent impact as a function of team size.</p

Effect of repeat collaboration on innovation performance of teams with similar team age.

<p>(a, b) Inventors. (c, d) Companies. To separate the aging effect of a team from that of repeat collaboration among its teammates, we grouped patents according to the quartiles of their team age (<i>A</i>). The <i>A</i>-range of each group is shown in the legend. For each <i>A</i>-group we further grouped patents according to their repeat collaboration number (<i>R</i>_l) and then calculated the average patent impact for each <i>R</i>_l-subgroup.</p

Supplementary Information from Degree heterogeneity and stability of ecological networks

Technical details in obtaining the key results presented in the main text

Distribution of normalized control centrality of several real-world networks (blue) and their randomized counterparts: rand-ER (red), rand-Degree (green), plotted in log-log scale.

<p>(a) Intra-organizational network of a manufacturing company <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0044459#pone.0044459-Cross1" target="_blank">[49]</a>. (b) Hyperlinks between weblogs on US politics <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0044459#pone.0044459-Adamic1" target="_blank">[50]</a>. (c) Email network in a university <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0044459#pone.0044459-Eckmann1" target="_blank">[51]</a>. (d) Ownership network of US corporations <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0044459#pone.0044459-Norlen1" target="_blank">[52]</a>. In- and out-degree distributions for each network are shown in the insets. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0044459#pone-0044459-t001" target="_blank">Table 1</a> for other network characteristics.</p

Real networks analyzed in the paper.

<p>For each network, we show its name and reference; number of nodes () and edges (); mean degree (); degree correlation () <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0044459#pone.0044459-Watts1" target="_blank">[4]</a>; and clustering coefficient () <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0044459#pone.0044459-Newman3" target="_blank">[53]</a>, respectively.</p

The impact of different attack strategies on network controllability with respective to the random attack.

<p> with represents the generic dimension of controllable subspace after removing a fraction of nodes using strategy-. The nodes are removed according to six different strategies. (Strategy-0) Random attack: randomly remove fraction of nodes. (Strategy-1 or 2) Random upstream (or downstream) attack: randomly choose fraction of nodes, randomly remove one of their upstream neighbors (or downstream neighbors). The results are averaged over 10 random choices of fraction of nodes with error bars defined as s.e.m. Lines are only a guide to the eye. (Strategy-3,4, or 5) Targeted attacks: remove the top fraction of nodes according to their control centralities (or in-degrees or out-degrees).</p