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These new matrices are generated by bootstrapping elements with replacement from the original matrices of network proximity.

Finally, for each year, we measure the correlation between the newly generated matrices of network proximity and market similarity.

Spearman partial Mantel correlations (Pearson correlations yield similar results) between the matrices of network proximity and market similarity of traded corporations in each year.

Moreover, to recover the transfer matrices of network coding at the destination node, a coefficient vector of k symbols is usually included in each of the transmitted packets [3].

We calculate partial Mantel correlations [24] (both Spearman and Pearson correlations) between the matrices of network proximity and market similarity while controlling for other proximity matrices given by market sector, geographic distance, board size, fraction of directors with financial expertise, and average stock price in the year (see the Appendix for further details).

We find that each of the positive correlations shown in Figure 2 is also higher than the expected correlation between network proximity and market similarity when we randomly generate new matrices of network proximity (see Figure S5 in Additional file 2).

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To measure this, we look at the rank of the weight matrices of networks trained with soft targets.

Given the adjacency matrices of networks, the structural difference of networks can be studied in many different ways.

While earlier studies attempted to apply clustering algorithms directly to the adjacency matrices of networks in order to partition network nodes into groups [ 10, 14], later studies have relied on graph partitioning algorithms or special purpose algorithms for identifying subnetworks of certain properties [ 5- 7, 11, 17, 18, 20].

Note that our matrix of network proximity takes into account all pairs of traded corporations.

The above process could be denoted as xt + 1 = Px t, where P represents the matrix of transition probability, which is column-normalized matrix of weighted adjacency matrix of network graph G.

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