depicts a number of the distributions used in the disparity filter [1] for various node degrees. We see that the family of distributions is quite rich and able to capture many different shapes of the local fractional edge weight distributions. However, Supporting Information Figure  S2 shows the local empirical cdfs for a random sample of nodes from the equities network (left panel), and the airline network (right panel). In the case of the equities network, all nodes have 873 neighbors (it is fully connected) and so a single distribution (thicker, pink-colored curve in figure) will be used by the disparity filter to determine significance. It is evident that the parametric distribution does not well-approximate the observed distributions. For the airline network we show the empirical cdfs of all nodes with degree 51, along with the distribution corresponding to the model of [1] . In this case the parametric distribution has similar shape to the empirical distributions; however, the shapes and scales of the empirical distributions vary, and it is evident that local information is being ignored. As a quantitative measure to show the statistical significance of local heterogeneity we performed a Kolmogorov-Smirnov test [2] for all nodes in the equities, airline, and art networks with 40 or more neighboring nodes. The Kolmogorov-Smirnov test is a statistical test to determine equality of distributions between a sample (the empirical cdf) and a reference distribution (the parametric distribution [1] would have used for the node). Supporting Information Table S1 summarizes the results. We see that in the equities and art networks all tests rejected the null hypothesis at the 5% and 1% significance levels indicating the parametric distribution does not capture the true distributions of fractional edge weight. For the airline network 99.4% and 98.3% of the tests rejected the null-hypothesis at the 5% and 1% significance levels. Again we see that there is statistically significant heterogeneity in the distribution of fractional edge weight among nodes with the same degree. Therefore, the use of the empirical cdf in LANS is justified as it captures this extra information that previously has been ignored. We evaluate network sparsification using LANS as a preprocessing step for a common task in network analysis, namely clustering nodes. The goal is to assign each node to a single cluster, such that nodes within the same cluster are "more similar" to each other than to nodes in other clusters. A common method for clustering the nodes of a network is with spectral clustering [3] . We perform clustering on the equities network sparsified using the LANS method with α = 0.003. In Figure 4 in the main text of the paper, we have colored the nodes of the equities network according to the cluster assignment found with spectral clustering with 22 clusters. The number 22 was chosen as the number of clusters by maximizing the likelihood of a Gaussian mixture model on the same data. We see that most clusters do correspond to known economic sectors. This agrees with previous work on clustering equities using correlations [4] .
Finally, there has been some work on defining backbones in the emerging field of multilayer network analysis REF .
Nonparametric Sparsification of Complex Multiscale Networks
PLoS ONE
PLoS ONE
Physics Medicine