We further demonstrate our method in larger scaled networks. As our identifiability theorem 2 suggests, the nonnegative values identify the compressed nodes. We investigate the node assignment accuracy for network size up to 100,000 nodes. Fix-StNMF and Adaptive-StNMF are implemented to compare with existing methods Undirected WNCut and Spectral as above. Node assignment accuracy (the mean followed by the standard deviation) from different noise settings is reported in Table 2. Noise levels are shown as direction or background noise. From the table, we can see that, for all network sizes, Adaptive-StNMF correctly assigns the vertices when the noise level is minimum (left panel of Table 2). This could not be achieved using other methods. Moreover, in large-scale networks, the Adaptive-StNMF can recover the compressed nodes with high accuracy, outperforming the competing methods (shown in the middle and right columns of Table 2).

Table 2:

Node Assignment Accuracy for Large-Scale Networks.

Noise Level: 0.0 /0.1Noise Level: 0.5 /0.7Noise Level: 0.9 /1.0
(accuracy)n $=$ 1kn $=$ 10kn $=$ 100kn $=$ 1kn $=$ 10kn $=$ 100kn $=$ 1kn $=$ 10kn $=$ 100k
Adaptive-StNMF 1. (7e-16) 1. (1e-16) 0.99 (1e-4) 0.99 (5e-4) 0.99 (1e-6) 0.99 (9e-4) 0.79 (7e-6) 0.86 (1e-5) 0.82 (2e-4)
Fixed-StNMF 0.71 (5e-2) 0.71 (1e-1) 0.65 (1e-2) 0.76 (8e-2) 0.65 (5e-3) 0.74 (7e-2) 0.69 (8e-2) 0.72 (7e-2) 0.70 (5e-2)
Undirected 0.55 (1e-2) 0.34 (2e-3) 0.34 (3e-3) 0.36 (7e-3) 0.34 (3e-3) 0.34 (2e-3) 0.32 (1e-2) 0.34 (3e-3) 0.34 (4e-3)
WNCut 0.53 (1e-1) 0.54 (2e-2) 0.52 (8e-2) 0.39 (3e-2) 0.37 (9e-3) 0.37 (2e-3) 0.35 (6e-2) 0.37 (9e-3) 0.34 (6e-3)
Spectral 0.89 (9e-2) 0.82 (1e-1) 0.87 (1e-1) 0.71 (2e-2) 0.63 (2e-2) 0.54 (1e-1) 0.36 (1e-1) 0.32 (2e-1) 0.38 (2e-2)
Noise Level: 0.0 /0.1Noise Level: 0.5 /0.7Noise Level: 0.9 /1.0
(accuracy)n $=$ 1kn $=$ 10kn $=$ 100kn $=$ 1kn $=$ 10kn $=$ 100kn $=$ 1kn $=$ 10kn $=$ 100k
Adaptive-StNMF 1. (7e-16) 1. (1e-16) 0.99 (1e-4) 0.99 (5e-4) 0.99 (1e-6) 0.99 (9e-4) 0.79 (7e-6) 0.86 (1e-5) 0.82 (2e-4)
Fixed-StNMF 0.71 (5e-2) 0.71 (1e-1) 0.65 (1e-2) 0.76 (8e-2) 0.65 (5e-3) 0.74 (7e-2) 0.69 (8e-2) 0.72 (7e-2) 0.70 (5e-2)
Undirected 0.55 (1e-2) 0.34 (2e-3) 0.34 (3e-3) 0.36 (7e-3) 0.34 (3e-3) 0.34 (2e-3) 0.32 (1e-2) 0.34 (3e-3) 0.34 (4e-3)
WNCut 0.53 (1e-1) 0.54 (2e-2) 0.52 (8e-2) 0.39 (3e-2) 0.37 (9e-3) 0.37 (2e-3) 0.35 (6e-2) 0.37 (9e-3) 0.34 (6e-3)
Spectral 0.89 (9e-2) 0.82 (1e-1) 0.87 (1e-1) 0.71 (2e-2) 0.63 (2e-2) 0.54 (1e-1) 0.36 (1e-1) 0.32 (2e-1) 0.38 (2e-2)

Notes: The first number shows the mean accuracy, and the number in parentheses shows its standard deviation from 100 trials. Noise level: $L1/L2$ denotes that the network is simulated with direction noise at $L1$ and background noise at $L2$.

Close Modal