Figure 4: Identifying the hidden node and reconstructing the connections among the other nodes.
From: Reconstructing direct and indirect interactions in networked public goods game
(a) Illustration of a hidden node. The hidden node (node 50 in red) interacts with five nodes (in orange), including direct interactions (solid lines) and indirect interactions (dashed lines). For clarity, the virtual self-loop of each node and interaction weights are not shown. The direct (red solid lines) and indirect (red dashed lines) interactions between the hidden node and its neighbors and neighbors’ neighbors can be inferred in terms of the standard variance (Equation (17)). However, the direct and indirect interactions cannot be distinguished because of missing the data of the hidden node. The time series of the other nodes except the hidden node are measurable. The interactions (orange lines) between the neighbors of the hidden node and the other nodes except the hidden node can be reconstructed accurately by simply ignoring the hidden node. However, indirect (orange dashed lines) and direct interactions (orange solid lines) still cannot be distinguished. The other connections (gray lines) can be accurately reconstructed and classified. (b) The standard variance of reconstructed interactions αi of each node. The five nodes who interact with the hidden node exhibits much larger values of α than the other nodes. (c) The reconstructed network except the hidden node. The gray lines are true existent links which are reconstructed successfully, and the red lines are false positive (fake) links which do not exist in the original network. (d) The success rate of reconstructing the WS small-world network in (c). In the WS small-world network, the network size N is 50 (including the hidden node), the average degree 〈k〉 = 2 and the rewiring probability is 0.1. The results are obtained by averaging over 10 independent realizations.
