PMC10173245_01
The original and generated graphs from Zachary's Karate network (Zachary). A The original graph, where vertex color indicates a category associated with degree, vertex size represents a category associated with centrality, and community areas are drawn. B The PGM-generated graph, if just categorical attributes and degree augmentation are used to generate the network. C The GRANDPA-generated graph, where attributes, community labels, and degree are used to generate the network. In Fig. 2C, we recaptured similar vertex attribute relationship and nearly identical communities in both overall number of communities, size of communities, and internal community structure
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Fig2
The plots of the three selected graphs are shown in Fig. 2. The original graph, with generated labels and communities, is shown in Fig. 2A. The first, in Fig. 2B, represents a PGM-generated graph with both attributes and augmented degree (=10). When we create a community label following the marked areas and feed this to the GRANDPA algorithm, we quickly recover a graph that is highly similar to the original (Fig. 2C).
The CCDF of the vertex degree between the original and generated networks for Zachary's Karate Club network (Zachary)
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Fig3
The overall degree distribution across the vertices is close to the original graph with  and shown in Fig. 3. The NRMSE between the degree distribution of the original and GRANDPA-generated graph with community augmentation is nearly identical, as shown in Fig. 3 with overall value .
The original and GRANDPA-generated graphs from our 2019 Medicare data. A The original graph, where vertex color indicates a physician's primary specialty. B The final GRANDPA-generated graph with community, degree, and linchpin centrality augmentation. In Figure 4B, we recovered the same number of communities and similar in-between and internal community structures compared to Figure 4A
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Fig4
The original graph is shown in Fig. 4A. The GRANDPA-generated graph with community, degree and linchpin centrality augmentation is shown in Fig. 4B. To this end, we trained a regression model on both the original graph and the final graph (Fig. 4) which aimed to predict if a physician's primary speciality was either hospitalist or internal medicine given the proportion of their neighbors with the same specialty.
The CCDF of the vertex degree between the original and generated networks for the 2019 Medicare Data from CMS
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Fig5
While the GRANDPA-generated graph with only attribute information and community augmentation produced a graph with matching communities, the degree distribution was poorly recovered (NRMSE=0.237; Fig. 5). The NRMSE between the CCDF of the final graph and the original was 0.051 (Fig. 5) and similarly the KS-test indicates that the observed degrees are unlikely to have arisen from the same distributions and thus model for the graph.
The calculated joint label probabilities between the original physician graph based on actual Medicare claims data and final GRANDPA generated graph. A Shows the returned vertex joint label probabilities and B shows the edge label probabilities
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Fig6
This relationship is shown in Fig. 6.