I am a mathematician looking for a survey/book on methods for inference of graph/network topology (structure). Specifically, the kind of problem I am looking to study is as follows:

Given a graph $G$ consider an unknown function $f$ such that $f(G)=y$ (for a real number $y$). Assume we have a collection of graphs ($G_1, G_2, \dots, G_n$) for which the values $f(G_k)=y_k$ are known. Given a proposed value $y_*$:

  • How can one reconstruct a graph $\hat{G}_*$ such that $f(\hat{G}_*)\approx y_*$?
  • How about confidence intervals or high-density regions in some space of graphs?

What I am not interested in is:

  • A book such as Durrett's Random Graph Dynamics which presents specific probabilistic models of graph generation but no inference.
  • Kolaczyk's Statistical Analysis of Network Data which focuses on inferring part of a network's topological descriptors but not the whole network.

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