Planting trees in graphs, and finding them back

In this paper we study detection and reconstruction of planted structures in Erdős-Rényi random graphs. Motivated by a problem of communication security, we focus on planted structures that consist in a tree graph. For planted line graphs, we establish the following phase diagram. In a low density region where the average degree λ of the initial graph is below some critical value λ_c=1, detection and reconstruction go from impossible to easy as the line length K crosses some critical value f(λ)ln(n), where n is the number of nodes in the graph. In the high density region λ>λ_c, detection goes from impossible to easy as K goes from o(√(n)) to ω(√(n)), and reconstruction remains impossible so long as K=o(n). For D-ary trees of varying depth h and 2≤ D≤ O(1), we identify a low-density region λ<λ_D, such that the following holds. There is a threshold h*=g(D)ln(ln(n)) with the following properties. Detection goes from feasible to impossible as h crosses h*. We also show that only partial reconstruction is feasible at best for h≥ h*. We conjecture a similar picture to hold for D-ary trees as for lines in the high-density region λ>λ_D, but confirm only the following part of this picture: Detection is easy for D-ary trees of size ω(√(n)), while at best only partial reconstruction is feasible for D-ary trees of any size o(n). These results are in contrast with the corresponding picture for detection and reconstruction of low rank planted structures, such as dense subgraphs and block communities: We observe a discrepancy between detection and reconstruction, the latter being impossible for a wide range of parameters where detection is easy. This property does not hold for previously studied low rank planted structures.