Section: New Results

An Impossibility Result for Reconstruction in a Degree-Corrected Planted-Partition Model

In a paper currently in preparation [45] , we consider a degree-corrected planted-partition model: a random graph on $n$ nodes with two equal-sized clusters. The model parameters are two constants $a,b>0$ and an i.i.d. sequence ${\left({\phi }_i\right)}_{i=1}^n$, with finite second moment ${\Phi }^2$. Vertices $i$ and $j$ are joined by an edge with probability $\frac{{\phi }_i{\phi }_j}{n}a$ whenever they are in the same class and with probability $\frac{{\phi }_i{\phi }_j}{n}b$ otherwise. We prove that the underlying community structure cannot be accurately recovered from observations of the graph when ${(a-b)}^2{\Phi }^2\le 2(a+b)$.