The non-backtracking random walk and its usage for vertex clustering

In case of sparse graphs, relation between the real eigenvalues of the non-backtracking matrix and those of the non-backtracking transition probability matrix is considered with respect to vertex clustering. For this purpose, the random walk along the non-backtracking graph is considered, the vertices of which are the bioriented edges, and the adjacency relation depends on whether the random walk goes through the oriented edges with the rule of ``not going back in the next step''. This is encoded in the non-backtracking matrix that is the adjacency matrix of the non-backtracking graph. The structural real eigenvalues of the transition probability matrix are related to the constant multiples of the non-backtracking one, the concordance of which indicates the existence of a sparse stochastic block model behind the graph. ``Inflation--deflation'' techniques are also developed for clustering the vertices of the original graph together with real world applications.