Random Tensors and Planted Cliques

The r-parity tensor of a graph is a generalization of the adjacency matrix, where the tensor’s entries denote the parity of the number of edges in subgraphs induced by r distinct vertices. For r = 2, it is the adjacency matrix with 1’s for edges and −1’s for nonedges. It is well-known that the 2-norm of the adjacency matrix of a random graph is O( √ n). Here we show that the 2-norm of the r-parity tensor is at most f(r) √ n log n, answering a question of Frieze and Kannan [3] who proved this for r = 3. As a consequence, we get a tight connection between the planted clique problem and the problem of finding a vector that approximates the 2-norm of the r-parity tensor of a random graph. Our proof method is based on an inductive application of concentration of measure.