We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on \$k\$ vertices. For \$k\geq4\$, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (\$k=3\$), which is approximately 0.79. We raise the question whether for some \$k\$ the exponent per edge can be below \$2/3\$, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalise to higher order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity.