Markov and Shi in their paper Simulating quantum computation by contracting tensor networks define the contraction complexity as follows (page 10):
The complexity of π is the maximum degree of a merged vertex during the contraction process. The contraction complexity of G, denoted by cc(G), is the minimum complexity of a contraction ordering.
The question is: what real-life resource do authors mean by this definition?
Consider the tensor contraction formula presented in the paper (page 8, formula 1). According to it, in order to calculate each of $2^{k+k^\prime}$ items of $f$ we are to perform $2^l$ multiplications which gives us $2^{k+k^\prime+l}$ per contraction i.e. the number of multiplications depends not only on the degree of contracted node and so it is probably not what authors mean.
Alternatively, the definition could be attributed to the number of writes to the memory holding the resulting tensor. It makes more sense but in my opinion is quite unusual.