  Temporal networks are widely used models for describing the architecture of
complex systems. Network memory -- that is the dependence of a temporal
network's structure on its past -- has been shown to play a prominent role in
diffusion, epidemics and other processes occurring over the network, and even
to alter its community structure. Recent works have proposed to estimate the
length of memory in a temporal network by using high-order Markov models. Here
we show that network memory is inherently multidimensional and cannot be
meaningfully reduced to a single scalar quantity. Accordingly, we introduce a
mathematical framework for defining and efficiently estimating the microscopic
shape of memory, which fully characterises how the activity of each link
intertwines with the activities of all other links. We validate our methodology
on a wide range of synthetic models of temporal networks with tuneable memory,
and subsequently study the heterogeneous shapes of memory emerging in various
real-world networks.
