  Counting the number of occurrences of small connected subgraphs, called
temporal motifs, has become a fundamental primitive for the analysis of
temporal networks, whose edges are annotated with the time of the event they
represent. One of the main complications in studying temporal motifs is the
large number of motifs that can be built even with a limited number of vertices
or edges. As a consequence, since in many applications motifs are employed for
exploratory analyses, the user needs to iteratively select and analyze several
motifs that represent different aspects of the network, resulting in an
inefficient, time-consuming process. This problem is exacerbated in large
networks, where the analysis of even a single motif is computationally
demanding. As a solution, in this work we propose and study the problem of
simultaneously counting the number of occurrences of multiple temporal motifs,
all corresponding to the same (static) topology (e.g., a triangle). Given that
for large temporal networks computing the exact counts is unfeasible, we
propose odeN, a sampling-based algorithm that provides an accurate
approximation of all the counts of the motifs. We provide analytical bounds on
the number of samples required by odeN to compute rigorous, probabilistic,
relative approximations. Our extensive experimental evaluation shows that odeN
enables the approximation of the counts of motifs in temporal networks in a
fraction of the time needed by state-of-the-art methods, and that it also
reports more accurate approximations than such methods.
