  Since its introduction in 2011, the partial information decomposition (PID)
has triggered an explosion of interest in the field of multivariate information
theory and the study of emergent, higher-order ("synergistic") interactions in
complex systems. Despite its power, however, the PID has a number of
limitations that restrict its general applicability: it scales poorly with
system size and the standard approach to decomposition hinges on a definition
of "redundancy", leaving synergy only vaguely defined as "that information not
redundant." Other heuristic measures, such as the O-information, have been
introduced, although these measures typically only provided a summary statistic
of redundancy/synergy dominance, rather than direct insight into the synergy
itself. To address this issue, we present an alternative decomposition that is
synergy-first, scales much more gracefully than the PID, and has a
straightforward interpretation. Our approach defines synergy as that
information in a set that would be lost following the minimally invasive
perturbation on any single element. By generalizing this idea to sets of
elements, we construct a totally ordered "backbone" of partial synergy atoms
that sweeps systems scales. Our approach starts with entropy, but can be
generalized to the Kullback-Leibler divergence, and by extension, to the total
correlation and the single-target mutual information. Finally, we show that
this approach can be used to decompose higher-order interactions beyond just
information theory: we demonstrate this by showing how synergistic combinations
of pairwise edges in a complex network supports signal communicability and
global integration. We conclude by discussing how this perspective on
synergistic structure (information-based or otherwise) can deepen our
understanding of part-whole relationships in complex systems.
