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Abstract Headbutting is a combative behavior most popularly portrayed and exemplified in the extant bighorn sheep (Ovis canadensis). When behaviorally proposed in extinct taxa, these organisms are oft depictedOvis‐like as having used modified cranial structures to combatively slam into one another. The combative behavioral hypothesis of headbutting has a long and rich history in the vertebrate fossil literature (not just within Dinosauria), but the core of this behavioral hypothesis in fossil terrestrial vertebrates is associated with an enlarged osseous cranial dome—an osteological structure with essentially no current counterpart. One confounding issue found in the literature is that while the term “headbutting” sounds simplistic enough, little terminology has been used to describe this hypothesized behavior. And pertinent to this special issue, potential brain trauma and the merits of such proposed pugilism have been assessed largely from the potential deformation of the overlying osseous structure; despite the fact that extant taxa readily show that brain damage can and does occur without osteological compromise. Additionally, the extant taxa serving as the behavioral counterpart for comparison are critical, not only because of the combative behaviors and morphologies they display, but also the way they engage in such behavior. Sheep (Ovis), warthogs (Phacochoerus), and bison (Bison) all engage in various forms of “headbutting”, but the cranial morphologies and the way each engages in combat is markedly different. To hypothesize that an extinct organism engaged in headbutting like an extant counterpart in theory implies specific striking:contacting surfaces, speed, velocity, and overall how that action was executed. This review examines the history and usage of the headbutting behavioral hypothesis in these dome‐headed fossil taxa, their respective extant behavioral counterparts, and proposes a protocol for specific behavioral terms relating to headbutting to stem future confusion. We also discuss the disparate morphology of combative cranial structures in the fossil record, and the implications of headbutting‐induced brain injury in extinct taxa. Finally, we conclude with some potential implications for artistic reconstructions of fossil taxa regarding this behavioral repertoire. 

We propose data-driven one-pass streaming algorithms for estimating the number of triangles and four cycles, two fundamental problems in graph analytics that are widely studied in the graph data stream literature. Recently, Hsu et al. (2019a) and Jiang et al. (2020) applied machine learning techniques in other data stream problems, using a trained oracle that can predict certain properties of the stream elements to improve on prior “classical” algorithms that did not use oracles. In this paper, we explore the power of a “heavy edge” oracle in multiple graph edge streaming models. In the adjacency list model, we present a one-pass triangle counting algorithm improving upon the previous space upper bounds without such an oracle. In the arbitrary order model, we present algorithms for both triangle and four cycle estimation with fewer passes and the same space complexity as in previous algorithms, and we show several of these bounds are optimal. We analyze our algorithms under several noise models, showing that the algorithms perform well even when the oracle errs. Our methodology expands upon prior work on “classical” streaming algorithms, as previous multi-pass and random order streaming algorithms can be seen as special cases of our algorithms, where the first pass or random order was used to implement the heavy edge oracle. Lastly, our experiments demonstrate advantages of the proposed method compared to state-of-the-art streaming algorithms.