Conflict and Computation on Wikipedia: a Finite-State Machine Analysis of Editor Interactions

What is the boundary between a vigorous argument and a breakdown of relations? What drives a group of individuals across it? Taking Wikipedia as a test case, we use a hidden Markov model to approximate the computational structure and social grammar of more than a decade of cooperation and conflict among its editors. Across a wide range of pages, we discover a bursty war/peace structure where the systems can become trapped, sometimes for months, in a computational subspace associated with significantly higher levels of conflict-tracking "revert" actions. Distinct patterns of behavior characterize the lower-conflict subspace, including tit-for-tat reversion. While a fraction of the transitions between these subspaces are associated with top-down actions taken by administrators, the effects are weak. Surprisingly, we find no statistical signal that transitions are associated with the appearance of particularly anti-social users, and only weak association with significant news events outside the system. These findings are consistent with transitions being driven by decentralized processes with no clear locus of control. Models of belief revision in the presence of a common resource for information-sharing predict the existence of two distinct phases: a disordered high-conflict phase, and a frozen phase with spontaneously-broken symmetry. The bistability we observe empirically may be a consequence of editor turn-over, which drives the system to a critical point between them.Comment: 23 pages, 3 figures. Matches published version. Code for HMM fitting available at http://bit.ly/sfihmm ; time series and derived finite state machines at bit.ly/wiki_hm

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space $\mathbb P^ r$, for all $r\geq 3$. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of dual hypersurfaces to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. These examples fit non--classical versions of de Jonqui\`eres transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan--Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision to the present knowledge.Comment: 56 pages. Some material added in section 1; minor changes. Final version to appear in Advances in Mathematic