bibkey: jianglimyaoye2011hodge
authors: Xiaoye Jiang, Lek-Heng Lim, Yuan Yao, Yinyu Ye
year: 2011
title: Statistical ranking and combinatorial Hodge theory
doi: 10.1007/s10107-010-0419-x
url: https://arxiv.org/abs/0811.1067
claim: "Positive edge weights specify the cochain inner product and its adjoint; Hodge decomposition splits gradients from cyclic residuals orthogonally."
strata_touched: []
license: citation-only
triage: anchor

Weighted graph cochains and currents

The primary preprint, arXiv:0811.1067v2, Definitions 4.2–4.3 and Lemma 4.4, pp. 16–18, specifies coboundaries, weight-dependent adjoints and the identity of two consecutive coboundaries with zero. Its divergence convention is minus the gradient adjoint. Theorems 4.7–4.8, pp. 18–20, supply the orthogonal decomposition; Theorem 5.1, pp. 21–22, applies the weighted normal equations to ranking. The inner product in equation (14) uses positive weights on the edges actually retained; zero weights denote absent edges.

For incidence columns equal to head minus tail, the consumer uses ${\delta }_0=B^{\mathsf{T}}$. With vertex inner product matrix $M$ and edge inner product matrix $W$, the adjoint is $M^{-1}BW$. Transport from a cochain $a$ to a current $j=Wa$ changes the edge metric to $W^{-1}$; cyclic currents satisfy $Bj=0$, whereas cyclic cochains satisfy $BWa=0$. For a one-dimensional graph with no two-cells, the coexact summand vanishes. A separately prescribed source potential equals the gradient projection only when their boundaries match.

These are classical intermediate inputs to the five-edge weighted seam lift and clock obstruction, Section 15. The paper’s ranking residual is not an independently supplied observer error, probability flux, clock cost or physical force.