We define the co-spectral radius of inclusions ${\mathcal S}\leq {\mathcal R}$ of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

Media and scholastic accounts describe a strong backlash against attempts to advance gay rights. Academic research, however, increasingly raises questions about the sharply negative and enduring opinion change that characterizes backlash among the mass public. How can we reconcile the widespread backlash described by the media with the growing body of academic research that finds no evidence of the opinion change thought to be its hallmark trait? We argue that rather than widespread opinion change, what appears to be backlash against gay rights is more consistent with elite-led mobilization—a reaction by elites seeking to prevent gays and lesbians from achieving full incorporation in the polity. We present evidence from what is widely considered to be a classic case of anti-gay backlash, the 2010 Iowa Judicial Retention Election. Analysis of campaign contribution data in Iowa versus other states between 2010 and 2014, and voter roll-off data exploiting a unique feature of the 2010 retention election supports this argument. The results simultaneously explain how reports of backlash might occur despite increased support for gay rights, and an academic literature that finds no evidence of backlash.

Genetic resistance to gastrointestinal worms is a complex trait of great importance in both livestock and humans. In order to gain insights into the genetic architecture of this trait, a mixed breed population of sheep was artificially infected with Trichostrongylus colubriformis (n=3326) and then Haemonchus contortus (n=2669) to measure faecal worm egg count (WEC). The population was genotyped with the Illumina OvineSNP50 BeadChip and 48 640 single nucleotide polymorphism (SNP) markers passed the quality controls. An independent population of 316 sires of mixed breeds with accurate estimated breeding values for WEC were genotyped for the same SNP to assess the results obtained from the first population. We used principal components from the genomic relationship matrix among genotyped individuals to account for population stratification, and a novel approach to directly account for the sampling error associated with each SNP marker regression. The largest marker effects were estimated to explain an average of 0·48% (T. colubriformis) or 0·08% (H. contortus) of the phenotypic variance in WEC. These effects are small but consistent with results from other complex traits. We also demonstrated that methods which use all markers simultaneously can successfully predict genetic merit for resistance to worms, despite the small effects of individual markers. Correlations of genomic predictions with breeding values of the industry sires reached a maximum of 0·32. We estimate that effective across-breed predictions of genetic merit with multi-breed populations will require an average marker spacing of approximately 10 kbp.