Analysis of the inference of ratings and rankings in complex networks using discrete exterior calculus on higher--order networks

The inference of rankings plays a central role in the theory of social choice, which seeks to establish preferences from collectively generated data, such as pairwise comparisons. Examples include political elections, ranking athletes based on competition results, ordering web pages in search engines using hyperlink networks, and generating recommendations in online stores based on user behavior. Various methods have been developed to infer rankings from incomplete or conflicting data. One such method, HodgeRank, introduced by Jiang em et al.~[Math. Program. bf 127, 203 (2011)], utilizes Hodge decomposition of cochains in higher--order networks to disentangle gradient and cyclical components contributing to rating scores, enabling a parsimonious inference of ratings and rankings for lists of items. This paper presents a systematic study of HodgeRank's performance under the influence of quenched disorder and across networks with complex topologies generated by four different network models. The results reveal a transition from a regime of perfect retrieval of true rankings to one of imperfect retrieval as the strength of the quenched disorder increases. A range of observables are analyzed, and their scaling behavior with respect to the network model parameters is characterized. This work advances the understanding of social choice theory and the inference of ratings and rankings within complex network structures.