Inferring Monopartite Projections of Bipartite Networks: An Entropy-Based Approach

Whenever detecting a relationship between any two entities is unfeasible—either because impractical (as in social systems where observing all interactions is impossible) or impossible (as in financial systems where direct connections are hidden)—a procedure to infer the presence of connections from available information must be devised.

This information is often encoded in a bipartite network, a representation that evidences co-behaviors of the entities under analysis. Examples abound: scientists who co-author papers, customers who purchase similar products, investors who hold similar stocks, or actors who appear in the same films. The fundamental challenge is determining when such co-occurrences are statistically significant indicators of underlying relationships.

The method presented in this research rests upon a foundational assumption: a large number of shared attributes proxies a significant node similarity, providing evidence of an otherwise undetectable connection between them. However, the crucial question is not simply whether nodes share attributes, but whether they share more attributes than would be expected by chance given the overall structure of the network.

More specifically, to infer the presence of a link between any two nodes, our method prescribes counting their number of shared attributes and comparing it with a statistical benchmark derived from an entropy-maximization approach. The benchmark represents what we would expect to observe in a random system with the same basic structural constraints. A critical innovation in our approach is the application of the False Discovery Rate (FDR) procedure—a sophisticated multiple hypothesis test that overcomes many limitations of more conservative criteria like the Bonferroni correction.

Remarkably, our algorithm can be employed to define a novel recommendation system that goes beyond simple popularity-based suggestions. An interesting problem concerning bipartite customer-goods networks is devising a procedure for recommending products to users that reflects genuine affinity rather than mere popularity.

Within our framework, this challenge can be addressed quite naturally by selecting those goods that have been "co-purchased" a significantly large number of times—that is, more than would be expected by chance. This provides suggestions tailored to each user's basket of preferences based on statistically validated patterns, rather than simply recommending the most popular items to everyone.

The key advantage of this approach is that it identifies products genuinely similar in terms of user preferences, filtering out spurious co-occurrences that arise merely from differences in product or user popularity. We demonstrate the effectiveness of this approach using the MovieLens dataset, showing that our entropy-based projection method successfully identifies meaningful movie similarities and user communities.