The increasing relevance of valued networks, which account for both the presence and intensity of relations, spans various disciplines, from life sciences to engineering and social sciences. However, comparing such networks remains a methodological challenge due to the lack of effective measures for quantifying differences. This paper introduces the ξ -distance, a novel metric designed to measure dissimilarity between valued networks. The proposed approach extends current graph comparison techniques by integrating both topological and relational intensity features into a single framework. Unlike existing measures, ξ -distance efficiently handles directed and weighted networks, employing a probabilistic framework based on vertex distance sequences (VDS). This approach is demonstrated through extensive simulations and real-world applications, including international trade and social interaction networks, where ξ reliably differentiates networks based on structural and relational characteristics. Moreover, the ξ -distance is computationally efficient, with an implementation that combines R and C++, ensuring accessibility and high performance. This metric not only addresses the limitations of previous methods but also opens new avenues for analysing multiplex, dynamic, and complex network systems, offering significant improvements in network comparison tasks across multiple domains.
Fabio Ashtar Telarico (Fri,) studied this question.