Balance with Memory in Signed Networks via Mittag-Leffler Matrix Functions

Structural balance is an important characteristic of graphs where edges can be positive or negative, or signed graphs, with a direct impact on the study of real-world complex systems. When a graph is not structurally balanced, it is important to know how much balance still exists. Although several measures have been proposed to characterize the degree of balance, the use of matrix functions of the signed adjacency matrix emerges as an up-and-coming area of research. Here, we take a step forward to use Mittag-Leffler (ML) matrix functions to quantify the notion of balance of signed graphs. We show that the ML balance index can be derived from first principles on the basis of a nonconservative diffusion dynamic and that it accounts for the memory of the system about the past, by diminishing the penalization that long cycles typically receive in other matrix functions. Finally, we demonstrate the important information in the ML balance index with both artificial signed networks and real-world networks in various contexts, ranging from biological and ecological to social ones.