The proposed measures based on algebraic topology, specifically the gradient, curl, and harmonic components derived from Hodge Decomposition, can be extended to weighted and directed higher-order networks by incorporating edge weights and directionality into the simplicial complex framework. In weighted networks, the edge weights can be integrated into the boundary operators, allowing for a more nuanced representation of interactions. For instance, the indicator vector for an edge could be modified to reflect the strength of the relationship, thus influencing the Hodge Decomposition results.
In directed networks, the definition of simplicial complexes can be adapted to account for the directionality of edges. This would involve redefining the boundary operators to reflect directed relationships, ensuring that the flow of information or interaction is accurately captured. The gradient, curl, and harmonic components could then be interpreted in the context of directed flows, providing insights into how information or influence propagates through the network.
Moreover, the extension to weighted and directed networks would allow for the exploration of more complex social dynamics, such as the impact of varying interaction intensities on tie strength and the role of directed ties in facilitating or hindering information diffusion. This could lead to the development of new measures that capture the intricacies of social interactions in real-world scenarios, enhancing the applicability of the findings to diverse fields such as sociology, economics, and organizational behavior.