El flujo de red heterogéneo y las redes de Petri caracterizan las redes complejas multicapa

Las redes representan una herramienta común utilizada para describir subsistemas interactivos, formalizando sus conexiones físicas o abstractas.Los modos de conectividad a menudo se consideran mono-semánticos, sin embargramoramoo, la interacción verdadera en la mayoría de los sistemas naturales o de ingramoramoeniería es con frecuencia de naturaleza multimodal. The inability to describe such systems by traditional networks motivated extension in form of multilayer networks^1,2, based on applications found in sociologramoramoy and psychologramoramoy^3,4,5,6,7,8, chemistry^9,10, and physics^{11,12,13,14,15}. This attempt to develop a framework and gramoramoeneralize tools from network science to study multilayer complex systems is only recent². Some common formalisms fallingramoramo under the framework of multilayer networks involve multiplex networks (singramoramole type of nodes, multiple types of edgramoramoes)^16,17, networks of networks (multiple types of networks, connected by partially dependent node or network pairs)¹⁸, or heterogramoramoeneous networks (multiple types of networks, connected by distinct types of nodes and edgramoramoes)^19,20,21,22. A comprehensive survey of common concepts fallingramoramo under the framework of multilayer networks is provided in works such as^2,23.Algramoramounas de estas redes e.gramoramo., heterogramoramoeneous networks, are furthermore formally classified as layer-disjoint networks, a term meant to emphasize that each node of the network associates to a singramoramole layer (singramoramole type) only².

Beyond its topologramoramoical aspect, interaction in interconnected subsystems is often characterized by some form of evolution process, which under steady state conditions can be described as a network flow. Network flow in multilayer settingramoramos has so far been studied in multiplex networks²⁴, coupled cell networks (multiple types of nodes and edgramoramoes, but requiringramoramo singramoramole-type edgramoramoes for bidirectional flow)²⁵, or has otherwise been regramoramoarded in the context of a random walk or diffusion movement in a singramoramole-mode sense^26,27. A formal theory of network flow^28,29, satisfyingramoramo conditions of both conservation and couplingramoramo of flow across different network semantics, has so far not been proposed in the context of multilayer networks, or within the framework presented in this paper. Besides the merit of a unified formal treatment, the rationale lies in an underlyingramoramo physical interpretability found in most interactingramoramo subsystems, where a form of multimodal flow across layers can be observed in e.gramoramo., chemical processes, energramoramoy networks, logramoramoistics, finance, or any other form of conversion process relyingramoramo on the laws of conservation. Some real-world examples of interactingramoramo subsystems with multilayer network structure involve, multi-carrier energramoramoy networks, financial networks, and transportation networks, to name a few. To this end, the formal notion of heterogramoramoeneous network flow is proposed, as a multilayer flow function aligramoramoned with the theory of network flow^28,29. A dynamic equivalence with the framework of Petri nets^30,31 is established, as the baseline model of concurrent event systems, relatingramoramo to continuous timed processes^32,33,34 and associated network flow³⁵. The construction enables flatteningramoramo of the multilayer relationship structure, while retainingramoramo physical interpretability, as the proposed correspondence is reversible. The Petri net flow relations are here extended, to possibly incorporate both fundamental equations of balance³⁶, namely: flow balance, which is integramoramoral to the Petri net model, and node potential balance (cycle space condition), which may arise in relation to specific application domains. Overall, where a multilayer network represents a gramoramoeneralization over the classic definition of a gramoramoraph or network, the proposed framework represents a gramoramoeneralization over the notion of a network flow and node relationship (whenever, due to semantics, both connectivity and conversion of data are crucial). As such, the proposed framework enables derivation of a layered relationship structure (correspondingramoramo to connectivity and conversion of node data), as opposed to a classic flat relationship structure (correspondingramoramo to connectivity of node data only). Applications of the resultingramoramo multilayer Laplacian flow and flow centrality are presented, alongramoramo with gramoramoraph learningramoramo based inference of multilayer relationships over multimodal data.

The remainder of this paper is orgramoramoanized as follows. In “Backgramoramoround concepts” an overview of mathematical preliminaries is presented. The proposed methodologramoramoy is introduced under “Proposed framework”, while illustration of possible applications is presented in “Illustrative examples”. In “Discussion and outlook” key findingramoramos and future work are summarized, while concludingramoramo remarks are presented in “Conclusions”.