Abstract
Interacting subsystems are commonly described by networks, where multimodal behaviour found in most natural or engineered systems found recent extension in form of multilayer networks. Since multimodal interaction is often not dictated by network topology alone and may manifest in form of cross-layer information exchange, multilayer network flow becomes of relevant further interest. Rationale can be found in most interacting subsystems, where a form of multimodal flow across layers can be observed in e.g., chemical processes, energy networks, logistics, finance, or any other form of conversion process relying on the laws of conservation. To this end, the formal notion of heterogeneous network flow is proposed, as a multilayer flow function aligned with the theory of network flow. Furthermore, dynamic equivalence is established with the framework of Petri nets, as the baseline model of concurrent event systems. Application of the resulting multilayer Laplacian flow and flow centrality is presented, along with graph learning based inference of multilayer relationships over multimodal data. On synthetic data the proposed framework demonstrates benefits of multimodal flow derivation in critical component identification. It also displays applicability in relationship inference (learning based function approximation) on multimodal time series. On real-world data the proposed framework provides, among others, multimodal flow interpretation of U.S. economic activity, uncovering underlying empirical steady state probability distribution, as well as inherent network (economic) robustness.
Highlights
The Petri net flow relations are here extended, to possibly incorporate both fundamental equations of balance[36], namely: flow balance, which is integral to the Petri net model, and node potential balance, which may arise in relation to specific application domains
The heterogeneous flow network enables derivation of a layered relationship structure, as opposed to a classic flat relationship structure. This enables a physical interpretation of models with complex interactions between different semantic domains, in the form of multilayer network flows
In this paper the formal notion of heterogeneous network flow is proposed, as a multilayer flow function aligned with the theory of network flow
Summary
To aid further understanding of the proposed framework, a brief introduction of background concepts is presented, including basic terminology and notation. An information network represents a mathematical object, establishing correspondence between a node set V and arc set AN , and a set of object types OT and relation types RT , respectively, such that reference to each element is preserved. The formal notion enables introduction of a heterogeneous network (Fig. 1b, left panel), where nodes and edges associate to one specific layer or type (Fig. 1b, right panel), such that there is more than one object type OT or relation type RT , as introduced (Definition 5). A network schema of a heterogeneous network corresponds to a topological projection of paths between node-layers (Fig. 1b, left panel), onto a set of composite relations between elements of object types (Fig. 1b, right panel).
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