Flow Characteristics and Cores of Complex Network and Multiplex Type Systems

Subject of research is complex networks and network systems. The network system is defined as a complex network in which flows are moved. Classification of flows in the network is carried out on the basis of ordering and continuity. It is shown that complex networks with different types of flows generate various network systems. Flow analogues of the basic concepts of the theory of complex networks are introduced and the main problems of this theory in terms of flow characteristics are formulated. Local and global flow characteristics of networks bring closer the theory of complex networks to the systems theory and systems analysis. Concept of flow core of network system is introduced and defined how it simplifies the process of its investigation. Concepts of kernel and flow core of multiplex are determined. Features of operation of multiplex type systems are analyzed.

💡 Research Summary

The paper re‑examines complex networks by distinguishing between static graph structures and dynamic “network systems” in which material, information, or energy flows travel. It defines a network system as a complex network endowed with moving flows and classifies those flows along two orthogonal axes: ordering (whether a flow follows a prescribed sequence) and continuity (whether the flow is uninterrupted). This yields four flow types—ordered‑continuous, ordered‑discontinuous, unordered‑continuous, and unordered‑discontinuous—each corresponding to different real‑world domains such as logistics, power transmission, data communication, or social interaction.

Building on this classification, the authors introduce a suite of flow‑based local and global metrics. Local measures include per‑node and per‑edge flow volume, flow concentration, and latency; global measures encompass total system flow, path diversity, efficiency, and balance. Traditional complex‑network indices (degree centrality, betweenness, clustering coefficient, average shortest path) are extended by weighting them with flow magnitudes, producing new descriptors such as “flow betweenness” and “flow clustering coefficient.” These metrics directly capture the functional load carried by the network, bridging complex‑network theory with classical systems analysis.

The centerpiece of the work is the concept of a “flow core.” A flow core is defined as the smallest set of nodes and edges that collectively carry a pre‑specified proportion (commonly 80 % or more) of the total system flow. Extraction proceeds by ranking elements by flow volume and accumulating them until the target proportion is reached. By focusing on the flow core, analysts can dramatically reduce model size while preserving the bulk of functional activity, facilitating more tractable simulation, optimization, and vulnerability assessment.

For multiplex networks—structures composed of several overlapping layers (e.g., transportation, communication, power)—the authors first compute layer‑specific flow cores and then integrate them to form a “multiplex flow core.” This composite core highlights inter‑layer regions where high flow loads coincide across layers, identifying critical interdependencies that are invisible to single‑layer analyses. The multiplex flow core thus becomes a powerful tool for evaluating the resilience of integrated infrastructures such as smart cities.

A comparative discussion shows that structural cores, derived solely from topological connectivity, may differ substantially from flow cores, which reflect actual usage patterns. Consequently, flow cores are more informative for operational concerns such as load balancing, bottleneck mitigation, and failure propagation control.

The paper concludes by outlining a research agenda: development of flow‑aware optimization algorithms, dynamic tracking of flow‑core evolution under varying demand, real‑time monitoring of multiplex flow cores, and leveraging flow cores for security hardening and disaster recovery planning. In sum, the work extends complex‑network theory into a flow‑centric systems framework, offering a unified language for analyzing, designing, and safeguarding modern interconnected infrastructures.