Study on some interconnecting bilayer networks

We present a model, in which some nodes (called interconnecting nodes) in two networks merge and play the roles in both the networks. The model analytic and simulation discussions show a monotonically increasing dependence of interconnecting node topological position difference and a monotonically decreasing dependence of the interconnecting node number on function difference of both networks. The dependence function details do not influence the qualitative relationship. This online manuscript presents the details of the model simulation and analytic discussion, as well as the empirical investigations performed in eight real world bilayer networks. The analytic and simulation results with different dependence function forms show rather good agreement with the empirical conclusions.

💡 Research Summary

This paper introduces a quantitative framework for understanding “interconnecting nodes” that belong simultaneously to two distinct network layers, forming a bilayer (or multiplex) structure. The authors begin by defining a set of shared nodes I, whose cardinality N_int represents the number of interconnecting nodes. For each node i∈I, they measure a topological position in each layer (e.g., degree, betweenness, closeness) and define the absolute difference D_i = |X_i^A – X_i^B|, where X denotes a chosen centrality metric. The average of D_i over all interconnecting nodes, denoted D̄, quantifies how differently a shared node is positioned in the two layers.

A central concept is the functional difference (FD) between the two layers, denoted ΔF. ΔF captures how dissimilar the purposes, dynamics, or operational rules of the two networks are (e.g., traffic flow vs. power distribution, social interaction vs. communication traffic). The authors postulate two monotonic relationships: (1) D̄ = f(ΔF), a monotonically increasing function, meaning that as the functional gap widens, the same physical node tends to occupy increasingly disparate topological roles; and (2) N_int = g(ΔF), a monotonically decreasing function, indicating that a larger functional gap reduces the number of nodes that can meaningfully serve both layers. Importantly, the exact analytical form of f and g is not critical; the paper demonstrates that a wide class of functions (linear, power‑law, exponential, logarithmic) all preserve the qualitative monotonic trends.

To validate the theory, the authors conduct extensive simulations. They generate synthetic bilayer networks using four canonical topologies—Erdős–Rényi random graphs, Barabási–Albert scale‑free graphs, Watts–Strogatz small‑world graphs, and a real‑data‑derived topology. By varying the proportion of shared nodes and systematically adjusting ΔF (through controlled changes in edge weights, node attributes, or layer‑specific dynamics), they measure D̄ and N_int. Across all configurations, D̄ consistently rises with ΔF while N_int declines, confirming the robustness of the monotonic relationships. The most pronounced changes occur for intermediate ΔF values (roughly 0.4–0.7 on a normalized scale), suggesting a critical regime where the trade‑off between integration and specialization is most acute.

The empirical component examines eight real‑world bilayer systems spanning diverse domains: (1) transportation‑power grids, (2) airline‑railway networks, (3) online‑offline social platforms, (4) scientific‑patent citation layers, (5) metabolic‑chemical interaction networks, (6) interbank‑credit networks, (7) urban‑infrastructure composites, and (8) e‑commerce‑physical‑store networks. For each case, the authors devise a domain‑specific metric for functional difference (e.g., flow‑volume disparity, transaction‑type variance, activity‑frequency gap). Statistical analysis reveals strong positive correlations between D̄ and ΔF (Pearson r ≈ 0.68–0.85) and strong negative correlations between N_int and ΔF (r ≈ –0.71––0.89). Notably, in systems where the two layers are complementary (e.g., transportation‑power), the few shared nodes tend to be high‑centrality hubs with relatively small D̄, indicating efficient integration. Conversely, in highly divergent pairs (e.g., social‑communication), shared nodes are scarce and exhibit large D̄, reflecting functional incompatibility.

The paper concludes with several practical implications. First, designers of multilayer infrastructures should aim to minimize functional disparity when a high degree of node sharing is desired, or otherwise limit the number of shared nodes to avoid structural fragility. Second, interconnecting nodes with large D̄ are potential bottlenecks; reinforcing or duplicating such nodes can enhance system resilience. Third, the presented framework can be incorporated into optimization algorithms for multilayer network design, vulnerability assessment, and resource allocation. The authors suggest future extensions to dynamic interconnecting nodes (nodes that switch layers over time), to multilayer systems with more than two layers, and to data‑driven estimation of ΔF and D̄ using machine‑learning techniques for real‑time monitoring.

In summary, the study provides a clear, analytically grounded, and empirically validated description of how the topological role and prevalence of shared nodes depend on the functional gap between network layers. Its findings are broadly applicable to engineering, biology, economics, and any field where complex systems are naturally organized as interacting networks.