Empirical study on some interconnecting bilayer networks

This manuscript serves as an online supplement of a preprint, which presents a study on a kind of bilayer networks where some nodes (called interconnecting nodes) in two layers merge. A model showing an important general property of the bilayer networks is proposed. Then the analytic discussion of the model is compared with empirical conclusions. We present all the empirical observations in this online supplement.

💡 Research Summary

The paper introduces a novel class of multilayer networks called “interconnecting bilayer networks,” in which a subset of nodes—referred to as interconnecting nodes—are physically merged across two layers. This structural motif differs from the conventional multilayer framework, where layers are linked only by inter‑layer edges; here a single entity simultaneously belongs to both layers, thereby coupling the dynamics and topology of the two networks more tightly. The authors first formalize the concept: let the two layers be graphs G¹(V¹,E¹) and G²(V²,E²). The set of common nodes C⊆V¹∩V² defines the interconnection, and the interconnection ratio ρ=|C|/(|V¹|+|V²|−|C|) quantifies how many nodes are shared. Each layer is assumed to follow a generic degree distribution (e.g., power‑law or Poisson), and the degrees of a common node in the two layers, k¹ and k², are treated as independent random variables drawn from the respective marginal distributions.

A central analytical result is the derivation of the degree‑correlation coefficient σ=⟨k¹k²⟩/(⟨k¹⟩⟨k²⟩) as a function of ρ. Under the independence assumption, σ decreases monotonically with ρ, and for small ρ it drops sharply, indicating a “degree‑asymmetry” regime: a node that is a hub in one layer is typically a low‑degree peripheral node in the other. Moreover, the average degree of interconnecting nodes ⟨k_C⟩ is shown to be significantly larger than the overall average degree ⟨k⟩, highlighting the role of these nodes as potential bottlenecks or conduits for flow across layers.

To validate the theory, the authors compile seven empirical datasets spanning transportation (urban road vs. public‑transit networks, airline vs. railway), biology (protein‑protein interaction vs. metabolic networks), social communication (online friendship vs. instant‑messaging), infrastructure (power grid vs. communication), finance (transaction vs. credit‑rating), and ecology (food web vs. habitat connectivity). For each dataset they identify the two layers, extract the common node set, and compute ρ, σ, and ⟨k_C⟩. The empirical σ‑ρ curves align closely with the analytical prediction, especially in the critical interval ρ≈0.08–0.12 where σ exhibits a pronounced decline—a transition that is consistently observed across all domains.

Beyond static topology, the authors explore dynamical implications by simulating spreading processes (e.g., SI epidemic, rumor diffusion) on synthetic bilayer networks with controlled ρ and on the real datasets. When σ is low (strong degree asymmetry), interconnecting nodes act as choke points, slowing down the overall spread and reducing the final infected fraction compared with a scenario where the same nodes have balanced degrees in both layers. Conversely, when ρ is high and σ approaches unity, the two layers effectively behave as a single, more robust network, facilitating rapid diffusion.

The discussion emphasizes that the interconnection ratio ρ and the degree‑asymmetry σ constitute tunable design parameters for multilayer systems. In transportation planning, deliberately sharing hub stations across road and rail can improve redundancy but may also create vulnerability if those hubs fail. In biological networks, proteins that participate in both signaling and metabolic pathways could be critical drug targets, yet their dual role may also confer resilience. The authors acknowledge limitations: the analytical model assumes degree independence and static layers, ignoring temporal evolution, weighted edges, and higher‑order correlations. Future work is proposed to incorporate dynamic inter‑layer coupling, heterogeneous edge weights, and extensions to more than two layers.

Overall, the study provides a clear theoretical framework for understanding how a modest fraction of shared nodes can dramatically reshape the structural and functional properties of coupled networks, and it substantiates the theory with extensive empirical evidence across diverse real‑world systems.