Impacts of bridging nodes on the epidemic activation mechanisms

TL;DR

This work investigates how a systematic addition of degree-2 bridging nodes attached to network hubs reshapes epidemic activation in power-law networks. Using SIS and SIRS dynamics on cores with a hub-enabled bridging layer, the authors show that linear hub attachment leaves activation mechanisms unchanged, while superlinear attachment induces strong hub-hub feedback that can localize activity and modify thresholds, with notable changes for $\gamma>3$ in SIRS where localization can dominate and thresholds vanish asymptotically. The analysis combines stochastic simulations with mean-field approaches based on the Hashimoto (non-backtracking) matrix and recurrent dynamical message passing, linking threshold behavior to spectral properties and to the inverse participation ratio of the hub activity. The results demonstrate a robust, regime-dependent role of bridging nodes: they leave maximum-$k$-core activation largely intact but can fundamentally alter SIRS dynamics by promoting localized, hub-pair activation via indirect feedback, with potential implications for other systems featuring refractory or bridging motifs.

Abstract

Bridging nodes, which connect critical components of a network, play an important role in maintaining structural integrity and facilitating communication within the network, representing indirect yet relevant connections. Epidemic triggering mechanisms in networks often involve long-range mutual activation of hubs, mediated by paths composed of low-degree nodes. While low-degree nodes are abundant in networks, their role in bridging central nodes in epidemic activation mechanisms has not been thoroughly analyzed. Starting with a backbone network with a power-law degree distribution, we investigate the role of adding degree-2 bridging nodes that are preferentially attached to hubs. Our findings reveal that bridging nodes can mediate an indirect feedback interaction between hubs that modifies the epidemic localization and activation mechanisms of the epidemic processes with recurrent infections. In particular, the collective activation observed in the presence of waning immunity, which produces a finite epidemic threshold in power-law networks with degree exponent $γ>3$, is altered to a localized activation with a vanishing threshold. Our results are supported by the non-backtracking matrix properties that emerge in the recurrent dynamical message-passing theory.

Figures (8)

• Figure 1: Schematic representation of the network model with bridges and drawbridges. (a) Bridging nodes of degree 2 (dark blue) are attached to the hubs of the core network (light blue). (b) Bridging nodes are split into two unconnected nodes of degree 1, preserving the number of edges in the network.
• Figure 2: (a) Degree distribution and (b) degree correlation analysis for a core network doped with a fraction $f=0.5$ of bridging nodes. The original core uses a UCM model with $N_\text{c}=10^6$ nodes, minimum degree $k_\text{min}=3$, degree exponent $\gamma=3.5$, and a rigid upper cutoff $k_\text{max} \sim N^{1/\gamma}$. Random attachment of bridging nodes ($\nu=0$) is compared with linear ($\nu=1$) and super-linear ($\nu=2$) preferential attachment rules. Dashed lines represent power-law decays with exponents $\gamma'=3.5$ and $\gamma'=2.25$, corresponding to the predictions for $\nu=1$ and $\nu=2$, respectively. Degree distributions were shifted to improve visibility.
• Figure 3: (a,b) Epidemic threshold and (c,d) IPR as functions of the core size for different attachment rules in the SIS dynamics. Random ($\nu=0$), linear ($\nu=1$), and super-linear ($\nu=2$) attachments are shown. The fraction of bridging nodes is $f=0.25$, and the core is a UCM network with (a,c) $\gamma=2.3$ or (b,d) $\gamma=2.8$, for $k_\text{min}=3$ and $k_\text{c}=2\sqrt{N_\text{c}}$. The line in (b) represents the decay determined by the activation of the largest hub of degree $\xi_\text{max}$, while in (c) it corresponds to the scaling $Y_4\sim N_\text{c}^{-(3-\gamma)/2}$, compatible with localization on the maximum $k$-core.
• Figure 4: Epidemic threshold (top) and IPR of the NAV (bottom) as functions of the core size $N_\text{c}$ for the SIS dynamics, considering different fractions $f$ of added bridging nodes. A super-linear attachment with $\nu=2$ is used. Left panels: the core is a UCM network with $\gamma=2.3$, $k_\text{min}=3$, and $k_\text{max}=2\sqrt{N_\text{c}}$. Right panels: the core is a UCM network with $\gamma=3.5$, $k_\text{min}=3$, and $k_\text{max}\sim N_\text{c}^{1/\gamma}$. Lines have the same meaning as in Fig. \ref{['fig:compara_nus']}.
• Figure 5: Epidemic threshold (top) and IPR of the NAV (bottom) as functions of the core size $N_\text{c}$ for SIRS dynamics with $\alpha=0.1$ and a super-linear attachment rule with $\nu=2$. The core is a UCM network with $k_\text{min}=3$ and degree exponent $\gamma=2.8$ with $k_\text{max}=2\sqrt{N_\text{c}}$ (left) or $\gamma=3.5$ with $k_\text{max}\sim N_\text{c}^{1/\gamma}$ (right). Dashed lines represent the scaling exponents obtained from a power-law regression of the IPR.