Monotone convergence of spreading processes on networks

TL;DR

This work establishes a rigorous, time-varying-parametric framework for Bass and SI spreading on networks by deriving master equations and proving monotone convergence of the macroscopic adoption/infection level to infinite-population limits. The authors show monotone increasing behavior of the network-level adoption with network size across complete, circular, and two-group complete networks, and provide explicit limiting ODEs: for complete networks the limit $f^{\rm compart.}$ satisfies $\frac{df}{dt}=(1-f)(p(t)+q(t)f)$, while for circles the limit reduces to a 1D lattice form, and for two-group complete networks to a pair of coupled equations governing $f_1$ and $f_2$. The top-down master-equation analysis simplifies convergence proofs and yields practical formulas and bounds, enabling rigorous assessment of optimal-control strategies with time-dependent network parameters.

Abstract

We analyze the Bass and SI models for the spreading of innovations and epidemics, respectively, on homogeneous complete networks, circular networks, and heterogeneous complete networks with two homogeneous groups. We allow the network parameters to be time dependent, which is a prerequisite for the analysis of optimal strategies on networks. Using a novel top-down analysis of the master equations, we present a simple proof for the monotone convergence of these models to their respective infinite-population limits. This leads to explicit expressions for the expected adoption or infection level in the Bass and SI models, respectively, on infinite homogeneous complete and circular networks, and on heterogeneous complete networks with two homogeneous groups with time-dependent parameters.

Figures (3)

• Figure 1: Monotone convergence of $f^{\rm complete}_{\rm Bass}$ (dashes) to $f^{\rm compart.}_{Bass}$ (solid). Here $\frac{q}{p} = 10$, $I^0= 0$, and $M=2,10,30,200$.
• Figure 2: Monotone convergence of $f^{\rm circle}_{\rm Bass}$ (dashes) to $f^{\rm 1D}_{\rm Bass}$ (solid). Here $\frac{q}{p} = 10$, $I^0= 0$, and $M=2,4,6,8$.
• Figure 3: Monotone convergence of $f^{\rm 2-groups}$ (dashes) to $f^{\rm 2-groups}_{\infty}$ (solid). Here $\frac{q_1}{p_1} =\frac{q_2}{p_2}=10$, $p_2=2p_1$, $I^0= 0$, and $M=2,8,20,80$.

Theorems & Definitions (40)

• Theorem 2.1: MOR-22
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3