Boundary Effects in the Diffusion of New Products on Cartesian Networks

Abstract

We analyze the effect of boundaries in the discrete Bass model on D-dimensional Cartesian networks. In 2D, this model describes the diffusion of new products that spread primarily by spatial peer effects, such as residential photovoltaic solar systems. We show analytically that nodes (residential units) that are located near the boundary are less likely to adopt than centrally-located ones. This boundary effect is local, and decays exponentially with the distance from the boundary. At the aggregate level, boundary effects reduce the overall adoption level. The magnitude of this reduction scales as~$\frac{1}{M^{1/D}}$, where~$M$ is the number of nodes. Our analysis is supported by empirical evidence on the effect of boundaries on the adoption of solar.

Figures (10)

• Figure 1: (A) The adoption probability of central ($f^{\mathbb{Z}^+}_{\rm central}$, blue solid) and boundary ($f^{\mathbb{Z}^+}_{\rm bdry}$, red dashes) nodes on the semi-infinite line, see \ref{['eq:local_effect-semi-infinite-line']}, as a function of $\sqrt{pq}t$. Here $\frac{q}{p} = 10$. (B) The ratio $R:={f^{\mathbb{Z}^+}_{\rm bdry}}/{f^{\mathbb{Z}^+}_{\rm central}}$ as a function of $\sqrt[4]{pq^{3}}t$ for $\frac{q}{p}=10$. (C) and (D): Same as (A) and (B) for $\frac{q}{p} = 10^3$.
• Figure 2: $R_{\min}(\widetilde{q})$, ploted on a semi-log scale.
• Figure 3: (A) $j^j \left(f^{\rm 1D}-f^{\mathbb{Z}^+}_{j}\right)$ as a function of $j$ ( ); $y$-axis is in log scale. Solid line is the fitted curve $e^{-\alpha}\beta^j$ with $\alpha \approx 6.0$ and $\beta \approx 3.7$. Dashed line is the upper bound $e^{-\left(p+\frac{q}{2}\right)t}\left(e\frac{q}{2}t\right)^j$ of Theorem \ref{['thm:1D_decay']}. Here, $p=0.01$, $q=0.1$, and $t = \frac{4}{q}$. (B) Fitted value of $\alpha$ as a function of $qt$ ( ). Dashed line is $(p+\frac{q}{2})t$. (C) Fitted values of $\beta$ as a function of $qt$ ( ). These values lie on the solid line $0.63qt\approx e\frac{q}{4}t$. Dashed line is the theoretical bound $e\frac{q}{2}t$.
• Figure 4: (A) The adoption probability of central ($f^{\mathbb{Z}^+ \times \mathbb{Z}}_{\rm central}$, blue solid) and boundary ($f^{\mathbb{Z}^+ \times \mathbb{Z}}_{\rm bdry}$, orange dashes) nodes on $\mathbb{Z}^+ \times \mathbb{Z}$, as a function of $\sqrt[3]{pq^2}t$. Here $\frac{q}{p} = 10$. (B) The ratio $R:=f^{\mathbb{Z}^+ \times \mathbb{Z}}_{\rm bdry}/f^{\mathbb{Z}^+ \times \mathbb{Z}}_{\rm central}$ as a function of $\sqrt[4]{pq^{3}}t$. (C) and (D): Same as (A) and (B) with $\frac{q}{p} = 10^6$. The curves are obtained from simulations of the discrete Bass model \ref{['eq:Bass-model-2D+']}.
• Figure 5: $f^{\rm D}-f^{B_D}$ as a function of $M = M_1^D$, on a log-log scale ( ). Here $\frac{q}{p}=10$ and $t=\frac{20}{q}$. Solid line is the fitted curve $\log(f^{\rm D}-f^{[1, \dots, M_1]^D}) \sim \alpha_D + \beta_D \log{M}$. (A) $D = 1$, $\alpha_1 \sim -1.02$, $\beta_1 \sim -1.00$. (B) $D = 2$, $\alpha_2 \sim -1.54$, $\beta_2 \sim -0.49$. (C) $D=3$, $\alpha_3 \sim -1.32$, $\beta_3 \sim -0.34$. .

Theorems & Definitions (31)

• Lemma 1: OR-10Bass-boundary-18
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 1
• Corollary 1
• Lemma 6
• Lemma 7
• Lemma 8