Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure

Fibich, Gadi; Gibori, Ro'i

doi:10.1287/opre.1100.0818

Cited by 40 publications

(70 citation statements)

References 35 publications

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“…For such products, it is reasonable to approximate the social network by a two-dimensional grid, in which each node represents a residential unit. Previous studies of the discrete Bass model on two-dimensional networks avoided boundary effects by imposing periodic boundary conditions [6,7,8]. In that case, all nodes are interior and are influenced from all sides by the adjacent nodes.…”

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confidence: 99%

Boundary Effects in the Discrete Bass Model

Fibich¹,

Levin²,

Yakir³

2019

SIAM J. Appl. Math.

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To study the effect of boundaries on diffusion of new products, we introduce two novel analytic tools: The indifference principle, which enables us to explicitly compute the aggregate diffusion on various networks, and the dominance principle, which enables us to rank the diffusion on different networks. Using these principles, we prove our main result that on a finite line, one-sided diffusion (i.e., when each consumer can only be influenced by her left neighbor) is strictly slower than two-sided diffusion (i.e., when each consumer can be influenced by her left and right neighbor). This is different from the periodic case of diffusion on a circle, where one-sided and two-sided diffusion are identical. We observe numerically similar results in higher dimensions.1 If the graph is undirected, then qi,j = qj,i. 2 qi,j = 0 if there is no edge from i to j. 3 See, e.g., (2.5b) and (2.6b).

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confidence: 99%

Boundary Effects in the Discrete Bass Model

Fibich¹,

Levin²,

Yakir³

2019

SIAM J. Appl. Math.

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“…Empirically, the Bass model captures the S‐curve for cumulative adoption of various products. Extensive empirical application of the model found typical values for the parameters to be p = 0.03/year and q = 0.38/year, with p < 0.01/year and q typically in the range 0.3–0.5/year .…”

Section: Modelling System Behaviormentioning

confidence: 99%

“…Empirically, the Bass model captures the S-curve for cumulative adoption of various products. Extensive empirical application of the model found typical values for the parameters to be p 5 0.03/year and q 5 0.38/year, with p < 0.01/year and q typically in the range 0.3-0.5/year [48]. The model has been widely used for forecasting the market penetration of durable goods, and estimating the rate of diffusion attributed to different theoretical aspects of external influence p and internal influence q [24].…”

Section: Dynamics Of Innovationmentioning

confidence: 99%

Modeling sustainability transitions on complex networks

Tran¹

2014

Complexity

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There has been renewed interest in sociotechnical systems in the context of transitioning to a more sustainable society. While gains have been made in the qualitative understanding of sustainable transitions and sociotechnical systems, these approaches have not been well-operationalized. Given the importance of meeting future energy and environmental policy targets, there is need to develop predictive techniques and more robust methods to quantify and analyze sociotechnical systems undergoing rapid change and uncertainty due to sustainability pressures. Sustainability transitions depend on largescale diffusion of technological and behavioral innovations on physical and virtual networked systems. Transitions can therefore be viewed as a subclass of diffusion phenomenon and subject to a range of mathematical and computational methods. We review, categorize, and critically assess methodological and theoretical approaches that integrate different aspects of sustainability, innovation, and complexity. We argue that these approaches should be adapted to improve our understanding of the behavior and dynamics of a broad range of sociotechnical systems to meet sustainability objectives. We therefore also make the conceptual link between complexity and sustainability as complimentary fields of research to inform policy and decision making to achieve more sustainable outcomes.

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“…Indeed, symmetry holds because changing the indexes of the bidders does not affect the revenue. Therefore, assuming differentiability, the averaging principle for functions (SI Text) yields [12] where F = F + e k P k i=1 H i . Substituting F in Eq.…”

Section: Applied Mathematicsmentioning

confidence: 99%

Averaging principle for second-order approximation of heterogeneous models with homogeneous models

Fibich

Gavious

Solan

2012

Proc. Natl. Acad. Sci. U.S.A.

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Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced by its average value. In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(e 2 ) equivalent to the outcome of the corresponding homogeneous model, where e is the level of heterogeneity. We then use this averaging principle to obtain new results in queuing theory, game theory (auctions), and social networks (marketing).homogenization | perturbation methods M athematical modeling is a powerful tool in scientific research.Typically, the mathematical model is merely an approximation of the actual problem. Therefore, when choosing the model to work with, one has to strike a balance between complex models that are more realistic and simpler models that are more amenable to analysis and simulations. This dilemma arises, for example, when the model contains a heterogeneous quantity. In such cases, a huge simplification is usually achieved by replacing the heterogeneous quantity with its average value. The natural question that arises is whether this approximation is "legitimate," i.e., whether the error that is introduced by this approximation is sufficiently small.Let us illustrate this with the following example, which is discussed in detail below. Consider a queue with k heterogeneous servers, whose expected service times are μ 1 , . . . , μ k . We want to calculate analytically the expected number of customers in the system, which we denote* by F(μ 1 , . . . , μ k ). Although an explicit expression for F(μ 1 , . . . , μ k ) is not available, there is a well-known explicit expression in the case of k homogeneous servers, which we denote by F homog:for the expected number of customers in the system iswhere μ is the average of {μ 1 , . . . , μ k }. More generally, let F(μ 1 , . . . , μ k ) denote the "outcome" of a heterogeneous model, letdenote the level of heterogeneity of {μ 1 , . . . , μ k }, and let F homog. (μ) denote the outcome of the corresponding homogeneous model. If the function F(μ 1 , . . . , μ k ) is differentiable, then it immediately follows thatTherefore, for a 10% heterogeneity level, the error of approximating F(μ 1 , . . . , μ k ) with F homog: ðμÞ is, roughly speaking, on the order of 10%. In many studies in different fields, however, researchers have noted that the error of this approximation is considerably smaller than e. Moreover, this observation seems to hold even when the level of heterogeneity is not small. In this study we show that these observations follow from a general principle, which we call the averaging principle. Specifically, we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(e 2 ) asymptotically equivalent to the outcome of the corresponding homogeneous model; i.e.,Thus, if the function F is also symmetric, the error of the approximation in Eq. 1 for a 1...

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Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure

Cited by 40 publications

References 35 publications

Boundary Effects in the Discrete Bass Model

Boundary Effects in the Discrete Bass Model

Modeling sustainability transitions on complex networks

Averaging principle for second-order approximation of heterogeneous models with homogeneous models

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