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A complex adaptive systems perspective of innovation diffusion: an integrated theory and validated virtual laboratory

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Abstract

In this paper we integrate the rich yet fragmented insights from the extensive literature on the diffusion of innovation into an elegant, coherent model. Using complex adaptive systems theory as the overarching framework, we integrate prior literature around three constructs: agents, interactions, and an environment. The integrated model is presented in both natural language and as an agent-based simulation model. A series of validation experiments instill confidence that our agent-based model (and similar others) can be used as a virtual research laboratory. We provide theoretical and methodological directions for future research.

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Notes

  1. We also repeated each simulation 30 times and found similar results.

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Authors and Affiliations

  1. Management Information Systems, Sauder School of Business, The University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Ning Nan

  2. Management Information Systems, Price College of Business, University of Oklahoma, 307 West Brooks, Norman, OK, 73019, USA

    Robert Zmud & Emre Yetgin

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  1. Ning Nan
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Correspondence to Ning Nan.

Appendices

Appendix A: Behavioral rules

1.1 A.1 Behavioral rules affecting both the adopter role and influencer role

  • Awareness stock update rule:

    Incremental awareness = (neighbor awareness − awareness ∗ neighbor tie strength) ∗ (1 − innovation arduousness) ∗ adoption history multiplier

    Awareness = awareness ∗ awareness forgetting factor + incremental awareness ∗ other-learning rate

    An agent’s awareness stock level is increased if this agent’s contacts hold higher awareness stock levels. Contacts’ awareness stock levels are adjusted by the tie strength of their relationships with the agent. The amount of awareness stock level increase for the agent is adjusted by innovation arduousness, and the agent’s adoption history, learning-through-others rate, and awareness forgetting rate.

  • Capability stock update rule:

    Incremental capability = random number ∗ the capability difference between the agent and the signaling neighbors that have higher capabilities

    Capability = capability ∗ adoption-history-multiplier + incremental capability ∗ other-learning rate

    An agent’s capability stock level is increased if this agent’s contacts hold higher capability stock levels. Contacts’ capability stock levels are adjusted by the tie strength of their relationships with the agent. The amount of capability stock level increase for the agent is adjusted by the agent’s learning-through-others rate, self-learning rate, and capability forgetting rate.

  • Motivation stock update rule:

    Incremental motivation = incremental value gained from adoption + (neighbor motivation − motivation ∗ neighbor tie strength)

    Motivation = motivation ∗ motivation forgetting factor + incremental motivation ∗ other-learning rate

    An agent’s motivation stock level is increased when the agent gains value from the innovation (if the agent is already an adopter), or if this agent’s contacts hold higher motivation stock levels. Contacts’ motivation values are adjusted by the tie strength of their relationships with the agent. The amount of motivation stock level increase for the agent is adjusted by the agent’s learning-through-others rate and motivation forgetting rate.

1.2 A.2 Adopter role behavioral rules

  • Adoption rule (for agents who currently have not adopted an innovation):

    Adoption is set to one IF awareness ≥ awareness threshold AND capability ≥ capability threshold AND motivation ≥ motivation threshold.

    Incremental value gained from adoption = random number ∗ adoption history multiplier ∗ (1 + proportion of adopters among contacts ∗ externality multiplier)

    Value gained from adoption = value gained from adoption + incremental value gained from adoption

    When an agent’s awareness, capability, and motivation stock levels are simultaneously above their respective thresholds, the agent adopts the innovation. This agent’s value gained from adoption is incremented. The amount of value gain increment is determined by the value gained by prior adopters, the agent’s capability, the innovation’s arduousness, and, if network externality effects are present, the proportion of the agent’s contacts who have adopted the innovation.

  • Disadoption rule (for agents who have currently adopted an innovation):

    Adoption is set to zero IF capability < capability threshold OR motivation < motivation threshold.

    Incremental value gained from adoption = 0

    If an agent’s awareness stock level is above the awareness threshold but the agent’s capability or motivation stock levels drop below their respective thresholds, the agent disadopts the innovation, and no longer accrues value from the innovation.

1.3 A.3 Influencer role behavioral rules

  • Signaling rule:

    Value gained from signaling = motivation ∗ adoption history multiplier ∗ (1 + proportion of adopters among contacts ∗ externality multiplier) IF awareness ≥ awareness threshold

    Agent signals IF value gained from signaling ≥ signaling threshold

    When an agent’s awareness of the innovation is above the awareness threshold, this agent’s value gained from signaling is determined by the agent’s motivation stock level, whether the agent has adopted the innovation and the time since such adoption, and, if network effects are present, the proportion of the agent’s contacts who have adopted the innovation. If this agent’s value gained from signaling is above the signaling threshold, this agent will transmit a signal to all of the agent’s contacts.

  • Signal amplifying rule:

    Specialist is set to one IF capability > the average capability of the agents within the same innovativeness category + 1.2 standard deviations

    Incremental stock = incremental stock ∗ exemplar multiplier IF influencer is an exemplar

    If an influencer’s capability stock level is sufficiently large (relatively to all other agents within the same innovativeness category), the influencer becomes a specialist and the influencer’s signals are amplified. If an influencer is randomly identified as an exemplar, the influencer’s signals are amplified. If an influencer holds a high dominance value, the agent is more likely to serve as a central node within the population, and to possess relatively higher tie strengths with contacts—both of which serve to amplify the agent’s signal strength.

Appendix B: Network structure

  • Setting up focal links:

    Number of focal links = number of focal agents ∗ (number of focal agents − 1)/ 2 ∗ focal network density

    The number of focal links is calculated as a function of the focal network density. For example, if the focal population is 50 and network density is 0.5, the number of focal links is set at 612 by using the formula specified above. To create a link, an agent is randomly selected to be the first node. This first node then selectively builds a link with a second node by biasing toward more dominant agents. The link creation procedure is repeated 612 times in order to build the necessary number of links in the focal population. The existence of links between focal agents therefore accounts for focal network centrality with more dominant agents serving as central nodes, as well as network density specified by the model user.

  • Setting up external links:

    Number of other links = number of other agents ∗ (number of other agents − 1)/ 2 ∗ other network density

    The number of external (other) links is calculated according to the external network density. Then, this existence of links between external agents is specified the same way as that of focal agents.

  • Setting up inter-population links:

    Number of inter-links = proportion of inter-links ∗ minimum (number of focal agents, number of other agents)

    The number of inter-population links is calculated according to the proportion of inter-links that exist between the focal and external populations. Then, the existence of inter-population links is specified, accounting for the innovativeness of the focal agents with more innovative focal agents being more likely to have links with external agents. The innovativeness of focal agents is defined based on Rogers’ (2003) five adopter categories: innovators, early adopters, early majority, late majority, and laggards. We randomly assigned a focal agent into an adopter category so that the distribution of these five adopter categories follows the distribution found by Rogers (2003). The more innovative a focal agent is, the more likely it would build a tie with an agent in the external population.

  • Calculating the tie strength:

    The strength of a tie between two agents is calculated according to their spatial proximity; the closer two agents are, the stronger tie they have. Then, the tie strength is adjusted depending on the distribution of the spatial distances between the agent pairs with the same innovativeness category.

Appendix C: Model variables

3.1 C.1 Global variables

L::

The number of clock ticks (i.e., iterations) in a simulation.

N::

The adoption duration of an agent, in terms of the number of clock ticks. 0 means that the agent has never adopted the innovation.

Number of focal links::

The number of links in the focal population.

Number of inter-links::

The number of links between the focal and external populations.

Number of other links::

The number of links in the external population.

3.2 C.2 Exogenous variables

Awareness forgetting factor::

A multiplier that refers to the rate at which agents’ awareness decreases with each iteration, increasing continuously from 0 to 1. 1 means that an agent’s awareness does not decrease at all. 0 means that an agent’s awareness is completely diminished.

Capability forgetting factor::

A multiplier that refers to the rate at which agents’ capability decreases with each iteration, increasing continuously from 0 to 1. 1 means that an agent’s capability does not decrease at all. 0 means that an agent’s capability is completely diminished.

Exemplar multiplier::

A multiplier that is used to amplify the influence of exemplars on agents that favor exemplars, increasing continuously from 1 to 2. 1 means that exemplars’ influence is not amplified at all, whereas 2 means that exemplars’ influence is doubled.

Externality multiplier::

A multiplier that is used to amplify the value gained from adoption due to network externalities, increasing continuously from 1 without an upper bound. 1 means that the innovation does not have any network externality effects.

Focal network centrality::

The centrality of the focal network, increasing continuously from 0 to 1.

Focal network density::

The density of the focal network, increasing continuously from 0 to 1.

Initial other adopters::

The proportion of initial adopters in the external population, increasing continuously from 0 to 1. 0 means that there are no initial adopters in the external population. 1 means that every agent in the external population is initially an adopter.

Innovation arduousness::

The arduousness of the innovation, increasing continuously from 0 to 1.

Motivation forgetting factor::

A multiplier that refers to the rate at which agents’ motivation decreases with each iteration, increasing continuously from 0 to 1. 1 means that an agent’s motivation does not decrease at all. 0 means that an agent’s motivation is completely diminished.

Number of clusters::

The total number of clusters in the focal and external populations.

Number of focal agents::

The number of agents in the focal population.

Number of other agents::

The number of agents in the external population.

Other network centrality::

The centrality of the external network, increasing continuously from 0 to 1.

Other network density::

The density of the external network, increasing continuously from 0 to 1.

Proportion of inter-links::

The proportion of the inter-population links, increasing continuously from 0 to 1. 0 means that there are no inter-population links. 1 means every agent in the focal or external population, depending on whichever is fewer, has an inter-population link.

3.3 C.3 Endogenous variables

Adoption history multiplier::

A multiplier that is used to dampen the impacts of the innovation, depending on how long an agent has been an adopter. It is calculated as (N+1)/N (see Appendix C.1).

Adoption::

The binary adoption state of an agent. 0 means that the agent is a non-adopter during the current tick of the clock, whereas 1 means that the agent is an adopter during the current clock tick.

Attentiveness::

The tendency of an agent to pay attention to its contacts’ signals. It ranges continuously from 0.1 to 0.8, depending on an agent’s innovativeness category. Agents that are more innovative have higher attentiveness values.

Awareness threshold::

The minimum awareness value of an agent for it to consider adopting the innovation. Once an agent’s awareness stock exceeds the awareness threshold, awareness of the innovation is maintained throughout the remainder of a simulation run.

Awareness::

The awareness stock of an agent, increasing continuously from 0 to 150. 0 means the state of unaware. 150 is an upper bound we found through sensitivity analyses; awareness values greater than 150 did not produce noticeable impact on the diffusion processes and outcomes. We therefore use 150 to represent the state of being fully aware.

Capability threshold::

The minimum capability value of an agent for it to adopt or keep the innovation.

Capability::

The capability stock of an agent, increasing continuously from 0 to 200. 0 means the state of no capability. 200 is an upper bound we found through sensitivity analyses; capability values greater than 200 did not produce noticeable impact on the diffusion processes and outcomes. We therefore use 200 to represent the state of being fully capable.

Dominance::

The relative dominance of an agent. It is calculated randomly from a gamma distribution with alpha and lambda values of 1.

Exemplar::

An agent’s binary state of being an exemplar. 0 means that an agent is not an exemplar, whereas 1 means that the agent is an exemplar.

Favor exemplar::

The extent to which an agent favors exemplars. It is calculated randomly from a normal distribution with a standard deviation of 1, in which the mean is shifted depending on an agent’s innovativeness category. The more innovative an agent is, the less likely it favors exemplars.

Gain::

A multiplier that is used to adjust the impacts of the innovation. It is calculated randomly from a normal distribution, in which the mean is shifted depending on the innovation’s arduousness.

Incremental awareness::

The incremental change in an agent’s awareness stock over each iteration.

Incremental capability::

The incremental change in an agent’s capability stock over each iteration.

Incremental motivation::

The incremental change in an agent’s motivation stock over each iteration.

Incremental value gained from adoption::

The value an agent gains from the innovation over each iteration (see Appendix A.2 for its calculation).

Innovativeness::

The innovativeness category (Rogers 2003) of an agent. It is an integer between 1 and 5, where 1 is the most innovative category (i.e., innovator) and 5 is the least innovative category (i.e., laggard).

Motivation threshold::

The minimum awareness value of an agent for it to adopt or keep the innovation.

Motivation::

The motivation stock of an agent, increasing continuously from 0 to 300. 0 means the state of no motivation. 300 is an upper bound we found through sensitivity analyses; motivation values greater than 300 did not produce noticeable impact on the diffusion processes and outcomes. We therefore use 300 to represent the state of being fully motivated.

Neighbor awareness::

The average awareness stock of an agent’s contacts that are signaling about the innovation.

Neighbor motivation::

The average motivation stock of an agent’s contacts that are signaling about the innovation.

Neighbor tie strength::

The average strength of an agent’s relationships with its contacts that are signaling about the innovation.

Other-learning rate::

A multiplier that is used to adjust the awareness, capability, and motivation that an agent gains from its contacts. It increases continuously from 0.05 to 0.8, depending on an agent’s innovativeness category. Agents that are more innovative have higher other-learning rates.

Proportion of adopters among contacts::

The proportion of adopters among an agent’s contacts. 0 means that none of an agent’s contacts is an adopter, while 1 means that all contacts of an agent are adopters.

Self-learning rate::

A multiplier that is used to adjust the capability that an agent gains from its own experience with the innovation. It increases continuously from 1.0 to 1.3, depending on an agent’s innovativeness category. Agents that are more innovative have higher self-learning rates.

Signaling threshold::

The minimum value an agent needs to gain from signaling, in order for the agent to start or keep signaling about the innovation.

Specialist::

An agent’s binary state of being a specialist. 0 means that an agent is not a specialist, whereas 1 means that the agent is a specialist.

Tie strength multiplier::

The strength of the relationship between two agents (see Appendix B for its calculation).

Value gained from adoption::

The cumulative value an agent gains from the innovation (see incremental value gained from adoption).

Value gained from signaling::

The value an agent gains from signaling about the innovation (see Appendix A.3 for its calculation).

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Nan, N., Zmud, R. & Yetgin, E. A complex adaptive systems perspective of innovation diffusion: an integrated theory and validated virtual laboratory. Comput Math Organ Theory 20, 52–88 (2014). https://doi.org/10.1007/s10588-013-9159-9

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