Editing Diffusion of innovations (section)

== Mathematical treatment ==
{{Main|Logistic function}}

The diffusion of an innovation typically follows an S-shaped curve which often resembles a [[logistic function]]. Roger's diffusion model concludes that the popularity of a new product will grow with time to a saturation level and then decline, but it cannot predict how much time it will take and what the saturation level will be. Bass (1969)<ref>{{Cite journal |last=Bass |first=Frank M. |date=1969 |title=A New Product Growth for Model Consumer Durables |url=http://dx.doi.org/10.1287/mnsc.15.5.215 |journal=Management Science |volume=15 |issue=5 |pages=215–227 |doi=10.1287/mnsc.15.5.215 |issn=0025-1909|url-access=subscription }}</ref> and many other researchers proposed modeling the diffusion based on parametric formulas to fill this gap and to provide a means for a quantitative forecast of adoption timing and levels. The [[Bass diffusion model|Bass model]] focuses on the first two (Introduction and Growth). Some of the Bass-Model extensions present mathematical models for the last two (Maturity and Decline). MS-Excel or other tools can be used to solve the Bass model equations, and other diffusion models equations, numerically. [[Mathematical programming]] models such as the [[S-d model|S-D model]] apply the diffusion of innovations theory to real data problems.<ref>{{cite journal | doi = 10.1287/opre.1110.0963 | title=Rating Customers According to Their Promptness to Adopt New Products | journal=Operations Research | date=2011 | volume=59 | issue=5 | pages=1171–1183 | first=Dorit S. | last=Hochbaum| s2cid=17397292 |author-link=Dorit S. Hochbaum}}</ref> In addition to that, [[agent-based models]] follow a more intuitive process by designing individual-level rules to model the diffusion of ideas and innovations.<ref name="Nasrinpour et al. 2016">{{Cite arXiv|last1=Nasrinpour|first1=Hamid Reza|last2=Friesen|first2=Marcia R.|last3=McLeod|first3=Robert D.|date=2016-11-22|title=An Agent-Based Model of Message Propagation in the Facebook Electronic Social Network|eprint=1611.07454|class=cs.SI}}</ref>

=== Complex systems models ===

[[Complex network]] models can also be used to investigate the spread of innovations among individuals connected to each other by a network of peer-to-peer influences, such as in a physical community or neighborhood.<ref name="citylab">[http://www.citylab.com/tech/2013/04/what-math-can-tell-us-about-how-technology-spreads-through-cities/5249/ What Math Can Tell Us About Technology's Spread Through Cities].</ref>

Such models represent a system of individuals as ''nodes'' in a network (or [[Graph (discrete mathematics)|graph]]). The interactions that link these individuals are represented by the edges of the network and can be based on the probability or strength of social connections. In the dynamics of such models, each node is assigned a current state, indicating whether or not the individual has adopted the innovation, and model equations describe the evolution of these states over time.<ref name="siam">[http://connect.siam.org/how-does-innovation-take-hold-in-a-community-math-modeling-can-provide-clues/ How does innovation take hold in a community? Math modeling can provide clues]</ref>

In threshold models,<ref>{{Cite journal | last1 = Watts | first1 = D. J. | title = A simple model of global cascades on random networks | doi = 10.1073/pnas.082090499 | journal = Proceedings of the National Academy of Sciences | volume = 99 | issue = 9 | pages = 5766–5771| year = 2002 | pmid = 16578874| pmc = 122850| bibcode = 2002PNAS...99.5766W| doi-access = free }}</ref> the uptake of technologies is determined by the balance of two factors: the (perceived) usefulness (sometimes called utility) of the innovation to the individual as well as barriers to adoption, such as cost. The multiple parameters that influence decisions to adopt, both individual and socially motivated, can be represented by such models as a series of nodes and connections that represent real relationships. Borrowing from social network analysis, each node is an innovator, an adopter, or a potential adopter. Potential adopters have a threshold, which is a fraction of his neighbors who adopt the innovation that must be reached before he will adopt. Over time, each potential adopter views his neighbors and decides whether he should adopt based on the technologies they are using. When the effect of each individual node is analyzed along with its influence over the entire network, the expected level of adoption was seen to depend on the number of initial adopters and the network's structure and properties. Two factors emerge as important to successful spread of the innovation: the number of connections of nodes with their neighbors and the presence of a high degree of common connections in the network (quantified by the [[clustering coefficient]]). These models are particularly good at showing the impact of opinion leaders relative to others.<ref>{{cite book|last1=Easley|first1=D|last2=Kleinberg|first2=J|title=Networks, Crowds and Markets: Reasoning about a Highly Connected World|url=https://archive.org/details/networkscrowdsma00easl_224|url-access=limited|date=2010|pages=[https://archive.org/details/networkscrowdsma00easl_224/page/n514 497]–531|publisher=Cambridge University Press|location=Cambridge|isbn=9780521195331}}</ref> [[Computer models]] are often used to investigate this balance between the social aspects of diffusion and perceived intrinsic benefit to the individuals.<ref name="mccullen">{{cite journal | doi = 10.1137/120885371 | title=Multiparameter Models of Innovation Diffusion on Complex Networks | journal=SIAM Journal on Applied Dynamical Systems | date=2013 | volume=12 | issue=1 | pages=515–532 | first=N. J. | last=McCullen| arxiv=1207.4933 }}</ref>