Università di Padova Dipartimento di Scienze Statistiche


Università di Padova

Innovation Diffusion Processes: Differential Models,
Agent-Based Frameworks and Forecasting Methods
Progetto di Eccellenza, 2007-2008
Fondazione Cassa di Risparmio di Padova e Rovigo


Innovation Diffusion Processes

Prof. Renato Guseo




Populations and Systems  UpGreen.gif

When approaching statistics we have all met the notion of “population” with reference to some of its relevant properties and global relationships such as means, percentages, probabilities and other functional relationships considered within given distributive contexts -simple or multiple-. One of the main features of the statistical meaning of population is its characterization as a “set” to which a vector of observable variables is associated. Many other elements are latent or residual. However, we observe that individual data, that is the level of relevant variables, may be partly due to a system of interpersonal relationships that connect between them the population components. Individuals’ change may be strictly due to the concrete possibility to communicate, through various languages, information that is relevant for many others. The specialized diffusion of human languages and the dual characterization of populations’ genes have been widely studied and confirmed (see for instance Cavalli-Sforza, 1996). The existence of language-based networks is an important feature of social systems and has much to do with collective learning. An interesting insight on this topic has been offered by the physicist Marchetti (1980): no abstract formulations but simple ideas to be read without prejudices. The absorption of a new idea, a discovery, a new technology by social systems, that cannot be considered as simple statistical populations, is based on the concept of innovation. Innovation implies a change in terms of production and consumption systems: while invention and discovery represent the potential change, innovation is the realization of such a change.

The diffusion of an innovation –technological or cultural- in a social system is characterized by specific dynamics (see Rogers, 2003) that depend on the structure of the system itself, on its internal rules and on external actions that introduce constraints and evolutionary modifications. With the term social system we intend a complex structure of socio-economic, biological or physical nature, whose typical feature is the ability of autoregulation based on language and information exchange. The physical level deals with entities that reflect different levels of matter aggregation. The languages that ensure connection among components are forces –gravity, nuclear, magnetic, etc.-. The biological level is based on the previous one and develops the languages of intracellular and intercellular communication and the more complex mechanisms of differentiation, reproduction and selection. The social level allows, through communication in strict sense, the definition of relationships and selection mechanisms giving rise to both competitive and cooperative dynamics. A short work by Couzin (2007) on “collective minds”, following a more technical one, Couzin (2005), may provide useful insights on the behaviour of individual agents within groups with no central control. Large-scale access to distributed information and local interaction with neighbours aid the emergence of a global adaptive behaviour, normally driven by few specialized agents. The “swarm rules” (bees, birds, fishes, locusts, ants, etc.) appear quite simple and typical of different creatures: information left by “tracks” and regulation of “local density” (or local pressure) may be considered the basis of such rules. Tracks are left by “innovators”, while information diffusion is driven by local density in time and space of “imitators”.




 Complex Systems and Aggregate Differential approaches  UpGreen.gif

The Complex Systems approach has a quite long history and is essentially based on the definition of local “transition rules”, whose evolution, mapped through computer simulations, detects the emergence of a global behaviour. See for instance Wolfram (1983) and Boccara (2004). The Complex Systems approach has been generally considered opposite to systems’ dynamics represented through differential equations. There are many scientific fields that have employed both these approaches in a separate way: among them, biology, epidemiology, physics, chemistry, systems theory, applied mathematics, economics, quantitative marketing, sociology, statistics,  etc.




Innovation Diffusion, Quantitative Marketing and Energy Sources   UpGreen.gif

USA quantitative marketing has given a notable contribution on aggregate differential models, since the early ‘70s. In innovation diffusion modelling, the Bass model (1969), BM, represents an essential reference. This is based on an extension of the Verhulst’s logistic equation and belongs to the more general family of Riccati equations. The Bass model separates for the first time in a formal way the sub-populations of “innovators” (leaders) and of “imitators” (followers): however this is a latent distinction, since the data observed just refer to the adoption of a susceptible agent without any other specification. A very important extension of the BM is due to Bass et al. (1994) with the Generalized Bass Model (GBM), which introduces the effect of a general intervention function, able to take into account the possible effect of exogenous variables on the diffusion process. In both cases the market potential (or carrying capacity) is considered constant over the entire innovation life cycle. There are thousands of successful applications of the Bass model: among these we recall 35 mm projectors, answering machines, color TV, mainframes, PC, drams, VCR, cell phones, etc. A particular use of the Bass model has been made in the energy context, crude oil in particular, see Guseo and Dalla Valle (2005); Guseo et al. (2007). In these works, estimated carrying capacities are considered as measures of specific URR (Ultimate Recoverable Resource, i.e. the total amount of resource recoverable along the whole production cycle) at regional and global level. Estimating the URRs has made possible to obtain an indirect estimate of reserves, avoiding the risk of reserves’ overestimation, often due to financial speculations. Production peaks and depletion times (based on suitable quantiles, such as t(0.90), are the obtained main results. Further work is in progress and applies BM and GBM to Natural Gas (methane) life cycle and to evolutionary dynamics of alterative energies, like photovoltaic, wind power, biomasses, biofuels, etc. See, for instance, Dalla Valle and Furlan (2011), Guidolin and Mortarino (2010), and Guseo (2011).



Complex Systems and Aggregate Differential Approaches: Dualism   (trailer)   UpGreen.gif

A research challenge recently faced at the University of Padova is the study of a plausible theoretical linkage between Complex Systems and Aggregate Differential models. Guseo and Guidolin (2008) confirm for the first time the existing dualism between a Cellular Automata model and its aggregate differential representation, obtained through a “mean field approximation”. Guidolin’s PhD dissertation (2008) and a  paper by Guseo and Guidolin (2009a) propose a co-evolutionary model based on two nested Agent-Based models, aimed at describing the evolution of latent  information network associated to a given innovation. This information network is the necessary condition for the real adoption process (nested model). Through a mean-field approximation a Riccati equation with closed-form solution has been obtained. This result allows to represent a dynamic version of the market potential and to account in a separate way for exogenous interventions (as in the GBM). Interesting applications of this model have been produced in the pharmaceutical sector (weekly sales data provided by IMS Health). Further progress may be found in a) the representation and interpretation of the slowdown effect (chasm, dip or saddle are similar effects) separating two waves of sales in product diffusion, in particular within new pharmaceutical drugs (see Guseo and Guidolin, 2011), b) the detection and modeling of network externality effects induced by network goods (telephone, fax, etc.) (see Guseo and Guidolin, 2010), and c) the analysis of sequential competition in the diffusion of similar products (Guseo and Mortarino, 2010, 2011).


References   UpGreen.gif

·       BASS, F.M. (1969). A new product growth model for consumer durables, Management Science, 15, 215-227.

·       BASS, F.M., KRISHNAN, T.V., JAIN, D.C. (1994). Why the Bass model fits without decision variables, Marketing Science, 13, 203-223.

·       BOCCARA, N. (2004). Modeling Complex Systems, Springer, New York.

·       CAVALLI SFORZA, L. (1996). Geni, Popoli e Lingue, Adelphi, Milano.

·       COUZIN, I. (2007). Collective Minds,  Nature, vol. 445, 15/2/2007, 715.

·       COUZIN, I., KRAUSE, J., FRANKS, N.R., LEVIN, S.A. (2005). Effective leadership and decision-making in animal groups on the move, Nature, vol. 433, 3/2/2005, 513-516.

DALLA VALLE, A., FURLAN, C. (2011).  Forecasting accuracy of wind power technology diffusion models across countries. International Journal of Forecasting, 27(2), 592-601. http://dx.doi.org/10.1016/j.ijforecast.2010.05.018

·       GUIDOLIN, M. (2008). Aggregate and Agent-Based Models for the Diffusion of Innovations, PhD Thesis, Scuola di Dottorato di Ricerca in "Economia e Management", ciclo XX, Università di Padova.

GUIDOLIN, M., MORTARINO, C. (2010). Cross-country diffusion of photovoltaic systems: modelling choices and forecasts for national adoption patterns, Technological Forecasting and Social Change, 77(2), 279-296,  http://dx.doi.org/10.1016/j.techfore.2009.07.003

·       GUSEO, R. (2004). Interventi strategici e aspetti competitivi nel ciclo di vita di innovazioni; Strategic Interventions and Competitive Aspects in Innovation Life Cycle ; Working Paper Series, N. 11, Department of Statistical Sciences, University of Padua, Italy.

·       GUSEO, R. (2007). How much Natural Gas is there? Depletion Risks and Supply Security Modelling. (submitted)  

·       GUSEO, R. (2011). Worldwide Cheap and Heavy Oil Productions: A Long-Term Energy Model. Energy Policy, 39(9),  5572-5577, doi: 10.1016/j.enpol.2011.04.060

·       GUSEO, R., CAPUZZO, S. (2006). A quando l'ultima stilla? Le serie storiche delle estrazioni alla base di nuovi scenari di previsione sul picco del petrolio, Ambiente Risorse Salute , n. 108 Marzo/Aprile, 6-14. (download)

·       GUSEO, R., DALLA VALLE, A. (2005). Oil and Gas Depletion: Diffusion Models and Forecasting under Strategic Intervention, Statistical Methods and Applications, vol. 14, 3, 375-387 doi: 10.1007/s10260-005-0118-6.

·       GUSEO, R., DALLA VALLE, A., GUIDOLIN, M. (2007). World Oil Depletion Models: Price Effects Compared with Strategic or Technological Interventions; Technological Forecasting and Social Change, 74(4), 452 - 469 doi: 10.1016/j.techfore.2006.01.004

·       GUSEO, R., GUIDOLIN, M. (2006). Cellular Automata and Riccati Equation Models for Diffusion of Innovations. (Short Paper), Atti della XLIII Riunione Scientifica della Società Italiana di Statistica, Torino 14-16/6/2006, Vol. Sessioni Spontanee, 103-106, CLEUP, Padova.

·       GUSEO, R., GUIDOLIN, M. (2007a). Cellular Automata with Network Incubation Period versus Perturbed Riccati Equation Models in Information Technology Innovation Diffusion. S.Co.2007, Fifth Conference, Complex Models and Computational Intensive Methods for Estimation and Prediction, Venice 6-8 september 2007, P. Mantovan, A. Pastore, S. Tonellato, (Eds.) Book of Short Papers, 272-77.

·       GUSEO, R., GUIDOLIN, M. (2007b). A Class of Automata Networks for Diffusion of Innovations Driven by Riccati Equations. Working Paper Series, N. 6, April 2007, Department of Statistical Sciences, University of Padua, Italy.

·       GUSEO, R., GUIDOLIN, M. (2008). Cellular Automata and Riccati Equation Models for Diffusion of Innovations. Statistical Methods and Applications, 17(3), 291 - 308 doi: 10.1007/s10260-007-0059-3.

·       GUSEO,R., GUIDOLIN, M. (2009a). Modelling a Dynamic Market Potential: A Class of Automata Networks for Diffusion of Innovations. Technological Forecasting and Social Change, 76(6), 806-820.   http://dx.doi.org/10.1016/j.techfore.2008.10.005.

·       GUSEO, R., GUIDOLIN, M. (2009b). Market potential dynamics in innovation diffusion: modelling the synergy between two driving forces. Working Paper Series, N. 10, May 2009, Department of Statistical Sciences, University of Padua, Italy (download)

·       GUSEO, R., GUIDOLIN, M. (2010). Cellular Automata with Network Incubation in Information Technology Diffusion. Physica A, Statistical Mechanics and its Applications, 389, 2422-2433. http://dx.doi.org/10.1016/j.physa.2010.02.007

·       GUSEO, R., GUIDOLIN, M. (2011). Market potential dynamics in innovation diffusion: modelling the synergy between two driving forces. Technological Forecasting and Social Change, 78(1), 13-24.  doi: 10.1016/j.techfore.2010.06.003

·       GGUSEO, R., MORTARINO, C. (2010). Correction to the paper “Optimal Product Launch Times in a Duopoly: Balancing Life-Cycle Revenues with Product Cost”. Operations Research, (Forthcoming), http://dx.doi.org/10.1287/opre.1100.0811 .

·       MARCHETTI, C. (1980). Society as a learning system; discovery, invention and innovation cycles revisited, Technological Forecasting and Social Change, 18, 257-282.

·       ROGERS, E.M. (2003). The Diffusion of Innovations, 5th Ed., Free Press, New York.

·       WOLFRAM, S. (1983). Statistical Mechanics of Cellular Automata, Review of Modern Physics, 55, 601-644.



Department of Statistical Sciences | University of Padova

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