Abstract
The diffusion of ideas is often closely connected to the creation and diffusion of knowledge and to the technological evolution of society. Because of this, knowledge creation, exchange and its subsequent transformation into innovations for improved welfare and economic growth is briefly described from a historical point of view. Next, three approaches are discussed for modeling the diffusion of ideas in the areas of science and technology, through (i) deterministic, (ii) stochastic, and (iii) statistical approaches. These are illustrated through their corresponding population dynamics and epidemic models relative to the spreading of ideas, knowledge and innovations. The deterministic dynamical models are considered to be appropriate for analyzing the evolution of large and small societal, scientific and technological systems when the influence of fluctuations is insignificant. Stochastic models are appropriate when the system of interest is small but when the fluctuations become significant for its evolution. Finally statistical approaches and models based on the laws and distributions of Lotka, Bradford, Yule, Zipf–Mandelbrot, and others, provide much useful information for the analysis of the evolution of systems in which development is closely connected to the process of idea diffusion.
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1 Knowledge, Capital, Science Research, and Ideas Diffusion
1.1 Knowledge and Capital
Knowledge can be defined as a dynamic framework connected to cognitive structures from which information can be sorted, processed and understood (Howells 2002). Along economics lines of thought (Barro and Sala-I-Martin 2004; Leydesdorff 2006; Dolfsma 2008), knowledge can be treated as one of the “production factors”, – i.e., one of the main causes of wealth in modern capitalistic societies (Tables 3.1–3.5).
According to Marshall (Marshall 1920) a “capital” is a collection of goods external to the economic agent that can be sold for money and from which an income can be derived. Often, knowledge is parametrized as such a “human capital” (Romer 1996, 1994a,b, 2002; Jaffe and Trajtenberg 2002). Walsh (1935) was one pioneer in treating human knowledge as if it was a “capital”, in the economic sense; he made an attempt to find measures for this form of “capital”. Bourdieu (1986); Coleman (1988), Putnam Putnam (1993), Becker and collaborators have further implanted the concept of such a “human capital” in economic theory (Becker and Murphy 1988; Becker 1996; Stiglitz 1987).
However, the concept of knowledge as a form of capital is an oversimplification. This global-like concept does not account for many properties of knowledge strictly connected to the individual, such as the possibility for different learning paths or different views, multiple levels of interpretation, and different preferences (Davis 2003). In fact, knowledge develops in a quite complex social context, within possibly different frameworks or time scales, and involves “tacit dimensions” (beside the basic space and time dimensions) requiring coding and decoding (Dolfsma 2008).
Key point Nr. 1Knowledge is much more than a form of capital: it is a dynamic framework connected to cognitive structures from which information can be sorted, processed and understood.
1.2 Growth and Exchange of Knowledge
Science policy-makers and scholars have for many decades wished to develop quantitative methods for describing and predicting the initiation and growth of science research (Price 1951, 1971; Foray 2004). Thus, scientometrics has become one of the core research activities in view of constructing science and technology indicators (van Raan 1997).
The accumulation of the knowledge in a country’s population arises either from acquiring knowledge from abroad or from internal engines (Nonaka 1994; Nonaka and Konno 1998; Nonaka and Takeuchi 1995; Bernius 2010). The main engines for the production of new knowledge in a country are usually: the public research institutes, the universities and training institutes, the firms, and the individuals (Dahlman 2009). The users of the knowledge are firms, governments, public institutions (such as the national education, health, or security institutions), social organizations, and any concerned individual. The knowledge is transferred from producers to the users by dissemination that is realized by some flow or diffusion of process (Dahlman et al. 2007), sometimes involving physical migration.
Knowledge typically appears at first as purely tacit: a person “has” an idea (Saviotti 1999; Cowan and Foray 1997). This tacit knowledge must be codified for further use; after codification, knowledge can be stored in different ways, as in textbooks or digital carriers. It can be transferred from one system to another. In addition to knowledge creation, a system can gain knowledge by knowledge exchange and/or trade.
In knowledge diffusion, the knowledge is transferred while subjects interact (Jaffe 1986; Antonelli 1996; Morone and Taylor 2010). Pioneering studies on knowledge diffusion investigated the patterns through which new technologies are spread in social systems (Rogers 1962; Casetti and Semple 1969). The gain of knowledge due to knowledge diffusion is one of the keys or leads to innovative products and innovations (Kucharavy et al. 2009; Ebeling and Scharnhorst 1985).
Key point Nr. 2An innovative product or a process is new for the group of people who are likely to use it. Innovation is an innovative product or process that has passed the barrier of user adoption. Because of the rejection by the market, many innovative products and processes never become an innovation.
In science, the diffusion of knowledge is mainly connected to the transfer of scientific information by publications. It is accepted that the results of some research become completely scientific when they are published (Ziman 1969). Such a diffusion can also take place at scientific meetings and through oral or other exchanges, sometimes without formal publication of exchanged ideas.Footnote 1
Key point Nr. 3Scientific communication has specific features. For example, citations are very important in the communication process as they place corresponding research and researchers, mentioned in the scientific literature, in a way similar to the kinship links that tie persons within a tribe. Informal exchanges happening in the process of common work at the time of meetings, workshops, or conferences may accelerate the transfer of scientific information, whence the growth of knowledge.
2 Qualitative Research: Historical Remarks
2.1 Science Landscapes
Understanding the diffusion of knowledge requires research complementary to mathematical investigations. For example, mathematics cannot indicate why the exposure to ideas leads to intellectual epidemics. Yet, mathematics can provide information on the intensity or the duration of some intellectual epidemics.
Qualitative research is all about exploring issues, understanding phenomena, and answering questions (Bryman 1988) without much mathematics. Qualitative research involves empirical research through which the researcher explores relationships using a textual methodology rather than quantitative data. Problems and results in the field of qualitative research on knowledge epidemics will not be discussed in detail here. However, through one example it can be shown how mathematics can create the basis for qualitative research and decision making. This example is connected to the science landscape concepts outlined here below.
The idea of science landscapes has some similarity with the work of Wright (1932) in biology who proposed that the fitness landscape evolution can be treated as optimization process based on the roles of mutation, inbreeding, crossbreeding, and selection. The science landscape idea was developed by Small (1997, 1998, 2006), as well as by Noyons and Van Raan (1998). In this framework,Scharnhorst (1998, 2001) proposed an approach for the analysis of scientific landscapes, named “geometrically oriented evolution theory”.
Key point Nr. 4The concept of science landscape is rather simple: Describe the corresponding field of science or technology through a function of parameters such as height, weight, size, technical data, etc. Then a virtual knowledge landscape can be constructed from empirical data in order to visualize and understand innovation and to optimize various processes in science and technology.
As an illustration at this level, consider that a mathematical example of a technological landscape can be given by a function C = C(S, v), where C is the cost for developing a new airplane, and where S and v represent the size and velocity of the airplane.
Consider two examples concerning the use of science landscapes for evaluation purposes:
(1) Science landscape approach as a method for evaluating national research strategies
For example, national science systems can be considered as made of researchers who compete for scientific results, and subsidies, following optimal research strategies. The efforts of every country become visible, comparable and measurable by means of appropriate functions or landscapes: e.g., the number of publications. The aggregate research strategies of a country can thereby be represented by the distribution of publications in the various scientific disciplines. In so doing, within a two-dimensional space,Footnote 2 different countries correspond to different landscapes. Various political discussions can follow and evolution strategies can be invented thereafter.
Notice that the dynamics of self-organized structures in complex systems can be understood as the result of a search for optimal solutions to a certain problem. Therefore, such a comment shows how rather strict mathematical approaches, not disregarding simulation methods, can be congruent to qualitative questions.
(2) Scientific citations as landscapes for individual evaluation Scientific citations can serve for constructing landscapes. Indeed, citations have a key position in the retrieval and valuation of information in scientific communication systems (Scharnhorst 1998; Egghe 1998; Egghe and Rousseau 1990). This position is based on the objective nature of the citations as components of a global information system, as represented by the Science Citation Index. A landscape function based on citations can be defined in various ways. It can take into account self-citations (Hellsten et al. 2006, 2007a,b; Ausloos et al. 2008), or time-dependent quantitative measures (Hirsch 2005; Soler 2007; Burrell 2007).
Key point Nr. 5Citation landscapes become important elements of a science policy (e.g., in personnel management decisions), thereby influencing individual scientific careers, evaluation of research institutes, and investment strategies.
2.2 Lotka and Price: Pioneers of Scientometrics
Alfred Lotka, one of the modern founders of population dynamics studies, was also an excellent statistician. He discovered (Lotka 1926) a distribution for the number of authors n r as a function of the number of published papers r, – i.e., \({n}_{r} = {n}_{1}/{r}^{2}\).
However, Derek Price, a physicist, set the mathematical basis in the field of measuring scientific research in recent times (Price 1963; Price and Gürsey 1975; Price 1961). He proposed a model of scientific growth connecting science and time. In the first version of the model, the size of science was measured by the number of journals founded in the course of a number of years. Later, instead of the number of journals, the number of published papers was used as the measure of scientific growth. Price and other authors (Price and Gürsey 1975; Price 1961; Gilbert 1978) considered also different indicators of scientific growth, such as the number of authors, funds, dissertation production, citations, or the number of scientific books.
In addition to the deterministic approach initiated by Price, the statistical approach to the study of scientific information developed rapidly and nowadays is still an important tool in scientometrics (Chung and Cox 1990; Kealey 2000). More discussion on the statistical approach will be given in sect. 3.6 of this chapter.
Key point Nr. 6Price distinguished three stages in the growth of knowledge: (a) a preliminary phase with small increments; (b) a phase of exponential growth; (c) a saturation stage. The stage (c) must be reached sooner or later after the new ideas and opportunities are exhausted; the growth slows down until a new trend emerges and gives rise to a new growth stage. According to Price, the curve of this growth is a S-shaped logistic curve.
2.3 Population Dynamics and Epidemic Models of the Diffusion of Knowledge
Population dynamics is the branch of life sciences that studies short- and long-term changes in the size and age composition of populations, and how the biological and environmental processes influence those changes. In the past, most models for biological population dynamics have been of interest only in mathematical biology (Murray 1989; Edelstein-Keshet 1988). Today, these models are adapted and applied in many more areas of science (Dietz 1967; Dodd 1958). Here below, models of knowledge dynamics will be of interest as bases of epidemic models. Such models are nowadays used because some stages of idea spreading processes within a population (e.g, of scientists), possess properties like those of epidemics.
The mathematical modeling of epidemic processes has attracted much attention since the spread of infectious diseases has always been of great concern and considered to be a threat to public health (Anderson and May 1982; Brauer and Castillo-Chavez 2001; Ma and Li 2009). In the history of science and society, many examples of ideas spreading seem to occur in a way similar to the spread of epidemics. Examples of the former field pertain to the ideas of Newton on mechanics and the passion for “High Critical Temperature Superconductivity” at the end of the twentieth century. Examples of the latter field are the spreading of ideas from Moses or Buddha (Goffman 1966), or discussions based on the Kermack–McKendrick model (Kermack and McKendrick 1927) for the epidemic stages of revolutions or drug spreading (Epstein 1997).
Epidemic models belong to a more general class of Lotka–Volterra models used in research on systems in the fields of biological population dynamics, social dynamics, and economics. The models can also be used for describing processes connected to the spread of knowledge, ideas and innovations (see Fig. 3.1). Two examples are the model of innovation in established organizations (Castiaux 2007) and the Lotka–Volterra model for forecasting emerging technologies and the growth of knowledge (Kucharavy et al. 2009). In social dynamics, the Lanchester model of war between two armies can be mentioned, a model which in the case of reinforcements coincides with the Lotka–Volterra–Gause model for competition between two species (Gause 1935). Solomon and Richmond (2001, 2002) applied a Lotka–Volterra model to financial markets, while the model for the trap of extinction can be applied to economic subjects (Vitanov et al. 2006). Applications to chaotic pairwise competition among political parties (Dimitrova and Vitanov 2004) could also be mentioned.
Relation among epidemic models, Lotka–Volterra models, and population dynamics models
To start the discussion of population dynamics models as applied to the growth of scientific knowledge with special emphasis on epidemic models, two kinds of models can be discussed (Fig. 3.2): (1) deterministic models, see Sect. 3.3, appropriate for large and small populations where the fluctuations are not drastically important, (2) stochastic models, see Sect. 3.4, appropriate for small populations. In the latter case the intrinsic randomness appears much more relevant than in the former case. Stochastic models for large populations will not be discussed. The reason for this is that such models usually consist of many stochastic differential equations, whence their evolution can be investigated only numerically.
Relationships between system size, influence of fluctuations, and discussed classes of models
Finally, let us mention that the knowledge diffusion is closely connected to the structure and properties of the social network where the diffusion happens. This is a new and very promising research area. For example, a combination can be made between the theory of information diffusion and the theory of complex networks (Boccaletti et al. 2006). For more information about the relation between networks and knowledge, see the following chapters of the book.
3 Deterministic Models
Below, 13 selected deterministic models (see Fig. 3.3) are discussed. The emphasis is on models that can be used for describing the epidemic stage of the diffusion of ideas, knowledge, and technologies.
Discrete (3) and continuous (10) models discussed in the chapter. Two continuous models account for the influence of time lag, three models are simple models of technological diffusion. Two models are simple epidemic models and two models are more complicated models. In addition, the basic logistic curve is discussed
3.1 Logistic Curve and Its Generalizations
In a number of cases, the natural growth of autonomous systems in competition can be described by the logistic equation and the logistic curve (S-curve) (Meyer 1994). In order to describe trajectories of growth or decline in socio-technical systems, one generally applies a three-parameter logistic curve:
where N(t) is the number of units in the species or growing variable to study; K is the asymptotic limit of growth; α is the growth rate which specifies the “width” of the S-curve for N(t); and β specifies the time t m when the curve reaches the midpoint of the growth trajectory, such that N(t m ) = 0. 5 K. The three parameters, K, α, and β, are usually obtained after fitting some data (Meade and Islam 1995). It is well known that many cases of epidemic growth can be described by parts of an appropriate S-curve. As an example, recall that the S-curve was also used for describing technological substitution (Rogers 1962; Mansfield 1961; Modis 2007), ca. 60 years ago.
However, different interaction schemes can generate different growth patterns for whatever system species are under consideration (Modis 2003). Not every interaction scheme leads to a logistic growth (Ausloos 2010). The evolution of systems in such regimes may be described by more complex curves, such as a combination of two or more simple three-parameter functions (Meyer 1994; Meyer et al. 1999).
3.2 Simple Epidemic and Lotka–Volterra Modelsof Technology Diffusion
As recalled here above, the simplest epidemic models could be used for describing technology diffusion, like considering two populations/species: adopters and non-adopters of some technology. Such models can be put into two basic classes: either broadcasting (Fig. 3.4) or word-of-mouth models (Fig. 3.5). In the broadcasting models, the source of knowledge about the existence and/or characteristics of the new technology is external and reaches all possible adopters in the same way. In the word-of-mouth models, the knowledge is diffused by means of personal interactions.
Schematic representation of a word-of-mouth model of technology diffusion. The number of adopters of technology increases by interpersonal interactions
Schematic representation of a broadcasting model of technology diffusion. The number of adopters of technology increases by mass media influence
(1) The broadcasting model (Fig. 3.4)Let us consider a population of K potential adopters of the new technology and let each adopter switch to the new technology as soon as he/she hears about its existence (immediate infection through broadcasting). The probability that at time t a new subject will adopt the new technology is characterized by a coefficient of diffusion κ(t) which might or might not be a function of the number of previous adopters. In the broadcasting model κ(t) = a with (0 < a < 1); this is considered to be a measure of the infection probability.
Let N(t) be the number of adopters at time t. The increase in adopters for each period is equal to the probability of being infected, multiplied by the current population of non-adopters (Mahajan and Peterson 1985). The rate of diffusion at time t is
The integration of (3.2) leads to the number of adopters: i.e.,
N(t) is described by a decaying exponential curve.
(2) Word-of-mouth model (Fig. 3.5)In many cases, however, the technology adoption timing is at least an order of magnitude slower than the time it takes for information spreading (Geroski 2000). This requires another modelization than in (1): the word-of-mouth diffusion model. Its basic assumption is that knowledge diffuses by means of face-to-face interactions. Then the probability of receiving the relevant knowledge needed to adopt the new technology is a positive function of current users N(t). Let the coefficient of diffusion κ(t) be bN(t) with b > 0. The rate of diffusion at time t is
Then
where \({N}_{0} = N(t = {t}_{0})\). N(t) is described by an S-shaped curve.
A constraint exists in the word-of-mouth model: it explains the diffusion of an innovation not from the date of its invention but from the date when some number, N(t) > 0, of early users have begun using it.
(3) Mixed information source model (Fig. 3.6) In the mixed information source model, existing non-adopters are subject to two sources of information (Fig. 3.6). The coefficient of diffusion is supposed to look like a + bN(t). The model evolution equation becomes
The result of (3.6) is a (generalized) logistic curve whose shape is determined by a and b (Mahajan and Peterson 1985).
Schematic representation of mixed information source model. The number of adopters increases by mass media influence and interpersonal contacts
(4) Time lag Lotka–Volterra model of innovation diffusion (Fig. 3.7)Let it be again assumed that the diffusion of innovation in a society is accounted for by a combination of two processes: a mass-mediated process and a process connected to interpersonal (word-of-mouth) contacts. Let N(t) be the number of potential adopters. Some of the potential adopters adopt the innovation and become real adopters. The equation for the he rate of growth of the real adopters n(t), in absence of time lag, is
where α denotes the degree of external influence such as mass media, β accounts for the degree of internal influence by interpersonal contact between adopters and the remaining population; μ is a parameter characterizing the decline in the number of adopters because of technology rejection for whatever reason.
Schematic representation of a Lotka–Volterra model with time lag. The model accounts for the time lag between hearing about innovation and its adoption
A basic limitation in most models of innovation diffusion has been the assumption of instantaneous acceptance of the new innovation by a potential adopter (Mahajan and Peterson 1985; Bartholomew 1982). Often, in reality, there is a finite time lag between the moment when a potential adopter hears about a new innovation and the time of adoption. Such time lags usually are continuously distributed (May 1974; Lal et al. 1988).
The time lag between the knowledge about the innovation and its adoption can be captured by a distributed time lag approach in which the effects of time delays are expressed as a weighted response over a finite time interval through appropriately chosen memory kernels (Karmeshu 1982) (see Fig. 3.7). Whence (3.7) becomes
Equation (3.8) reduces to (3.7) when the memory kernels K i ∗ (t) (i = 1, 2, 3) are replaced by delta functions.
Two generic types of kernels are usually considered (Lal et al. 1988):
in which ν − 1 is some characteristic time scale of the system.
The number of potential adopters N(t) changes over time. Several possible functional forms of N(t) are used (Sharif and Ramanathan 1981):
Equation (3.12) represents an approximation for short- and medium-term forecasting since for t large, N(t) grows without bound, as in Keynes (1930). Equations (3.13) and (3.14) are useful in long-term forecasting as N(t) has an upper limit. Such forms for N(t) are valid within a deterministic framework.
However, a stochastic framework (see below) is more appropriate when the carrying capacity N(t) is governed by some stochastic process, as when the influence of socioeconomic and natural factors are subject to “random” or hardly explainable fluctuations. In such systems, N(t) can be time-dependent: for example, N(t) ∼ N 0(1 + εcos(ωt)) where ε < < 1 and the periodicity takes into account the influence of some (strong) cyclic economic factors. In presence of a strong stochastic component, N(t) can be stochastic: \(N(t) = {N}_{0} + \xi (t)\), where the noisy component is ξ(t) and N 0 is the average value of the so-called carrying capacity (Odum 1959).
Key point Nr. 7Time lags between observations and decisions lead to complicated dynamics. Perform some preliminary careful analysis of system behavior based on time lags before making a decision.
3.3 Price Model of Knowledge Growth: Cycles of Growthof Knowledge
The Price evolution model of scientific growth ignited intensive research (Fernandez-Camo et al. 2004; Szydlowski and Krawiez 2001) (see Fig. 3.8). This model is in fact a dialectical addition to Kuhn’s idea (Kuhn 1962) about the revolutionary nature of science processes: after some period of evolutionary growth, a scientific revolution occurs. Price considered the exponential growth as a disease that retards the growth of stable science, producing narrower and less flexible specialists.
Key point Nr. 8An interesting result of the research of Price can be read as follows:if a government wants to double the usefulness of science, it has to multiply by about eight the gross number of workers and the total expenditure of manpower and national income.
The unreserved application of the Price model faces several difficulties:
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Many scientific products which seem to be new are not really new
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Creativity and innovation can be confused (Plesk 1997; Amabile et al. 1996)
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Creative papers with new ideas and results have the same importance as trivial duplications (Magyari-Beck 1984)
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Two things are omitted:
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Quality (whatever that means, but it is an economic notion) of research
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The cost or measure of complexity.
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Diagram of relationships between Price model and its modifications. The presence of time lags can lead to much complication in the evolution dynamics of a scientific field
In answer to this, Price formulated the hypothesis that one should be studying only the growth of important discoveries, inventions, and scientific laws, rather than both important and trivial things. In so doing, one might expect that any of such studied growth will follow the same pattern.
A generalized version of the Price model for the growth of a scientific field (Szydlowski and Krawiez 2009; Price 1956) is based on the following assumptions: (a) the growth is measured by the number of important publications appearing at a given time; (b) the growth has a continuous character, though a finite time period T = const is needed to build up a result of the fundamental character; (c) the interactions between various scientific fields are neglected. If, in addition, the number of scientists publishing results in this field is constant, then the rate of scientific growth is proportional to the number of important publications at time t minus the time period T required to build up a fundamental result. The model equation is
where α is a constant. The initial condition x(t) = ϕ(t) is defined on the interval [ − T, 0].
Let the population of scientists be varying and consider the evolution of the average number of papers per scientist. In general, instead of the linear right-hand side (3.15), a non-linear model can be used:
where f(t − T) is a homogeneous function of degree one. The simplest form of such a function is a linear function. Let n(t) represent the rate of growth of the population of scientists and write L(t) = exp[n(t) t]. For simplicity, let the population of scientists grow at the constant rate \(n = \frac{1} {L} \frac{dL} {dt}\) and let \(z = x/L\). Then the evolution of the number of papers written by a scientist has the form
If n = 0 and T = 0, the Price model of exponential growth is recovered. Equation (3.17) is linear, but a cyclic behavior may appear because of the feedback between the delayed and non-delayed terms.
3.4 Models Based on Three or Four Populations: Discrete Models
(1) SIR (Susceptible-Infected-Removed) model (Fig. 3.9)In 1927, Kermack and McKendrick (1927) created a model in which they considered a fixed population with only three compartments: S(t), the susceptibles; I(t), the infected; R(t), the recovered, or removed.
SIR (susceptibles S, infectives I, recovered R) model of intellectual infection with influxes of susceptibles and infectives to the corresponding scientific ideas
Following this idea, Goffman (1966); Goffman and Newill (1964) considered the stages of fast growth of scientific research in a scientific field as “intellectual epidemics” and developed the corresponding scientific research epidemic stage based on three classes of population: (i) the susceptibles S who can become infectives when in contact with infectious material (the ideas); (ii) the infectives I who host the infectious material; and (iii) the recovered R who are removed from the epidemics for different reasons (Fig. 3.9).
The epidemic stage is controlled by the system of differential equations
where μ and ν are the rates at which the new supply of susceptibles and infectives enter the population. A necessary condition for the process to enter the epidemic state is \(\frac{dI} {dt} > 0\). Then
is the threshold density of susceptibles, i.e., no epidemics can develop from time t 0 unless S 0, the number of susceptibles at that time, exceeds the threshold ρ: the epidemic state cannot be maintained over some time interval unless the number of susceptibles is larger than ρ through that interval of time. As I increases, ν ∕ I converges to 0 and ρ converges rapidly to γ ∕ β.
In Goffman (1966), Goffman evaluated the rate of change of infectives ΔI ∕ Δt. From the system equations, it is difficult to determine I(t). Yet in the epidemic stage, the behaviour of I(t) is exponential. For small t close to t 0, I(t) can be expanded into a power series: \(I(t) = {C}_{0} + {C}_{1}t + {C}_{2}{t}^{2} + \ldots {C}_{n}{t}^{n} + \ldots \) such that the approximate rate of ΔI ∕ Δt can be obtained. On the basis of this rate and the raw data, the development and peak of some research activity can be predicted, – under the assumption that the research is in an epidemic stage.
(2) SEIR model for the spreading of scientific ideas (Fig. 3.10)The SIR epidemic models can be further refined by introducing a fourth class, E, i.e., persons exposed to the corresponding scientific ideas (Fig. 3.10). Such models are discussed in Bettencourt et al. (2008, 2006); they belong to the class of so-called SEIR epidemic models. One typical model goes as follows
where S(t) is the size of the susceptible population at time t, E(t) is the size of the exposed class, I(t) is the size of the infected class. These individuals have adopted the new scientific idea in their publications. Finally, R(t) is the size of the population of recovered scientists, i.e., those who no longer publish on the topic. The size of the entire population is: \(N = S + E + I + R\). An exit term is assumed to be very small, and because of this, t is included in the recovered class. N grows exponentially with rate λ. The parameters of the model are: β, the probability and effectiveness of a contact with an adopter; 1 ∕ κ, the standard latency time, (in other words, the average duration of time after one has been exposed but before one includes the new idea in one’s own publication); 1 ∕ γ, the duration of the infectious period, thus how long one publishes on the topic and teaches others; ρ, the probability that an exposed person has multiple effective contacts with other adopters.
SEIR model of intellectual infection with influxes of susceptibles and infectives to the corresponding scientific ideas, thus extending the SIR model by including a class of scientists exposed (E) to the specific scientific ideas
This simple model can incorporate a wide range of behaviors. For many values of the parameters λ,β, κ, γ and ρ, the infected class grows as a logistic curve. For large values of the contact rate β or recruitment λ, I(t) grows nearly linearly, as indeed has been found empirically for some research fields (Bettencourt et al. 2008).
Key point Nr. 9Epidemic models are the best suited for describing the expansion stage of a process growth.
(3) SI discrete model for the change in the number of authors in a scientific field (Fig. 3.11) With the goal of predicting the spreading out of scientific objects (such as theories or methods), Nowakowska (1973) discussed several epidemic discrete models for predicting changes in the number of publications and authors in a given scientific field. With respect to the publications, the main assumption of the models is that the number of publications in the next period of time (say, 1 year) will depend: (i) on the number of papers which recently appeared, and (ii) on the degree at which the subject has been exhausted. The numbers of publications appearing in successive periods of time should first increase, then would reach a maximum, and as the problem
Schema of a discrete SI evolution model of the number of authors of scientific papers. The model takes into account that several scientists stop their work in a scientific field; it can be due to different reasons as for example death or losing interest in particular questions
becomes more and more exhausted, the number of publications would decrease.
Let it be assumed (Fig. 3.11) that if at a certain moment t the epidemics state is (x t , y t ) (x t is the number of infectives (authors who write papers on the corresponding research problems), y t is the number of susceptibles), then for a sufficiently short time interval Δt, one may expect that the number of infectives x t + Δt will be equal to \({x}_{t} - a{x}_{t}\Delta t + b{x}_{t}{y}_{t}\Delta t\), while the number of susceptibles y t + Δt will be equal to y t − bx t y t Δt; a and b being appropriate constants. Let the expected number of individuals who either die or recover, during the interval (t, t + Δt), be ax t Δt, and let bx t y t Δt be the expected number of new infections. The equations of this model are:
Note here that such discrete models are useful for the analysis of realistic situations where the values of the quantities are available at selected moments (every month, every year, etc.).
(4) Daley discrete model for the population of papers (Fig. 3.12)Daley (1967) investigated the spread of news as follows: individuals who have not heard the news are susceptible and those who heard the news are infective. Recovery is not possible, as it is assumed that the individuals have perfect memory and never forget. The Daley model can be applied also to the population of papers (Nowakowska 1973) (see Fig. 3.12). For Δt = 1 (year), the Daley model equation reads
where x 1, x 2.... are the numbers of papers on the subject which appear in successive periods of time, b and N being parameters. The expected number x t + 1 of papers in year t + 1 is proportional to the number x t of papers which appeared in year t, and to the number \(N - {x}_{1} - {x}_{2}\cdots - {x}_{t} = N -{\sum \nolimits }_{i=1}^{t}{x}_{i}\). N is the number of papers which have to appear in order to exhaust the problem: the problem under consideration may be partitioned into N sub-problems, such that solving any of them is worth a separate publication; these subproblems are solved successively by the scientists. The b and N parameters may be estimated by the method of least squares, e.g. from a given empirical histogram. A parameter characterizing the initial growth dynamics in the number of publications can also be introduced: τ = bN. Therefore, (3.26) can be used for short-time prediction, even when the corresponding research field is in the epidemic stage of its evolution.
Daley model for evolution of population of papers on problems in a scientific field. The exhausting of the scientific field is taken into account
(5) Discrete model coupling the populations of scientists and papers (Fig. 3.13)A discrete model coupling the populations of scientists and papers can be considered (Fig. 3.13); it depends on four parameters: N, a, b and c. N as above denotes the number of sub-problems of the given problem; a is the probability that a scientist working on the subject in a given year abandons research on the subject for whatever reasons; b is the probability of obtaining a solution to a given subproblem by one scientist during one year of research; c denotes the coefficient of attractiveness of the subject. The basic variables of the model are: u t , the number of scientists working on the subject in year t, and x t , the number of publications on the subject which appear in year t.
Discrete model for the joint evolution of populations of scientists and papers. The attractiveness of the field, the exhaustion of the field, and the possibility for declining interest for working in the scientific field are taken into account through adequate rate parameters
The model equations are
The equation for the number u t + 1 of scientists working on the subject in year t + 1 tells that in year t + 1, the expected number of scientists working on the subject will be the number of scientists working on the subject in year t, u t , minus the expected number of scientists who stopped working on the subject, au t , plus the expected number of scientists, cx t , who became attracted to the problem by reading papers which appeared in year t. The equation expressing the number of publications in year t + 1 tells us that x t + 1 equals the number of subproblems that were solved in the year t. The probability that a given subproblem will be solved in year t by a given scientist equals b. Then the probability of the opposite event, i.e. a given scientist will not solve a particular problem, equals 1 − b. As there are u t scientists working on the subject in year t, the probability that a given subproblem will not be solved by any of them is \({(1 - b)}^{{u}_{t}}\). Consequently, the probability that a given subproblem will be solved in year t (by any of the u t scientists working on the subject) is equal to \(1 - {(1 - b)}^{{u}_{t}}\). Next, in year t there remained \(N -{\sum \nolimits }_{i=1}^{t}{x}_{i}\) subproblems to be solved. The expected number of subproblems solved in year t is equal to the product which gives the right-hand side of (3.28).
N.B. It is assumed, that the waiting time for publishing of the paper is one year. A more realistic picture would be to assume that the unit of time is not 1 year, but 2 years, or that the publication has some other time delay.
Key point Nr. 10In many cases, the data is available as one value per week, or one value per month, or one value per 3 months, etc. For modeling and subsequent short-range forecasting, so-called discrete (time) models are thus very appropriate.
3.5 Continuous Models of the Joint Evolution of Scientific Sub-Systems
(1) Coupled continuous model for the populations of scientists and papers: Goffman–Newill modelThe Goffman–Newill model (Goffman and Newill 1964) (Fig. 3.14) is based on the idea that the spreading process within a population can be studied on the basis of the literature produced by the members of that population. There is a transfer of infectious materials (ideas) between humans by means of an intermediate host (a written article). Let a scientific field be F and SF a sub-field of F. Let the number of scientists writing papers in the field F at t 0 be N 0 and the number of scientists writing papers in SF at t 0 (the number of infectives) be I 0. Thus, \({S}_{0} = {N}_{0} - {I}_{0}\) is the number of susceptibles; there is no removal at t 0, but there is removal R(t) at later times t. The number of papers produced on F at t 0 is N 0 ′ and the number of papers produced in SF at this time is I 0 ′. The process of intellectual infection is as follows: (a) a member of F is infected by a paper from I′; (b) after some latency period, this infected member produces ‘infected’ papers in N′, i.e. the infected member produces a paper in the subfield SF citing a paper from I′; (c) this ‘infected’ paper may infect other scientists from F and its sub-fields, such that the intellectual infection spreads from SF to the other sub-fields of F.
Schema of Goffman–Newill model for the evolution of a scientific field. Scientists are attracted to a sub-field after being intellectually infected by papers from the sub-field
Let β be the rate at which the susceptibles from class S become ‘intellectually infected’ from class I. Let β′ be the rate at which the papers in SF are cited by members of N who are producing papers in SF. As the infection process develops, some susceptibles and infectives are removed, i.e. some scientists are no longer active, and some papers are not cited anymore. Let γ and γ′ be the rates of removal of infectives from the populations I and I′ respectively, and δ and δ′ be the rates of removal from the populations of susceptibles S and S′. In addition, there can be a supply of infectives and susceptibles in N and N′. Let the rates of introduction of new susceptibles be μ and μ′, i.e. the rates at which the new authors and new papers are introduced in F, and let the rates of introduction of new infectives be υ and υ′, i.e. the rates at which new authors and new papers are introduced in SF. In addition, within a short time interval a susceptible can remain susceptible or can become an infective or be removed; the infective can remain an infective or can become a removal; and the removal remains a removed. The immunes remain immune and do not return to the population of susceptibles. If, in addition, the populations are homogeneously mixed, the system of model equations reads
The conditions for development of an epidemic are as follows. If as an initial condition at t 0, a single infective is introduced into the populations N 0 and N′ 0, then for an epidemic to develop, the change of the number of infectives must be positive in both populations. Then, for \(\rho = \frac{\gamma -\upsilon } {\beta }\) and \(\rho \prime = \frac{\gamma \prime-\upsilon \prime} {\beta \prime},\) the threshold for the epidemic arises from the conditions βSI′ > γI − υ and β′S′I′ > γ′I′ − υ′, such that the threshold is
The development of epidemics is given by the equation \(\frac{dI} {dt} = D(t)\). The peaks of the epidemic occur at time points where \(\frac{{d}^{2}I} {d{t}^{2}} = 0\), while the epidemic’s size is given by I(t → ∞).
(2) Bruckner–Ebeling–Scharnhorst model for the growth of n subfields in a scientific fieldThe evolution of growth processes in a system of scientific fields can be modeled by complex continuous evolution models. One of them, the Bruckner–Ebeling–Scharnhorst approach (Bruckner et al. 1990) (Fig. 3.15), is closely related to several generalizations of Eigen’s theory of prebiotic evolution and is briefly discussed here (see also Ebeling et al. 2006). In 1912, Lotka (Lotka 1912) published the idea of describing biological epidemic processes, like malaria, as well as chemical oscillations, with the help of a set of differential equations. These equations, known as Lotka–Volterra equations (Lotka 1925; Volterra 1927), are used to describe a coupled growth process of populations. However, they do not reflect several essential properties of evolutionary processes such as the creation of new structural elements. Because of this, one has to consider a more general set of equations for the change in the number x i of the scientists from the ith scientific subfield (a Fisher–Eigen–Schuster kind of model), i.e.,
Schema of Bruckner–Ebeling–Scharnhorst model of evolution of n scientific sub-fields. Self-reproduction and decline of subfields as well as field mobility are taken into account
The model based on (3.33) describes the coupled growth of n subfields, of a scientific discipline. Three fundamental processes of evolution are included in (3.33) : (a) self-reproduction: students and young scientists join the field and start working on corresponding problems. Their choice is influenced mainly by the education process as well as by individual interests and by existing scientific schools; (b) decline: scientists are active in science for a limited number of years. For different reasons (for example, retirement) they stop working and leave the system; (c) field mobility: individuals turn to other fields of research for various reasons or maybe open up new ones themselves.
The reasoning to obtain (3.33) goes as follows. The general form of the law for growth of the ith subfield is supposed to be
By separation, f i = w i x i , one obtains the replicator equation
Notice that when w i = const, the fields are uncoupled, i.e., there is an exponential growth in science. Otherwise, w i itself is a function of x and of various parameters, but can be separated into three terms according to the above model assumptions, i.e.,
Equation (3.33) is thus obtained from (3.35) and (3.36) for B ij = 0, k 0 = 0. To adapt this model to real growth processes, it can be assumed that the coefficients A i , D i , and A ij themselves are functions of x i :
Each of the three fundamental processes of change is represented in (3.33) with a linear and a quadratic term only. For example, the terms A i 1 and D i 1 account for cooperative effects in self-reproduction and decline processes respectively, while D i 0 accounts for a decline, because of aging. The contributions A ij 0 assume a linear type of field mobility behavior for scientists analogous to a diffusion process. On the other hand, the terms A ij 1 represent a directed process of exchange of scientists between fields. The best way to obtain these parameters is to estimate them for specific data bases using the method of least squares.
Key point Nr. 11The Bruckner–Ebeling–Scharnhorst model does not belong to the class of epidemic models which are best applicable only for describing the expansion stage of a process. The Bruckner–Ebeling–Scharnhorst model is an evolution model: it describes all stages of the evolution of a system.
4 Small-Size Scientific and Technological Systems: Stochastic Models (Fig. 3.16)
The movement of large bodies in mechanics is governed by deterministic laws. When the body contains a small number of molecules and atoms, stochastic effects such as the Brownian motion become important. In the area of scientific systems, the fluctuations become very important when the number of scientists in a certain research subfield is small. This is typical for new research fields with only a few researching scientists.
Several examples of stochastic models for the description of the diffusion of ideas or technology and the evolution of science are: (a) the model of evolution of scientific disciplines with an example pertaining to the case of elementary particles physics (Kot 1987); (b) stochastic models for the aging of scientific literature (Glänzel and Schoepflin 1994); c) stochastic models of the Hirsch index (Burrell 2007) and of instabilities in evolutionary systems (Bruckner et al. 1989); (d) models of implementation of technological innovations (Bruckner et al. 1996), etc. (Braun et al. 1985). In the following, see Fig. 3.16, two probabilistic and two stochastic models are discussed. Some attention is devoted to the master equation approach as well.
Hierarchy of stochastic models discussed in this chapter
4.1 Probabilistic SI and SEI Models
Epidemiological models of differential-equation-based compartmental type have been found to be limited in their capacity to capture heterogeneities at the individual level and in the interaction between individual epidemiological units (Chen and Hicks 2004). This is one of the reasons to switch from models in which the number of individuals are in given known states to models involving probabilities. One such model (Kiss et al. 2000) captures the diffusion of topics over a network of connections between scientific disciplines, as assigned by the ISI Web of Science’s classification in terms of Subject Categories (SCs). Each SC is considered as a node of a network along with all its directed and weighted connections to other nodes or SCs (Kiss et al. 2000, 2005). As with epidemic models, nodes can be characterized in a medical way. SCs that are susceptible (S) are either not aware of a particular research topic or, if aware, may not be ready to adopt it. Incubating SCs (E) are those that are aware of a certain topic and have moved to do some research on problems connected with this topic. Infected SCs (I) are actively working and publishing in a particular research topic.
Two probabilistic models, i.e., (i) the Susceptible-Exposed-Infected (SEI) model (Fig. 3.17) and (ii) a simpler Susceptible-Infected (SI) model (Fig. 3.18), are thereby only discussed.
Schema of the probabilistic SEI model for epidemics in a network connecting scientific disciplines
Schema of the probabilistic SI model for epidemics in a network connecting scientific disciplines
(1) Susceptible-Exposed-Infected (SEI) modelThe SEI model equations for the evolution of the node state probabilities are given by (Kiss et al. 2000):
where 0 ≤ I i (t) ≤ 1 denotes the probability of node i being infected at time t (likewise for S i (t) and E i (t)). The directed and weighted contact network is represented by A ij = rΓ ij with Γ ij = (w ij ) i, j = 1, . . . , N denoting the adjacency matrix that includes weighted links; r is the transmission rate per contact and 1 ∕ γ is the average incubation or latent period.
This set of equations states that an increase in the probability E i of a node i being exposed to an infection is directly proportional to the probability S i of node i being susceptible and the probability I j of neighbouring nodes j being infected. The number of such contacts and the per-contact rate of transmission are incorporated in A ij . Likewise, E i decreases if exposed/infected nodes become infected after an average incubation time 1 ∕ γ. The number of infected SCs at time t, according to the model, can be estimated as I(t) = ∑ i I i (t). Since \({S}_{i}(t) + {E}_{i}(t) + {I}_{i}(t) = 1\), for each t > 0, (3.38)–(3.40) are readily understood, in view of (3.39).
(2) Susceptible-Infected (SI) modelThe above SEI model can be simplified to an SI model when the possibility of an exposed period is excluded, i.e,. if \(\frac{d{E}_{i}(t)} {dt} = 0\). The equations for this simpler SEI model are reduced to
where the probability I i of a node i being infected and infectious only depends on the probability S i of the node i being susceptible. The comparison of both models with available data shows (Kiss et al. 2000) that while the agreement at the population level is usually much better for the SEI model, for the same pair of parameters, the agreement at the individual level is better when the simpler SI model is used.
4.2 Master Equation Approach
(1) Stochastic evolution model with self-reproduction, decline, and field mobility There exists a high correlation between field mobility processes and the emergence of new fields (Bruckner et al. 1990). This can be accounted for by a stochastic model (see Fig. 3.19), in which the system at time t is characterized by a set of integers N 1, N 2, …, N i , …, N n , with N i being, e.g., the number of scientists working in the subfield i, which is considered now as a stochastic variable. The three fundamental types of scientific change mentioned in the discussion of the Bruckner–Ebeling–Scharnhorst model (see above) here correspond to three elementary stochastic processes with three different transition probabilities:
Schema of the master equation model of evolution of scientific fields in presence of self-reproduction, decline, and field mobility
-
(a)
For self-reproduction, the transition probability is given by \(W({N}_{i} + 1\mid {N}_{i}) = {A}_{i}^{0}{N}_{i} = {A}_{i}^{0}{N}_{i} + {A}_{i}^{1}{N}_{i}({N}_{i} - 1)\).
-
(b)
The transition probability for decline is \(W({N}_{i} - 1\mid {N}_{i}) = {D}_{i}^{0}{N}_{i} + {D}_{i}^{1}{N}_{i}({N}_{i} - 1)\).
-
(c)
The transition probability for field mobility is \(W({N}_{i} + 1,{N}_{j} - 1\mid {N}_{i}{N}_{j}) = {A}_{ij}^{0}{N}_{j} + {A}_{ij}^{1}{N}_{i}{N}_{j}\).
The probability density \(P({N}_{1},\ldots,{N}_{i},{N}_{j},\ldots,t)\) is given by the so-called master equation
which can be solved analytically only in some very special cases (van Kampen 1981).
(2) The master equation as a model of scientific productivityThe productivity factor is a very important ingredient in mathematically simulating a scientific community evolution. One way to model such an evolution is through a dynamic equation which takes into account the stochastic fluctuations of scientific community members productivity (Romanov and Terekhov 1997) (Fig. 3.20). The main processes of scientific community evolution accounted for by this model are, beside the biological constraints (like the self-reproduction, aging of scientists, and death), their departure from the field due to mobility or abandon of research activities. Call a the age of an individual and let a scientific productivity index ξ be in incorporated into the individual state space; both a and ξ are being considered to be continuous variables with values in [0, ∞]. The scientific community dynamics is described by a number density function n(a, ξ, t), – another form of scientific landscape, which specifies the age and productivity structure of the scientific community at time t. For example, the number of individuals with age in [a 1, a 2] and scientific productivity in [ξ1, ξ2] at time t is given by the integral \({\int \nolimits \nolimits }_{{a}_{1}}^{{a}_{2}}{ \int \nolimits \nolimits }_{{\xi }_{1}}^{{\xi }_{2}}da\ d\xi \ n(a,\xi,t)\).
Schema of the master equation model for scientific productivity
A master equation for this function n(a, ξ, t) can be derived (Romanov and Terekhov 1997):
where w(a, ξ, t) denotes the departure rate of community members. If x(t) is a random process describing the scientific productivity variation and if p a (x, t∣y, τ) (with τ < t) is the transition probability density corresponding to such a process, then
The transition rate, at time t from the productivity level ξ, J(a, ξ, t) is by definition: \(J(a,\xi,t) ={ \int \nolimits \nolimits }_{-\xi }^{\infty }d\xi \prime\ \chi (a,\xi,\xi \prime,t)\). The increment ξ′ may be positive or negative. The balance equation for n(a, ξ, t) reads as follows
The term on the right-hand side, [1 − J(a, ξ, t)Δt]n(a, ξ, t), describes the proportion of individuals whose scientific productivity does not change in [t, t + Δt]; the integral term describes the individuals whose scientific productivity becomes equal to ξ because of increasing or decreasing in [t, t + Δt]; the last term corresponds to the departure of individuals due to stopping research activities or death. After expanding \(n(a + \Delta t,\xi,t + \Delta t)\) around a and t, keeping terms up to the first order in Δt, one obtains the master equation (3.43).
As the master equation is difficult to handle for an elaborate analysis, it is often reduced to an approximated equation similar to the well-known Fokker–Planck equation (Risken 1984; Hänggi and Thomas 1982; Gardiner 1983). The approximation goes as follows. Let
where the brackets denote the average with respect to the conditional probability density \({p}_{a}(\xi + \xi \prime,t + \Delta t\mid \xi,t)\). In addition, the following assumptions are made: (i) μ1, μ2 < ∞; μ k = 0 for k > 3; (ii) n(a, ξ, t) and χ(a, ξ, ξ′, t) are analytic in ξ for all a, t and ξ′. The additional assumption μ k = 0 for k > 3 demands the productivity to be continuous in the sense that as Δt → 0, the probability of large fluctuations ∣ξ′∣ must decrease so quickly that < ∣ξ′∣3 > → 0 more quickly than Δt.
When the above assumptions hold, the function n satisfies the equation (Romanov and Terekhov 1997):
If w = 0, (3.47) is converted to the well known Fokker–Planck equation. (3.47) describes the scientific community evolution through a drift along the age component and a drift and diffusion with respect to the productivity component. The diffusion term characterized by the diffusivity μ2 takes into account the stochastic fluctuations of scientific productivity conditioned by internal factors (such as individual abilities, labour motivations, etc.) and external factors (such as labor organization, stimulation system, etc.). The initial and boundary conditions for (3.47) are: (a) n(a, ξ, 0) = n 0(a, ξ), where n 0(a, ξ) is a known function defining the community age and productivity distribution at time t = 0; and (b) n(0, ξ, t) = ν(ξ, t) where the function ν(ξ, t) represents the intensity of input flow of new members at age a = 0 being set ν(ξ, 0) = n 0(0, ξ). In addition, n(a, ξ, t) → 0 as a → ∞.
The general solution of equation (3.47) with the above initial condition (a) and boundary condition (b) is still a difficult task. However, for many practical applications, a knowledge of first and second moments of distribution function n(a, ξ, t) is sufficient. Equation (3.47) can be solved numerically or can be reduced to a system of ordinary differential equations (Romanov and Terekhov 1997).
Finally, two additional problems that can be treated by the master equation approach can be mentioned:
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Age-dependent models where the birth and death rates connected to the selection are age-dependent (Ebeling et al. 1986, 1990)
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The problem of new species in evolving networks (Ebeling et al. 2006). On the basis of a stochastic treatment of the problem, the notion of ‘innovation’ can be introduced in a broad sense as a disturbance and/or an instability of a corresponding social, technological, or scientific system. The fate of a small number of individuals of a new species in a biological system can be thought to be mathematically equivalent to some extent to the fate of a new idea, a new technology, or a new model of behavior. The evolution of the new species can be studied on evolving networks, where some nodes can disappear and new nodes can be introduced. This evolution of the network can change significantly the dynamic behavior of the entire system of interacting species itself. Some of the species can vanish in a finite time. This feature can be captured effectively by the master equation approach.
Key point Nr. 12In deterministic cases, the system is robust against fluctuations: it follows some trajectory and the fluctuations are too weak to change it. When the fluctuations are important, then different trajectories for the evolution of the system become possible. To each trajectory, a probability can be assigned. This probability reflects the chance that the system will follow the corresponding trajectory. The collection of the probabilities leads to a probability distribution which can be calculated, in many evolutionary cases, on the basis of the master equation approach.
5 Space-Time Models: Competition of Ideas – Ideological Struggle
A further level of complication is to include spatial variables explicitly in the above models describing the diffusion of ideas. At this stage of globalization of economies, with several of its concomitant features, like idea, knowledge, and technology diffusion, to consider the spatial aspect is clearly a must. A large amount of research on the spatial aspects of diffusion of populations is already available. As examples of early work, papers by Kerner (1959); Allen (1975); Okubo (1980), and Willson and de Roos (1993) can be pointed out. From the point of view of diffusion of ideas and scientists, the previously discussed continuous model of research mobility (Bruckner et al. 1990) has to be singled out. Moreover, the model presented below is closely connected to the space-time models of migration of populations developed by Vitanov and co-authors (Vitanov et al. 2009a,b). In addition, a reproduction-transport equation model (see Fig. 3.21) can be discussed.
Relations between space-time models discussed in this chapter
5.1 Model of Competition Between Ideologies
The diffusion of ideas is necessarily accompanied by competition processes. One model of competition between systems of ideas (ideologies) goes as follows (Fig. 3.22). Let a population of N individuals occupy a two-dimensional plane. Suppose that there exists a set of ideas or ideologies \(P =\{ {P}_{0},{P}_{1},\ldots,{P}_{n}\}\) and let N i members of the population be followers of the P i ideology. The members N 0 of the class P 0 are not supporters of any ideology; in some sense, they have their own individual one and do not wish to be considered associated with another one, global or not. In such a way, the population is divided in n + 1 sub-populations of followers of different ideologies. The total population is: \(N = {N}_{0} + {N}_{1} + \ldots {N}_{n}\). Let a small region ΔS = ΔxΔy be selected in the plane. In this region there are ΔN i individuals holding the ith ideology, \(i = 0,1,\ldots,n\). If ΔS is sufficiently small, the density of the ith population can be defined as \({\rho }_{i}(x,y,t) = \frac{\Delta {N}_{i}} {\Delta S}\).
Schema of the space-time model for describing competition between ideologies
Allow the members of the ith population to move through the borders of the area ΔS. Let \(\vec{{j}}_{i}(x,y,t)\) be the current of this movement. Then \((\vec{{j}}_{i} \cdot \vec{ n})\delta l\) is the net number of members of the ith population/ideology, crossing a small line δl with normal vector \(\vec{n}\). Let the changes be summarized by the function C i (x, y, t). The total change of the number of members of the ith population is
The first term in (3.48) describes the net rate of increase of the density of the ith population. The second term describes the net rate of immigration into the area. The r.h.s. of (3.48) describes the net rate of increase exclusive of immigration.
Let us now specify \(\vec{{j}}_{i}\) and C i : \(\vec{{j}}_{i}\) is assumed to be made of a non-diffusion part \(\vec{{j}}_{i}^{(1)}\) and a diffusion part \(\vec{{j}}_{i}^{(2)}\) where \(\vec{{j}}_{i}^{(2)}\) is assumed to have the general form of a linear multicomponent diffusion (Kerner 1959) in terms of a diffusion coefficient D ik
Let some of the followers of the ideology P i be capable of and interested in changing ideology: i.e., they can convert from the ideology P i to the ideology P j . It can be assumed that the following processes can happen with respect to the members of the subpopulations of the property holders: (a) deaths: described by a term r i ρ i . It is assumed that the number of deaths in the ith population is proportional to its population density. In general r i = r i (ρν, x, y, t; p μ), where ρν stands for (\({\rho }_{0},{\rho }_{1},\ldots,{\rho }_{N}\)) and p μ stands for \(({p}_{1},\ldots,{p}_{M})\) containing parameters of the environment; (b) non-contact conversion: in this class are included all conversions exclusive of the conversions by interpersonal contact between the members of whatever populations. A reason for non-contact conversion can be the existence of different kinds of mass communication media which make propaganda for whatever ideologies. As a result, members of each population can change ideology. For the ith population, the change in the number of members is: ∑ j = 0 n f ij ρ j , f ii = 0. In general, f ij = f ij (ρν, x, y, t; p μ); (c) contact conversion: it is assumed that there can be interpersonal contacts among the population members. The contacts happen between members in groups consisting of two members (binary contacts), three members (ternary contacts), four members, etc. As a result of the contacts, members of each population can change their ideology. For binary contacts, let it be assumed that the change of ideology probability for a member of the jth population is proportional to the possible number of contacts, i.e., to the density of the ith population. Then the total number of “conversions” from P j to P i is a ij ρ i ρ j , where a ij is a parameter. In order to have a ternary contact, one must have a group of three members. The most simple is to assume that such a group exists with a probability proportional to the corresponding densities of the concerned populations. In a ternary contact between members of the ith, jth, and kth population, members of the jth and kth populations can change their ideology according to P i = b ijk ρ i ρ j ρ k , where b ijk is a parameter. In general, a ij = a ij (ρν, x, y, t; p μ); b ijk = b ijk (ρν, x, y, t; p μ); etc.
On the basis of the above, the C i term looks as follows (for more research of these types of population models see (Dimitrova and Vitanov 2000, 2001a,b)):
and the model system of equations becomes
The density of the entire population is \(\rho ={ \sum \nolimits }_{i=0}^{n}{\rho }_{i}\). It can be assumed that it changes in time according to the Verhulst law (but see the note after (3.56)!)
where C(ρν, x, y, t; p μ) is the so-called carrying capacity of the environment (Odum 1959) and r(ρν, x, y, t; p μ) is a positive or negative growth rate. When pertinent sociological data are available, the same type of equation could hold for any ith population with a given r i .
First, consider the case in which the current \(\vec{{j}}_{i}^{(i)}\) is negligible, i.e., \(\vec{{j}}_{i}^{(i)} \approx 0\) (no diffusion approximation). In addition, consider only the case when all parameters are constants. The model system of equations becomes
for
Let plane-averaged quantities and fluctuations (linear or nonlinear) be enough relevant. Let q(x, y, t) be a quantity defined in an area S. By definition, a plane-averaged quantity is \(\overline{q} = \frac{1} {S} \int \nolimits \nolimits {\int \nolimits \nolimits }_{S}dxdy\ q(x,y,t)\). Call the fluctuations Q(x, y, t) such that \(q(x,y,t) = \overline{q}(t) + Q(x,y,t)\). If the territory is large and within the stationary approximation, S can be assumed to be large enough such that each plane-averaged combination of fluctuations vanishes, such that \(\overline{{Q}_{i}} = \overline{{Q}_{i}{Q}_{j}} = \overline{{Q}_{i}{Q}_{j}{Q}_{k}} = \cdots = 0\). In addition to S being large and ∫ ∫ S dxdyΔQ k assumed to be finite, it can be also assumed that \(\overline{\Delta {Q}_{k}} = \frac{1} {S} \int \nolimits \nolimits {\int \nolimits \nolimits }_{S}dxdy\Delta {Q}_{k} \rightarrow 0\).
On the basis of the above (reasonable) assumptions, it is possible to separate the dynamics of the averaged quantities from the dynamics of fluctuations. As a result of the plane-average of (3.53), the following equations for the dynamics of the plane-averaged densities are obtained
Instead of (3.55) we can write an equation for \({\overline{\rho }}_{0}\) from the kind of (3.56). Then the total population density \(\overline{\rho }\) will not follow the Verhulst law.
Equations (3.55) and (3.56) represent the model of ideological struggle proposed by Vitanov et al. (2010). There is one important difference between the Lotka–Volterra models (Lotka 1912; Volterra 1927), often used for describing prey-predator systems, and the above model of ideological struggle. The originality resides in the generalization of usual prey-predator models to the case in which a prey (or predator) changes its state and becomes a member of the predator pack (or prey band), due to some interaction with its environment or with some other prey or predator. Indeed, it can be hard for rabbits and foxes to do so, but it can be often the case in a society: a member of one population can drop his/her ideology and can convert to another one.
In order to show the relevance of such extra conditions on an evolution of populations, consider a huge (mathematical) approximation – it might be a drastic one in particular in a country with a strictly growing total population. (Recall that the growth rate r could be positive or negative or time-dependent). Let r be > 0 and let the maximum possible population of the country be C. Consider more convenient notations by setting \(\overline{\rho } = N\); \({\overline{\rho }}_{0} = {N}_{0}\); \({\overline{\rho }}_{i} = {N}_{i}\) and assume that the binary contact conversion is much stronger than the ternary, etc. conversions. The system equations become
Reduce the discussion of (3.57) and (3.58) to a society in which there is the spreading of only one ideology; therefore, the population of the country is divided into two groups: N 1, followers of the “invading” ideology and N 0, people who are at first “indifferent” to this ideology. Let only the non-contact conversion scheme exist, as possibly moving the ideology-free population toward the single ideology; thus f 10 is finite, but b 10 = 0. Let the initial conditions be \(N(t = 0) = N(0)\) and \({N}_{1}(t = 0) = {N}_{1}(0)\). The solution of the system of model equations is
like the Verhulst law, but
with
in which Φ is the special function \(\Phi (z,a,v) ={ \sum \nolimits }_{n=0}^{\infty } \frac{{z}^{n}} {{(v+n)}^{a}}\); ∣z∣ < 1.
The obtained solution describes an evolution in which the total population N reaches asymptotically the carrying capacity C of the environment. The number of adepts of the ideology reaches an equilibrium value which corresponds to the fixed point \(\hat{{N}}_{1} = C{f}_{10}/({f}_{10} - {r}_{1})\) of the model equation for \(\frac{d{N}_{1}} {dt}\). The number of people who are not followers of the ideology asymptotically tends to \({N}_{0} = C -\hat{ {N}}_{1}\). Let C = 1, f 10 = 0. 03, and \({r}_{1} = -0.02\), then \(\hat{{N}}_{1} = 0.6\), which means that the evolution of the system leads to an asymptotic state in which 60% of the population are followers of the ideology and 40% are not.
Other more complex cases with several competing ideologies can be discussed, observing steady states or/and cycles (with different values of the time intervals for each growth or/and decay), chaotic behaviors, etc. (Vitanov et al. 2010). In particular, it can be shown that accepting a slight change in the conditions of the environment can prevent the extinction of some ideology. After almost collapsing, some ideology can spread again and can affect a significant part of the country’s population. Two kinds of such resurrection effects have been found and described as phoenix effects in the case of two competing ideologies. In the phoenix effect of the first kind, the equilibrium state connected to the extinction of the second ideology exists but is unstable. In the phoenix effect of the so-called second kind, the equilibrium state connected to extinction of the second ideology vanishes. In fine, the above model seems powerful enough to discuss many realistic cases. The number of control parameters seems huge, but that is the case for many competing epidemics in complex systems. However, it was observed that the values of parameters can be monitored when enough data is available, including the time scales (Vitanov et al. 2010).
Key point Nr. 13Space-time models are very appropriate for modeling migration processes such as the spatial migration of scientists, besides the diffusion of ideas through competition without strictly physical motion.
5.2 Continuous Model of Evolution of Scientific Subfields: Reproduction-Transport Equation
The change of subject of a scientist can be considered as a migration process (Bruckner et al. 1990; Ebeling and Scharnhorst 2000). Let research problems be represented by sequences of signal words or macro-terms \({P}_{i} = ({m}_{i}^{1},{m}_{i}^{2},\ldots,{m}_{i}^{k},\ldots,{m}_{i}^{n})\) which are registered according to the frequency of their appearance, joint appearance, etc., respectively, in the texts. Each point of the problem space, described by a vector \(\vec{q}\), corresponds to a research problem, with the problem space consisting of all scientific problems (no matter whether they are under investigation or not). The scientists distribute themselves over the space of scientific problems with density \(x(\vec{q},t)\). Thus, there is a number \(x(\vec{q},t)d\vec{q}\) working at time t in the element \(d\vec{q}\). The field mobility processes correspond to a density change of scientists in the problem space: instead of working on problem \(\vec{q}\), a scientist may begin to work on problem \(\vec{q\prime}\). As a result, \(x(\vec{q},t)\) decreases and \(x(\vec{q\prime},t)\) increases. This movement of scientists (see also Fig. 3.23) can be described by means of the following reproduction-transport-equation:
In (3.62), self-reproduction and decline are represented by the term \(w(\vec{q}\mid x)\ x(\vec{q},t)\). For the reproduction rate function \(w(\vec{q}\mid x)\), one can write
The local value of \(a(\vec{q})\) is an expression of the rate at which the number of scientists on field \(\vec{q}\) is modified through self-reproduction and/or decline, while \(b(\vec{q},\vec{q\prime})\) describes the influence exerted on the field \(\vec{q}\) by the neighbouring field \(\vec{q\prime}\). The field mobility is modeled by means of the term \(\frac{\partial } {\partial \vec{q}}\left (f(\vec{q},x) + D(\vec{q}) \frac{\partial } {\partial \vec{q}}x(\vec{q},t)\right )\). In most cases, (3.62) can only be solved numerically. For more details on the model, see Bruckner et al. (1990).
Schema of the reproduction-transport equation model of joint evolution of scientific fields
6 Statistical Approaches to the Diffusion of Knowledge
Solomon and Richmond (2001, 2002) have shown that the systems of generalized Lotka–Volterra equations are closely connected to the Pareto–Zipf probability distribution. Since such a distribution arises among other distributions and laws connected to the description of the diffusion of knowledge, it is of interest to discuss briefly the diffusion of knowledge within statistical approach studies. Lotka was its pioneer; a large amount of research has followed. Just as examples, one can mention the work of Yablonsky and Haitun on the Lotka law for the distribution of scientific productivity and its connection with the Yule distribution (Yablonsky 1980, 1985; Haitun 1982), where the non-Gaussian nature of the scientific activities is emphasized. Interesting applications of the Zipf law are also presented in (Li 2002). The connection to the non-Gaussian distributions concepts of self-similarity and fractuality have been applied to the scientific system in (Katz 1999) and (van Raan 2000). Several tools for appropriate statistical analysis are hereby discussed. At the center of the discussion Lotka law shall receive some special attention (see Fig. 3.24).Footnote 3
Statistical laws and their relationships as discussed in the chapter
As part of this discussion on the statistical approach, the analysis of the productivity of scientists can be considered. The information connected to new ideas is thought to be often codified in scientific papers. Thus, the statistical aspects of scientific productivity is of practical importance. For example, the Lotka law reflects the distribution of publications over the set of authors considered as the information sources. Bradford law describes the distribution of papers on a given topic over the set of journals publishing these papers and ranked according to the order in the decrease of the number of papers on a given topic in each journal. These laws have a non-Gaussian nature and, because of this, possess specific features such as a concentration and dispersal effect (Yablonsky 1980): for example, it is found that there is a small number of highly productive scientists who write most of the papers on a given topic and, on the other hand, a large number of scientists with low productivity.
In order to give an example of the connection between the deterministic and statistical approaches, remember that the Goffman–Newill model, discussed here above, presents a connection between the number of scientists working in a research area and the number of relevant publications. In Bettencourt et al. (2008), it was found that the number of new publications scale as a simple power law with the corresponding number of new authors: ΔP = C(ΔT)α where ΔP and ΔT are the new publications and the new authors over some time period (for an example 1 year). C is a normalization constant, and α is a scaling exponent. It has been demonstrated (Bettencourt et al. 2008) that the latter relationship provides a very good fit to data for six different research fields, but with different values of the scaling exponent α. For α > 1, a field would grow by showing an increase in the number of publications per capita, i.e., in such a research field, the individual productivity increases as the field attracts new scientists. A field with α < 1 has a per capita decrease in productivity. This can be a warning signal for a dying subject matter. It would be interesting to observe whether the exponent α is time-dependent, as is the case in related characterizing scaling exponents of financial markets (Vandewalle and Ausloos 1997) or in meteorology (Ivanova and Ausloos 1999). Policy control can thus be implemented for shaking α, thus the field mobility.
Key point Nr. 14There exist two different kinds of statistical approaches for the analysis of scientific productivity: (i) the frequency approach and (ii) the rank approach. The frequency approach is based on the direct statistical counting of the number of corresponding information sources, such as scientists or journals. The rank approach is based on a ranking of the sources with respect to their productivity. The frequency and the rank approaches represent different and complementary reflections of the same law and form.
6.1 Lotka Law: Distributions of Pareto and Yule
Pareto (Chen et al. 1993) formulated the 80/20 rule: it can be expected that 20% of people will have 80% of the wealth. Or it can be expected that 80% of the citations refer to a core of 20% of the titles in journals. The idea of the rule of Pareto is very close to the research of Lotka who noticed the following dependence for the number of scientists n k who wrote k papers
In (3.64), n 1 is the number of scientists who wrote just one paper and k max is the maximal productivity of a scientist.
where N is the total number of scientists. If we assume that k max → ∞ and take into account the fact that \({\sum \nolimits }_{k=1}^{\infty }1/{k}^{2} = \pi /6\), we obtain a limiting value for the portion of scientists with the minimal productivity (single paper authors) in the given population of authors: \({P}_{1} = {n}_{1}/N \approx 0.6\). Then, if the left and the right hand sides of (3.64) are divided by N, the frequency expression for the productivity distribution is: \({p}_{1} = 0.6/{k}^{2}\); \({\sum \nolimits }_{k=1}^{\infty }{p}_{k} = 1\). Equation (3.64) is called Lotka law, or the law of inverse squares: the number of scientists who wrote a given number of papers is inversely proportional to the square of this number of papers.
It must be noted that, like many other statistical regularities, Lotka law is valid only on the average since the exponent in the denominator of (3.64) is not necessarily equal to two (Yablonsky 1980). Thus, Lotka law should be considered as the most typical among a more general family of distributions:
where α is the characteristic exponent of the distribution, n 1 is the normalizing coefficient which is determined as follows:
Then the distribution of scientific output, (3.66), is determined by three parameters: the proportion of scientists with the minimal productivity p 1, the maximal productivity of a scientist k max , and the characteristic exponent α. If one of these parameters is fixed, it is possible to study the dependence between two others. Let us fix k max in (3.67). Then, we obtain the proportion of “single paper authors” p 1 as a function of α: p 1(α). When (3.67) is differentiated with respect to α, one can show that the corresponding derivative is positive for any α : dp 1(α) ∕ dα > 0. On the basis of a similar analysis of the portion of scientists with a larger productivity p k (α) as a function of α, we arrive at the conclusion: the increase of α is accompanied by the increase of low-productivity scientists. This means that when the total number of scientists is preserved the portion of highly productive scientists will decrease.
Let us show that the Lotka law is an asymptotic expression for the Yule distribution. In order to obtain the Yule distribution, one considers the process of formation of a collection of publications as a Markov-type stochastic process. In addition, it is assumed that the probability of writing a new paper depends on the number of papers that have been already written by the scientist at time t: the probability of the transition into a new state on the interval [t, t + Δt] should be a function of the state in which the system is at time t. Moreover, the probability of publishing a new paper during a time interval Δt, p(x → x + 1, Δt) is assumed to be proportional to the number x of papers that have been written by the scientists, introducing an intensity coefficient λ: p(x → x + 1, Δt) ∝ λxΔt. After solving the corresponding system of differential equations for this process, the following expression (the Yule distribution) for the probability p(x ∕ t) of a scientist writing x papers during a time t is obtained (Yablonsky 1980):
The mean value of the Yule distribution is x t = exp(λt). Let us take into account the fact that every scientist works on a given subject during a certain finite random time interval [0, t] which depends on the scientist’s creative potential, the conditions for work, etc. With the simplest assumption that the probability of discontinuing work on a given subject is constant at any time, one obtains an exponential distribution for the time of work of any author in the scientific field under study: \(p(t) = \mu \exp (-\mu t)\), where μ is the distribution parameter. The time parameter t which characterizes the productivity distribution, (3.68), is a random number. Then in order to obtain the final distribution of scientific output observed in the experiment over sufficiently large time intervals, (3.68) should be averaged with respect to this parameter t which is distributed according to the exponential law:
After integrating (3.69), the distribution of scientific output reads
where \(B(x,\alpha + 1) = \Gamma (x)\Gamma (\alpha x + 1)/\Gamma (x + \alpha + 1)\) is a Beta-function, Γ(x) ≈ (x − 1)! is a Gamma-function, and \(\alpha = \mu /\lambda \) is the characteristic exponent. For instance, if α ≈ 1 then \(p(x) = 1/[x(x + 1)]\). Let us assume that x → ∞ and apply the Stirling formula. Thus, the asymptotics of the Yule distribution (3.70) is like Lotka law (3.66) (up to a normalizing constant): \(p(x) \propto \Gamma (\alpha + 1)\alpha /{x}^{1+\alpha }\).
6.2 Pareto Distribution, Zipf–Mandelbrot and Bradford Laws
For large enough values of the total number of scientists and the total number of publications, we can make the transition from discrete to continuous representation of the corresponding variables and laws. The continuous analog of Lotka law, (3.66), is the Pareto distribution
which describes the distribution density for a number of scientists with x papers; x 0 is the minimal productivity x 0 < < x < < ∞, a continuous quantity.
Zipf law is connected to the principle of least effort (Zipf 1949): a person will try to solve his problems in such a way as to minimize the total work that he must do in the solution process. For example, to express with many words what can be expressed with a few is meaningless. Thus, it is important to summarize an article using a small number of meaningful words. Bradford law for the scattering of articles over different journals is connected to the success-breeds-success (SBS) principle (Price 1976): success in the past increases chances for some success in the future. For example, a journal that has been frequently consulted for some purpose is more likely to be read again, rather than one of previously infrequent use.
In order to obtain the law of Zipf–Mandelbrot, we start from the following version of Lotka law : \({n}_{x} = C/{(1 + x)}^{1+\alpha }\), where x is the scientist’s productivity, α is a characteristic exponent, C is a constant which in most cases is equal to the number of authors with the minimal productivity x = 1, i.e., to n 1. On the basis of this formula, the number of scientists r who are characterized by productivity x r < x < k max (k max is the maximal productivity of a scientist) reads
Depending on the value of x r , r can have values \(1,2,3,\ldots \) and in such a way the scientists can be ranked. If all scientists of a scientific community working on the same topic are ranked in the order of the decrease of their productivity, the place of a scientist who has written x r papers will be determined by his/her rank r. When the productivity of a scientist x r is found from (3.72) as a function of rank r, the relationship
This is the rank law of Zipf–Mandelbrot, which generalizes Zipf law: \(f(r) = c{r}^{-\beta };\) \(r = 1,2,3,\ldots \), where c and β are parameters. Zipf law was discovered by counting words in books. If words in a book are ranked in decreasing order according to their number of occurrences, then Zipf law states that the number of occurrences of a word is inversely proportional to its rank r.
Assuming that in Lotka law the exponent takes the value α = 1 and that in most cases C = n 1, one has \({x}_{r} = {n}_{1}/(r + a)\), where \(a = {n}_{1}/{k}_{max}\), r ≥ 0. Integration of the last relationship yields the total productivity R(n) of all scientists, beginning with the one with the greatest productivity k max and ending with the scientist whose productivity corresponds to the rank n (the scientists are ranked in the order of diminishing productivity; the rank is assumed to be a continuous-like variable):
This is Bradford law. According to this law, for a given topic, a large number of relevant articles will be concentrated in a small number of journals. The remaining articles will be dispersed over a large number of journals. Thus, if scientific journals are arranged in order of decreasing published articles on a given subject, they may be split to a core of journals more particularly devoted to the subject and a shell consisting of sub-shells of journals containing the same numbers of articles as the core. Then the number of journals from the core zone and succeeding sub-shells will follow the relationship \(1 : n : {n}^{2} : \ldots \).
Key point Nr. 15The Zipf–Pareto law, in the case of the distribution of scientists with respect to their productivity, indicates that one can always single out a small number of productive scientists who wrote the greatest number of papers on a given subject, and a large number of scientists with low productivity. The same applies also to scientific contacts, citation networks, etc. This specific feature (so-called hierarchical stratification) of the Zipf–Pareto law reflects a basic mechanism in the formation of stable complex systems. This can/must be taken into account in the process of planning and the organization of science.
7 Concluding Remarks
Knowledge has a complex nature. It can be created. It can lead to innovations and new technologies, and on this base, knowledge supports the advance and economic growth of societies. Knowledge can be collected. Knowledge can be spread. Diffusion of ideas is closely connected to the collection and spreading of knowledge. Some stages of the diffusion of ideas can be described by epidemic models of scientific and technological systems. Most of the models described here are deterministic, but if the internal and external fluctuations are strong, then different kinds of models can be applied taking into account stochastic features.
Much information about properties and stability of the knowledge systems can be obtained by the statistical approach on the basis of distributions connected to the Lotka–Volterra models of diffusion of knowledge. Interestingly, new terms occur in the usual evolution equations because of the variability and flexibility in the opinions of actors, due to media contacts or interpersonal contacts, when exchanging ideas.
The inclusion of spatial variables in the models leads to new research topics, such as questions on the spreading of systems of ideas and competition among ideas in different areas/countries.
In conclusion, the epidemiological perspective renders a piece of mosaic to a better understanding of the dynamics of diffusion of ideas in science, technology, and society, which should be one of the main future tasks of the science of science (Wagner-Döbler and Berg 1994).
Notes
- 1.
For example, at Gordon Research Conferences, it is forbidden to take written notes and to quote participant interventions later.
- 2.
For example, take the scientific disciplines and the number of publications as axes.
- 3.
Let us mention a curious and interesting fact connected to statistical indicators. Very interesting is the conclusion in Gao and Guan (2009) that the scale-independent indicators show that in the fast growing innovation system of China, research institutions financed by the government play a more important role than the enterprises.
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Thanks to the editors of the book for inspiring us into writing this work. The authors gratefully acknowledge stimulating discussions with many wonderful colleagues at several meetings of the ESF Action COST MP-0801 ‘Physics of Competition and Conflict’.
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Vitanov, N.K., Ausloos, M.R. (2012). Knowledge Epidemics and Population Dynamics Models for Describing Idea Diffusion. In: Scharnhorst, A., Börner, K., van den Besselaar, P. (eds) Models of Science Dynamics. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23068-4_3
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