Non-Linear Models of Social Systems
This article presents a non-linear modelling framework to capture the intricate dynamics of
social systems. Within this framework it is possible to capture both gradual and abrupt
transitions in social systems. In the context of social dynamics, the relevance of
self-organising behaviour and complexity theory are highlighted. Social systems are
essentially complex structures capable of self-organisation, exhibiting a rich variety of
features like bifurcation, hysteresis and multiple equilibria. Conventional linear approach is
too simplistic a framework to capture and deal with such phenomena. A number of specific
models are discussed. These describe the qualitative changes in a variety of social systems
they simulate and also explain the forces that govern their evolutionary behaviour.
KARMESHU, V P JAIN
I
Introduction
M
odelling as a theoretical framework has become an
essential feature of scientific inquiry. In its mathematical sophistication, it is capable of capturing the
dynamics of the complex behaviour of a variety of systems. The
origin and development of mathematical modelling framework
is based on the Newtonian paradigm which dominated scientific
thinking for nearly three centuries, the mechanical vision being
paramount in comprehending reality. These systems are linear
in character and, in principle, analysable into their constituent
parts: the dynamics of the system can be understood on the basis
of machine like ‘causal mechanisms’ [Davies 1987]. The success
of the paradigm in dealing with natural systems motivated social
scientists to extend the framework to study and analyse social
systems as well. However, the approach which came to be known
as ‘social physics’ did not show the same promise for social
system analysis, even though it had been used to great effect in
physical systems. It was realised that the linear approach which
implies that the effects or outcomes are proportional to the causes,
was too simplistic an assumption to underpin the complexity of
social processes.
The primary reason for such dissatisfaction was the inability
of these mechanical linear systems to account for the freedom
of choice the individuals enjoy and exercise in social systems.
Social scientists sought recourse to investigating statistical regularities present in social phenomena, e g, income distribution
resulting in Pareto distribution. This change in perspective was
driven by the developments in statistical physics. Unpredictability
at the individual level, combined with large numbers, produced
the regularities and patterns in the aggregate behaviour in living
systems as well, as had been observed in physical systems. In
social systems it allowed researchers to aggregate individual
choices into a macro system and to maintain the connection
between them. The mathematical arguments were based on the
application of central limit theorem which states that under
appropriate normalisation conditions the sum of a large numbers
of random variables tends to Gaussian distribution.
To be meaningful, these regularities should reveal the underlying conditions, factors and mechanisms which affect the state
of the world we observe. Mere description of the patterns of
regularities, no matter how meticulous and adequate, nor any
3678
congruent abstraction, provide any substantial insight unless they
also bring out the underlying processes at work. The efforts to
study statistical regularity gave a new boost to modelling social
phenomena and attempts were made to locate the explanatory
mechanisms for such patterns. These attempts resulted in the
development of stochastic models which structure the dynamics
of the system in probabilistic terms. A major flaw perceived in
much of the stochastic models of regression type was that they
were devoid of real social and economic content as too much
of what is of value was swept up into the stochastic component
of the model.
Even though the Newtonian equations were very general in
nature, appropriate for both linear and non-linear phenomena,
researchers generally avoided the investigation of non-linear
systems and resorted to linear approximation. For much of the
history of science, mathematics was limited to the study of linear
systems and scientists dismissed complex system behaviour simply
as annoying aberrations only to be pushed aside by adopting a
reductionist approach. All formal approaches to the study of nonlinear systems have become possible mostly due to the advent
of the awesome power of computing technology. The decisive
change over the past three decades or so has been to recognise
that nature is ‘relentlessly non-linear’ [Stewart 1989]. Non-linear
phenomena dominate a lot more of the physical and the social
world than was ever imagined. The exploration of non-linear
phenomena, over the past several decades, had a profound and
dramatic impact on scientific thinking and on the notions of
simplicity and complexity [Zellner, Kaeuzenkamp and Mc Aleer
2001]. In sharp contrast to conventional belief, it was observed
that simple non-linear deterministic equations might produce
unsuspected richness and variety of behaviour. What was
even more startling was that complex and seemingly chaotic
behaviour gave rise to ordered structures, to subtle and beautiful
patterns.
Over the past few decades, what scientists, in many and diverse
fields of study, have come to appreciate is that such non-linearities
are intrinsic to nature’s coherence and manifest in a variety of
behaviour. In fact, in the past few decades the boundaries that
delineate various disciplines of study have become fuzzy. Factors
contributing to this development are many, but a particularly
striking one is the emergence of the new area of complexity
studies which has brought together scientists from different areas.
Researchers have felt encouraged to move freely from one field
Economic and Political Weekly
August 30, 2003
Figure 1: Evolution of Number of Adopters
800
Figure 2: Solid Curve, with Singularity at Critical Point, d0 = dc = 1,
Corresponds to ((n(∞ )/N) = 0 for d ≤ 1 and
(n(∞ )/N) = 1–
1
d
for d>1
Adopters
600
400
n (∞)/N)
1.0
200
0
0
2
4
6
8
10
dc
Time
1
to another, gainfully employing tools and techniques developed
in one field for tackling problems in others.
The theory of complexity, as a new scientific paradigm, offers
a way of understanding which reveals that uncertainty, instability
and unpredictability are essential features of evolutionary processes of natural and social systems. The theory of complex
behaviour is variously understood within the framework of
synergetics, self-organisation, dissipative structures, catastrophe
theory and the like [Haken 1983, Nicolis and Prigogine 1977].
These fields are essentially non-linear and stochastic in character
and involve the interplay of fluctuations and determinism, exhibiting a rich variety of features like bifurcation, hysteresis and
multiple equilibria. The focus of enquiry has shifted from equilibrium and stasis to disequilibrium and change [Allen 1982].
The dissipative structures, in a state far from equilibrium,
describe how microscopic and local interactions give rise to
complex macroscopic structures. These systems show that state
variables in a system can change discontinuously, producing
phase transitions. The phenomena of ‘threshold’ and ‘critical
mass’ play a decisive role in such discontinuous jumps. The
system leaps abruptly and spontaneously into a new state only
at a precise point, where the system is in some kind of a suspended
animation between stability and total dissolution. It is impossible,
in practice, to predict which of the many potential forms it will
actually assume. But, once a pathway has been chosen and the
new structure comes into being its fate is sealed and determinism
dominates. These phase transitions may characterise a change
from order to disorder and vice versa.
These structures are able to not only maintain themselves in
a stable state far from equilibrium, but also to evolve. Selforganising systems have several characteristics which govern
their dynamics. They are, invariably, open rather than closed
systems, capable of exchanging energy and information with their
environment. Another essential feature of complex systems is
their adaptive nature. Species adapt to their changing environment. Birds are known to unconsciously organise themselves into
a patterned flock. People organise themselves into an economy
and markets adjust to changing prices, technological advances,
change in consumer preferences, etc. Such systems are, therefore,
not static but dynamic structures, in a state of continual flux and
acquire a distinct identity [Weidlich 2000].
The study of dynamical systems of the self-reinforcing or
autocatalytic type has been extensively studied in physical,
chemical and biological systems. There is considerable evidence
to show that social systems also display far from equilibrium
behaviour analogous to natural systems and have been appropriately termed as ‘dissipative social systems’ [Harvey and Michael
Economic and Political Weekly
August 30, 2003
2
3
4
d
1986]. “Indeed, as the new perception of reality reveals, social
systems themselves are biologically driven emergent processes,
whose deepest structures arise as evolutionary phenomena”
[Smith 1977].
We now go on to examine, in some detail, a number of specific
models which describe qualitative changes in the systems they
simulate, and also explain the forces that govern their evolutionary behaviour. The models discussed in this article are primarily
confined to temporal evolution of the examined systems. However,
spatio-temporal aspects of such an evolution, though extensively
addressed in the literature, do not form part of the article.
II
Diffusion of Innovation in a Social Group
An innovation is defined as an idea, practice or object that is
perceived as new by an individual. “This newness of an innovation may be expressed in terms of knowledge, persuasion or
a decision to adopt” [Rogers 1995]. Given the comprehensive
nature of the term innovation, it is not surprising that researchers
from varied disciplines like sociology, geography, economics,
marketing and management have contributed to the development
of modelling innovation diffusion process in order to understand
its dynamics. Some of the prominent themes that have attracted
the attention of scholars pertain to diffusion of technology,
culture, opinion, information, news, rumours and product. The
major focus of diffusion models centres around the exploration
of the temporal growth curve of the number of adopters of the
innovation, which emerges typically as S-shaped curve (Figure 1).
The basic innovation diffusion model is based on the twin
processes of (i) mass-mediated source as an external influence
and (ii) interpersonal interaction as an internal influence. Much
of the existing modelling framework regarding innovation diffusion is deterministic in nature [Mahajan and Peterson 1985].
In most of the cases, however, a stochastic model would be more
appropriate to describe the process. The need for a stochastic
perspective arises due to inherent uncertainties both at the micro
level, e g, heterogeneity of population and at the aggregate level
in the form of structural (intrinsic) and parametric (environmental) stochasticities. A new feature which accounts for loss
of interest (discontinuation) in an innovation has been incorporated
to make the model more realistic. These models are non-linear
in character because of the inherent mechanism of the interpersonal contact interaction [Karmeshu and Jain 1997].
We give here the stochastic formulation of the model for the
growth of the number of adopters n(t) in a social system of size N.
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3680
Figure 3: Bimodal Probability Mass Function of Adopters for
Different Values of Parameters with α = 0.001,
µ = 12α
α and N=100
(a)
0.04
β = 0.0460
0.03
0.02
0.01
0
Steady state probability mass function
The innovation diffusion process can be structured as a birth and
death process with transition probabilities
Pr{n→n+1in (t,t+δt)} = λnδt+o(δt) (n≥0)
Pr{n→n –1 in (t,t+δt)} = µnδt+o(δt) n(≥1)
(1)
where the transition rates are given by
λn= α(N – n) + β(n/N)θ(N-n), µn= µn; 0 ≤ n ≤ N
(2)
The parameter a and β measure respectively the transition
intensity of the mass-mediated source and the propagation intensity through interpersonal contacts. The parameter m characterises
loss of interest and m –1 represents the mean duration of time an
adopter remains continually active. The constant parameter q
signifies the state dependent propensity of the influence process
through interpersonal interaction. Various models of innovation
diffusion in the literature can be obtained as special cases of the
generalised model as given in eq (1), depending on the value
of the parameters m and q [Karmeshu, Sharma and Jain 1992].
The generalised model eq (1) with the assumption of m = q
and θ = 1 reduces to the generic model of new product diffusion
known as the Bass model (1969). It is pertinent to note that the
Bass model has been extensively used to study the product life
cycle and forecasting. Recently, attempts have been made to
incorporate population heterogeneity in innovation diffusion
models by treating parameters as random variables. These models
capture saddles which correspond to sequence of peaks and
troughs appearing in multi-model life cycle curves [Karmeshu
and Goswami 2001, Goswami and Karmeshu 2003]. The various
shades of the innovation diffusion models have been adopted and
applied successfully to examine innovation diffusion patterns in
a wide variety of areas. The significant case studies relate to patent
violation suit filed by Polaroid against Kodak, diffusion of high
yielding varieties of seeds and legal and shadow diffusion in the
context of software piracy. These examples can, of course be
multiplied. The model has also been adapted to determine the
optimal time for introduction of successive generation of new
technologies [Mahajan and Muller 1996].
Software companies take strong exception to software piracy.
Conner, Reavis and Richard (1991) have reasoned that software
piracy may not harm the economic interest of firms, where a
software product derives its value from the user base in the form
of network externality. An interesting observation emerged from
the study of Givon, Mahajan and Muller (1995) with respect to legal
and shadow diffusion for particular software products. They
found from the data sets analysed that the word of mouth mechanism, through both legal and shadow diffusion, contributed equally.
The generalised model with µ ≥ 0 and q =1 reduces to
Bartholomew’s model of diffusion of information. The interesting feature of this model is the emergence of the critical phenomenon in the absence of mass mediated process, i e, a = 0.
When the control parameter d (= Nb/µ) reaches threshold point
dc = 1 the system undergoes a phase transition (for deterministic
system) from no information diffusion to macroscopic diffusion
or no adoption level to macroscopic adoption level (Figure 2).
In the stochastic version of the model, the system experiences
a transition from disordered state to ordered state in terms of
entropy per mean spreader as one crosses the threshold point
[Karmeshu 2001]. Such a model has important implication for a
new social group to become a viable entity. All efforts below the
threshold point (critical point) get nullified and the social group
fails to sustain itself. But once the threshold point has been reached
there is a sudden spurt in the membership of the group and the
group takes off.
0.024
β = 0.0477
(b)
0.016
0.008
0
0.032
β = 0.0490
(c)
0.024
0.016
0.008
0
0
20
40
60
Number of adopters
80
100
The generalised model with m ≠0 and q=2 reduces to innovation
diffusion with multiple adoption levels which can be related to
stochastic cusp catastrophe and brings out a rich variety of
features like bimodality, sudden jumps and hysteresis. The model
is in sharp contrast to Bartholomew’s model of innovation diffusion (1982) where (in the absence of mass mediated process)
one only observes bifurcation in the form of threshold phenomenon as the structure of the model yields only two steady states.
We present in Figure 3 a case of bimodal distribution for
various values of β depicting transition from high probability
with low adoption (Figure 3a) to high probability with high
adoption level (Figure 3c).
The important policy implication from the model is a pointer
to designing control policies so as to ensure parameter trajectories
Economic and Political Weekly
August 30, 2003
to remain outside or within the cusp depending on the level of
adoption. Timing is crucial here. In David’s phrase: “There are
only narrow windows in which policies are effective”.
III
Positive Feedback Economics
Economists have begun to realise that the notion of positive
feedback in economic systems needs a reformulation of economic
theory [Arthur 1988]. Conventional economic theory is based
on the notion of ‘negative feedback’ in the form of diminishing
returns that ensures a predictable and a stable equilibrium position
for the economy. However, positive feedback is destabilising and
serious doubts have been cast on the unambiguous nature of
‘optimal behaviour’ and strategic rationality leading to
maximisation of some ‘potential function’ (profit for companies
and utility for consumers) in the form of attaining a unique
equilibrium, analogous to most efficient use and best allocation
of resources. It is not surprising that many social scientists,
particularly economists, now believe that economies can be
viewed as complex adaptive systems. They have made considerable headway in applying the framework of self-organisation and
non-linear dynamics to study the technologically-driven, contemporary economies. Such models offer considerable insight into
the dynamics of niche acquisition and self-reinforcing ‘lock-in’
mechanism.
With the advent of high technology industries like software,
pharmaceuticals and telecommunications, the scenario has dramatically changed. The success of these industries is shaped by
‘bandwagon effect’ triggered by increasing returns due to positive
feedbacks in the form of ‘niche acquisition’ and self-reinforcing
‘lock-in’ mechanism. These products require enormous expenditure on research and development to start manufacturing. But
once the production starts, for instance, in the case of a software
programme, it costs virtually nothing to replicate. In the real
world, a small fortuitous early lead, due to, for example, unexpected early sales, may help a particular brand or a firm (from
among the many early competitors) to capture a significant market
share. These small random events can accumulate and get amplified
by positive feedbacks to determine the eventual outcome, a
general tendency of success to breed further success. A key
concept in bandwagon market success story is the notion of
attaining the ‘critical mass’, in terms of gaining a sizable market
share, which provides the necessary breakthrough. Once the
critical mass is reached, the demand for the product becomes
subject to bandwagon effect. On the contrary, bandwagon products and services that fail to reach the critical mass (e g, picture
phone by AT&T in the early 1970s) have little success, and may
even fail altogether.
The bandwagon phenomenon is most discernible in the high
technology areas like microelectronics, computers and telecommunications which operate in network patterns that generate
economies of scale in the guise of ‘externalities’. Several technologies which have had spectacular success on the basis of
network externalities are ATM machines, fax, VCR and, of
course, the internet. Bandwagon benefit accrues to a user as a
result of others’ using the same product he or she uses; the value
of a network technology like the internet to a user increases in
proportion to the number of users. “The consumer then enjoys
a ‘rational exuberance’ as the user set expands” [Rohlfs 2001].
Bandwagon markets, usually, have multiple equilibria: the
Economic and Political Weekly
August 30, 2003
outcome selected may not be the best one, and worse, the inferior
technology may get locked-in into a dominant position which
is then difficult to dislodge as was the case in the success story
of VHS video recorder, even though it was widely believed that
Beta Max was technically superior. Both the systems entered the
market initially, with more or less the same market share. Sheer
luck and corporate manoeuvring tilted the scale in favour of VHS,
even though, initially, the fortunes fluctuated and the outcome
was any body's guess. Positive feedbacks on early gains accumulated enough of an advantage and helped consolidate the
position for VHS which captured virtually the entire VCR market.
Yet, at the outset, it would have been impossible to predict which
of the two possible equilibrium points would be selected. What
is even more pertinent is to note that the choice of the market
did not represent the best economic outcome. Several cases like
Qwerty keyboard and Windows operating system can be cited
to support such a proposition (Rohlfs 2001).
Arthur (1988) has adapted Weidlich’s model [Weidlich and
Haag 1983] of opinion formation in the context of market share
of American and German cars labelled as A and B. The dynamics
of the model is governed by the transition probabilities:
p AB ( x ) = ν exp[(δ + kx )(N − x )]
(3)
pBA ( x) = ν exp[− (δ + kx )(N + x )]
where 2N denotes the size of the market, v represents the frequency of switches. Here δ and k correspond to preferences bias
and conformity effect respectively.
It can be demonstrated that with small k, centralisation dominates and the stationary density function turns out to be unimodal.
As the parameter k increases, the transition from unimodal to
bimodal distribution (depicting the relative preferences of one
type of car over others) occurs.
Arthur (1988) has developed new insights into the crucial role
of positive feedbacks in the economy. He has demonstrated that
positive feedback makes the economy operate as a non-linear
system. The positive feedback mechanism manifests in the form
of increasing returns giving rise to a rich variety of properties
in the form of multiple equilibria, unpredictability, history
dependence, lock-in, inefficiency and asymmetry, where conventional economic theory simply fails.
IV
Replacement Dynamics and Technology
Substitution
The models of innovation diffusion considered earlier treated
an innovation in isolation. However, innovations are invariably
perceived as new alternatives by individuals, organisations and
societies. In this sense, an innovation can be treated as a replacement of an idea, product or technology by another. Joseph
Schumpeter, the noted economist, had advanced his famous
hypothesis of the process of “creative destruction” in the context
of technology substitution. The innovation induced ‘market power’,
sooner or later, peters out to be followed by a new burst of
innovations to put the system back into dynamic motion and the
new repetition of the process thus continues. Several examples
can be thought of to illustrate such a phenomenon: replacement
of agricultural activity by industrial activity, soap by detergent,
the Bessemer process by the open hearth for the production of
steel and carriage by automobiles. In fact history is replete with
such examples.
3681
Figure 4: Transition from Unimodal Distribution to Bimodal
Distribution as the Conformity Parameter k Increases
Figure 5: Successive Replacement of World Primary Sources
0.90
P(x)
101
Natural Gas
Coal
Wood
0.70
f /(1–f)
0.50
0.30
0.10
10–1
0.0
x
1.0
Fisher and Pry (1971) have advanced a replacement dynamics
model to study technology substitution. They examined replacement processes in a large number of industrial units and provided
significant insight into the behaviour of such a phenomenon.
Fisher and Pry postulated their hypothesis in terms of logistic
growth equation to describe market penetration of many new
products and technologies. The replacement hypothesis can be
structured as:
ln[ f (t ) /(1 − f (t ))] = α + β t
(4)
where f(t) and 1 – f(t) denote proportional market shares of
the new technology and the old technology, respectively. The
parameters α and β (>0) correspond to the intercept and the
replacement parameter. Fisher and Pry have verified this model
with the help of data sets of a large number of industrial replacement cases and the fits are remarkably good. They find that the
graph of ln[ f (t ) /(1 − f (t ))] against time comes out to be a
straight line. It is pertinent to note that the rationale for the
replacement hypothesis can be obtained from a model of selforganisation based on Lotka-Volterra equations [Batten 1982,
Karmeshu, Bhargava and Jain 1985]. It was observed that if a
new product or a process can manage to capture market share
of the order of 10 to 15 per cent, it will establish the lead and,
in all likelihood, completely dominate the market eventually.
Motivated by these findings Marchetti and Nakicenovic (1979)
and Marchetti (1991) investigated the world energy use and the
phenomenon of substitution on the basis of the logistic growth
model (Figure 5).
Technological and social evolution, asserts Montroll (1978)
is the consequence of a sequence of replacements of one technique
(idea, tradition, artefact) by another. The replacement hypothesis
has been extended to study the dynamics of the growth of
scientific knowledge within the Kuhnian framework as a selforganising system [Karmeshu and Jain 1995].
V
Urbanisation as a Replacement Process
Hermann and Montroll observed in the context of industrial
transformation of an economy that the proportion of agricultural
workers declines while that of non-agricultural workers increases,
corresponding to replacement hypothesis [Montroll 1987]. They
plotted on semi log paper the data of the ratio of non-agricultural
workers to agricultural workers for Sweden and the US for 100
years. It is remarkable the graph obtained turned out to be a
straight line. Treating urbanisation as a replacement of rural
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Nuclear
10–2
1850
1900
1950
Source: Marchetti and Nakicenovic 1979.
Sollus
2000
0.01
2050
Figure 6: Pattern of World Urbanisation based on UN Estimates
of Urban population U(t) and R(t)
0.2
0.0
–0.2
In [U(t)/R(t)]
–1.0
Fraction F
100
–0.4
–0.6
–0.8
–1.0
1940
1960
1980
2000
Year
Source: UN Population Projection (1980).
population by urban population, Karmeshu (1988) studied the
replacement dynamic process in connection with UN projections
(1980) of world urbanisation over the period 1950-2000 which
turns out to be a straight line (Figure 6).
Jain and Karmeshu (1982) have extended the idea of the
replacement hypothesis in the context of urbanisation in India.
The replacement hypothesis has been validated by Rao, Karmeshu
and Jain (1989) from cross-sectional time-series data from 11
countries and the fits are found to be extremely good. The
replacement hypothesis appears to be very promising as a model
of urbanisation. Inter country variations in the growth of
urbanisation are explained on the basis of the structure of the
economies governed by sectoral composition of the national
output. It is remarkable that the regression fits are so good even
Economic and Political Weekly
August 30, 2003
Figure 7: Resilience in Urbanisation Pattern in Japan
Urbanisation (per cent)
100
80
60
40
20
0
1921
1931
1941
1951
Year
1961
1971
1981
though the data is drawn from countries from a wide geographical
range, cultural and political diversity, and covers almost threequarter of the last century.
What is of even greater interest is the resilience in the stability
pattern exhibited in the evolutionary dynamics of urbanisation
which follows a stable path, structured by the pattern of development as an inevitable outcome. In the eventuality of a shock in
the form of an exogenous variable dislocating the economy,
urbanisation may be deflected from its path in the form of a slide
down or slide up. Japan exhibited such a feature during the period
1940-50. The Japanese economy suffered a setback during the
second world war due to nuclear bombing of Hiroshima and
Nagasaki, and after the shock was absorbed the time evolution
path of urbanisation restored itself as if the intervening period
simply acted as a fluctuation from the stable growth path. This
phenomenon is depicted in Figure 7.
The urbanisation pattern has emerged as the inevitable outcome, as the unfolding of the dynamics of the system. Any change
in the desired urbanisation scenario would, therefore, need a
structural change and would bear fruit only by a profound redefinition of the development paradigm adopted.
VI
Arms Race Model between Nations
Richardson’s arms race model, a pioneering work in the area,
describes the dynamics of armament build-up between two or
more nations antagonistic to each other [Richardson 1960]. Each
country seeks to build its military strength to match the strength
of its rivals. Given this basic premise, if a country takes the
initiative in the armament build-up then action-reaction pattern
ensues on a quid-pro-quo basis. However, the process cannot go
on indefinitely because armament build-up is subject to resource
constraints.
The dynamics of arms race between two nations can be modelled
in terms of differential equations for arms expenditure x1(t) and
x2(t) of nations 1 and 2 as
dx1
= a1 x2 − b1x1 + c1
dt
dx2
= a2 x1 − b2 x2 + c2
(5)
dt
The positive parameters a1 and a2 represent action-reaction
coefficients, b1 and b2 (≥ 0) denote the fatigue terms as strains
on the resources of the economy. The parameters c1 and c2 (≥ 0)
characterise the grievance terms which signify mutual distrust.
From the analysis of the model, one can obtain the condition
Economic and Political Weekly
August 30, 2003
b1 b2 > a1 a2 under which arms race stabilises. This model has
been refined by several researchers to make it more realistic.
Bhaduri (1981) has employed the model to investigate different
policy strategies like domination, deterrence, etc, on the outcome
of the arms race. Another variant of the model takes into account
not only the current expenditure, but also the stockpile (inventory)
as determinants of arms race [Karmeshu, Lal and Rao 1991].
One significant feature of arms race between super powers is
its asymmetrical dynamics. As noted by Huxley (1994): “Armament has come to play a vital part in western economies, particularly in the American economy, which depends completely on
the expenditure by the government of huge amount of money
on the manufacture of armament”. Thus, the preservation of the
system and armament build-up have become organic needs to
develop and sustain the economy.
Richardson's model is restricted in the sense that it cannot depict
bounded behaviour other than fixed points or periodic oscillations. For example, incorporating discrete time delays in the
response of one nation to another can exhibit deterministic chaos
[Campbell and Mayer-Kress 1997]. Saperstein (1984) reformulated the model to predict the outbreak of war in terms of discrete
Richardson type model. The model can be described as
xt+1 = 4ayt (1 – yt )
yt+1 = 4bxt (1 – xt )
(6)
where the variables x and y represent the armament expenditures
as a proportion of total resources of the respective countries X and
Y. As Saperstein observes: “The transition to chaos was associated
with unpredictable behaviour, crisis – unstable arms races, and
therefore with an increased risk for an outbreak of war”.
VII
Modelling Dynamics of Domestic Political Conflict
The evolution of domestic political conflict has been modelled
extensively by researchers in mathematical terms. These studies
can be broadly classified into two theoretical approaches on the
basis of the underlying variables. One approach is based on the
notion of 'relative depravation' [Gurr 1980], which links various
kinds of depravation induced discontent and political and social
conflict. The other approach centres round organisation of discontent to be the crucial factor [Tilly 1978]. Karmeshu, Jain and
Mahajan (1990) have formulated a model which integrates these
approaches into a synthetic framework.
The model has been structured to capture the dynamics of the
political conflict process in terms of the antagonistic interplay
of the regime (ruling group) and the challengers (hostile group)
engaged in open confrontation. These conflicts manifest at different levels and with varying intensities in a society. Various
social, economic and political factors may give rise to such a
conflict situation if they jeopardise the cooperative behaviour
and consensus around which a social system is organised and
sustained amicably. The model has been formulated in terms of
coupled non-linear differential equations:
•
HR = eR HC – φR HR
•
HC = d + eC HR – φC HC + rHR HC – cHC f (HR)
(7)
In this model HR and HC denote the hostility levels of the
regime and the challengers respectively. The discontent parameter d measures the extent of relative deprivation of the hostile
3683
Figure 8: Level of Challengers Hostility in Steady State vs
Discontent Parameter d*
H
R
P1
branch to the higher branch QP1R. The increase in hostility level
in this case is a clear departure from the normal situation in the
sense that the situation cannot be retrieved through the same path
(history dependence). The hostility level can be brought back
to the lower level branch only by reducing the parameter d* to
a much lower level d1 than the one that prompted the jump
initially (d2). It means that once the hostility situation gets out
of bound, it needs efforts out of proportion to restore the initial
position.
Concluding Remarks
P
Q1
o
d1
d2
d
Figure 9: Level of Challenges Hostility towards Regime
Depicting Successive (scaled) Discontinuous Jumps
group. The parameter φ, e, r and c represent the coefficients corresponding to the inhibiting factor, the exacerbating propensity,
reinforcing effect and cost per unit of collective action organised
by the hostile group. The cost per unit of collective action f (HR)
has been examined to be proportional to H Rn (n ≥ 0).
The model is posited in the dynamical system theory framework
to work out possible “growth trajectories” of the hostility level
in the system. The model formulated is a generalised model in
which the various outcomes of the repression/dissent nexus has
been explained as special cases of the same fundamental process.
The model brings out a rich variety of features including hysteresis in the form of discontinuous jumps in the level of hostility.
When multiple steady states exist in some situations, corresponding to the behaviour of hostility ( H C* ) with respect to
grievance parameter d*, it can be shown that the lower branch
OQ1P and the upper branch QP1R for H C* correspond to steady
states. The branch PQ is also a steady state but unrealisable. It
is interesting to observe that hostility level H C* increases in
response to increase in the grievance term d* but remains
confined to the lower branch OQ1P. The counter measures taken
by the regime also remain at the lower level. But, once the
parameter reaches the critical point (overloading the tolerance
threshold) d 2*, the level of hostility H C* leapfrogs from the lower
3684
Realising that the world we live in is essentially non-linear
need not necessarily lead to the irresistible conclusion that
complexity theory has all the promise, as is commonly thought,
of becoming the grand theory of everything. Many social science
phenomena depict stable and predictable behaviour where traditional theoretical framework may be gainfully employed as an
analytical tool with substantial pay-offs. However, the new science
of complexity and non-linear dynamics does enrich our understanding of systems involving complex interactions that defy
conventional approaches to scientific enquiry. The notion of
‘emergence’ as a property of non-linear systems has radically
altered our vision of reality and the scientific world view. The
new understanding that the problems of social systems may be
the deeper problems of the system’s complexity has far-reaching
implications from the perspective of social science policy.
A significant feature of non-linear modelling of social system
in general is that it helps organise and clarify verbal concepts
and notions. This in turn can help us ‘analyse’ in ‘detail’ the
often unsuspected implications of commonly used theories.
We remark in passing that dynamical processes with three or
more non-linearly interacting variables may exhibit chaotic phenomenon and possibly defy verbal and cognitive comprehension.
The fact that many social systems fall under this genre underscores the inevitability of sophisticated mathematical framework
for the study of their evolution. -29
Address for correspondence:
Dr Karmeshu
School of Computer and System Sciences
Jawaharlal Nehru University
New Delhi - 110 067
Dr V P Jain
Department of Economics
School of Correspondence Courses
University of Delhi, Delhi 110 007
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