Scope and Limits of Predictions by Social Dynamic Models: Crisis,
Innovation, Decision Making
Péter Érdi1,2
1
: Center for Complex Systems Studies, Kalamazoo College, 1200 Academy Street, Kalamazoo, MI
49006, USA and
2
: Department of Biophysics, KFKI Res. Inst. Nuclear and Particle Physics,
Hungarian Academy of Sciences, Budapest, Hungary. Email: perdi@kzoo.edu
Abstract
Models of social dynamics grew up from methods used on the physical and life sciences. Different
types of growth processes, among others those which lead to finite time singularities are reviewed. The
scope and limits of the analogy between neurological disorders (such as epilepsy) and financial crises
are analyzed by exploiting the concept of dynamical diseases. Governing rules of social dynamics
phenomena can be extracted from collected empirical data, and they have some predictive power. Data
for evolving patent citation networks, and annual budget changes are analyzed for generating
dynamical models.
Classification codes
Keywords:
social dynamics, feedback, causality, predictability, financial crisis, patent citations, innovation, budget
changes, power law distribution
1. Introduction
Sociodynamics, as a discipline grew up from the application of the theory of dynamical systems. (For
illustration, Montroll, 1978, Helbing 1995, Weidlich, 2002, 2006). The theory became a predominant
paradigm, and materialized the philosophical concept of causality by mathematical tools. This concept
tells that causes imply effects, consequently the present state determines the future. The general idea
behind sociodynamics is that we are able to characterize the state of a social system at a fixed point,
and there are rules, which govern the change of state of a system. If a rule acting on the actual state is
deterministic, there is only “one” future. In case of probabilistic rules, different futures could be
predicted with certain probabilities.
Dynamical models are able to describe both temporal phenomena (which assume perfect mixing of
“players”, i.e. spatial homogenity) and spatiotemporal processes (which lead to homogenization in
closed systems, and pattern formation – including waves – in open systems). Models of boundless and
bounded growths, of business and other economic cycles and of the emergence of chaotic patterns are
now extensively used. Complex systems theory offers a new perspective to understand the mechanism
of the emerging patterns. Due to the consequence of natural and social crises, such as earthquakes,
tsunamis, crises of financial markets, the occurrence of rare large (“extreme”) events are now in the
focus of extensive mathematical analysis (Sornette 2003, Albeverio et al. 2006).
In this paper, with the intention to follow the spirit of Mainzer (2009), who discussed the challenges of
complexity we further analyze the scope and limits of predictability in social sciences. As Mainzer
says, the research program to develop social dynamics is not physicalism. The assumption is that by
using the concepts of dynamical systems, spatiotemporal patterns observed in natural and
socioeconomical systems are supposed to be subject of causal explanations. One difference, what
should be mentioned here is, that while the rules behind the changes are supposed to be fixed in nature,
rules governing social dynamics can be subjects of change.
Some ingredients of complex systems (Érdi, 2007) are reviewed first (Section 2) by emphasizing the
role of feedback, circular causality and super-exponential growth related to uncompensated positive
feedback. We will argue that large stock market crashes are social analogues of epileptogenesis.
Social networks can be derived as the discrete, somewhat more microscopic approaches to continuous
models; the nodes might be people, organizations, communities, documents, etc. Dynamic network
models have two extreme forms: (i) propagation on networks, (ii) evolution of networks. An
application for this latter case is given in Section 3, where the dynamics of innovation is analyzed. How
did innovative technology develop? Are we in the position to predict the emergence of new
technological fields? Are there any generative rules for the evolution of technology and innovation and
can we extract them from data?
Patent systems should promote innovation. Advances in computer technology and the availability of
large patent databases have recently made possible to study many aspects of the patent system
quantitatively. Since patents and the citations between them can be seen as a growing network,
techniques from network theory can usefully be applied to analyze the patent citation networks. The
“nodes” of the network are patents and the “links” are citations of one patent. For each patent it is
required to cite all relevant prior art related to the patented invention. We were particularly interested in
the inverse problem: what is the best rule for describing the evolution of a given network, or a set of
networks, in a well-defined mathematical framework? (Csárdi et al, 2009) By extracting the rules
which govern the dynamics, there are some possibilities to make predictions.
Section 4 is about the dynamics of annual government budgeting. Current empirical data show that
frequency distributions characterizing budget change in the United States and elsewhere strongly
deviate from the Gaussian distribution. Year to year budget changes (adjusted for inflation) from many
nations are all distributed as power functions. We are building a model framework which takes into
account such assumptions as: (i) limited attention of political institutions; (ii) friction by collective
decision rules, resistance to change, (iii) error accumulation or strong signals, (iv) prioritized
preferences. Such kind of model frameworks have predictive power, at least in statistical sense.
2. Some Ingredients of Complex Systems
What are simple systems?
Before telling anything about complex systems, we should answer this question. Here are some
characteristic properties of simple systems:
 Single cause and single effect
 A small change in the cause implies a small change in the effects
 Predictability
Common sense thinking and problem solving often adopts the concept of`”single cause and a single
effect”. It also suggests that small changes in the cause imply small changes in the effect. It does not
literally mean (as sometimes it is mentioned) that there is a linear relationship between the cause and
the effect, but it means that the system's behavior will not be surprising, its is predictable. With a
somewhat more technical terminology, small changes in the parameters (or in the structure of the
system) do not qualitatively alter the system's behavior, i.e. the system is “structurally stable”.
Circular causality, feedback, growth processes
Circular
causalit
y
in
Figure 1. Illustration of linear and circular causality.
essence is a sequence of causes and effects whereby the explanation of a pattern leads back to the first
cause and either confirms or changes that first cause. The concept itself had a bad reputation in
legitimate scientific circles, since it was somehow related to use “vicious circles” in reasoning. It was
reintroduced to science by cybernetics (Wiener, 1948) emphasizing feedback. The concept of circular
causality is reflected also in the theory of reflexivity, an approach promoted in economics by George
Soros (1988).
In a feedback system there is no clear discrimination between “causes” and “effects”, since the output
influences the input. In particular, feedback between the rate of change (x') and the state (x) controls
the dynamics.
Growth processes describe the increase of different physical, chemical, biological, economical, etc.
state variables, such as size, chemical concentration, population density, price, etc. Assuming that other
variables do not influence the dynamics, we might set a polynomial differential equation:
x'=x
m
(1)
Different types of growth processes can be described by models relying on different assumptions for
the velocity of the specific process (Fig. 2.).
Figure 2. Unbounded growth processes: infinite and finite-time singularities
The solution of the equation with m>1 is
x t =x 0 t c−
t
tc
− 1/ m− 1
If m>1 The equation itself implements large, “higher-than-linear” (HTL) positive feedback, which
seems to be a general mechanism behind finite time singularities. It ensures that the “instantaneous
doubling time” tends to zero after a finite period. Finite time singularity roughly speaking means that a
dynamical variable gets an infinite value during finite time. This phenomenon is qualitatively different
from the exponential growth, when infinite value can be attained during infinite time only.
In terms of application, there are two types of approaches. If we build assumption-driven models (as
we did by setting Equation 1), and ask what can we say about the properties of the model (“about the
solution of the equation”), we speak about “direct problems”. Another set of problems is data-driven,
and asks for finding out the generating rules. The magic bullet of social dynamic modeling might be
related to the sophisticated combination of inverse and direct models.
From finite-time singularity to crash
The lack of the stabilizing effects of negative feedback mechanisms may lead to catastrophic
consequences. If there are no mechanisms to compensate for the effects of HTL positive feedback, the
processes lead to finite time singularities. Positive feedback seems to be a general mechanism
(Sornette, 2002) behind eruption of earthquakes, stock market crashes, hyperinflation, and epileptic
seizures.
Conventional economic theory, which is based on the equilibrium between demand and supply, says
that decreasing demand is compensated by increasing supply. Complex systems approach, however,
suggests that partially due to our susceptibility to imitate each others behavior, there is a period when
both demand and supply increase, which finally lead to singularities. Equilibrium theory works well
when the negative feedback effects expresses its stabilizing effect to the increase due to positive
feedback. While in “normal situations” the activities of “buyers” and “sellers” neutralize each other, in
“critical situations” there is a cooperative effect due to the imitative behavior (“everybody wants to buy
it since everybody else has already bought it”). So, the positive feedback is HTL. Super-exponential
increase (due to the irrational expectations) cannot be continued for “ever” due to the unstable nature of
this process and the increase is unsustainable. Consequently, it should be followed by a compensatory
process (i.e. stock market crash).
Uncompensated Positive Feedback: Epilepsy and Financial Crisis
Onset of two types of financial crises, namely stock market crash and hyperinflation (Sornette et al,
2003) are generated by the positive feedback between the actual and the expected growth rate. Large
stock market crashes are social analogues of big earthquakes and epileptogenesis (as all analogies, this
also has scope and limits). Sornette's view is that the stock market crash is not induced by single local
events (such as a raise in the interest values, or other governmental regulations), but due to the
unsustainable velocity of price increase. This speculative increase will take the system more and more
unstable situation. Finally, the market collapses due to any small disturbance. Unsustainable velocity of
price increase resembles to over-excitation of the epileptic brain. Indeed, the existence of vicious
circle-type mechanism characterizes both epilepsy and financial crisis.
The expression, “Seizures Beget Seizures”, for explaining the progression of epilepsy goes back to Sir
William Gowers, a British neurologist in the nineteenth century. Complex systems research puts
epilepsy into a broader perspective: As a complex systems disorder, epilepsy poses challenges for
identifying single mechanisms or processes that are required for epileptogenesis or progression. An
attempt to identify a list of specific molecular or cellular mechanisms responsible for progression of
epilepsy is likely to be difficult because emergent phenomena such as network synchronization are not
likely to be universally or linearly related to the specific defect (Sutula, 2004, see also Ben-Ari et al.
2008).
The stable dynamic operation of the brain is based on the balance of excitatory and inhibitory
interactions. The impairment of the inhibitory synaptic transmission implies the onset of epileptic
seizures. Epileptic activity occurs in a population of neurons when the membrane potentials of the
neurons are “abnormally” synchronized. Both experiments and theoretical studies suggest the existence
of a general synchronization mechanism in the hippocampal CA3 region. Synaptic inhibition regulates
the spread of firing of pyramidal neurons. In experimental situations inhibition may be reduced by
applying drugs to block (mostly) GABA-A receptors. (GABA is the most important inhibitory
transmitter). If inhibition falls below a critical level, the degree of synchrony exceeds the threshold of
normal patterns, and the the system's dynamics switches to epileptic pattern. Collective phenomena
occurring in neural networks, such as the case of disinhibiton-induced epilepsy have been studied
successfully by combined physiological and computational methods. (An excellent book for
summarizing the basic physiological mechanisms and computational models of epilepsy, see Soltesz
and Staley. 2008).
How economics can exploit the analogy? There are at least two fields, where the developed methods
can be transferred; detecting of precursors and new therapeutic strategies.
Methods for epileptic seizure prediction are subject of extensive research. While it is true that “...none
of the current epileptic seizure prediction methods can widely be accepted...” (Liu et al, 2009), still the
methods developed can be transferred from computaional neurology to economics. The analysis of
EEG curves based on dynamical approach offered methods to separate the different qualitative regions
before and during a seizure. The challenge is to define good measures. Recurrence plot analysis, see
http://www.recurrence-plot.tk/ (11 February, 2010), is a method to analyze complex dynamics. A measure
(called recurrence rate) proved to be useful (Li et al 2004) to separate the different qualitative regions
before and during a seizure. While by visual inspection only two stages (inter-ictal and ictal periods)
can be discriminated, the analysis helped to show the existence of a third stage, the pre-ictal period,
which separated the two stages previously mentioned. For details and figures see the original paper.
Another method for detecting precursors is Sornette's method based on the phenomenon called logperiodicity. He found that on a refined scale for different systems (from acoustic energy to stock
market price) the dynamics leading to 'finite-time singularity'' is modulated with log-periodicity. This
log-periodic behavior is characteristic for such kinds of critical events, when it is driven by slowly
accumulating, endogenous processes. (Of course, there are events which are responses to exogenous
shocks) From the technical point of view, for the stock market crash, the simple price equation showing
β
finite time singularity, log [p t ]=A+B t c− t , should be replaced by
log [p t ]=A+B t c− t
β
[
1+C⋅ cos ω⋅ log
T − c− t
T
]
.
Very formally, log-periodicity reflects the fact that critical exponents, more generally dimensions can
be complex, (of course, here “complex” means such kind of numbers which when squared give
negative values). The method was used to see that endogenous price increase due to positive feedback
mechanisms of speculative herding should be followed by crashes. This way the log-periodicity serves
as a signature of crash, since it precedes the actual drawdown.
Therapy
Epilepsy, and some other neurological and psychiatric disorders can be understood as dynamical
diseases. Such diseases are characterized by sudden changes in qualitative dynamic behavior (Belair et
al 1995). We offered (Aradi and Érdi. 2006) a new way of drug discovery by using dynamical models.
Efficient drugs (“therapeutic interventions”) should shift the system from pathological dynamical states
to normal ones.
What we learned from neuroscience is that there are some pathological mechanisms, which by blocking
the inhibitory effects, induce a vicious circle of disinhibition. “...This may result in aberrant activity
triggering epileptiform discharges...” (Rigo et al, 2002). A large class of antiepileptic drugs acts to
break the vicious circle, i.e. to restore the inhibition by acting on GABAergic synaptic transmission.
Subprime mortgage crisis is being seen now as a typical example of vicious circle:
http://www.viswiki.com/en/Virtuous_circle_and_vicious_circle (February 26, 2010). The self-regulated market
collapsed. What to do with it?
“...Now we're in the therapy phase. And what therapy are we using? Well, it's very interesting because
we're using two quite contradictory courses of therapy. One is the prescription of Dr. Friedman Milton Friedman, that is - which is being administered by the Federal Reserve: massive injections of
liquidity to avert the kind of banking crisis that caused the Great Depression of the early 1930s. I'm fine
with that. That's the right thing to do. But there is another course of therapy that is simultaneously
being administered, which is the therapy prescribed by Dr. Keynes - John Maynard Keynes - and that
therapy involves the running of massive fiscal deficits in excess of 12 percent of gross domestic
product this year, and the issuance therefore of vast quantities of freshly minted bonds. There is a clear
contradiction between these two policies, and we're trying to have it both ways. You can't be a
monetarist and a Keynesian simultaneously - at least I can't see how you can, because if the aim of the
monetarist policy is to keep interest rates down, to keep liquidity high, the effect of the Keynesian
policy must be to drive interest rates up.” (Niall Ferguson, 2009).
Each analogy has its limits of course. What we certainly learned from systems theoretical perspective
that both the healthy brain and a healthy macroeconomics have cycles. In both cases the cycles are
mostly endogenous (of course, self-sustained oscillation occurs in open systems only). Furthermore, in
both cases we see pathological rhythms (including arhythmia, i.e chaotic patterns).
Neo-Keynesian pioneers (such as Michael Kalecki and Nicholas Kaldor) in the 1930s suggested to use
nonlinear dynamical systems to explain the endogenous generation of business cycles. These cycles are
natural part of a working economy. The soul of the Kaldor model is that investment and saving are
nonlinear functions of income. The mathematical version of the original model leads to limit cycle
behavior, i.e. self-regulated oscillations, other variants of the Kaldor and other post-Keynesian models
may lead to chaotic patterns (for a review, see Rosser 2006). While chaos has benevolent functional
consequences in the neural information processing, it does not seem to be desirable in economic
systems. Therapeutic strategies should maintain the “natural” oscillations (superimposed to the growth
processes) built-in our actual world economical system. (Of course, economics is a moral science,
normative models of economics describe how a system should behave. See also Aruka, 2007).
There is some more analogy between economical crises and neurological and psychiatric diseases. The
world-wide economic crisis can be interpreted as collective panic disorder, as a group of German
psychiatrists suggested (Sperling et al. 2008). Based on this assumption the same authors in another
paper (Sperling et al, 2009), suggested therapeutic strategies. Steps of cognitive behavioral therapy can
be applied for both the banking system and the general public (including both media and individual
persons).
Social dynamics: inductive versus deductive
Francis Bacon (1561-1626) suggested that the only way of acquiring scientific knowledge is “true and
perfect Induction”. Whatever successful this program has been (nobody denies the role of thorough
observation and data collection), Bacon underestimated the role of the creative imagination, and the
formation of testable (falsifiable, using Karl Popper's terminology) hypotheses.
As it is well known, Kepler's theory relied on Tycho Brahe's observations, as searched for simple
algebraic relations for the motion of celestial bodies in defense of Copernicus' heliocentric concept, and
derived his laws empirically.
In the first law Kepler corrected Copernicus' hypothesis about the circular motion of the planets, stating
that their orbits are ellipses, and the Sun's center of mass is at one of the two foci of the ellipse. The
second law gives some quantitative characterization: The planets move so that the line between the Sun
and the planet (i.e. the radius vector) sweeps over equal areas in equal intervals of time in any place of
the orbit. Both the first and the second laws refer to the individual motion of the planets. The third law,
however, makes quantitative connection between the motions of two planets, by finding relationship
3/2
between the periods and the mean distances of the orbits from the Sun. Specifically, T ~ R is a
scaling law1, where T is the period of a planet's orbit, and R is the length of its semi-major axis.
It is interesting to note that Sornette's log-periodicity is also based on scaling laws. Discrete
scale-invariance governs the dynamics of the stock market.
1
Einstein explained the logical status of the Kepler's laws (Einstein 1927):
“But the principal thing is that these laws have reference to motion as a whole, and not to the question
of how there is developed from one condition of motion of a system that which immediately follows it
in time. They are, in our phraseology of today, integral laws and not differential laws.”
In Einstein's remark the term “motion as whole” means that Kepler's laws describe the orbit globally.
Kepler did not know the cause of the motions, i.e. he was not able yet to predict the motion of a planet
from its actual position and velocity. To put it another way, there were no known local rules to
calculate the motion. By defining the law of universal gravitation Newton was able to derive (and
slightly modify) Kepler's data-driven global, integral laws by deductive, local, differential rules. Based
on his theory it was possible to make predictions which led to the discovery of Neptune and Pluto.
How to apply this dynamical approach to social sciences? A lot of data has been accumulated in social
sciences, but the overwhelming majority of them have been evaluated and interpreted in a static or
equilibrium perspective. The a priori knowledge of the rules, which basically govern the dynamic
evolution of the systems is difficult: social dynamics is in the post-Keplerian and pre-Newtonian age
(and it may remain there). Still, at least in certain cases, we try to predict our social Neptunes and
Plutos2.
In the next two sections, sketches of examples are given for developing dynamical models by
combining inductive and deductive approaches.
3. Dynamic Modeling of Innovation: Development of the Patent Citation Network
Technological development is reflected in patents and in the citations between them. The network of
patents connected by citations is an imperfect but faithful representation of technological development.
A patent citing another implies that the cited patent represents a piece of previously existing knowledge
that the citing patent builds upon. Consequently, a directed link between patents can be interpreted as
an idea-flow.
Nassim Nicholas Taleb (Taleb 2005) analyses Benoit Mandelbrot's vision on applying the
concept of fractals to financial fluctuations: “This would make BM the true Kepler of the social
sciences. The analogy to Kepler is at two levels, first in the building of insights rather than mere
circuitry, second because you can step on his shoulders – the title of Kepler or “Newton of the social
sciences” is one so many thinkers with grand ideas have tried to grab (Marx for one aimed at being the
Newton of the sciences of man). I am not in the business of defining genius, but it seems to me that the
mark of a genius is the ability to pick up pieces that are fragmented in people’s mind and binding them
together in one, a meta-connection of the dots.”
2
Patent citation networks are special cases of evolving networks. The evolution of a network involves
the addition (and in the more general case even deletion) of edges and vertices. Gábor Csárdi
developed a mathematical framework to solve the ”inverse problem”, which asks: what is the rule
which gives the best description of the evolution of a given network, or a set of networks. (Csárdi et al
2007, 2009).
Modeling framework: states, and kinetics
How to define a mathematical model framework for describing the temporal evolution of the patent
network?
1. State variables. The first, obviously somewhat arbitrary step is to decide about the state variables. In
our model, each patent is described by two variables:
 k, the number of citations it has received up to the current time step, and
 l, the age of the patent, which is simply the difference between the current time step (as measured
in patent numbers) and the patent number. Because a given patent may cite more than one other
patents, several citations may be made in one time step.
These two variables define what we call the “attractiveness” of a patent, A(k,l), which determines the
likelihood that the patent will be cited when the next citation is made.
2. Temporal change. The evolution of the network is modeled by a discrete time, discrete space
stochastic dynamic system. Time is measured in patent number units, so that each “time step”
represents the citations made by a single patent.
In every time step the probability that an older patent will be cited is proportional to the older patent's
attractiveness multiplied by the number of citations made in that time step. We found that this simple
model gives a very good approximation of the observed kinetics of the growth of the patent citation
network.
The A(k,l) function determines the evolution of the network. It describes the average citation
preferences of the citing patents (the inventors and patent examiners in reality). We measured and
analysed A(k,l) for the United States patent system during the time period covered by our data and
found first that the parametrization by k and l consistently describes the average kinetics of the patent
citation network. Of course, there are many hidden relationships behind patent citations like
social relationships between patentees, the evaluations of the significance
of
the cited patent by the patent examiner and the technological relationship
between the citing and cited patents that our probabilistic approach does
not
capture. The way in which these “microscopic dynamics” are translated into the average behavior that
we observe remains an open question.
Direct and inverse problems can be defined, as Figure 3 shows.
Figure 3.The direct and the inverse problems. In the direct problem we
have the A(·,·) kernel function (and maybe other kernel functions for more
general networks) and we use a kernel-based network generator
algorithm to create an artificial network. The question is what kind of
structures can we obtain from a given class of kernel functions.
In the inverse problem, we examine a real network assembled from
observations. The kernel-based measurement algorithm takes this as the
input and gives a possible kernel function as the output. If the
measurement algorithm is good then the output kernel function is good,
i.e. it creates the observed network with high probability when used in a
kernel-based generator.
The inverse problem can be solved by the intuitive (frequentist) and the better founded maximum
likelihood methods.
Figure 4. Sections from the in-degree and age based maximum likelihood fitted kernel function for the
US patent citation network. Both plots have logarithmic axes.
The obtained free-form kernel can be well fitted with a function where the effects of degree and age are
separated, A d,l =Ad d ⋅ Al l . The degree-dependent part is the usual preferential attachment form:
A d d =d α +a , of course with a different exponent and different a parameter. We fitted the agedependent part with a double Pareto function, this is a unimodal function with an initial power-law
increase and a power-law decrease in the tail:
Al l =
{
1
tp
1
tp
β p− 1
if l≤ t p
α p− 1
if l>t p
}
About 96% of the goodness of the free-form model is preserved with the fitted form. Consequently, the
preferential attachment times double Pareto form is a very good model if one wants to model the
network based on in-degree and age.
Change in the Dynamics
Rules in social dynamics might be subjects of change. So far we assumed that a single kernel function
is valid during the whole development of the network. We showed that the dynamics of the patent
network changed around 1990 (supposedly) because of the introduced legal changes. We handled the
time dependence of the kernel function by using a sliding time-window of 500,000 patents, i.e. separate
kernel functions were estimated for each time window.
A model based on in-degree and age was studied. We used the maximum likelihood fitting method to
α
fit the parameters of a predefined shape: A d,l = d +a ⋅ Al l , where Al(l) is a double Pareto form.
Stratificatio
n
meant
that
the
more
citable
patents are
increasingl
y
more
citable than
the
less
citable
patents.
More than
400 classes
and six big
categories
were
Figure 5.Time evolution of the stratification parameter using in-degree and
defined.
age.
The
six
categories
are Chemical, Computers and Communications, Drugs and Medical, Electrical and Electronic,
Mechanical and Others: Stratification proved to be higher in some categories than in others, i.e.
.
Computers & Communication and Drugs and Medical Patent citation network is highly assortative
(i.e. say Chemical cites Chemical). In these categories a “few” highly important patents receive the
majority of citations.
Based on the assumption that emergence of new technological fields can be detected from nonassortative citatitons, recently, with Kinga Makovi and Péter Volf (in preparation) we started a new
analysis. The dynamics of the community structure of the non-assortative patents – those who have
citations outside their own industry – allows us to make predictions about the near future by describing
the evolution of clusters in terms of growth, shrinking, splitting and merging, which might be the
elementary processes found by Palla et al (2007). While the patent citation network is an open system
with continuous interaction with its environment, the scopes and limits of predictions are different from
that of the natural sciences. Patent laws, habits of patent examinees, the pace of economic growth and
many other factors influence the development of the patent network, and vice versa, the creation of
patents changes the environment. This limits the scope of predictions, which can be made to only a
statistical description: we can predict only patterns of behavior, but not specific events.
4. Dynamic Modeling of Public Policy Decision Making Processes
General remarks
Institutional decision making mechanisms can be analysed in terms of their information processing,
and/or as an input/output system. Annual budgeting has been studied well. Input data are supposed to
come from independent sources, and assumed to be characterized by Gaussian distributions. The
measured output shows a deviation from the Gaussian distribution, so the general problem is to uncover
the dynamic mechanism of the transformation during decision making. By knowing this mechanism we
might have some method to make predictions for the future of the national budgets, at least in a
statistical sense.
Lessons from political science
Bryan Jones and Frank Baumgartner have published a set of papers and books focusing on annual
budget changes (Jones and Baumgartner 2005). The budget is a scalar number annually (the sum of
many elements, i.e. categories). Each year a new point is generated , and the difference is the budget
change (adjusted to the inflation). So technically, data on annual budget changes define a set of positive
and negative numbers, as values of some random process.
Leptokurtic distribution of percentual budget changes were observed in a broad range of settings: small
increases and small decreases of budgets and budget components are the most frequent, but time to
time large increases and cut-offs are observed as well. Most frequency distributions appeared linear on
diagrams with double log axes, which strongly suggests a special case of kurtosis, the power-law
distribution.
Dynamic mechanisms leading to power laws are one of the hot topics in the theory and practice of
complex systems (for a short review, see Érdi 2007, 6.3.2.). In our case a plausible hypothesis is that
the skew distribution emerges due to the existence of institutional friction.
Figure 6. Category Midpoints of Percentage Budget Real Percent Change
Model framework
Traditional pure incremental models of budget changes (and their improved versions, Padgett 1980)
lead to Gaussian distributions. In a more advanced model (Jones at al 2001) the key was the recognition
that humans are disproportionate processors of information - that is, they are incapable of adjusting
actions directly to the incoming information that has implications for action. There are a variety of
reasons for this inability, but the most important is the inability of people to pay attention to multiple
streams of information at the same time. Instead, they juggle among income streams, with emotional
arousal being the central adjustment mechanism. This leads to disjoint and episodic shifts in attention
to the information streams they face. Punctuations in processing capacity, and hence in any action
taken, is a direct result of coping with numerous streams of information in a complex world.
In political systems, there are good reasons to expect that this tendency may be magnified.
Democracies have built-in error correction mechanisms that if policymakers are not appropriately
responding, they may be replaced. Predictive models were built based on the assumptions listed:
 limited attention of political institutions
 friction by collective decision rules
 resistance to change
 error accumulation
 prioritized preferences
Jones and Baumgartner (2005) developed a single threshold model to account for the strongly
leptokurtic budget output distributions they observed. Errors, in the sense of a mismatch between
information inputs and policy outputs, would accumulate in a policy area until the errors exceeded a
threshold, after which policymakers would attend to the problem and initiate action. Simulations based
on the model suggested considerable agreement, but problematic mismatches in some cases. In a
cooperation between political scientists and complex systems modelers, László Zalányi made further
improvements on the model (Jones et al 2009).
A dynamic model with an error-accumulating threshold-based nonlinearity and an infinite memory of
earlier signals was defined. A single threshold was set: if the accumulated input is below the threshold,
the response is small, if it is above than large.
The variables and parameters of the model can be summarized as follows:
Rt: response variable
St: input signal, (Gaussian random variable)
C: threshold
λ: efficiency parameter or friction
β: amplification parameter.
Error accumulation (incorporated as sum of the signals) is a key feature of the model. 0 < λ < 1, the
efficiency below the threshold C is lower than one, but definitely should be lower than β. This is one of
the characteristic point of the model: below the threshold there are smaller reaction than it is above the
threshold. The model has two versions. Error accumulation can or cannot be reset to zero,
Rt = β⋅ S t if ∑ S t' >C
λ⋅ β⋅ S t otherwise
where k is the time of the resetting (k=1 means no reset).
{
}
Simulations results for two situations (without and with resetting) are shown in Fig. 7. Appropriate
choice of the λ, β, C parameters makes this model reach the actual kurtosis values observed on real
data. C = 1, λ = 0.4 and β = 1 were chosen.
Figure 7. A leptokurtic distribution as a simulated response,
comparing to a Gaussian distribution with and without
resetting the counter
What is the difference between the two models? The counter spends a lot of time above the threshold
and also below in the first case. The higher the threshold the narrower the output distribution
(increasing the kurtosis) but the rarer the “overshooting”, i.e. the big changes. The resulted output
distribution is a sum of two Gaussians. In the second case the output is much narrower as we restrict
the “time” what the counter can spend above the threshold, and hence the weight of the wider Gaussian
(β) in the sum is smaller. However, there is one more difference. It is hardly seen that the output
distributions are asymmetric (and hence they are not exactly sums of Gaussians, but can be
approximated so very well). This asymmetry comes from the requirements of threshold crossing
direction: it can only be crossed in the positive direction to switch into the frictionless state. In
summary, the single threshold model cannot explain the higher frequency of the occurrence of some
extreme events.
A three-threshold model
The institutional decision making is obviously a multi-stage process. The simplest way to implement
the series of processes into a model is to increase the number of thresholds. Different frictions for
negative and positive counter (i.e. demand) were defined. Also it was necessary to smooth the positive
side of the output distribution which can be remedied by introducing a second, upper threshold for the
real overshooting reactions and a specific friction for the intermediate regime. This model gives
different responses in the negative counter depending on the sign of the actual input: the response is
different to an increase than to a decrease demand. By incorporating these modifications a much finer
fit to the budget change distributions was obtained.
C1=-4
C2=1.2
lower threshold
middle threshold
C3=3
λ1n=0.02
λ1p=0.03
λ2p=0.3
β=1
T
upper threshold
efficiency for the region of negative accumulated sum
efficiency for the region of positive accumulated sum
efficiency between the two positive thresholds
and
denotes the time of last resetting to 0 of the accumulated signal.
The response:
Rt =
{
λ1p⋅ β⋅ S t
∑ s t' >C 3 ,T=t
λ2p⋅ β⋅ S t if C 3 ∑ st' >C 2
λ 1n⋅ β⋅ S t if ∑ st' 0 ∧ S t 0
λ1n⋅ β⋅ S t if ∑ st' <C 1 ,T=t< 0
if C 2 ∑ st' 0 ∨ 0 ∑ s t' >C 1 ∧ S t
β⋅ S t if
0
}
Fig. 8 shows the US budget data with the fitted model distribution. The fittings were performed as
follows: the six-dimensional parameter space was numerically explored and the set of parameters
which gives the maximum likelihood of the dataset was selected. The three-threshold model fits are
better than that of the former models. It reproduces generally the shape, the asymmetric feature of the
distribution of the data, and gives significant probability to extreme events. However, the model is not
perfect: both the mean and standard deviations of the simulated distributions are bigger then the
momenta of the datasets. The reason of this deviation partially comes from the features of the model: it
is a continuous one which has to fit to a rather sparse dataset. There are several ways to extend or
modify the threshold models. To make the next step one has to investigate the decision making
procedure of the different countries in details due to the natural limitation of the annual budget data
sizes.
Figure 8. Transformation of the Gaussian
input signals to skew distribution fitted to the
US budget data. The upper figure shows the
tenfold enlarged input Gaussian sample
distribution and the budget distribution
histogram. The lower figure shows the
measured data histogram and the fitted model
output histogram.
Further questions
Dynamical modeling strategy offers a methodological tool to set and test hypotheses about the
budgeting process. Many open questions left to be answered. Can we design institutions that are more
efficient? Do we want to? Can we get direct measurements of decision making costs? Why do budgets
produce power laws when virtually all other distributions (such as GDP) are less extreme?
5. Conclusions
Dynamic systems theory proved to be very efficient in modeling phenomena of the physical and life
sciences. Local rules have the power to make casual explanations and predictions. Rules behind
different types of growth processes were reviewed. As Sornette argued, finite time singularities are
rather general models of different types of crises. Epileptic seizures and certain types of financial crises
are the results of uncompensated positive feedback. A little bit more is suggested: the concept of
“dynamical diseases” can cautiously be used for understanding undesired macroeconomic
irregularities, and offering therapeutic interventions.
As Kepler and Newton combined data-driven inductive and concept-driven deductive approaches,
governing rules of social dynamics phenomena might be extracted from collected empirical data. Such
kinds of rules might have some predictive power. We showed two examples. First, a project based on
data for evolving US patent citation networks were mentioned. The analysis led to the construction of a
stochastic kinetic model of the development of the citation network. We believe that it is possible to
identify technological branches at various times and predict the emergence of new technologies based
on the clustering of the citations. Second, comparative data for the annual budget changes are analysed
for generating dynamical models for the institutional decision making (i.e. policy making) mechanisms.
The basic phenomenon, valid over different countries, is that budget change distributions deviate from
the Gaussian. Since it seems to be very plausible (but data are missing) that the input to the policy
making system is Gaussian, the systems transforms the Gaussian distribution to either a leptokurtic, or
most probably a power law distribution. Institutional frictions and error-accumulation phenomena are
taken into account by setting a set of thresholds. This model framework also has some predictive
power.
Acknowledgments
Thanks for motivation to Prof Yuji Aruka: this paper grew out from a lecture presented at Chuo
University in October 2008. The “patent citation project” described is a cooperative effort with Gábor
Csárdi, Katherine Strandburg and Jan Tobochnik, recently with Kinga Makovi and Péter Volf. The
“budget change” project is collaborative work with Bryan D. Jones, Frank Baumgartner, and László
Zalányi. Thanks to Mihály Bányai for technical help. General support from the Henry Luce Foundation
is acknowledged.
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