Table of Contents
Representing Heterogeneity in Complex Feedback System Modeling:
Computational Resource and Error Scaling
Nathaniel Osgood.
Department of Civil and Environmental Engineering, MIT’
1 Introduction
Heterogeneity often plays an important role in shaping complex system behavior. If the evolution of
system components is characterized by non-linear relationships with respect to some factor that varies
over those components, it is not in general possible to capture system reference modes or behavior of
interest using simulations containing only aggregate representations of that factor.
The representation of heterogeneity is not only necessary for characterizing endogenous system
behavior; it is often critical for assessing the relative effectiveness of policy decisions at even the most
qualitative level. [Shephard Zeckhauser 1982] describes the essential role heterogeneity plays in
determining impact of many health policies. Heterogeneity has a strong impact on decision making in
domains as diverse as educational admissions, environmental management, electoral strategizing,
insurance, energy regulation, and many other areas.
Within the context of this paper, we will focus on cases where a modeler seek to understand and better
Manage some system that involves a real-world population of system components (e.g. people, cars,
towns) that is heterogeneous with respect to some attributes. For a health-oriented model to study the
impact of tobacco policy, this might be a population’s age and gender, weight, and smoking status. For
a model aimed at understanding the environmental impact of emissions-control standards, the relevant
attributes might be vehicle age and emissions profile. For students ina university, we might describe
their length of time at matriculation, area of specialization and possible (for diversity concems), gender
and economic or ethnic background. Ina model involving spatial effects, we may wish to characterize
interacting regions by geographical location or using some other coordinate system (distance froma
central city, etc.). Some of the attributes of interest — such as age, weight, family income level, vehicle
age and emissions profile— may be continuous, while others are discrete (categorical), such as gender
and area of academic specialization. In other cases, the appropriate designation of an attribute as
discrete or continuous may not be clear (e.g. smoking status, ethnic background).
This paper briefly surveys three common representations of heterogeneity in complex systems modeling
frameworks and analyzes the tradeoffs associated with these approaches on computational resources and
systemenor. Because these approaches for modeling heterogeneity are applicable to a wide variety of
modeling tools, the paper characterize the approaches in a generic manner.
The next section of the paper provides a brief motivation for capturing heterogeneity in complex system
modeling, including discussion of both the mathematical need to capture heterogeneity in the context of
nor: linear system evolution and a simple example which demonstrates the importance representing
* Address comespondence and reprint requests to Nathaniel Osgood, 77 Massachusetts Avenue Rm 1-376, Cambridge, MA
02139. Email: nosgood@mit.edu
heterogeneity in a stylized simplification of a health policy model. The following two sections briefly
describe three popular ways in which heterogeneity can be captured in systems modeling frameworks.
This is followed by a simple analysis of the computational resources required by each approach. The
penultimate section analyzes the scaling of exor with the number of distinct types of heterogeneity that
need to be captured. The conclusion summarizes these results and discusses their implications for
modeling.
2 Motivations for Representing Heterogeneity
2.1 Implications of Non-linearity for Aggregation
One of the most important motivations for the representation of heterogeneity in a model is non-linearity
of system evolution with respect to system state. It is well understood in the system dynamics
commumity that non-linearity of this sort is widespread and effects greatly complicate the analytic study
of systems by preventing easy decomposition of a system into readily understood and pieces or extemal
inputs into eigenfunctions, and that non-linear systems therefore require the use of numerical integration
of the broader system to gain an accurate picture of system evolution. What is less widely appreciated
inthis community is that non-linearity in the evolution of elements of a population (e.g. people, cars,
etc.) often requires that models of such a system represent important components of heterogeneity
among the members of that population if they wish to accurately capture qualitative system behavior.
Consider a complex deterministic system that includes a population of n components of the same
general class (e.g. people, cars, etc.), each of which exhibits some system state comprised of d
characteristics (for example, people with different ages, weights, genders and smoking status, or cars
with different levels of fuel efficiency and age). We terma given member of this population x (where
¥ is dx1) and take the general case vadisctinosttiymathentt det eats
population is govemed (as above) by a shared but possibly non-linear function f: x = £(%)
Now consider the evolution of the entire population X = {%}; weuse X to denote Ww" Fromthe
above, it follows directly that E(X) = E( £(X)). Consider now the situation for a nor-linear function f.
For sucha function, in general f (4+) £(&)+ £(6) and thus
¥ f&) “%
#f oT = £(E(X)).
E(f(X))=E
In spite of this difficulty, it is common to encounter aggregated non-linear models that attempt to
characterize system evolution using just the mean states or attributes of the entire population or broad
divisions thereof. An example with which the author was associated (and a motivation for the example
below) is described in [Tobacco Policy Model] Many of the models suffering from this problem are of
high quality and have made their way into prominent model collections ([Kaibab] [Flowers] [Epidemic]
[Resources] [Bass Diffusion]). Perhaps the most familiar type of neglect in the treatment of
heterogeneity can be found in the widespread tendency to represent of a non-linear stochastic system
using the mean trajectory of the system, and the calculation of non-linear effects on the basis of this
mean trajectory.
As will be shown below, the consequence of ignoring heterogeneity in non-linear systems is inaccuracy
and — in some important cases —failure to capture important qualitative components of system behavior.
2.2 A Simple Example
In this section, we examine a very simple but realistic example of a non-linear system and demonstrate
how significant systematic inaccuracy can result when simulation is conducted with respect to
attribute values.
As the basis for this example, we take the characterization of smoking behavior — a subject that has
served as the basis for several policy-oriented system dynamics models [Tobacco Policy Model]
[Roberts, Homer et al]. Ina subdivision common to their field, [Tobacco Policy Model] characterize the
population into three categories based on current and past smoking status: Never smokers, current
smokers, and former smokers. Following this model, we make use of a Markovian model structure
where individuals in a given smoking category have a certain probability density of changing their
smoking behavior over time and transitioning to a new smoking state. Thus, never smokers are modeled.
as initiating smoking (“uptaking’”) with probability density u, current smokers have a probability density
c of quitting (“‘ceasing’) smoking, and former smokers exhibit probability density r of relapsing into
smoking.
The resulting system can be characterized as a classical set of 3 first-order linear differential equations:
_ {a 0 0
—=/u -q r]|x
at 0 q-r
N
Where xX is the state vector consisting of the count of never, current and former smokers (X=| C |).
F
N
For simplicity of this example, we will assume an initial state vector of | 0 | (ie. that all of the
0
population starts in the never-smoker state).
It is worth remarking that while the differential equation is linear with respect to the smoking status state
vectors, it is non-linear when both the state vector and (individually dictated) transition probabilities are
taken as variables. More specifically, while the behavior of an individual i is described by a % = AX
(where that contents of A are dictated by that individual’s values of u,r,q), that of individual j may be
described by a different equation x = Bx, (where AB), because the transition probabilities for each
individual may differ” Thus E(x + X,) = E(AX + BX,). In general, unless the two individuals have
? Ttis widely recognized in the health community that individuals vary widely in likelihood of changing smoking-rdated
behavior.
identical transition \ probebilities (and thus A=B), there is no constant matrix C such that
f,%})- XK o(354)- CE({X, x; }) for all individual states ¥ and %.
Given this non-linearity, the remainder of the section will examine the implications of heterogeneity for
this particular system as it occurs in two parameters and affects two respective types of systems
behavior. In particular, we will examine the effects of heterogeneity in initiation rates and its effect on
dynamic behavior, and the implications of heterogeneity in relapse rates and its effect on the system
steady state. In both cases, we demonstrate the systematic discrepancies that result from simulation or
analytic solutions that assume system transition rate averages (i.e. which model the system at an
aggregate level, assuming that the aggregate parameters (u, r, q) are associated with the system-wide
averages for these parameters).
Consider first the case of initiation behavior. As represented in this stylized model, the stock of never
smokers is associated with a single (out) flow. The behavior is thus the familiar one of a first-order
delay, consisting of the mean interarrival time of a Poisson process. Considering for the moment this
component of the system in isolation. We have
dN
—=-uN
dt
The solution is the declining exponential, where c is a constant to be determined by the initial
conditions:
N on cet
To emphasize the dependence of N on parameter u, we will write this as N, below.
Consider a simple case of a population of No=1000 in which half the population (500 individuals) have
uptake probability density .05%/month and half 1.5%/month. We will contrast the difference between
the exact solution to this simple situation and the solution that arises from neglecting the heterogeneity
in the population and modeling it as a homogeneous population with uniform uptake transition.
probability density 1% (a number identical to the mean population transition density for the population).
As demonstrated by Figure 1, the numerical behavior between the two cases diverge. In particular, the
use of an aggregated transition probability in the model yields a systematically pessimistic estimate of
the number of individuals remaining as current smokers.
The systematic nature of this difference arises directly from the system heterogeneity. It reflects the fact
that the population of never-smokers is changing over time, and in particularly is being rapidly drained
of the “high risk’ population with higher uptake rate, leaving an increasingly dense population of
individuals at low risk of smoking. Within a homogeneous population (as simulated by the model using
aggregated parameters), the aggregate probability of uptake would remain constant over the lifetime of
the simulation, and a considerably larger quantity would begin smoking, leaving fewer as remaining
nonsmokers. In the presence of sufficiently large heterogeneity, the qualitative behavior of these two
approaches may differ (with the heterogeneous model continuing for a prolonged period with a sizeable
fraction of the “low-risk” population remaining never-smokers, while the never-smoker population in
the aggregate model is rapidly and uniformly depleted).
The difference also reflects the underlying mathematics of the scenario: The analytic solution for the
count of never smokers N(t) as a function of time depends non- linearly on the value of u. Given that the
N,+N,
exp function is convex, basic calculus and Simpson’s rule tell us that N<5>. a
NS for agg(red) vs. disagg (green)
aa bo 80 to 120 740 760 780 200
!
Figure 1: Count of Never Smokers under Assumption of Mean Transition Probabilities (Red) and Heterogeneous
Transition Probabilities Green)
The discussion above has focused on the dynamics associated with a particularly simple subcomponent
of the model, exhibiting only exponential decay. Because of the simplified nature of the model — in
which the never smoker stock is only drained and never replenished — it is clear that despite the
significant differences in uptake rate between an aggregated and aggregated model, the asymptotic
behavior of the never smoker stock in both models is identical: The stock depletes to zero over time.
We now tum to examine how choice of an aggregated model introduces systematic inaccuracies into our
understanding of the steady-state behavior of the system.
N
Consider again the system of differential equations shown above. Foran initial population vector | 0
0
exhibiting homogenous transition parameters (u,r,q) the stylized model gives a steacly state population
distribution of Nj Fea 7 in other words, a situation in whicha fraction of the population
r+q
consists of current smokers, and. ss consists of former smokers. It bears mentioning that these
r+q
fractions are non-linear in both r and q, and thus we would not expect the steady state of mean transition.
values to be the same as the mean of the steady state of the full distribution of transition values. To
quantify this difference, suppose that we have a heterogeneous population in which one half of the
population is at “high risk” of relapse whenever they are former smokers, and the other half is at low
risk whenever they have ceased smoking. We will now contrast the implications of modeling sucha
population using a model assuming a uniform relapse rate across the entire population (a relapse rate
equal to the mean of the actual relapse rate over the entire population) , and a disaggregated model
which accurately characterizes the relapse behaviors of the high-risk and low-risk groups by following
each separately. Noting that only one of the steady-state populations of current or former smokers needs
to be specified to determine the other, we will henceforth focus our attention on the current smoker
count.
In this stylized model, the exact steady-state behavior of the heterogeneous system can be obtained by
simulating the two populations independently and combining the results. Thus, we if have half of the
population (No/2 “high risk” individuals) associated with relapse risk r+o. and the other half of the
individuals with relapse risk ra, we arrive at an steacly state in which the number of current smokers at
. r+m-a’
steady-state [SS SS
8 No@rqra)(r+qua)
homogenous rate of relapse uniform across the population would yield a steady-state estimate for the
current smoker count of Ne Thus, despite the fact that both the aggregated and exact estimates
made exhibited identical mean relapse rates, a systematic exor of
2
lue N, ___© __ig introduced between the two estimates (respectively) of the size
(r+ q)(r+ q+a)(r+ q-a)
of the steady-state current smoker population.
Given that r,q, and « are all positive, this difference will be positive as long as u<{r+q). Thus, givena
real-world Markovian system such as that shown, a model that ignored heterogeneity in relapse rates
would consistently over-estimate the number of individuals who remain as current smokers and under-
estimate those who successfully remain former smokers.
The value of the difference in magnitude depend heavily on the values the transition probabilities;
Figure 2 shows the absolute error expressed as a function of the fractional heterogeneity (a/r). The
proportional error (error in the estimate of the current smoker population as a fraction of the current
smoker population) will generally be several times higher than that shown in these figures.
By contrast, a model built around the assumption of a
Absolute Error as a Function of Fractional Heterogenedy int
0.16
0.14
0.12
01
0.08
0.06 y
0.04 a
0.02
0 02 04 06 08 1
kractionalAlpha
Figure 2: Inacouracy of Steady-State Estimate of Current Smoker of Population as a Function of the Fractional
Population Relapse Rate Heterogeneity (a /r). The Total is Expressed as Fraction of Population for Sample Data with
q7sL
This section has demonstrated that for a very simple system that is non-linear with respect to population
parameters, making use of a model that fails to capture heterogeneity of the system with respect to
model parameters can introduce potentially significant errors into the expectations of system evolution.
It is only through the disaggregation of models of such system with respect to key attributes that close
fidelity of model simulation to real-life system evolution can be expected. The body of this paper
examines the accuracy and computational costs associated with three approaches to representing
heterogeneity.
3 Three Techniques for Representing Heterogeneity
In this section, we briefly survey three distinct means for capturing system heterogeneity within a
model. Suppose that we have an underlying population of size n of some general class of component
(people, cars, etc.) in the system of interest. Suppose further that each of these population members is
associated with d state elements (termed here “characteristics” or “‘attributes”) the heterogeneity of
which we believe likely to have a strong impact on the understanding system behavior and to impact
policy selection: For example, a population of people with distinct age, gender, weight, and smoking
status (which we take to be categorical for simplicity’s sake). There are three primary ways to
characterize heterogeneity in such frameworks:
e Using Stocks Disaggregated by Attribute Value. In this case, the user represents components
holding different ranges of attributes values using distinct stocks in the model. The framework
maintains distinct levels for each stock.? Each such level represents the quantity of that
particular type of component within the system. Although its exact form differs in different
3 Note that we confine this discussion to stocks, but the use of arraying extends to other variables. Other types of
formulations can either be represented as stocks (e.g. delayed fixed) or need not be represented at all as part of system state
vector, and can simply be computed from the state vector.
software packages, arraying is frequently used as a convenient means of disaggregating stocks
and auxiliary variables according to attribute value (for discrete attributes) or range (for
continuous attributes). This technique allows for convenient representation of common names
and common (vectored) equations shared between instances of an arrayed stock, but is
conceptually the same as explicit disaggregation.
e Agent-Based Disaggregation. Agent-based modeling (of which we consider micropopulation
models [Ackerman] a subset) is a general modeling technique that represents the set of
components of a systemusing a set of agents. These agents are associated with sets of user-
defined attributes shared between member of a population of a particular class of component.
For example, to draw on the examples introduced above, an agent might represent a particular
person in the population, and associated with particular values for the attributes of current
smoking status, and other characteristics that might influence likelihood of initiation, relapse and
cessation (such as age, gender, physiologic responsiveness to nicotine, etc.). An agent to be used
ina study of vehicle pollution might represent a particular vehicle, with attributes of age and
tailpipe emissions profile. In general, a particular agent can have arbitrarily many such.
attributes.
e UsingCo-flows. Co-flows are a common technique for capturing important statistics
(population means) arising from heterogeneity with respect to population characteristics without
the burden of disaggregation.
Each of these techniques will be briefly described below.
3.1 Population Stocks Disaggregated by Attribute Value
The representation of heterogeneity using disaggregated stocks is the most straightforward and likely the
most common of the techniques for representing heterogeneity in traditional system dynamics models.
It is also the one on which this paper will focus the greatest attention.
We can choose to represent this population within stocks disaggregated according to these attributes. In
order to accomplish this process, we create a collection of stocks, each occupying some particular
volume in d dimensional space. Each dimension of this space is associated with a particular attribute;
the attribute (state element) values of each member of the population can be conceptualized as
coordinate for this member in this d dimensional space.
Within this representation, the level of each such stock will represent the number (or, altematively, the
fraction) of individuals in the population whose attribute values in d are such that they fall within the
volume of interest. In other words, the stock counts the number of individuals in the population that
share particular sets of values for every one of the attributes (age, gender, weight and smoking status).
For continuous attributes and dimensions (e.g. age, weight), a particular stock will count members of the
population whose state with respect to that characteristic falls into a particular contiguous range of
values for that attribute (e.g. people ages 50 through 59) and who also have the appropriate values
specified required for the other attributes to fall in this category.
For discrete attributes (e.g. gender and categorical smoking status), a stock might represent population
members with a particular value for that attribute (e.g. females) and appropriate values for all other
attributes. Altematively, a stock might be associated with all population members who belong to a
particular countable set of values for that attribute (e.g. current and former smokers) and appropriate
values for all other stocks.
The representation discussed here is also very well suited to changes in population size and composition.
Appearance and disappearance of population members can be readily represented. For example, births
ina living population can be represented as flows into stocks representing the youngest members of the
population, with appropriate other attributes; mortality in a living population model simply requires
outflows from the appropriate stocks. Population members who change their attribute values (e.g.
members of the human population who age, gain or lose weight or change smoking status, or cars that
age or degrade) will participate in flows between stocks with appropriate values. (For example, the
aging of a single individual i aging from the 50-59 year old stock to the 60-69 year old and with other
values (gi, wi, S) for gender, weight and smoking status remaining constant, could be represented as a
flow of size one that increments the stock pop) ¢,g,.,s and decrements the stock popip.59,¢.,5 )+
It can readily be appreciated that while the framework is conceptually simple, very general and easy to
implement via techniques like arraying, the number of distinct volumes in d dimensional space — and
thus the number of distinct stocks — will increase rapidly according to the number of attributes of interest
and the number of distinct ranges that must be broken out in this way. This observation will play a key
role in the analysis below.
3.2 Co-Flows
Co-flows are system structures forma third important approach for the representation of the impacts of
continuous heterogeneity in traditional system dynamics models. Unfortunately, they are also less
general than stock disaggregation, which limits their use to certain special cases.
The general structure of coflows is shown in Figure 3. Within this approach, we characterize population.
means for some attribute or function of interest by keeping track the values of that attribute or function
for each of the inflows and outflows into stocks. The next two sections introduce the fundamental
coflow mechanisms, discusses tradeoffs on the applicability of the approach.
lambda
Deployed Lightbulbs
Lightbulbs being Lightbulb Failures
Deployed
Watts per New Mean Power Needs of
Lightbulbs Deployed Lightbulbs
Power Needs of
Power Needs of Deployed Lightbulbs Poe Needs af
Lightbulbs being Lightbulbs that Fail
Deployed
Figure 3: Example Co-Flow
3.2.1 Co-Flow Basics and Example
Suppose we have a new, higher-efficiency but higher-cost lightbulb design. Suppose we are interested
in understanding the impacts of this greater lightbulb efficiency on aggregate lighting efficiency across n
existing lamps in a warehouse that whose new lightbulb deployments will include some fraction of the
new lightbulb (with the fraction depending on the policies in effect).
As depicted in Figure 3, we thus have a stock of deployed lightbulbs (namedDeployed Lightbulbs” in
Figure 3) whose inflow corresponds to the introduction of the new, higher-efficiency lightbulbs
(“Lightbulbs being Deployed’) and whose outflow is simply bumt-out lightbulbs (“Lightbulb Failures”
in Figure 3). Suppose further that we treat the chance of failure in a lightbulbs as a Poisson process
(and thus independent of age), where the incidence rate 1 (Figure 3’s “lambda”) of failure is identical for
the original and new lightbulb types.
While we could certainly represent the population of lightbulbs using stocks disaggregated by age of
lightbulb, the structure of the processes are sufficiently simple that characterization of the basic
dynamics of aggregate lighting efficiency does not require it. In particular, because the structure of the
problem pemits us to characterize the energy efficiency of inflows and outflows at any given point in
time, we can calculate the energy efficiency of the aggregate stock of lightbulbs at any point in time as
well.
The coflow is a mechanism that allows the modeler to take advantage of situations such as this one,
where it is possible to maintain aggregate statistics for a heterogeneous population without explicitly
disaggregating that population. The coflow does this by maintaining two stocks: One representing the
population size (“Deployed Lightbulbs” in Figure 3), and the other the aggregate total for the quantity of
interest in the population (“Power Needs of Deployed Lightbulbs” in Figure 3). By keeping track of the
quantities of interest (energy use) on the inflows and outflows of the aggregate stock, we can at any time
divide one stock by the other and obtain the population mean (“Mean Power Needs of Deployed
Lightbulbs” in Figure 3) for the quantities.
The co-flow methodology generalizes naturally to the case where there are several attributes of interest
on which the statistic whose mean value we wish to calculate depends. In this case, we may be
interested in a particular statistic (the total energy use), but we could construct a similar a separate co-
flow to calculate the other quantities of interest (such as average age, etc.)
3.2.2 Co-Flow Tradeoffs
By virtue of their elegant and computationally economical representation of complex heterogeneous
systems, co-flows forma very valuable component of a system dynamics modeler’ s toolkit. Co-flows
allow modelers access to important aggregate statistics on systems without the need for expensive
disaggregation.
Unfortunately, co-flows also have strict limitations in applicability. Firstly, because they deal with
means over attributes in a population co-flows are really only applicable for continuously varying
attributes. Heterogeneity with respect to categorical (discrete) attributes is not easily characterized via
co-flows.
Secondly, co-flows are limited for use in calculating statistics on attributes with respect to which system.
evolution is invariant or linear over that attribute. This limitation reflects the need to forthe modeler to
specify both the intemal dynamics of the population stocks and the attribute statistics of population.
outflows. Coflows are very difficult or impossible to employ to characterize mean population attributes
10
when the population members change attribute values over time in a way that depends on residence time
in the stock or the attribute values themselves. For example, while it is possible to model population
aging using a co-flow (because the process applies uniformly over all ages of the population), it is not in
general possible to maintain a coflow that calculates average weight for a human population over time,
for the dynamics of change in weight are not linear in the time spent in the stock or mean weight.
In most co-flows, the mix of heterogeneous elements in the inflows is easily characterizahle; as a result,
calculation of the mean characteristics of the inflows is straightforward. Calculation of the mean
characteristics of outflows is only possible in the presence of special assumptions. The most common
case of this (seen in the example above) is where the population is “well mixed”, and the population
members exiting the population are associated with attribute values whose mean value is equal to the
mean of the attribute values of the population of the whole. It should be clear that this is a highly
restrictive assumption; in general, the exit likelihood for a population member will depend on the length
of time that member has been in the stock or on the value of the attribute of concen.
For example, in the co-flow example in the previous section, the use of a co-flow was made possible by
the fact that the Poisson failure rate 4 was identical for both the high-efficiency and low- efficiency
lightbulbs. With real lightbulbs, the likelihood of bumout for a given lightbulb may vary significantly
with energy efficiency — and perhaps with age. In this general case, it would not be possible to calculate
outflow rates or the characteristics of those exiting the stock without knowing the proportion of the
stocks occupied by high-efficiency and low- efficiency lightbulbs — and thus the use of disaggregation of
the stock of lightbulbs by one or more attributes.
While co-flows are valuable modeling tools, their limitations in applicability greatly restrict their use in
representing heterogeneity. For systems in which heterogeneity is important, the rates of stock outflows
and rates of change in attribute values by members of the population is frequently directly dependent on
attribute values, and not just on the mean of the attribute over some population.
3.3 Stocks Disaggregated by Agent
Regardless of modeling framework, one of the most frequent means of representing heterogeneity over a
population consists of maintaining distinct information on the attributes associated with each member of
the population. For example, consider a system in which four companies are competing, and where
attributes such as employee and customer count, level of efficiency, bank balance, etc. form the
dominant causal factors in success. A system dynamics model using agent-based disaggregation might
maintain distinct stocks for each attribute of each company (e.g. company A employee count, company
A customer count, etc.). Altematively, most system dynamics frameworks would permit the
representation of an arrayed stock for each of the attributes of those corporations. In an explicitly agent-
based modeling environment (such as SWARM, Repast, or Netlogo), the agents representing such.
companies would be represented by objects (associated with classes or structures), each possessing
attribute values represented as instance variables. A state equation model created in Matlab might
represent agents as a collection of vectors of state variables, each indexed by company.
The agent-based approach is very straightforward; as will be shown below, it also scales effectively to
large number of attributes. ‘The approach does, however, require some accommodation for
representation of very large populations. In particular, for cases in which the population being modeled
by the agents is very large (for example, the population of a country), the computational expense may
exceed the statistical benefits of attempting to simulate the full population. For sucha simulation, the
agent- population can be to be downsampled. from the real-world, thus forming a “micropopulation”’
11
Where needed, statistical analysis of data from this approach can be made statistically rigorous and.
attractive through use of bootstrap [Bootstrap] iterations. For the sake of generality and in line with our
desire to examine asymptotic behavior of the techniques, we will assume the use of the downsampling in
the analysis in Section 3.3. For example, to characterize smoking behavior in a population (such as that
modeled ina simple fashion in Section 2.2), computational and data demands would likely prevent the
characterization of the smoking behavior associated with each individual in the entire population.
In contrast to co-flow or attribute- based approaches, the representation of heterogeneity in agent-based.
models requires no special techniques: It emerges from the description of agents as entities with
attributes and the use of agents to characterize specific population members. In contrast to these other
approaches for representing heterogeneity, there is no need to quantize the attribute values for agents
into ranges for placement into discrete stocks: Each agent represents the data needed for characterizing
its behavior to whatever level of precision desired.
What is clear is that an agent-based approach to representing heterogeneity requires a very substantial
investment of computational resources for even the most conceptually simple models characterizing
behavior in a large population. The sections below quantify this investment and contrast it to what is
required in models which represent heterogeneity using attribute-based disaggregation.
3.4 Representational Choice in Two Popular Frameworks
While establishing criteria for the choice of frameworks lies outside the scope of this paper, it is worth
noting that modeling frameworks vary broadly in the degree of choice and convenience they offer to
modelers in capturing heterogeneity. A few words are in order about systems likely to be of greatest
familiarity to the audience: Agent-Based modeling and System Dynamics.
As would be expected, agent-based frameworks permit great expressiveness in characterizing agent-
based disaggregation, but offer only the most incidental support (if any) for other approaches. While
recourse to general-purpose programming mechanisms in principle allow the use of all techniques
described for representing heterogeneity, use of altematives to agent-based disaggregation is not
facilitated by the frameworks and in many ways runs counter to natural programming practice within
agent-based frameworks.
System dynamics models offer modelers a richer set of choices, by permitting highly accessible means
of representing simple forms of all three representations discussed. Unfortunately, within current
system dynamics frameworks the agent-based disaggregation approach is somewhat rigid, and only
grudgingly accommodates fluctuating populations: Regardless of whether an explicit or implicit
amraying strategy is used, the size and structure of the model is “hard-wired” to the size of the population
being modeled. This rigidness somewhat limits the appeal of agent-based disaggregation in system
dynamics models, making it challenging to represent populations with birth and death processes (and.
especially those in which agents dynamically merge or split). System dynamics packages could add
significant modeling expressiveness by permitting dynamically adjustable array dimensions.
4 Computational Resource Demands
The sections above have discussed the mechanics of representing heterogeneity in two modeling
paradigms. In the ideal world, the techniques discussed above would be used to disaggregate the
representation of a system to whatever degree desired for system simulation. Unfortunately,
computational resources in the form of performance and space are limited, and simulation models
exhibit great appetites for both. As a result, the modeler is forced to compromise precision to throttle
12
computational resource demands. This section demonstrates the means by which these paradigms
compromise accuracy in order to accommodate limits in computational resource availability. The
results of this analysis will play a role in comparing the relative advantages of these two modeling
approaches when provided with a particular level of computational resources.
In the case of attribute-based disaggregation, the compromises occur in the form of using less finely
disaggregated stocks. It is important to emphasize that this “lumping” is made statically — and the
scheme for associating stocks with volumes in d-space is frequently made independent of the knowledge
of the frequency distribution of the population at hand over the space of attribute values.
In the case of agent-based disaggregation, the means by which accuracy is compromised to ensure
computational feasibility is rather different. Rather than ignoring differences among certain groups of
the population, agent-based models will downsample a population to bring the simulated population of
agents to manageable size. This approach (which the analysis below will term as sampling) loses
accuracy by the fact that particular population members may be omitted from the downsampled.
population. It is important to emphasize that this downsampling is typically performed dynamic (during
simulation) and typically draws uniformly from all elements of the population. As a result, the
downsampled population typically represents a uniformly random sample of the overall population.
Given that there is a practical tradeoff between computational resources and accuracy, it is worth
considering the magnitude of the computation resource demands imposed by each approach. To this
end, this section of the paper derives simple formulas for the relationships between computational
processing time and storage space required for each paradigm. These relationships are not precise, but
express how the resource demands vary for different sizes of populations and attribute values, as well as
different levels of lumping and downsampling.
The next section tums to examine how the error associated with each paradigm varies with the size of
the problem. The results of both of these sections help to suggest regimes in which each paradigm has
strategic advantages. This may help guide the choice of one technique over the other by shedding light
on loss of accuracy required under each technique in the presence of a fixed amount of computational
resources.
Because of the contrasting representations involved in capturing heterogeneity, the computational
resources vary with respect to different model parameters. Because both simulation approaches allocate
data in memory and then update each of these data items in each timestep, performance is directly
proportional to the amount of space required — and both scale according to the same functional form.
over model parameters.
e Agent-based Disaggregation: Computational resource demands in agent-based disaggregation
vary with the number of attribute fields that must be updated across the population of agents
within each timestep. If there are D attributes and N members of the population, performance
and space usage scale with © (ND). If forthe sake of speed or space we downsample the
population by a factor of s (essentially sampling the population such that one out of every s
members of the population are included), then the number of operations and amount of memory
required scales with (2).
s
e Coflows. In those specialized cases where a co-flow based approach can be applied, the amount
of space and number of expression evaluations required to represent D attributes in a population
13
of size N is much smaller. Within a co-flow framework, one stock is required for the population
count itself, and one stock for each linearly independent statistic required for a given attribute. If
fur is the count of independent statistics required, this gives a total of © (1+ f,, ). Inrealistic
situations, a modeler will typically seek a report on at least one statistic per attribute, (so as to
report on the population statistics) and thus fj; > D. For most practical models, the number of
statistics reported per attribute will be a small constant and will not vary with D; in other words
fir <aD for somea > 1. Thus co-flows can be realized with time and space 0(D) g
e Attribute-based Stock Disaggregation. As noted in Section 3.1, the use of attribute-based
stock disaggregation is widespread in the systems modeling community. Understanding the
scaling of computational resource demands with respect to this approach is thus particularly
important. In this technique, a stock is arrayed — either literally or figuratively — with a separate
array index for each attribute dimension. If there are D dimensions and dimension i has d,
D
subdivisions, the total count of stocks to be calculated is (14): Consider that any non-
ied
D D
trivial dimension i will have d,>2. Thus, IIa >[[2=2°. As aresult, we have a space and
1 i=
D
compsiona time demas ofo [ [4 =2,(2?) This suggests that the computational time
isl
and space rises exponentially in D.
5 Error Scaling Analysis
For modeling of populations that are large and exhibit a large degree of heterogeneity, it is frequently
infeasible to rm a simulation as complete as one would otherwise wish. This constraint is particularly
binding for stochastic model that require the Monte Carlo simulations of large sets of realizations. In
such cases, acceptable performance is traditionally achieved by lowering the precision with which the
heterogeneity in population is represented. We have seen that representation of heterogeneity is
important for understanding system behavior, and lowered precision in capturing such heterogeneity
within a model typically leads to the divergence of model behavior from the behavior of the same model
if system heterogeneity was fully captured.
We have seen above that different modeling approaches capture information on system heterogeneity
using different mechanisms. It is therefore to be expected that the effects of lowered precision may lead.
to significantly different levels of error in different modeling frameworks. In this section, we consider
how the enors associated in estimating a population statistic within the different modeling frameworks
scale with the computational resources required by the simulation.
In particular, we consider the enor associated with totaling a statistic over the attributes of the
population represented in the model. For simplicity, we assume that the computation of the statistic for
each takes time ©(n). Forexample, we may wish to calculate the sum or mean or variance of the age
of the population being modeled.
Tn order to focus the discussion, we assume that the number of attributes to be represented is a constraint
inthe analysis below. That is, we assume that we represent a certain number (D) of characteristics
within the model to some degree, and simply vary the degree to which a given attribute is represented.
14
Suppose we now have a fixed level of computational resources at our disposal. We now consider how
accuracy scales for each of the techniques for capturing heterogeneity.
5.1 Co-Flows
For the special case in which co-flows can be used to capture the statistic of interest (e.g. where the
population is well-mixed with respect to the statistic of interest and the dynamics and attributes of entry
and exit can be adequately captured), there is no need to disaggregate the system to represent
heterogeneity. Computational resource demand is linear in D, and the statistic of interest can be
calculated exactly over the population as one of the co-flow calculations.
5.2 Attribute-Based Stock Disaggregation
For attribute-based stock disaggregation, the computational resource demands depend only on the
number of dimensions D and the level of detail with which the modeler represents each dimension (the
mumber of width Ax of divisions into which each dimension is divided). In order to analyze the scaling
behavior of the binning approach, we make the simplifying assumption that all dimensions (attributes)
are continuous and are subdivided into segments of uniform size ( Ax ), thus dividing the attribute space
as a whole into equal-sized hypercubes (“bins”) of dimensionality D equal to the number of attributes
being modeled. This restriction is less onerous than one might think: Given that the pattems of
heterogeneity in the underlying population (and thus the density of the population across the attribute
space) will in general be varying dynamically over the course of the simulation, the most advantageous
decomposition of attribute space into stocks is frequently not clear a priori. Moreover, any attribute
dimension can be linearly scaled to accord with the uniform size restriction.
15
Given these assumptions, the amount of error associated with calculating the population statistic varies
2
as © [e* ) . The derivation of this mirrors the calculation of quadrature enor bounds in basic calculus.
It is notable here that despite the sharp scaling with D, there is no scaling with N: We maintain the same
mumber of stocks regardless of the size of the population.
Scaling of Relative Binning Error
with Computational Resources
1 Dimension
@ 2 Dimensions
03 Dimensions
04 Dimensions
Error relative to 1D, 10 Resource
BR
on
10
5
0 —_—
1Dirension
x 8
Computational Resources
Figure 4: Scaling of Error in Campuiation of Population Statistics using Attribute-Based Disaggregation. Scalingis
depicted over Computational Resources and Count of Dimensions of Heterogeneity
Intuitively, the formula for exror means that as the number of dimensions to be represented doubles, the
new errors the square root of the old enor (for example, while the error in computing the statistics
might be 10* for 2 dimensions, it will be 107 for 4 dimension, and 107 for 8 dimensions). Looked at in
another way, the relationship between dimensions and error means that for any dimension D greater than
two, a given increase in computational resources will not be met by a proportional decrease in exor.
Consider an increase in computational resources by a factor of a (say, 1%). Now consider the
fractional decrease R(c, D) in exrorthat results
E(c,D) 2
,D)=1- =0| ((1
RED gE eT o|Cra)
Figure 5 illustrates the scaling of R(.01,D) with D. Forjust one dimension (D=1), a 1% increase in
computational power permits a 2% decrease in error. For two dimensions, the fractional decrease in
earor is very close to proportional to the increase in computational resources. For dimensions four and
above, an increase in computational power yields a considerably smaller decrease in error.
16
Proportional Decrease in Error for 1% Increase in
Computational Resource
Fractional Decrease in
Eror
J
88
ao e
—————
1 2 3 4 5 6 7 8 9
Dimension D
Figure 5: Scaling of Decrease in Error Associated frama 1% Increase in Computational Power with Number of
Attribute Dimensions (Attribute-Based Disaggregation)
As shown in Figure 6, the scaling of error over dimension also means that the increase in computational
power required to halve the observed error rises exponentially with the count of heterogeneity
D
dimensions D. R(c,D)=.5 yields a= 2? -1.
Required Computational Resource
Increase to Halve Error for Binning
25
N
oO
b
oO
in Computation
bh
ul
ul
Required Relative Increase
Oo
4 5 6 7 8 9
Dimension
1
all is
3
_
2
Figure 6: The Difficulty of Decreasing Error in Attribute-Based Disaggregatian Approach to Representing
Heterogeneity: For Dimensions Higher than 3, Halving Errar Requires Increasing Considerably Mare than Doubling
Computational Power.
17
5.3 Agent-Based Stock Disaggregation
For those cases in which agent-based disaggregation is being used, the primary means of lowering
computational resource requirements is through downsanpling. Using this method, we can forma
bootstrap population by sampling with replacement from the full population of individuals, and then run
the simulation on this bootstrap population.*
Now oonsider the estimate of the total of some population statistic over the true (real-world) population.
Within the model, we estimate this total by computing a statistic over the population of agents in the
model.
In this case, the standard enor of the sample mean of this statistic over the downsampled population is
e = -o( | seeing nest for enor forte ttl verte pation
i
of x3 ]-0(08). It is important to stress that in contrast to the case of attribute-based stock
disaggregation, the enor scales with the size of the population being simulated and not with the count of
attributes being considered. Intuitively, the formula above means that if we halve the number of
samples (agents) that we consider for a given population (and thus double the downsampling factor s),
the eor goes up by a factor of /2, or approximately 1.4 Conversely, in order to reduce the size of the
exror by a factor of 2 for a given sized population, we need to simulate 4 times as many samples (in
other words, reduce s by a factor of 4).
Now oonsider a case where we have a fixed amount c of computational resources available. We further
assume that demand for computational resources rises linearly with the number of agents whose rules
are computed and linearly with the count of attributes that require updating for each such agent, and that
it is too expensive to compute the evolution of the entire population. As a result, to remain using the
same level of computational resources, the level of downsampling s required varies as (2). Given
c
these assumptions, the error for the total over the population rises as
2()-0[ nt -o[n/2 This implies that for the regime in which our computational
resources are insufficient to handle a full sample of the population, error in computing a statistic over the
population rises linearly with population size N, with the square root of the dimension count and only
inverse to the square root of the computational resources that are available.
In contrast to the case for attribute-based disaggregation, the scaling of error with computational
resources is invariant in the number of dimensions involved. Increasing computational resources by a
factor of «. reduces error by a factor of a . Conversely, halving the enor in such a situation requires
* We consider here only the error for the trivial case where we compute statistics using only a single bootstrap realization,
leaving to a forthcoming paper the more general case of repeated bootstrap samples drawn from the population.
18
increasing computational resources by a factor of four—independent of the number of dimensions
involved.
Scaling of Relative Sampling Error with Computational Resources
(Independent of Dimension)
12
a 1
Sn os
g 0.
B38 06 =
é Boa ar —
2 02
ui
0
10 20 30 40 50 60 70 80 90
Computational Resources
Figure 7: Scaling of Relative Error with Respect to Increase in Camputational Resources (Agent-Based
Disaggregation). The Results Shown are Independent of Count of Attribute Dimensions.
6 Conclusion
The results presented above are scaling relationships and are not meant to numerically estimate the exact
amount of work required for any method; there are likely to be significant constant factors hidden within.
the scaling notation that have significant performance implications for a given size of problem.
Nonetheless, the results above can provide substantial insights into tradeoffs between the methods.
Generality | Computational Enror
Resource Demand For computational
For dimension count resourrces(©), dimension.
©), Population Size(N) | count (D), Population
Size(N)
Agent-Based High ND
= ec
Attribute-based | High Q(2 % 2
fi ti , oO E D )
Co-flows Low o( D) (0) ()
What is clear is that as the number of dimensions rises, agent-based disaggregation methods rapidly
dominate both the accuracy and performance of attribute-based disaggregation approaches. In
particular, we have seen that for the attribute-based stock disaggregation, the size of estimation error
rises rapidly as dimensions increase. Decreasing enors by a given factor requires an increase in
19
computational power that is exponential in the number of dimensions D. This reflects the familiar
“curse of dimensionality”.
By contrast, the error associated with agent-based disaggregation is independent of the number of
dimensions of heterogeneity being considered. For populations large enough to prevent direct
simulation, decreasing errors by a certain factor «. requires increasing computational resources by «” —
independent of the dimensionality of the model.
While attribute- based disaggregation fares poorly for large number of dimensions of heterogeneity,
attribute-based methods have an advantage for models of low dimension and large population. In
particular, the costs associated with agent-based representations rise linearly with population size while
the resource demands of attrilbute-based methods are invariant with respect to population size, as the size
of the population to be simulated (N) rises, attribute-based methods will slowly gain an edge in
accuracy, performance and storage requirements over agent-based methods.
Broadly speaking, we can say that attribute-based stock disaggregation models are best suited to
problems with low number of attribute dimensions (say, less than 3 or 4) or very large population size,
while agent-based models using simple downsampling to throttle computation are well adapted to
models that exhibit important heterogeneity with respect to medium or high numbers of attribute
dimensions and population sizes within a few orders of magnitude of that which can be simulated.
directly. Co-flow based techniques offer a highly precise and computationally frugal approach for
capturing heterogeneity in those specialized instances in which few statistics on attribute values are
required, in which the population is well-mixed with respect to the statistic of interest and where the
dynamics and attributes of entry and exit can be adequately captured.
Before concluding, it is worth noting that this paper has explored just two of many tradeoffs that require
consideration in the selection of a modeling technique to capture heterogeneity. Discussion of these
other considerations lies outside the scope of paper, but it is worth mentioning a few of them. The first
of these is data availability. Partly to achieve reliable sample sizes in descriptive statistics, secondary
data is sometimes available only following attribute-based disaggregation (see, for example, [Census
Projections]); in many other cases (such as in many modem health data sets, [Cument Population
Survey] [NHIS] and particularly in longitudinal studies such as [NHANES]), individual sample data is
available to researchers. The form in which data is received can strongly influence the convenience —
and sometimes the feasibility — of different schemes for representing heterogeneity. Although the
computational error resulting from representing heterogeneity inthe scheme most closely matched to
the data may be higher than for other techniques, selection of this technique can lower the risk of error in
data manipulation and provide the modeler with additional time to refine the model and increase overall
model accuracy. A second additional consideration involves the selection of a modeling framework. As
touched on in Section 3.4, modeling frameworks often differ in their expressiveness with respect to
different schemes for representation of heterogeneity. If considerations such as other characteristics of a
system, software accessibility, tools or client requirements play to the favor of one type of modeling
framework, the modeler must consider the impact of framework choice on the representation of
heterogeneity, and must balance the tradeoffs described here with these other considerations. Finally,
the desired scheme for model calibration must be consistent with the approach chosen for representing
heterogeneity. For example, agent-based models simulating large populations require considerably
different model calibration techniques than do highly aggregated models of the same populations.
Given that model calibration can require considerable computational resources and significant impact
model enor, when planning a model, the modeler would do well to at the outset to jointly select a
calibration method and a technique for representing heterogeneity.
20
7 Bibliography
[Ackerman] Ackerman, E. Simulation of micropopulations in Epidemiology: A series of tutorials
illustrated by coronary heart disease models. Tutorial I: Simulation: An introduction, IntJ Biomed.
Comput, 36 (1994) 229-238.
[Bass Diffusion] Sterman, John. Bass Diffusion Model. Section 9.3.3. of Business Dynamics. Boston:
McGraw-Hill Higher Education. 2000.
[Bootstrap] B. Efron and R. J. Tibshirani. Introduction to the Bootstrap. Chapman and Hall, New Y ork,
1993.
[Census Projections] U.S. Bureau of the Census, Population Division, 1996. Population Projections of
the United States by Age, Sex, Race, and Hispanic Origin: 1995 to 2050 -- Middle Series Vital Rate
Inputs. Available at:http://www.census.gov/population/www/projections/natvital. html.
[Current Population Survey] U.S. Bureau of the Census. Current Population Survey [Computer file].
ICPSR version. Washington, DC: U.S. Dept of Commerce, Bureau of the Census [producer], 1994.
Amn Arbor, MI: Inter-university Consortium for Political and Social Research [distributor], 1995.
Available at: http://www.icpsr.umich.edu.
[Epidemic] Sterman, John. Modeling Acute Infection. Section 9.2.2 of Business Dynamics. Boston:
McGraw-Hill Higher Education. 2000.
[Flowers] Ford, A. S-Shaped Growth. Chapter 6 of Modeling the Environment. Washington DC: Island
Press, 1999.
[Kaibab] Ford, A. The Kaibab Deer Herd. Chapter 16 of Modeling the Environment. Washington DC:
Island Press, 1999.
[NHANES] National Health and Nutrition Examination Survey I Epidemiologic Followup Study.
Atlanta, GA: Centers for Disease Control and Prevention; 1992. Available
at:ftp://ftp.cdc. gov/pub/Health_ Statistics/ NCHS/Datasets/NHEFS/NHEFS92.
[NHIS] National Health Interview Survey. Health Promotion and Disease Prevention Supplement
[Computer File]. ICPSR version. Washington DC: U.S. Dept. of Health and Human Services, National
Center for Health Statistics; 1991
[Resources] Ruth, M. and Hannon, B. Managing Open Access Resources. Chapter 25 in Modeling
Dynamic Economic Systems. New York: Springer-Vedag, 1997.
[Tobacco Policy Model] Tengs, T., Osgood, N. and Lin, T. Public health impact of changes in smoking
behavior: results from the Tobacco Policy Model. Medical Care. 2001 Oct;39(10):1131-41.
[Roberts, Homer et al] Roberts, E., Homer, J., Kasabian, A., and Varrell M. A Systems View of the
Smoking Problem Perspective and Limitations of the Role of Science in Decision-Making. Int J. Bio-
Medical Computing (13) (1982) 69-86.
[Shepard Zeckhauser 1982] Shepard, D.S. and Zeckhauser, RJ. The Choice of Health Policies with
Heterogeneous Populations. In Economic Aspects of Health Fuchs V.R., ed. Chicago: University of
Chicago Press, 1982, 255-297.
21
[Shepard Zeckhauser 1980] Shepard, D.S. and Zeckhauser, RJ. Long-Term Effects of Interventions
to Improve Survival in Mixed Populations. Journal of Chronic Diseases 33, 1980, 413-33
22
Back to the Top
CC BY-NC-SA 4.0