Notes on Network Effects:
AgentBased Computational Economics
 Last Updated: 7 March 2007
 Site Maintained By:

Professor Leigh Tesfatsion

tesfatsi AT iastate.edu

Syllabus for Econ 308
 References:
 W. Brian Arthur,
"Inductive Reasoning and Bounded Rationality: The El Farol Problem",
American Economic Review (Papers and Proceedings)
84, 1994, 406411.
 David F. Batten, Chapter 3:"Sheep, Explorers, and Phase Transitions"
(pdf preprint  no figures, 203K),
pages 81115 in Discovering Artificial Economics: How Agents Learn and
Economies Evolve, Westview Press, Boulder, Colorado, 2000, ISBN:
0813397707.
 Note: Unfortunately this book is now out of print. However, if you
are willing and able to handle a rather large download, the entire Batten book (figures included) in
pdf can be accessed at
here (pdf,17MB).
 B. Bollobás, Random Graphs, Academic Press, London, 1985.
 Andy Clark, Chapter 6: "Emergence and Explanation," pages 103128 in
Andy Clark, Being There: Putting Brain, Body, and World Together
Again, MIT Press, Cambridge, MA, 1998.
 P. Erdös and A. Rényi, "On the Evolution of Random Graphs,"
Publications of the Mathematical Institute of the Hungarian Academy of
Sciences, Vol. 5 (1960), 1761.
 Stuart Kauffman, Origins of Order, Oxford University Press, 1993.
 Steven H. Strogatz,
"Exploring Complex Networks",
Nature, Volume 410, No. 6825, March 8, 2001.
 Duncan J. Watts and Steven H. Strogatz,
"Collective Dynamics of `SmallWorld' Networks",
Nature Volume 393, No. 6684, 18 June 1998, pp. 440442.
 Duncan J. Watts, Small Worlds: The Dynamics of Networks
Between Order and Randomness, Princeton Studies in Complexity, Princeton
University Press, Princeton, N.J., 1999.
 Allen Wilhite, "Bilateral Trade and `SmallWorld' Networks",
Computational Economics,
Volume 18, No. 1, August 2001, 4964.
Key Issues
1. Distinguishing between "simple" and "complex" economic systems (Batten,
pp. 8588)
Batten proposes the following distinction between simple and complex economic
systems: "By and large, simple economic systems are homogeneous and
weakly interactive. Individual beliefs and expectations must be
sufficiently uniform, and the level of interactions sufficiently weak or
trivial, for us to be able to predict collective patterns of behavior with
any confidence. ... By way of comparison, complex economies are
heterogeneous and strongly interactive. They lie beyond the
point where individual behavior can be discerned with any degree of
confidence (see Figure 3.1). ... A complex economy is never additive.
It behaves quite differently from what we'd expect by simply adding up these
pairs, trios, and quartets of interaction. Selforganizing economies are not
additive; they're emergent. This is precisely why some of the
collective behaviors can't be predicted in advance."
This discussion by Batten blends the idea of "predictability" with
possible factors (heterogeneous expectations, strong interaction effects,
nonadditivity) that might make prediction difficult. For reasons
clarified below, a more rigorous and useful way to define the
difference between "simple" and "complex" economic systems would seem to be
to focus on the degree of predictability per se. A secondary issue then
arises: Which system features (if any) systematically affect the degree of
predictability? Proposed definitions for "simple" and "complex" systems
along these lines are outlined below.
Suppose a dynamic economic system can be simulated subject to controlled
experimental conditions. That is, starting from any feasible initial state,
the system can be run multiple times and its observed outcomes recorded.
Call the economic system simple if, starting from any given initial
state, the observed outcomes of multiple runs of the system have a central
tendency distribution, in the sense that there exists a unique
"attractor" (most likely outcome) around which the observed outcomes tend to
cluster. In this case, the system structure (the initial state) determines a
reliable "point prediction" for the system outcome: namely, the unique
attractor corresponding to the initial state.
 Example of a Simple System: W. Brian Arthur's El
Farol Bar Problem (Batten, Chapter 2, pp. 6267).
 Suppose a group of 100 people in Santa Fe are fond of attending the El
Farol bar on Thursday nights, when Irish music is featured. Space is
limited, however, and each person would prefer to stay home rather than go to
El Farol if he expects a crowd above sixty. The diabolical aspect of this
prediction problem is that homogeneous expectations can never be correct. If
everyone expects El Farol to be crowded, all 100 people will stay home
and the bar will not be crowded. If everyone expects El Farol
not to be crowded, all 100 people will go to the bar and it will be
crowded.
 Arthur (1994) constructs an agentbased computational model to
study how agents might coevolve their expectations in this situation. Each
of his 100 computational agents uses a classifier system to evolve a
predictor for bar attendance. The experimentally observed bar attendance
rates tend to cluster tightly around the 60 percent level (see Figure 2.4,
p. 65). This 60 percent outcome is supported by a continually evolving
ecology of heterogeneous expectations, not by uniform expectations. This
contradicts Batten's assertion (pp. 8586), quoted above, that individual
beliefs and expectations must be "sufficiently uniform" in order for the
outcome of collective behavior to be predicted with any accuracy.
In contrast, call an economic system complex if, starting from
any given initial state, the observed outcomes of multiple runs of the system
have a spectral distribution with two or more distinct attractors
around which the observed outcomes tend to cluster. In this case, the system
structure in and of itself does not determine a reliable point
prediction of the system outcome.
 Examples of Complex Systems: Systems with path dependence
and lockin effects, such as labor markets with adaptive job search, and
economic systems with technological innovations whose
value increases with the number of users. Such systems will be the focus of
Section VII of the course.
2. Is the global economy becoming strongly interactive?
(Batten, p. 98)
Batten points to the following phenomena which he claims are indeed
causing the global economy to become strongly interactive: Rising levels of
international trade; Increased "foreign direct investment" (i.e., increased
foreign ownership of domestically located productive assets); Rise of
"multinational" (i.e., multicountry owned) corporations.
On pages 9698 he discusses an example to support his thesis. He asks:
Who actually stands to gain most from increased beef exports from America to
Japan? He visualizes the possible relationships by means of signed digraphs
(Figures 3.3 and 3.4).
NOTE: The positive weights in Figures 3.3 and 3.4 on the outgoing
link from X (the yen/dollar exchange rate) to A (export of American
beef to Japan) and G (export of American grain to Japan) are not plausible.
A higher yen/dollar exchange rate X means that more yen are now
needed to buy each dollar's worth of American goods. All else equal, this
should cause Japanese consumers to lower their demand for imported
American goods since the yen price of these goods will now be higher.
Below I have combined and extended Batten's digraphs to a single signed
digraph that more fully reflects Batten's accompanying text discussion. The
following assumptions are made for illustration only, to illustrate Batten's
points about possible complicating effects. (It would be interesting to see
actual empirical data related to these effects).
 Japanese cattle are raised entirely on American grain;
 Some Americabased cattle ranches have Japanese owners, and these
ranches are the primary beneficiaries of an increase in exports of beef from
America to Japan;
 Some cattle ranches in Japan have American owners;
 Grain farms in America are entirely owned by Americans;
The nodes of the modified digraph are as follows:
 JConsBeef=Japanese consumption of beef (total)
 JConsBA=Japanese consumption of beef imported from America
 JConsBJ=Japanese consumption of beef produced in Japan
 ProdBeefJ=Production of beef in Japan
 IncJRJ=Income of Japaneseowned cattle ranches in Japan
 IncARJ=Income of Americanowned cattle ranches in Japan
 ExpABeefJ=Export of Americaproduced beef to Japan
 ProdBeefA=Production of beef in America
 IncARA=Income of Americanowned cattle ranches in America
 IncJRA=Income of Japaneseowned cattle ranches in America
 ExpGrainAJ=Export of grain from America to Japan
 ProdGrainA=Total production of grain farms in America
 IncGrainA=Income of Grain Farms in America
 X = Yen per dollar exchange rate = 2 yen/dollar
 Dollar price of American beef = $5/pound
 Japanese tariff on American beef = 3 yen/pound
 Yen price of American beef BEFORE tariff cut = 13 yen/pound
 Yen price of American beef AFTER tariff cut = 10 yen/pound

JConsBeef

/\ /\
+1   +1
 
+1 \/ 1 \/ +1 
X< JConsBA  JConsBJ > ProdBeefJ
>  
1  +1 +1  +1 
  
\/ IncJRJ IncARJ
ExpABeefJ  
 +1 \/ +1 \/
 
+1  ExpGrainAJ
 
 
\/ +1 
 +1 \/
ProdBeefA > ProdGrainA
 
+1  +1  +1 
  \/
\/ \/ IncGrainA
IncARA IncJRA
Suppose that the U.S. government convinces Japan to eliminate the tariff on
imports of American beef into Japan. Assuming this cut is immediately passed
through to the retail level, it will decrease the yen price of American beef
relative to the yen price of Japanese beef for Japanese consumers from 13 yen
per pound to 10 yen per pound. All else equal, this should lead to an
increase in JConsBA (Japanese consumption of beef imported from
America). The issue is who ultimately gets helped and who ultimately gets
hurt by this change? In particular, is America better off or worse off
overall as a result of this Japanese tariff cut?
Batten claims that standard "partial equilibrium" international trade
models tend to focus only on the chain consisting of the positively weighted
links connecting JConsBA to ExpABeefJ to ProdBeefA to IncARA. This chain
predicts that a tariff cut causing an increase in JConsBA will ultimately
increase the income IncARA of Americanowned cattle ranches, a good thing for
America. Batten points out, however, that complicated network effects make
it difficult to predict the overall impact of this tariff cut on the welfare
of American citizens.
Specifically, the increase in JConsBA due to a cut in the yen price of
American beef for Japanese consumers will have a negative impact on JConsBJ
(Japanese consumption of beef produced in Japan) as Japanese consumers now
substitute American beef for Japanese beef. This, in turn, will have a
negative effect on ProdBeefJ, the production of beef in Japan. This will
decrease the income of cattle ranches in Japan, hence in particular
it will decrease the income IncARJ of Americanowned cattle ranches in
Japan.
In addition, the reduction in ProdBeefJ will decrease ExpGrainAJ (the
export of grain from America to Japan) since not as much grain will be needed
in Japan to feed cattle. This decrease in the amount of American grain
needed to feed cattle in Japan will not be fully offset by an increase in the
amount of American grain needed to feed cattle in America since at least part
of any increase in beef exports to Japan will represent a switch of already
existing beef production from American consumers to Japanese consumers.
Consequently, the overall production ProdGrainA of grain in America will
decline, which will decrease the income IncGrainA of American grain
farmers.
NOTE: Realistically, for Japanese consumers to convince American
beef producers to sell more beef to them in the short run, before American
beef production can be increased to fully meet this increase in demand, the
Japanese consumers will have to bid up the dollar price of American beef
(thus discouraging American demand for American beef). However, the tariff
cut gives Japanese consumers some leeway to do this while still ensuring they
pay a lower yen price for American beef than before the tariff cut.
For example, assuming the exchange rate X stays fixed at 2 yen per dollar,
even if Japanese consumers bid up the dollar price of American beef from
$5/pound to $6/pound, they will only pay 12 yen per pound of American beef,
which is still lower than the 13 yen per pound they paid prior to the tariff
cut. Note that any such rise in the dollar price of American beef would
decrease the welfare of American beef consumers.
Finally, the income earned by cattle ranchers in America will increase due to
the increase in ProdBeefA, the production of beef in America, but the
overall increase in income to cattle ranches in America will be divided
between IncARA and IncJRA, i.e., between the incomes of American and
Japaneseowned cattle ranches in America.
The ultimate effect of the tariff cut on the welfare of American citizens
(grain farmers, cattle ranchers, and beef consumers) is therefore
ambiguous without further information regarding the timing and
magnitudes of the indicated production and income changes.
QUESTIONS: How can these complicated network effects be more
rigorously modeled? By an equationbased model? By an agentbased model?
By some other approach? What are the advantages and drawbacks of each
approach?
4. What type of phase transition do random graphs undergo as connectivity
increases? (Batten, pp. 99104)
Notes:The graph theory terminology used below is explained above in
the "Basic Concepts from Graph Theory" portion of these notes. For a more
rigorous and detailed treatment of random graphs, see Bollobás (1985),
Strogatz (2001), and Watts and Strogatz (1999). The first researchers to
study how the connectivity of a random graph changes as a function of the
number of its links were Erdös and Rényi (1960).
For any graph G, let R(G) denote the ratio of its size (number of links)
to its order (number of nodes), and let M(G) denote the size (number of
links) of its largest component. Batten notes that the plot of M against R
for a sequence of random graphs G_0, G_1, G_2,... with steadily increasing R
values reveals that the random graphs undergo a phase transition as R passes
through the critical value R=0.50. Specifically, as depicted in Figure 3.6
(p. 103), the random graphs transit from being "nearly unconnected" to
"nearly connected" at the critical value R=0.50, in the sense that the value
of M undergoes a sudden dramatic increase.
M

 _  
 
M=Size of 
largest 
component 

 

 
 
______________________________________________ R
0.50
R = Ratio of Links to Nodes
Batten describes the construction of the random graph sequence G_0, G_1,
G_2,... as follows. Let N denote some suitably large integer (Batten selects
N=20). Start with a completely disconnected graph G_0 having N nodes and 0
links. Choose two nodes at random from the nodes of G_0 and connect them
with a link. The resulting graph has N nodes and one link  call this graph
G_1. Next, choose two nodes at random from the nodes of G_1 and connect them
with a link  call this graph G_2. And so forth.
For each graph G_i in the sequence G_0, G_1, G_2,..., plot M(G_i) against
R(G_i). Batten claims that the resulting plot will lie along an Sshaped
("sigmoid") curve as depicted in Figure 3.6 (p. 103). At first the value of
M will increase at a slow but slightly accelerating rate. However, as R
approaches the critical value 0.50, the value of M will suddenly exhibit a
substantial increase. As R continues to increase beyond 0.50, the value of M
will increase at an ever diminishing rate as it approaches an upper bound.
NOTE: What is the upper bound of M? In Batten's description of
his construction procedure (pp. 99100), the sequential selection of nodes
for linking appears to be "with replacement" each time, which would permit
multiple selections of the same nodes. In particular, selfloops and
multiple links between the same two nodes would be permitted. In this case,
however, the upper limit of M would be infinite since additional links could
always be incorporated. This is not consistent with Batten's series of
figures (Figure 3.5(a)(e), pp. 100102), in which no selfloops or multiple
links are exhibited. Moreover, it is not consistent with Batten's Figure
3.6, in which the upper bound of M is depicted as being 380 = 20*19.
Consequently, although Batten does not clarify this, it would appear that
selflooping is ruled out in his construction procedure, and that at most two
links are permitted between any two nodes. Consistent with Figure 3.6, the
upper bound of M would then be N*[N1], the number of distinct ways in which
two nodes can be selected from among a collection of N nodes (with order
taken into account).
5. Do socioeconomic systems also undergo some kind of phase transition as
connectivity increases? (Batten, pp. 104105)
Note: The graph theory and smallworld network terminology used below
is explained above in the "Basic Concepts" portion of these notes. A more
rigorous and detailed treatment of the smallworld network findings discussed
below can be found in Strogatz (2001), Watts and Strogatz (1998), Watts
(1999), and Wilhite (2001).
Batten points to several suggestive examples of socioeconomic systems
exhibiting phase transitions as connectivity increases: for example, the
formation of community common interest groups (e.g., voting blocs, political
parties, unions, clubs); an audience watching a performer; traffic on city
highways. He notes that he will discuss the traffic example in detail in his
later Chapter 6.
At the time he wrote his book, Batten was apparently unaware of the
attempts by Duncan Watts and Steven Strogatz to rigorously address his
question through an analysis of small world networks (Watts and Strogatz,
1998; Watts, 1999). These authors explored a simple family of graphs G(p)
that could be tuned by a parameter p to range all the way from a regular
graph G(0) at p=0 (every node has the same number of links) to a random graph
G(1) at p=1 (all links are randomly generated). For intermediate values of
p, they discovered a range of "small world networks" G(p) displaying a number
of interesting properties. The networks exhibited global reachability
(similar to random graphs) yet had high degrees of local neighborhood
clustering (similar to regular graphs).
More precisely, as detailed in Strogatz and Watts (1998, pp. 440441), these
smallworld networks G(p) arise for an intermediate range of p values where
the characteristic path lengths L(p) of the networks are almost as small as
for a random graph yet the clustering coefficients C(p) of the networks are
much greater than for a random graph. The small L(p) values result from the
introduction of a few "short cut" links that connect nodes that otherwise
would be very far apart. For small p, each short cut has a highly nonlinear
effect on L(p), contracting the distance not just between the directly
connected pair of nodes but also between their immediate neighborhoods, and
between neighborhoods of neighborhoods, and so forth. In contrast, for small
p, a link removed from a clustered neighborhood to form a short cut has at
most a linear contracting effect on C(p). Thus, for small p, an increase in
p causes L(p) to drop very rapidly while C(p) decreases only slowly.
After discovering this smallworld form of network connectivity, Stogatz
and Watts (1998) investigated its empirical significance. They found that
smallworld network architectures appear to characterize a wide variety of
realworld networks as disparate as the neural network of the nematode worm
Caenorhabditis elegans, the power grid of the western United States,
and the collaboration pattern of motion picture actors. They conjectured
that one reason for the ubiquity of smallworld network architectures might
be enhanced signalpropagation speed, computational power, and
synchronizability.
A number of ACE researchers have begun to consider smallworld networks
in relation to economic processes. For example, Wilhite (2001) uses an ACE
model of a bilateral exchange economy to explore the consequences of
restricting trade to smallworld trade networks.
Specifically, Wilhite focuses on the tradeoff between market efficiency
and transaction costs under four types of trade networks: (a) completely
connected trade networks (every trader can trade with every other trader);
(b) locally disconnected trade networks consisting of disjoint trade groups;
(c) locally connected trade networks consisting of trade groups aligned
around a ring with a onetrader overlap at each meeting point; and (d)
smallworld trade networks constructed from the locally connected trade
networks by permitting from one to five randomly specified shortcut trade
links between members of nonneighboring trade groups. Given each type of
trade network, traders endowed with stocks of two goods seek out feasible
partners, negotiate prices, and then trade with those who offer the best
deals.
Wilhite's key finding is that smallworld trade networks provide most of
the marketefficiency advantages of the completely connected trade networks
while retaining almost all of the transaction cost economies of the locally
connected trade networks. His findings also suggest that there exist
microlevel incentives for the formation of smallworld trade networks, since
the traders who use this type of network tend to do well relative to the
traders who do not.
As Strogatz (2001) notes, however, for social networks it is not enough
to focus on the implications of a fixed or parametrically varied network
architecture. Feedback mechanisms are at work in social networks that
reduce the ability to predict system performance solely on the basis of
initial structural conditions. For example, in social networks, the
participants (nodes) might be able to preferentially determine their
relationships (links) over time by means of adaptive choice and refusal of
partners. In such cases, interaction networks might continually coevolve
over time through the operation of intricate feedback loops: who I partner
with today depends on the behaviors expressed by myself and my partners in
past interactions, and how I behave today depends on whom I partnered with in
past interactions.
The issue of evolving interaction networks will constitute one of the main
topics of Section VII of the course, which focuses on the ACE modelling
of markets with strong learning and network effects.
6. Does learning through interaction promote cooperation? (Batten, pp.
109111)
Batten asserts (p. 111) that "learning by interacting proves to be an
adaptive process out of which a desire for cooperation emerges." As will be
seen in Section VII of this course, this is too simplistic. Certain
structural conditions promote cooperation; others do not. For example, basic
market structural conditions (e.g., costs, capacities,...) and biases
inherent in market protocols (e.g., who gets to bid first) can systematically
advantage buyers over sellers, or vice versa, encouraging aggressive and
predatory behaviors on the part of the advantaged group.
Glossary of Basic Concepts
 Coevolutionary Learning (Batten, p. 81):
 "What agents believe affects what happens to the economy, and in
turn, what happens to the economy affects what agents believe. We call (this
positive feedback loop) coevolutionary learning."
 Fallacy of Composition (Batten, pp. 8182):
 "What seems to be true for individuals isn't always true for society
as a whole. Conversely, what seems to be true for all may be quite false for
any one individual." (p. 81)
 Example: If ONE farmer succeeds in producing a
bumper corn crop, he will likely be rewarded with a higher income (relative
to other farmers). However, if EACH farmer succeeds in producing a bumper
corn crop, the most likely outcome will be an aggregate excess supply of corn
a the current corn price, a sugsequent drop in the corn price, and
lower incomes for all corn farmers.
 "(Fallacy of Composition is) a fallacy in which what is true of a
part is, on that account alone, alleged to be also necesarily true of the
whole." (p. 82)
 Simple vs. Complex Economic Systems (Batten, pp. 8586):
 This distinction is taken up in detail as a "Key Issue," below.
 Phase Transition (Batten, p. 86):
 An abrupt marked change in the pattern of behavior exhibited by the
state of a dynamical system. For example, a smooth flow of traffic might
suddenly degenerate into stopandgo traffic, or a system exhibiting large
temperature fluctuations might suddenly transit to a constant temperature
state. Phase transitions can occur due to the intrinsic dynamics of a system
(e.g., a traffic accident) or in response to external effects (e.g., a
deliberate change in the value of an exogenously set system parameter).
 Batten (p. 86) identifies a phase transition as an "unexpected"
change, but this is not the meaning of "phase transition" as typically used
in mathematics. "Abrupt" changes need not be "unexpected" changes.
 Emergent Property (Batten, p. 88; Clark, Chapter 6):
 Roughly, a system consisting of a collection of agents is said to
exhibit an emergent property if it develops a global (systemwide)
property that cannot be predicted solely on the basis of the inherent
properties of the agents themselves. The idea is that the emergent property
depends on the interaction pattern among the agents, not just on their
intrinsic structural aspects. Consequently, the emergent property cannot be
deduced simply by "adding up" what is known about the individual agents,
considered one by one.
 Example: A system consisting of a container of
gas molecules (the "agents") exhibits a temperature (the "emergent
property").
 Andy Clark devotes Chapter 6 ("Emergence and Explanation") to a
detailed discussion of this controversial concept. On page 112 he defines a
collective variable to be a variable that tracks a pattern resulting
from the interactions among multiple elements of a system. He calls a
phenomenon emergent if it is best understood by attention to the
changing values of a collective variable.
 Clark goes on to say: "Different degrees of emergence can now be
identified according to the complexity of the interactions involved.
Multiple, nonlinear, temporally asynchronous interactions yield the
strongest forms of emergence; systems that exhibit only simple
linear interactions with very limited feedback do not generally
required understanding in terms of collective variables and emergent
properties at all. Phenomenon may be emergent even if they are
under the control of some simple parameter, just so long as the role
of the parameter is merely to lead the system through a sequence of
states themselves best described by appeal to a collective variable
(e.g., the temperature gradient that leads a liquid through a
sequence of states described by a collective variable marking the
varying amplitude of the convection rolls)... Emergence, thus
defined, is linked to the notion of what variables figure in a good
explanation of the behavior of a system."
 Connectivity (Batten, p. 89):
 Roughly, the connectivity of a system consisting of a
collection of agents is the degree to which the system agents interact with
one another. A connectivity measure would generally take into account two
aspects of this interaction pattern: (1) Which agents are interacting
with one another?; and (2) What is the strength (frequency,
regularity, impact,...) of these interactions? As Batten stresses,
connectivity is a fundamental aspect of both living and nonliving systems.
 Network:
 A network is a finite collection of entities together with a
specified pattern of relationships among these entities. Three main
tools have been used for the quantitative study of networks: graph theory;
statistical and probability theory; and algebraic models. In Chapter 3,
Batten uses basic concepts from graph theory to discuss network issues for
socioeconomic systems.
 Basic Concepts from Graph Theory (Batten, pp. 92105; Watts,
Section 2.2, pp. 2540):
 A graph is a set of nodes (or vertices) together
with a set of links (or edges) that connect various pairs of
nodes.
 The number of links joining a node n to other nodes (including possibly
itself) is called the degree of node n.
 A graph is called
regular if each of its nodes has the same number of links as any other
node, i.e., if each node has the same degree. A graph is called
random if its links are generated in some random fashion. The number
of nodes of a graph is referred to as the order of the graph and the
number of links of a graph is referred to as the size of the graph.
 The minimum number of links that must be traversed to travel from a node
n to another node n' is called the shortest path length or
distance between n and n'. A graph is connected if any node
can be reached from any other node by traversing a path consisting of only
finitely many links, or equivalently, if the shortest path length between any
two nodes is a welldefined finite number. Otherwise the graph is
disconnected.
 If a graph is disconnected, its nodes can be partitioned into two or more
subsets in which there are no connecting links between the nodes in the
different subsets. The maximal connected subsets of a graph are called its
components. For a connected graph, the graph itself is its only
component.
 A graph is called a directed graph, or digraph, if
each of its links is a directional arrow (or arc) between nodes that
indicates a direction to the nodal relationship. A graph is called
undirected if all of its links are undirected. A (di)graph is
referred to as a weighted (di)graph if each of its links has
associated with it a number indicating the strength of the link.
 A signed digraph is a weighted digraph in which the weights
are either +1 or 1. Consider a directed link from some node n to another
node n'. A weight of +1 on this link indicates that node n and node n' move
in the same direction: that is, an increase in node n will cause an
increase in node n', and a decrease in node n will cause a
decrease in node n'. Conversely, a weight of 1 on the link between
node n and n' indicates that these nodes move in opposite directions: that
is, an increase in node n will cause a decrease in node n', and
a decrease in node n will cause an increase in node n'.
 Terminological Note: In graph theory, a link connecting a node
back to itself is called a "loop." Batten uses "loop" in a more general way
to refer to any collection of one or more nodes in a digraph whose directed
links result in a circular pattern of links, e.g., from n to n' to n'' and
then back to n. To avoid confusion below, a link connecting a node to itself
will be referred to as a selfloop and a circular pattern of links
will be referred to as a closed chain.
 A closed chain in a signed digraph that has a +1 weight on all of
its links is deviation amplifying; it constitutes a special case of a
positive feedback loop. A closed chain in a signed digraph that has a
1 weight on all of its links is deviation counteracting; it
constitutes a special case of a negative feedback loop.
 Basic Graph Theory Concepts Related Specifically to SmallWorld
Networks (Relevant for Batten's discussion, pp. 99105 and Key Issue 5,
below):
 Following Watts (1999, pp. 2633), the average distance between pairs
of nodes in a connected graph is referred to as the graph's "characteristic
path length." More precisely, the characteristic path length L(G) of
a connected graph G is the median of the means of the shortest path lengths
connecting each node to all other nodes.
 A graph is called simple if it has no selfloops and no
multiple links between the same pair of nodes. The "clustering coefficient"
of a simple graph measures the extent to which nodes linked to any given node
n are also linked to each other. Or in other words, are the friends of my
friends also my friends?
 To be more precise about this, given a node n of a simple graph G,
define the neighborhood of n to be the collection of all nodes that
are adjacent (directly linked) to n, not including n itself. If this
neighborhood contains k nodes, then k*[k1]/2 is the maximum possible number
of links that could exist among nodes in this neighborhood. Given any simple
graph G, and any node n of G, the clustering coefficient for node n is
defined to be the fraction C_n of possible links that actually occur in the
neighborhood of n; and the clustering coefficient C(G) for G is
defined to be the average of C_n over all of the nodes n of G.
 As defined by Watts and Strogatz (1998), a smallworld network
is a connected simple graph exhibiting two properties:
 Large Clustering Coefficient: Each node of G is linked to a
relatively wellconnected set of neighboring nodes, resulting in a large
value for the clustering coefficient C(G);
 Small Characteristic Path Length: The presence of shortcut
connections between some nodes results in a small characteristic path length
L(G).
 In summary, then, smallworld networks have both local connectivity and
global reach.
 Isotropic Random Graph (Batten, p. 103):
 Following Kauffman (1993, p. 307), an isotropic random graph
is a random graph such that the probability an edge joins any pair of points
is equal.
 Explorers vs. Sheep (Batten, pp. 106109):
 Batten uses explorers (or innovators) to describe one
extreme of the learning spectrum  inductive learners who are always
actively searching for new possibilities. He uses sheep (or
imitators) to describe the other extreme of the learning spectrum,
deductive learners who prefer to remain with the status quo.
 In more neutral terms, Batten recognizes that successful decision making
typically involves two distinct behavioral modes: (1) Exploration:
Actively seeking to learn more about your decision environment through
information collection and experimentation; and (2) Exploitation:
Making use of the information already in your possession. In systems
science, this is known as the dual control problem  having to learn
about your problem environment even as you seek to optimize your situation
within this environment.
Copyright © 2007 Leigh Tesfatsion. All Rights Reserved.