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CURRENT RESEARCH IN SOCIAL PSYCHOLOGY
Note From the Editor
CURRENT RESEARCH IN SOCIAL PSYCHOLOGY
Volume 1, Issue 2, October 24, 1995
Submitted September 8, 1995
Resubmitted October 16, 1995
Accepted October 16, 1995
Assessing Fundamental Power Differences in Exchange
Networks: Iterative GPI
Michael J. Lovaglia University of Iowa
John Skvoretz University of South Carolina
Barry Markovsky University of Iowa
David Willer University of South Carolina
ABSTRACT
Networks have been discovered for which Network Exchange Theory
(NET Markovsky, Willer and Patton 1988; Lovaglia, Skvoretz,
Willer and Markovsky 1995) fails to provide tenable
predictions. Here we elaborate NET to create a more general
method. We show not only when and where exchange networks break
into simpler substructures, but propose rules to decisively
classify networks and substructures as strong, weak, or equal
power. In doing so, we advance general heuristics for power
development in exchange networks and demonstrate the promise of
an approach using reciprocal comparison of general heuristics,
formal theory, and computer simulation.
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INTRODUCTION
Exchange networks must first be classified by power type
before an accurate prediction of power distribution in the
network can be made (Markovsky, Skvoretz, Willer, Lovaglia and
Erger 1993). Here we classify networks as strong, weak, or
equal power. In strong power networks, high power actors can
use power with impunity. That is, over a series of exchange
opportunities, they come to control nearly all available
resources. By contrast, in weak power networks, power use by
high power actors results in countervailing changes in exchange
relations, changes that moderate future power use. Power in
these networks reaches a stable equilibrium in which high power
actors maintain a reliable, though moderate, advantage. That
equilibrium point can be accurately predicted for positions in
a wide variety of weak power networks (Skvoretz and Willer
1993; Lovaglia, Skvoretz, Willer and Markovsky 1995). Finally,
in networks of equal power, no actor has an exchange advantage;
thus, no resource differentiation is predicted in them.
The Graph-theoretical Power Index (GPI) method developed by
Markovsky, Willer and Patton (1988) and Markovsky, Skvoretz,
Willer, Lovaglia, and Erger (1993) uses a path-counting
algorithm to identify how advantaged one actor is in comparison
to another. A position's GPI value is calculated by counting
non-intersecting paths of different lengths leading away from
it, with odd-length paths adding advantage, even-length paths
taking away advantage.
Consider the network in Figure 1 suggested by Noah Friedkin
(personal communication). Position A has a single 1-path to B,
a 2-path to C (the 2-path to D would intersect with the first
2-path at B and so is not counted), a 3-path through B and C to
D, and a 4-path through B, C, D, and ahead to the other C.
Adding 1 to the GPI index for the 1-path and 3-path while
subtracting 1 for the 2-path and 4-path yields a GPI value of 0
for position A. In contrast, position B has four 1-paths, a 2-
path through C to D, and a 3-path through C and D to the other
C. Thus its GPI value is 4 - 1 + 1 = 4. (See Markovsky et al.
1988 for details of GPI analysis.)
Figure 1. Friedkin Network and GPI Values
C 0
/ \ / \
A - B - D 0 - 4 - 3
\ / \ /
C 0
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When GPI values differ for two positions, one has a strong
power advantage over the other. Markovsky et al. (1988, Axiom
2) assume that actors seek exchange with partners whose GPI
value is lower than theirs. Or, if all partners have a GPI
value equal to or greater than an actor, the actor is assumed
to seek exchange with the weakest partner(s) available. How-
ever, exchange is possible only when an actor and a partner
mutually seek exchange with each other. Hence, if an actor and
a partner do not mutually seek each other, that tie is broken.
When such broken ties cause networks to break into subnetworks,
GPI is iteratively applied to resulting subnetworks.
Using Axiom 2 to analyze the Friedkin network, C actors will
seek exchange with D, but not B. The network breaks into an A-B
dyad and a C-D-C 3-line network. GPI equals 1 for the positions
in the dyad. In the 3-line, D's GPI equals 2, whereas C's
equals 0. The exchange seek assumption then applies to these
new GPI scores. C actors seek exchange with B, but not with D,
thus leaving D isolated from the rest of the network. Iterating
GPI again returns the dyad and 3-line. The analysis cycles
indefinitely from one iteration to the other. Thus, the
Friedkin network cannot be classified as strong power.
Nevertheless, simulation using Markovsky's X-Net program shows
that B and D are in fact strong power positions (Markovsky in
press describes the simulator).
The anomalous Friedkin network has a relatively easy
solution, a modification of the exchange seek assumption.
Markovsky et al. (1988) assume that C actors will see exchange
with D while avoiding B because B is more powerful than D.
However, whenever a strong power advantage exists, low power
actors eventually lose nearly all available resources.
Intuitively, it matters little to a disadvantaged actor whether
the difference in GPI scores is large or small. As such, a
better specification of the exchange seek assumption is:
Revised Exchange Seek Assumption (Axiom 2)
Actors seek exchange with those less powerful than they are.
If no actors with less power are available, actors seek
exchange with actors of equal power. If, however, no actors
of equal power are available, actors seek exchange with more
powerful actors.
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Applying the revised exchange seek assumption to the
Friedkin network, C actors seek both B and D and the network
does not break into subnetworks. We classify it as a strong
power network because GPI values differ for related actors.
Although this new axiom thus satisfactorily resolves the
anomaly of the Friedkin network, exploring its implications
soon revealed other networks that challenged GPI analysis.
Heuristics in the Construction of Test Networks
It is difficult to find networks that test theoretical
advances. GPI, for instance, was in use for six years before
the Friedkin network was discovered. Here we develop a general
method of iterating GPI by building up complex networks from
simpler structures, using heuristics about the way power
develops through exchange (Willer and Willer 1995). Then, we
use GPI analysis to see whether it indicates strong power. When
a discrepancy occurs between GPI and heuristic analysis, we
simulate the network using the X-Net program. In all cases thus
far, the X-Net simulator and the analyses that employ the
following heuristics have agreed.
Heuristic 1: Adding a relation between a low strong-power
position and a high strong-power position does not change
the type of power of any position in the network.
Heuristic 2: Adding a relation between two high strong-
power positions does not change the type of power of any
position in the network.
Heuristic 3: Adding a relation between two low strong-power
positions creates a weak or equal power structure.
Heuristic 4: Adding a relation between weak or equal power
positions cannot create a strong power structure.
Heuristic 5: Breaks occur between high strong-power
positions or between high strong-power positions and
equal or weak power positions, but not between equal or
weak power positions.
(Cf. Willer and Willer 1995 for heuristics 1, 2 and 3.)
Our explorations uncovered many networks for which GPI
analysis produced repeating cycles of subnetworks that would
not allow for the classification of positions as strong power
in any simple way, even though simulation and heuristic
analysis suggested that strong power was present. Further
analysis of the problem was necessary that, when performed,
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yielded a general method for the GPI analysis of networks. The
method decomposes a network into strong power, weak power, and
equal power components. The heuristics and computer simulation
then serve as checks on the method's results in particular
cases.
Iterating GPI
The general method for iterating GPI to decompose complex
networks uses the following 7 rules:
1. Iterate GPI using the new exchange seek assumption until a
stable solution--wherein GPI values of all positions
remain the same in two consecutive iterations--or a repeat-
ing cycle of solutions emerges. A stable solution ends
analysis.
2. (a) When a stable solution appears, inequalities between
connected positions indicate strong power.
(b) When a repeating cycle of solutions appears, draw the
network that includes all relations across iterations in
the cycle except when (i) breaks occur in every iteration
of a repeating cycle and (ii) a position has a GPI advan-
tage over other connected positions in every iteration of
a repeating cycle. In (i), the breaks are considered
permanent and are not redrawn. In (ii), the advantaged and
disadvantaged positions form a strong power component that
breaks off permanently. Then, reiterate GPI on the redrawn
network until a stable solution or repeating cycle of
solutions appears.
3. Re-apply rule 2 until the redrawn network is identical to
the previous application's redrawn network or until a re-
peating cycle of redrawn networks appears.
Rules 1 - 3 above identify most strong power structures in
exchange networks. (A computer program for analyzing networks
using these rules is available from John Skvoretz.) However,
computer simulation reveals that some structures harbor strong
power differences not identified by the first three rules. For
example, consider the 7p40 network in Figure 2 below. (We
started labeling networks sequentially for each size. 7p40 is
the 40th network with 7 positions.)
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Figure 2. 7p40 Network
A - B - C - B - A
\ /
D - D
All positions in the 7p40 network have GPI values of 1. We
would not classify this network as strong power, nor would we
predict any breaks in negotiations between network positions.
However, computer simulations and the heuristics tell a
different and convincing story. The B positions in the 5-line,
A-B-C-B-A, are high strong power. In addition, each B is
connected to one member of the D-D dyad. The heuristics reveal
that D actors will initiate a break from the B actors. D actors
will prefer to exchange equally with each other rather than
exchange at a disadvantage with B actors. In turn, B actors are
indifferent to exchange with D actors because B actors have low
strong-power alternatives to exploit. X-Net simulation confirms
that a break will develop between B and D actors, and that B
actors have a strong power advantage over A and C actors.
The following rules correctly decompose networks such as
7p40 that have strong power hidden within them. The rules work
by breaking down networks to their core structures to insure
that no lurking potential for strong power remains undetected.
4. Look for a "stem-dyad" in networks and subnetworks that
have not been identified as strong power. A stem-dyad
is understood to be a position of degree 1 (i.e., con-
nected to only one other actor) and the position con-
nected to it. The position connected to the degree 1
position has the potential to be high power. Thus we call
it the high power position in the stem-dyad; and, we call
the degree 1 position the low power position in the
stem-dyad.
5. Remove the stem-dyad from the network and examine the
residual network.
6. (a) If the residual is strong power and if the high
power position in the stem can reconnect to a low power
position in the residual, then the original structure is
strong power.
(b) If the residual is strong power and if the high power
position in the stem can reconnect only to high power
positions in the residual, then the stem breaks from the
residual as an equal power dyad.
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(c) If the residual breaks into strong and weak power
components while the high power position in the stem
connects to a low strong-power position in the residual,
then the network breaks where the residual breaks.
7. For remaining structures not identified as strong power,
re-apply steps 4 - 6. Continue until no stem dyads remain
attached to a larger structure not yet identified as
strong power. Then reconnect all relations among structures
not identified as strong power.
Rules 4 - 7 allow networks of any size to be broken down to an
easily analyzed core structure. For example, consider a 7-
actor line A-B-C-D-E-F-G. First remove the stem-dyad, A-B,
leaving a five-actor line. (Note that removing F-G instead has
the same result.) If the power type of a five-actor line is
unknown, remove the stem-dyad, C-D, from it. The remaining core
structure, E-F-G, is a 3-line, the prototypical strong power
structure. Hence, we conclude that the 7-Line is a strong power
structure. Now try the 7p40 network. Removing the A-B stem-dyad
results in a 5-actor T structure, which breaks into a strong
power 3-line and a dyad (Markovsky et al. 1988). Therefore, by
rule 6c and through symmetry, 7p40 breaks into a 5-actor line
and a dyad.
Rules 1 - 7 should identify all strong power structures--at
least all those in networks of 7 or fewer positions. (More
complex networks may require more subtle analysis.) We have
used them to analyze more than 200 6-position and 7-position
networks without finding predictions at odds with simulations
or other forms of analysis. Having thus identified and broken
out strong power structures, remaining networks can be analyzed
using the probability tree method of Markovsky et al. (1993) to
determine whether weak power exists in them. The method can
therefore be said to classify the fundamental power type
(strong, weak, or equal) of all positions in all exchange
networks. Exact resource point predictions at equilibrium can
then be made using the method of Lovaglia et al. (1995).
Pending empirical confirmation, this iterative GPI method
appears to solve a fundamental, though narrow, problem in
network exchange theory. Moreover, the heuristics developed as
tools in the solution have general implications. For example,
in exchange networks it seems impossible to gain a strong power
advantage by opening channels of exchange to weak or equal
power network members. Rather, strong power can only be
achieved by cutting off the alternative exchange opportunities
of one's partners (oppression), or by establishing new
connections to isolated individuals outside the network
(colonization).
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More generally, we are developing a procedure for conducting
social exchange research. We compare the results of analyses
using general heuristics, formal theory, and computer
simulation. Although these analyses are related to the extent
that they make similar predictions, all begin from quite
different foundations. Those networks for which analyses differ
thus become the target sites of future research. Hence, one can
systematically analyze all networks of a certain size, culling
anomalous networks for further study. As such, locating
interesting test networks may no longer be a hit-or-miss
endeavor requiring years of ancillary study, thus bringing
closer the goal of analyzing networks of the size and
complexity found in naturally occurring social situations.
In addition, the increasing complications of the GPI have
led us to develop an independent method for determining whether
networks are strong, weak, or equal power. After examining over
two hundred networks, the iterative GPI and the new method make
identical predictions, thereby validating GPI and adding a new
dimension to our evolving general procedure. Consequently, we
will be able to compare the predictions of two independent
formal theories with each other and with general heuristics and
computer simulations to find interesting exchange networks.
REFERENCES
Lovaglia, Michael J., John Skvoretz, David Willer, and Barry
Markovsky. 1995. "Negotiated Exchanges in Social Networks."
Social Forces 74:1.
Markovsky, Barry. In press. "Developing an Exchange Network
Simulator." Sociological Perspectives.
Markovsky, Barry, John Skvoretz, David Willer, Michael J.
Lovaglia and Jeffrey Erger. 1993. "The Seeds of Weak Power: An
Extension of Network Exchange Theory." American Sociological
Review 58:197-209.
Markovsky, Barry, David Willer, and Travis Patton. 1988. "Power
Relations in Exchange Networks." American Sociological Review
53:220-236.
Skvoretz, John and David Willer. 1993. "Exclusion and Power: A
Test of Four Theories of Power in Exchange Networks." American
Sociological Review 58:801-818.
Willer David, and Robb Willer. 1995. "Exchange Network Dynamics
and Structural Agency." Paper presented to the Annual Meeting
of the American Sociological Association, Washington, DC,
August.
ABOUT THE AUTHORS
Michael J. Lovaglia is an Assistant Professor of Sociology at
the University of Iowa and a co-editor of Advances in Group
Processes. His research interests include the sociology of
science and theory construction as well as status and power.
John Skvoretz is a Carolina Distinguished Professor of
Sociology at the University of South Carolina, co-editor of
Connections, and associate editor of Journal of Mathematical
Sociology. His research concerns the principles of social
structure as they are exhibited in interaction systems and
social networks.
Barry Markovsky is a Professor of Sociology at the University
of Iowa where he is Director of the Center for the Study of
Group Processes and the Iowa Workshop on Theoretical Analysis.
He is Co-Editor of Advances in Group Processes and Deputy
Editor of Social Psychology Quarterly. In addition to exchange
networks, his current research looks at social influences on
paranormal beliefs.
David Willer in Professor of Sociology, Universty of South
Carolina. His research focuses on testing and extending
Elementary Theory.
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