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Note From the Editor

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


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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   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 

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 

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 

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 

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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.


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 

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, 


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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