Social Networks 14 (1992) 309-323
North-Holland
309
New directions for network exchange
theory: strategic manipulation
of network linkages ’
Robert K. Leik
Unicersity of Minnesota, Minneapolis, MN, USA
Current theories of network power have considerable precision, and experimental methods for
studying power are highly developed. The problem posed in this paper is that the very strength
of the theory and methods is also a weakness. By constraining any form of strategic action by
experimental subjects to bargaining within existing network configurations, one of the most
powerful forms of strategy is denied: negotiating changes in the network itself. A beginning
theory of strategic agency is developed which generates a series of hypotheses and demonstrates
some implications for network power if links can be added or deleted.
Considerable consensus has been achieved regarding how structure
affects network power relationships. Various experimental methods
have evolved into relatively strandardized procedures for examining
network power. The work displays a maturity often lacking in other
areas of inquiry, with a much closer tie between formal theory and
experimental testing of that theory than is to be found in most social
science literature. However, our understanding of crucial aspects of
network power may have been constrained by the very theory and
methods that have proven so productive. The intent of this paper is
not to negate the important work accomplished to date, but rather to
suggest expansion beyond current boundaries.
The problem posed here is that current theory and methods have
eliminated an important class of strategic actions on the part of the
Correspondence fo: Professor R.K. Leik, Department of Sociology, University of Minnesota,
Minneapolis, MN 55455, USA.
’ I wish to thank David Wilier for extensive comments on an earlier draft of this paper and some
of the ideas incorporated in this version. Joseph Galaskiewicz and David Knoke made insightful
suggestions as well.
0378-8733/92/$05.00
0 1992 - Elsevier Science Publishers B.V. All rights reserved
310
R. K. Leik / New directions for tzeiwurk exchange theory
subjects. Typically, the experimenter establishes a particular network
configuration by imposing physical barriers, electronic barriers or
lower profit potential between selected dyads. Once a network is
established, the only bargaining that can take place (hence the only
realm in which subjects can attempt to exert power) is in exchanges
within the predefined network linkages.
Elsewhere, I have suggested that ‘bargaining’ be used to refer to
working out the terms of a particular deal (so many of my X for so
many of your Y), whereas ‘negotiation’ be used in the broader sense
of the working out the rules under which bargaining occurs (Leik
1991). Although there are many rules incorporated into the typical
experimental situation, 2 both implicit and explicit, the present focus
will be on rules governing who is allowed to bargain with whom.
Current methods place control over those rules entirely in the experimentalist’s hands; current theory predicts the consequences of bargaining in such fixed structures very well.
A desirable first step in expanding network power theory will be to
incorporate formally the possibility of gaining power through manipulating the linkages themselves, hence altering the power potential of
one’s position, While our methodology has not allowed subjects to
negotiate changes in the structure itself, our relatively precise models
of power within static structures may well provide the basis for
expanding theory to dynamic structures, including operating both
within and on the structure. D. Willer (personal communication, 1991)
has suggested we might call operating 011 the structure ‘strategic
agency’, and operating ~~t~i~ the structure ‘strategic action’. The
exact terminology choosen may not be crucial, but implications for
network power are indeed crucial.
Basic modes of strategic agency
Two modes of strategic agency will be analyzed here: adding links and
deleting links. There are certainly other strategies, such as negotiating
which position one occupies or what rules the network operates
’ Network rules include, but are not limited to, who can bargain with whom, constraints
timing of that bargaining,
relevance of tradition, laws or interested third parties constraining
bargaining,
public or private character of the bargaining
and so forth.
on
the
R.K. Leik / New directions for network exchange theory
311
under. These options will be explored in future papers. For now,
linkage changes seem not readily incorporated into existing theory.
As Willer (1992) has reminded us, network exchange theory derives
from two main sources: power-dependence
theory (Emerson 1962,
1964) and elementary theory (Willer and Anderson 1981; Willer 1987).
A key concept in both approaches is ‘alternatives’. Alternatives for
any given node can effectively be equated to the number of links that
node has with other nodes in the network iff the potential joint profit
for any pair of linked nodes is identical across all links in the network.
The following scope conditions, common to many current experiments, specify such equivalence for the simplest experimental situation. The last condition is equivalent to e = 1 as defined by Markovsky
et al. (1988).
Condition
Condition
Condition
Condition
Condition
1.
2.
3.
4.
5.
The number of nodes in the network is constant.
Only linked nodes can engage in exchanges.
All resource flows must be dyadic.
The joint profit for any exchange is constant.
Each node can engage in no more than one exchange
per round.
Such conditions rule out variable profit across linkages (Leik et al.
1975; Stolte and Emerson 1977; Cook and Emerson 19781, multiple
node exchanges, exchanges through an intermediary (Marsden 1982),
and more complex forms of exchange. However, those forms can be
reintroduced once dynamics of simpler forms are understood.
Although the models differ, all attempt at formal models of network power relate in some way to linkages (Cook et al. 1983, 1986;
Willer 1987; Markovsky et al. 1988; Stolte 1990). Thus manipulation of
linkages constitutes manipulation of alternatives, generating the possibility of basic shifts in power.
Postulate 1. A position of lower power can gain power by establishing one or more links to other nodes.
This postulate restates the linkage-alternative
equivalence and the
theoretical dependence of power on alternatives. Conversely,
Postulate 2. A position of higher power may lose power if lower
power nodes are able to establish mutual links.
312
R. K. Leik / New directions for network exchange theory
Various gambits may be suggested by these postulates: e.g. labor
unions, a lower-power strategy that constitutes coalition formation
(Cook and Gillmore 1984) vs. trade barriers or exclusive contracts
(higher power strategy). The postulates are suggested as relevant for
any linkage changes, however, regardless of motive, target, etc.
Four node networks
Willer’s network a is a convenient first case, shown here as Fig. l(a).
This is the same network used by Cook and Emerson (1978: 736, Figs.
2(a) and 2(b) except for links between the A’s (the position letters are
reversed in the Willer diagram compared with Cook and Emerson’s
diagram). GPI values (Markovsky et al. 1988) would be 3 for B and 0
for each A. For convenience of comparison, those values will be
normalized by their sum so that relative shifts in power are more
easily shown. The normalized values are shown at each node.
Consider first the implication of allowing one of the A’s to add a
link to another A. 3 How and at what cost need not concern us at this
point, since the focus is on changes in the power potential of the
positions. Figure l(b) shows the new network, with initial normalized
values of GPI. That is now the same as Willer’s Network f. Earlier
work by Markovsky et al. (1988) and by Leik (1991) suggests that such
a net would bifurcate into two balanced dyads, since the A, nodes
would learn that they could escape the higher demands of B by
dealing only with each other. 4 That would leave B effectively without
alternatives in any dealings with A,. The addition of one link, therefore, reduced the network to four nodes of equal power. Weak-power
network probability calculations ala Markovsky et al. (1991) suggest a
’ The approach
taken in Leik (19911, involved having to drop one link when one was added
elsewhere.
Although that may be a more likely result of some types of negotiation,
the power
implications
of changing linkages are made clear if adding and deleting are initially treated
independently.
4 Bifurcation
of a network is likely if all nodes in each subset of the network find their most
attractive
alternatives
(or only alternative)
to be within their own subset. In Fig. l(b), for
example, the At’s would find dealing with each other preferable
to dealing with B because B has
more alternatives
and would expect a greater share of profits. B would find dealing with A,
perferable
to dealing with the A,‘s because A, is totally dependent
upon B whereas the A,‘s are
not. A, has no alternatives
to B. Consequently,
B would tend to seek A, who would reciprocate,
and the A,‘s would tend to seek and reciprocate
each other, resulting in two equi-powered
dvads.
R.K. Leik / New directions for network exchange theory
0
a.
1
313
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Fig. 1. Adding
.b.
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ualuor
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Network
narmallred
an A-B
A.
remaining
power ordering
of B > A, > A,, but experimental
data
show very infrequent
dealings between B and the AI’s CD. Willer,
personal communication,
1991).
314
R. K. Leik / New directions for network exchunge theoq
It is crucial to understand why the earlier power differential has
been negated. In the initial network, only one exchange per round
could occur because B must be involved and could be involved in only
one. The new link has enabled two exchanges per round. Therefore,
neither of the A, nodes need be excluded, and their realization of
that fact will mean that B must deal with A,. With only four nodes,
two exchanges per round means full participation in the network. The
change in structure has resulted in a change in available resources.
Using the common 24 profit points per round, the initial network
allowed only 24 points among four nodes. The changed network
allows 48 points. That is the maximum a four-node network can utilize
under these rules.
It appears that adding other links (from A z to either A,) produces
no further power shifts. One possible link creates Fig. l(c), which is
likely to result in B and the three-link A, more or less ignoring each
other in order to deal with lower power alternatives. That is, the
network may degenerate into a circle structure of equal power,
because weak-power network calculations suggest the high power B
and the three-link A, will not seek exchange with each other. Adding
the last possible link (between A 1 and A 2, creates Cook and Emerson’s
totally balanced structure (their Fig. 2(b)). The following definitions
will be useful:
Definition I. NX is the maximum number of exchanges possible in
any round for a given network of fixed size and structure, under the
rules of the network.
Definition 2. MX(n) is the maximum number of exchange that can
occur in any round for a fully connected network of size n under
the rules of the network.
For the e = 1 and dyadic exchange rules, it is easily seen that NX is
determined by the structural linkages while MX equals the number of
nodes div 2 (i.e. the integer portion of nodes/2). Although a systematic analysis of NX vs MX is beyond the scope of this paper, the ratio
NX/MX represents how many nodes must be excluded in any given
round, hence it should relate to the the rapidity with which networks
converge to theoretical power limits (see Brennan 1981). A tentative
hypothesis, implied by the previous discussion, is:
Hypothesis 1. If NX < MX, then adding any link which increases
NX will decrease overall power differential in the network.
R.K. Leik / New directions for network exchange theory
In addition,
a second tentative
hypothesis
conclusions about additional links.
summarizes
315
the
above
Hypothesis 2. Adding links beyond those needed to achieve
MX will not produce a further decrease in power differential
network.
NX =
in the
These are tentative
hypotheses
because as yet they have not been
systematically
applied to larger networks or those with uneven numbers of nodes. For networks of even numbers of nodes, NX= MX
means all nodes can participate
fully each cycle. For networks of odd
size, NX = A4X still leaves at least one node excluded each cycle
regardless of how effectively people bargain.
Before considering
Willer’s five-person
networks,
note that the
network in Fig. l(b) is the same as Willer’s Network f. Adding a link
to one known structure produced another known structure. Given a
fixed network size, the probable consequences
of adding or deleting
links should be deducible from current knowledge of the fixed networks those changes will produce.
Would disadvantaged
actors in the original network capitalize on an
added link? In the laboratory, assuming that actors operate according
to the Markovsky et al. (1988: 223) assumptions,
it is likely that a
rearrangement
of power would follow relatively soon after restructuring the linkages (cf Leik 1975). In long established power system with
well ingrained norms and relationships,
however, it may take more
than mere opportunity
to over come the sense that the occupant of a
previously greater power position deserves a disproportionate
share of
profit (see Meeker’s 1971 discussion of a status-congruent
norm) or
that one is committed to previous partners (Cook and Emerson 1978,
Tallman et al. 1991). Norms and values will not be treated here, but
one aspect of norms will be addressed later.
The only other four-person
system in Willer’s examples
is his
Network b. That network is easily derived from his Network f by
deleting a link between
B and one of the D’s. As in Network f,
NX = MX and the network size is even. One would expect, then, that
any attempt by the B’s to deal unequally with each other would result
in each turning primarily to the A’s, creating two consistent dyads of
equal power and a bifurcated
network. It is a reasonable
conjecture
R. K. Lrik
316
/ New dircrtions
for network exchange theory
that even if NX is less than MX, deleting a link will not increase
power differential
unless it results in decreased
NX. In short,
Hypothesis 3. Deleting a link will result in increased
tial only if it results in a reduction in NX.
power differen-
The GPI indexes for Wilier’s Network f are not the same for all
nodes, implying power differential,
although they are the same for
Network b, implying power equality. Nevertheless,
it is likely that both
networks will bifurcate, producing power equality.
Three new scope conditions are needed for the next hypotheses:
Condition 6. Actors have complete and accurate information
about
all network linkages.
Condition 7. Actors understand
and use the principles of network
power (ability to expound them is irrelevant).
Condition 8. Actors try to maximize their own power.
Based on hypotheses
following hypotheses.
Hypotheses
add one or
Hypotheses
delete one
1 and 3 and conditions
6-8, we can derive the
4. An occupant of a low power position
more new links to the network.
5. An occupant of a high power position
or more existing links in the network.
will prefer
to
will prefer
to
In essence, low power positions need alternatives
while high power
actors prefer to isolate those dependent
upon them.
The three added conditions are most important. Without sufficient
information
and the savy to utilize it, neither the weak nor the strong
will be able to preceive the advantage of linkage changes. Regarding
information,
Willer (19921 states that it “. . . has been shown not to be
a condition of power distribution
or development”.
Yet experiments
which only allow bargaining
within fixed structures
do not allow
strategic agency. Information
would be crucial to plotting optimal
structural
changes and to timing the moves to be made (Leik and
Gifford 1985; Leik 1991).
‘Savy’ is used above rather than ‘intelligence’ because understanding the complexities
of network power requires
a great deal of
R. K. Leik / New directions for network exchange theory
317
experience
and/or
a strong grasp of the principles of the structural
basis of power. Thus an aspect of current methodology
which may
seriously constrain both theory and results is that the typical subject
spends only an hour or two in the laboratory, providing barely enough
time to learn to bargain within the existing structure.
How much
experience would an average subject need to become astute regarding
the strategic
agency? It may be unrealistic
in terms of research
funding, but an ideal experiment
may need to hire subjects for many
weeks of regular laboratory sessions.
Five-node network
For Willer’s five-node networks (c and g), adding one or more links is
rather more complex than for the four-actor networks. Space does not
permit a full discussion. Network c is shown in Fig. 2(a). Markovsky et
al. (1988) analysis showed that the network would typically bifurcate
into a power differentiated
triad and an equal power dyad. That
network,
incidentally,
is already at NX= MX. There can be two
separate
exchanges
in any given round, and five nodes can never
support more than two under the e = 1 rule.
Adding a single link to this network can occur in four ways:
connecting
the A’s (Fig. 2(b)), connecting
an A with C (Fig. 2(c)),
connecting an A with D (Fig. 2(d)), or connecting B with D (Fig. 2(e)).
Normalized
GPI values are shown in each case. The variance of the
those values is also shown for each network. Assuming that the GPI is
providing correct indications of relative power, and that the variance
is an appropriate
index of power differentiation,
the results are most
interesting.
In the initial structure, B is in an advantaged position, especially if
the system bifurcates.
Had the 2(a) network bifurcated,
then B’s
advantage
would decrease
if the B-D link is formed; but if the
original structure had not bifurcated,
the B-D link would moderately
increase relative power for B, C and D. However, 2(e) is likely to
bifurcate into the same A-B-A
and C-D subsystems likely for 2(a). D
might initially favor forming a B-D link, but the end result is likely to
be no change in power throughout
the network.
B would benefit considerably
from an A-D link (Fig. 2(d)), as
would D. GPI for the A linked to D falls to zero before and becomes
R.K. Leik / New directions for network exchange theory
318
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.17 .33
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probrblo
rlr
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mrdcovslcy,
lt al.
I
D .17
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0
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.43
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II
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C.
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0
R.K. Leik / New directions for network exchange theory
319
zero after adding the link, but by increasing the overall power differential of the network, the link would effectively put that A in a weaker
position. As astute A would certainly object to such a link. This
transformation
of the original structure shows the greatest power
differential of all variations considered.
Adding an A-A link (Fig. 2(b)) works to the disadvantage of B and
D while augmenting the positions of C and both A’s. If the original
structure had not bifurcated, such a link increases overall power
differentiation, but if it had bifurcated, then the A-A link would work
to make power more equal. Clearly, such a link would be opposed by
B and D and desired by C and the A’s who formed it.
Incidentally, if the A-A link was formed, and a B-D link was then
added, the result would be Willer’s Network g, which has B very
slightly advantaged and the other four nodes equal. Since the A’s
would presumably favor linking to each other, and B and D would
likely share dissatisfaction at the link, B and D could recoup losses by
linking and undoing what the A’s had accomplished.
An A-C link (Fig. 2(c)) leaves amount of power differentiation
basically the same compared with the original network after bifurcation, but increases differentiation compared with the network before
bifurcation. C gains by the A-C link, but the newly linked A losses the
advantage. Presumably that A would object to the link while C would
seek it.
The preceding discussion should make evident that moving from a
four-node net to a five-node net greatly increases the possible transformations and the complexity of strategy involved in those transformations even when only one additional link is considered. One clear
implication of the preceding analysis is that adding one or more links
in an odd-size network may either increase or decrease power differential when the network is at NX = MX.
Only one possible five-person network has NX < MX. That is a
single central node linked to the other four nodes which are not
linked to each other. The central node would be very powerful.
Adding any single link would set NX and MX equal and, according to
Hypothesis 1, greatly diminish power differentiation,
although the
central node would still have a power advantage.
320
R.K. Leik / New directions for network exchange theor);
Adding vs deleting
Little has been said about deleting links, but the power implications of
deleting can be deduced by working backward from the link-adding
examples. However, there is an interesting problem regarding adding
vs. deleting. If a node is allowed to add a link, that increment
in
alternatives is likely to be seen as a positive thing. On the other hand,
taking away a link has a kind of punishing or attacking character. If it
were a node’s only link, deleting it would mean isolation. The more
personal the contact, the more punishing would be its removal (see
Molm 1991, on the larger negative impacts of punishment on satisfaction compared with the positive impacts of reward).
So long as there are no major differences
in the implications
for
power advantage, it seems likely that from the standpoint of a sense of
decency or fair play, actors will prefer adding links to deleting them,
especially if they perceive that they are interacting with other individuals rather than ‘faceless’ corporations,
governments,
etc. Even if
deleting one link is necessary in order to forge another (as in Leik
19911, the node that is ‘dropped’ is likely to resent it. In short, the
emotional implications of manipulating
linkages are not symmetric for
adding vs. deleting.
Hypothesis 6. Power implication
negotiate
being equal, actors will prefer
for adding links than to negotiate for deleting links.
to
‘Real world’ competitors,
of course, frequently
attempt to break up
competing coalitions. Whether experimental
subjects could be brought
to use a deleting strategy even when it is within the rules of the
experiment
may well depend upon the scenario of the experiment,
such as whether the other nodes are seen as persons or as generalized
competitors.
Both scenarios and instructions for experiments
designed
to test propositions
about strategic agency will have to consider such
motivational
problems carefully.
Some remaining
issues
Although there
been addressed
are many aspects
here, two warrant
of strategic agency with have not
brief attention. The first concerns
R.K. Leik / New directions for network exchange theory
321
network size. Experiments necessarily deal with small networks. For
relatively few nodes, the implications of adding or deleting a single
link can be dramatic. As network size increases, however, while mean
network density remains constant, a single change should have less
impact on overall power differentiation. We might expect:
Hypothesis 7. As network
size increases, assuming constant mean
density, more successive linkage changes will be needed for any
node to experience a given degree of change in relative power.
There are undoubtedly notable exceptions to Hypothesis 7, in that
certain critical potential links may exist in quite large networks. The
intent of the hypothesis is ‘in general’ or ‘on the average’. Although
we do not create large networks in laboratories, a theoretical tie
between network exchange theory and macro analysis of network
power will have be accomodate network magnitude.
A related aspect of network size is illustrated by considering a node
that is connected to all other nodes in the network. For small
networks, if rules and time permit, that node could engage in exchange with each other node every round. As the size of the network
increases, however, limiting factors of time and energy, if not formal
network rules, will prohibit the node’s participation in all of the
possible exchanges. Thus MX becomes a practical problem in large
networks vs. a rules problem in small experimental networks. At some
point, adding links per se will have absolutely no value.
We do know, however, that being linked to the ‘right’ others is
more valuable than just being linked to others in general. This is a
different sense of linkage, compared with that of network exchange
theory. A powerful other is a nemesis when pure bargaining is
considered, but can be a great aid as a broker (see Marsden 1982) or
patron. As network exchange theory moves toward more macro formulations, it must consider both the diminishing utility of added
linkages and the increasing utility of the ‘right’ linkages.
The second problem concerns how linkage changes can be brought
about. If a new link has to be purchased in some sense, then more
powerful hence wealthy nodes can dominate structural change, using
it to their own advantage. Power will beget power. If new links are
earned in some way, existing power has less advantage, while if new
links depend only upon agreement between those to be linked, then
322
R.K. Leik / New directions
fornetwork exchange theory
existing power will face a proliferation
of linkages that deplete its
positional advantage.
Put in these terms, the question of network dynamics becomes a
political question. Principles or rules which govern adding or deleting
links will determine
the extent to which power structures
are self
perpetuating
and enhancing vs. random consequences
of change processes vs the source of their own destruction.
Unless the theory on
which experiments
are based is explicit on these questions, results are
likely to be contradictory.
Conclusions
The genius of Emerson’s early work was to allow the experimenter
to
vary social structure in the laboratory
to determine
its consequences
for social power. By fixing structure in any given experiment,
however,
network power theorists have ruled out the most powerful source of
power gain: strategic agency. If a single position is allowed the right to
affect linkage changes, that position can alter its power. If only
adjacent linkages are available for change, the power advantage may
be quite small. If all linkages can be changed, however, the position
can eventually dominate the network. If all positions have the same
opportunity,
then issues of differential
strategy become paramount.
Strategic agency is critical to a theory of network power. This paper
has urged expanding our theories to consider such agency. The task is
large and difficult, but the prospects are exciting.
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