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SOCIAL SCIENCES
A three-degree horizon of peace in the
military alliance network
Weihua Li,1 Aisha E. Bradshaw,2 Caitlin B. Clary,2 Skyler J. Cranmer2*
INTRODUCTION
The tendency of political organizations to ally likely dates back to the
earliest permanent human settlements and certainly predates the
modern political state by at least two millennia (1). The formation
of modern military alliances is multifaceted: States ally to counter
common threats, to bolster one another’s security, and even to help
facilitate peace between potential rivals (2, 3). Military alliances are
typically understood as a tool for securing defensive cooperation
and fostering peace between direct allies (4–9). Despite the long
history of the practice of forming alliances, the effect of the alliance
network’s structure on bilateral peace is poorly understood.
The past century has seen the advent of alliances meant to exist in
times of peace as well as war, and the corresponding increase in the
interconnectedness of the alliance network has been dramatic. The
system has gone from 8 individual alliance ties in 1900 to 1115 in
2000, including bilateral and multilateral defense pacts.
Much research has shown that allies are less likely to fight one another (6, 10), but we lack any understanding of higher-order effects.
That is, are states whose only connection is a common ally—those
separated by two degrees—less likely to fight? What about pairs
connected at three or more degrees? To answer this question, we examine the conflict propensities of indirectly allied states through the
concept of path length. Within the alliance networks, direct allies have
a path length of one degree, the allies of allies have a path length of
two degrees, and so on (fig. S1 and tables S1 to S3). That is, if state i is
allied to j, j is allied to k, and k is allied with l, then the path length
between i and l is three. Figure 1 shows the concept of distance in a
network—the shortest number of links required to connect one state
to another—and the focal question of bilateral conflict.
Many states that have never signed a formal defensive pact have at
some point experienced the pacifying effect of higher-order alliances.
One example is Turkey and Iran, who did not enter into conflict at
any level from 1965 to 1979, during which period they were indirectly
1
LMIB, BDBC and School of Mathematics and Systems Science, Beihang University,
Beijing 100191, China. 2Department of Political Science, The Ohio State University,
Columbus, OH 43210, USA.
*Corresponding author. Email: [email protected]
Li et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
connected in the alliance network at two degrees of separation. After
losing this connection in 1980, disputes arose between the neighbors,
reaching a peak in 1987 when Turkey was mediating the conflict between Iran and Iraq and became involved itself in a militarized dispute
with fatalities.
In another case, Spain and its former colony Morocco had several
disputes with one another in the 1970s, including an incident in 1972
when the states deployed naval forces to exchange fire after failing to
reach a new fishery agreement. However, the states have maintained
peaceful relations since they established an indirect alliance at three
degrees of separation beginning in 1981. Using state-of-the-art statistical models for network data, we investigate the importance of path
length and identify the horizon within which indirect alliances contribute to bilateral peace.
Defensive alliances are formal promises “to assist a partner actively
in the event of an attack on the partner’s sovereignty or territorial integrity” (11). Hence, they signal a basic level of agreement and affinity
between the allied states. Although alliance partners may not agree on
every issue, an alliance is unlikely to be sustained if the partners have
large differences in their preferences with respect to the security
environment. That is, the basic condition that must be met for a
defensive alliance to form and persist is that the allied parties must
agree that major changes to the status quo are in neither of their best
interests at that time. We use this as our operational understanding of
an alliance.
Although some scholars have argued that alliances can actually increase the risk of conflict incidence or escalation, by drawing more
actors into ongoing conflicts or by encouraging states to behave more
recklessly than they otherwise would have (12–14), there are compelling reasons to discount these arguments. First, alliances are sometimes formed to address already tense security situations, making a
positive association between alliances and wars potentially spurious
(10). Further, recent evidence suggests that defensive pacts (the alliances with which our study is concerned) are associated with less
conflict, not more (15). On balance, we therefore expect defensive
alliance ties to be associated with a reduced risk of conflict. Our
study starts from this premise about direct alliance ties and considers
the ramifications of indirect ties with the expectation that these ties
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States form defensive military alliances to enhance their security in the face of potential or realized interstate conflict. The network of these international alliances is increasingly interconnected, now linking most of the states in a
complex web of ties. These alliances can be used both as a tool for securing cooperation and to foster peace between direct partners. However, do indirect connections—such as the ally of an ally or even further out in the alliance network—result in lower probabilities of conflict? We investigate the extent to which military alliances
produce peace between states that are not directly allied. We find that the peacemaking horizon of indirect alliances
extends through the network up to three degrees of separation. Within this horizon of influence, a lack of decay in
the effect of degrees of distance indicates that alliances do not diminish with respect to their ability to affect peace
regardless of whether or not the states in question are directly allied. Beyond the three-degree horizon of influence,
we observe a sharp decline in the effect of indirect alliances on bilateral peace. Further investigation reveals that the
community structure of the alliance network plays a role in establishing this horizon, but the effects of indirect
alliances are not spurious to the community structure.
2017 © The Authors,
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SCIENCE ADVANCES | RESEARCH ARTICLE
?
(War)
(War)?
Fig. 1. Illustration of degrees of separation in the alliance network. Examples of
alliances at one to four degrees of separation are presented, with the focal states
embedded within the global alliance network. The paths linking the focal pair of
states are in red, whereas the rest of the states in the network are in light yellow.
Our research investigates the probability of conflict between states at different
degrees of separation in the alliance network.
should reduce the risk of conflict but should have a less pronounced
effect than direct alliances.
A basic argument about the transitive properties of positive affinity
in social networks holds that the friend of my friend should be my
friend or at least not my enemy. If an actor i has enough in common
with another actor j to form a positive bond, and j has enough in
common with a third actor k to form a positive bond, then this would
suggest that actors i and k have a fair amount in common. They may
not be friends, but enmity between the two would be surprising. We
can then ask how far this logic extends. If the ally of my ally should, at
minimum, not be my enemy, what about the ally of my ally of my
ally? Unlike existing studies, which have examined the effects of direct
alliances on the conflict propensities of states and have used measures
like alliance portfolio similarity to consider the consequences of shared
security preferences (16), our study is designed to capture the effects of
the emergent complexity inherent in the international alliance
network. This approach is different from that of studies testing classic
hypotheses about the relationship between direct alliance ties and conflict between states. We both test the general expectation that indirect
alliance ties have an effect on conflict propensity and probe the radius
within which indirect connections can effectively foster peace.
If states i and k both agree substantially with j on the security status
quo, this would suggest that i and k should also have a fair amount in
common, and that they should agree with one another, at least in
broad strokes. A naive expectation drawn from this argument could
be that a chain of allies, however long, should lead to peaceful relations
between the members of the chain. That is, allies of allies of allies would
be expected to behave peaceably, as would allies of allies of allies of allies
and so on. This simple transitive logic leads to our first network hypothesis about indirect alliance effects: The ability of higher-order alLi et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
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(War)?
(War)?
liances to suppress conflict is constant, regardless of the degree of the
connection.
The network of international defensive alliances in the post–World
War II period forms a giant cluster (or it can be said to be “percolated”): The web of alliance connections includes nearly all actors in
the system, and there are few isolated clusters of alliances. This means
that most of the states are connected to each other indirectly at some
degree. If the first network hypothesis were on target, then it should be
the case that conflict is quite rare between any states having one or
more defensive alliances. However, the increasing density of the alliance network has not been accompanied by a reduction in military
conflict (17).
To square the transitive argument about higher-order alliances
with the empirical reality of alliance network percolation and the
lack of near-complete global peace, we return to our consideration
of the three states: i allied to j allied to k. If i and k agree with one
another on the security status quo, then they might well decide to
ally with each other. This is the sort of transitive agreement on security
that we would expect to lead to triadic closure—i, j, and k all being
allied—and broad multilateral alliances. Not only has this pattern
of multilateral alliances been posited in the literature, but it also has
robust empirical support (17, 18). This pattern of alliances has a strong
effect on conflict between the states under consideration: Not only
is the probability of military conflict lower between allies (6, 10, 19),
but also having allies in common further lowers the probability of
conflict.
If trilateral (or multilateral) agreement on the status quo tends to
lead to trilateral (or multilateral) alliances and the lower probability of
conflict between allies that accompanies them, then the fact that i and
k are not directly allied sends a distinctly different signal than if they
were. It might be the case that the ij alliance pertains to one aspect of
the status quo and the jk alliance to another, meaning that the ik pair
might lack the common ground to form an alliance between them.
Yet, although i and k may not have sufficient common ground to form
an alliance, it seems relatively unlikely that their interests and attitudes
toward the security status quo would be so diametrically opposed as to
lead to conflict between the two of them. Thus, we expect a secondorder alliance like that between i and k to reduce the probability of
military conflict between the two of them relative to a pair of states
that share no alliance connections.
The same dynamic applies to third- and fourth-order alliances. If
the lack of a direct alliance between states that share a common ally
carries information, then we can expect that every time the order of
the alliance is increased, the two states under consideration have progressively less and less in common in terms of their attitudes toward
the security status quo. One can think of this like a game of telephone,
with each additional degree of separation introducing some decrease
in the amount of agreement between the originating state and the terminating state. This dynamic takes on the form of a decay: the extent
to which originating state i and some other state in the chain of alliance connections p agree on the security status quo decays, possibly
nonlinearly, because the distance between those two states in the alliance network increases. As the degree of agreement on the status quo
decreases with each step, the probability of conflict between the two
states sees a corresponding increase. It is important to note that this
increase is not relative to the conflict propensity of two unconnected
states but relative to two states that are directly allied. The theorized
decay in the peacemaking ability of alliances at higher degrees results
in our second network hypothesis that the probability of conflict is
SCIENCE ADVANCES | RESEARCH ARTICLE
RESULTS
Context of the study
We identify the network horizon within which indirect alliances—
those at degrees of separation greater than one—contribute to bilateral peace and probe the drivers of the observed pattern. We seek to
distinguish between two main possibilities for the effects of indirect
alliances. Our premise is that states form defensive alliances because
they see their security interests as fundamentally aligned with those
of their partners.
We would expect that states’ interests align closely with those of
their allies but less closely with those of more distant connections. If
this is the case, then the first possibility is that the effect of alliances
will gradually decline as the degree of separation increases before
eventually disappearing. Such a pattern has been found in empirical
studies of social networks (22–24). Alternatively, it may be that the
peacemaking influence of indirect alliances is driven by the structure
of the overall alliance network and the communities that form within
it. If this is the case, then the boundaries of communities of states
more tightly connected to each other than to the rest of the network
will define the horizon of peacemaking in the alliance network. Such
a process would produce an effect on conflict for low-order indirect
alliances before rapidly losing influence between more distantly
connected states. This second possibility implies that the observed
effect of indirect alliance ties on conflict likelihood could be spurious,
driven by community effects that drive down the probability of conLi et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
flict for closely connected states. Following our main results, we perform a mediation analysis to weigh these two possibilities.
Our application has an important difference from previous studies
on the role of network connection: International conflict requires the
states involved to be able to engage one another militarily. There are
barriers to war that are difficult for most states to surmount. Chief
among these barriers is physical distance. Some degree of proximity
is usually necessary for states to develop conflicts of interest serious
enough to escalate to violence. Only an extreme minority of states
are capable of projecting power over considerable distances, making
proximity a requirement for the act of war in most cases; Bolivia
and Angola could not fight a war with each other even if they wanted
to. Moreover, physical contiguity is usually necessary for war, because
few states are capable of projecting force over one or more of their
neighbors. This is all to say that the goals and expectations outlined
above apply to pairs of states sufficiently proximate to engage in war
rather than the less interesting set of states for which conflict is structurally unlikely.
We use the temporal exponential random graph model (TERGM)
(25–28) for longitudinal network data to trace the impact of indirect
alliances on bilateral conflict from 1965 to 2000 (see Materials and
Methods for model details). A serious military conflict is coded as
any incident that involved the direct and deliberate use of violence
by the armed forces of one state against another, and cases were
drawn from version 3.0 of the Militarized Interstate Dispute (MID)
data set gathered by the Correlates of War (COW) project (29). We
restrict our analysis to MIDs of levels 4 and 5, because those are the
hostility levels in which actors in at least one country make the deliberate choice to deploy military violence against another.
To build the conflict network for each time step, we linked a pair
of states if they engage in a new serious military dispute in that year.
The alliance network was constructed from the Alliance Treaty Obligations and Provisions (ATOP) data set (11). We connect a pair of
states in the alliance network if they have at least one active defense
pact in that given year. We then create indicators for the path distance between all pairs of states in the network. These indicators
serve as our key predictors. We include these indicators in our
network model together with well-documented controls found to influence conflict behavior: joint democracy (30), geographic contiguity
(29), combined national capability ratio (31), and trade dependence
(32). Details on the methods, data, and measures are presented in
Materials and Methods. Figure 2 shows the interplay between the
alliance and conflict networks.
Besides accounting for alliances and control covariates as described
above, we also include endogenous factors that reflect the structural
features of the outcome network. These endogenous factors must be
specified in the data-generating process to avoid omitted variable bias
and misspecification. The existence of a k-star in the conflict network
implies that the central state is in conflict with k other states (33). We
use the alternating k-star statistic proposed by Snijders et al. (34), in
which the weights have alternating signs in a geometrically decreasing
fashion. Triangles describe a situation where states i and j are in conflict, although they both are at war with another state k simultaneously. This configuration is unlikely to occur in the international
conflict network, according to the logic that “the enemy of my enemy
is my friend.” If state i is fighting state j, and state j is fighting state k,
then it is unlikely states i and k will fight each other (27). We include
the geometrically weighted edgewise shared partner (GWESP) statistic
for closed conflictual triads in our model (34).
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lower for dyads closer to each other in the alliance network than for
dyads connected through higher degrees of separation.
However, these processes do not function independently of regional dynamics. Specifically, past research has shown that alliance dynamics interact strongly with geographic contiguity (20). Contemplating
this persistent interaction, it is difficult to imagine two contiguous
states at any location in an alliance chain having the same probability
of conflict as a similarly located noncontiguous pair, and there are very
few states, however close their geographic distance, that have the
power to project serious conflict over another state. Thus, it is reasonable to account for power projection capabilities by including a dichotomous measure of state contiguity. We hypothesize that the effect of
degrees of separation in the alliance network is strongly interactive
with geographic contiguity; the effect is expected to be stronger for
contiguous states.
Last, our analysis also considers the effects of the communities that
states form within the alliance network. Communities here refer to
sets of countries that are joined together in tightly interconnected
groups, between which the connections are sparse (21). This sort of
community structure, which has been studied extensively in social
and biological networks, is also observed in the alliance network. It
is also worth noting that a state can simultaneously participate in several multinational alliances, but this view of network communities
places each state into exactly one community, as identified by a specific partition algorithm. Members of a community share a greater
common interest in the security status quo than outside states do.
The community also provides forums through which members can
strive for peaceful dispute settlements. States belonging to different
communities may have opposing interests and are more likely to resort to force in the face of international disputes. Thus, we expect that
the community structure of the alliance network contributes to the
peacemaking effect of indirect alliances.
SCIENCE ADVANCES | RESEARCH ARTICLE
Alliance
Conflict
Alliance
Conflict
Alliance
Conflict
CSTO
Arab League
NATO
NATO
Rio Pact
OAS
Warsaw Pact
Warsaw Pact
NATO
OAS
ECOWAS
Arab League
Arab League
ECCAS
ANAD
0.25
0.06
0.20
Probability of conflict
Rate of dispute onset
Apart from k-stars and triangles, we examine a more complex endogenous effect of the outcome network: the four-cycle, which is the
configuration that occurs when dyads ij, jk, kl, and li are connected
but dyads ik and jl are not. In our outcome network, this would produce a subgraph structure, where states i and j, j and k, k and l, and l
and i are fighting each other, but states i and k, and j and l are not in
conflict. The four-cycle accounts for a pattern of relations, where the
existing ties in the conflict network might affect the chances of a new
conflict (35).
0.15
0.10
0.05
0.00
Li et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
0.04
0.02
0.00
1
The three-degree peacemaking horizon
Our statistical analysis reveals that both direct and indirect alliances of
up to three degrees have a strong suppressive effect on conflict. This
relationship is borne out in the raw data (Fig. 3 and fig. S2): Across all
years in the period of observation, pairs of states that were unallied
(directly or indirectly) experienced higher rates of conflict than pairs
with direct or indirect alliances. This holds true for both physically
contiguous and noncontiguous states, though we naturally see a higher conflict rate among contiguous states because contiguous pairs are
most likely to develop serious conflicts of interest and have the ability
to fight.
Alliance
Absent
Present
2
3
4
5
Inf
Degrees of separation
1
2
3
4
Degrees of separation
Fig. 3. Conflict onset by degree of separation for geographically contiguous
states in the alliance network. Pairs of states that are not connected at any degree
are denoted as infinite (“Inf”), and conflict onsets are observed between contiguous
states at up to five degrees of separation. (Left) Rate of conflict onset from the raw
data, with the pink horizontal line indicating the average rate for contiguous dyads.
The first three degrees of separation see below-average conflict onset rates. (Right)
Dyads at degrees of separation ranging from one to four are sampled, and their respective conflict probabilities conditional on the rest of the network are computed. The plot
shows the probability of conflict when an alliance is present or absent between a contiguous pair of states with 95% confidence intervals. The probability of conflict drops
substantially between contiguous states indirectly allied up to three degrees.
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Fig. 2. The interplay between the international alliance and conflict networks. (Top) Networks in 1965, 1980, and 2000. States are plotted as nodes in the network, and
the node color indicates the number of alliances of a given state, with darker nodes having more allies. Red ties represent conflicts in all plots, whereas alliance ties are in
different colors across time. Names of communities are given by the multinational alliance to which most of the members in the community belong. On the world map of
1980, with the alliance network in light orange, we illustrate sample states in higher-order alliances. Spain and Italy, joined together by the United States, are at two degrees of
separation. Central African Republic and Sudan, allies at three degrees of separation, are connected by France and Djibouti. CSTO, Collective Security Treaty Organization;
ECOWAS, Economic Community of West African States; ECCAS, Economic Community of Central African States; ANAD, Agreement of Non-Aggression and Defense Assistance.
SCIENCE ADVANCES | RESEARCH ARTICLE
Li et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
Table 1. TERGMs of the conflict network with geographic contiguity
interacted with alliances, 1965–2000. All models are estimated with
bootstrap-corrected pseudolikelihood using 1000 replications. Ninety-five
percent confidence intervals are presented in brackets. Coefficients and
confidence intervals displayed in bold are statistically significant at the
P = 0.05 level.
Variable
Edges
Model 1
Model 2
−7.99
(−8.24 to −7.77)
−8.06
(−8.28 to −7.85)
Alternating k-stars (2)
1.11 (0.98 to 1.22) 1.11 (0.98 to 1.23)
Four-cycle
0.56 (0.44 to 1.16) 0.55 (0.44 to 1.08)
GWESP (0)
−0.30
(−0.51 to −0.13)
−0.31
(−0.52 to −0.13)
Joint democracy
−0.66
(−0.91 to −0.44)
−0.69
(−0.94 to −0.49)
Direct contiguity
4.34 (4.10 to 4.64) 4.39 (4.16 to 4.68)
Capability ratio
−0.17
(−0.22 to −0.12)
−0.16
(−0.22 to −0.12)
Trade dependence
−0.18
(−0.52 to −0.05)
−0.16
(−0.43 to −0.04)
One-degree alliance
0.96 (0.44 to 1.42)
Two-degree alliance
0.17 (−0.31 to 0.61)
Three-degree alliance
0.44 (−0.03 to 0.91)
Four-degree alliance
−0.26 (−1.02 to 0.26)
Contiguity ×
One-degree alliance
−1.49
(−1.92 to −0.99)
Contiguity ×
Two-degree alliance
−0.84
(−1.45 to −0.27)
Contiguity ×
Three-degree alliance
−1.28
(−2.10 to −0.54)
Contiguity ×
Four-degree alliance
0.13 (−0.85 to 1.15)
Within community
0.70 (0.26 to 1.10)
Neighboring communities
0.63 (0.27 to 0.99)
Contiguity ×
Within community
−1.16
(−1.54 to −0.73)
Contiguity ×
Neighboring communities
−1.43
(−2.02 to −0.87)
pute onsets as the dependent variable (38). As the MID codebook indicates, some disputes at hostility level 4 do not involve governmentto-government conflict; cases such as the seizure of a fishing vessel
belonging to a citizen of another state, which are much less likely to
involve decisions from, or authorization by, key executives, are included in the data set.
These disputes, which Gibler and Little (38) refer to as “protestdependent MIDs,” are caused by different processes and are unlikely
to escalate to major warfare. These cases targeting private citizens may
introduce additional measurement bias and are not in accordance with
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Among all alliance pairs, geographically contiguous pairs have a
much higher rate of conflict than noncontiguous pairs. This is as
expected because those pairs are the most at risk, and alliances are
sometimes used to maintain peace between rivals. Most interesting,
however, is the fact that the conflict rate drops markedly among alliances within three degrees: states allied with one, two, or even three
degrees of separation enjoy lower conflict rates than other pairs.
Hence, allies within three degrees of separation, whether contiguous
or not, form the most peaceable strata of state pairs in the international system.
Our results show that indirect alliances promote peace up to three
degrees of separation among geographically contiguous states in the
alliance network but have no effect at higher degrees of separation.
That is, allies, the allies of allies, and the allies of allies of allies enjoy
lower rates of military conflict than would be expected otherwise. In
the TERGMs fit to the data, we see a substantial interaction effect between one-degree (that is, direct) alliances and geographic contiguity,
where the coefficient is −1.49 (−1.92 to −0.99; Model 1, Table 1), and
the coefficient for the main effect of one-degree alliances is 0.96, which
suggests that contiguous states at one degree of separation in the alliance network tend to be less likely to experience interstate conflict.
Similarly, the coefficient for the main effect of two-degree alliances
is 0.17, whereas its interaction with contiguity is a statistically reliable
−0.84 (−1.45 to −0.27), and the main effect for three-degree alliances is
0.44, whereas its interaction is a statistically reliable −1.28 (−2.10 to
−0.54). A more detailed model interpretation is presented in fig. S3.
It is worth noting that indirect alliances have even stronger suppressive effects on conflict than joint democracy (coefficient, −0.66), which
has been consistently identified as a key, if not the key, factor for bilateral peace (36).
Examining the predicted probabilities of conflict produced by this
model further clarifies the results (Fig. 3): For contiguous states, the
probability of conflict decreases by a substantial margin for alliances of
one to three degrees, but this effect is lost beyond three degrees; for
noncontiguous states, there is no discernible effect. The predicted
probabilities of conflict are affected not only by the alliance network
but also by control factors such as democracy, trade, and capability
ratio. Together, these results indicate that an alliance (direct or indirect) reduces the risk of conflict between states separated by up to
three degrees, but no farther (see fig. S4 for model goodness of fit and
table S4 for summary statistics).
The suppressive effect of indirect alliances does not lose potency as
the degrees of separation are increased, up to the horizon of three
degrees. Although it would seem natural that the peacemaking effect
of these connections should wane as the degrees of separation between
two indirect allies increase, we observe a consistent effect across three
degrees, followed by a sharp drop-off in the pacifying effect at four
degrees of separation and higher. Thus, although our results bear
some resemblance to previous studies that have identified “three
degrees of influence” in social networks of individuals (22–24), these
earlier explanations cannot account for the lack of decay within the
horizon of influence and the sharp drop-off we observe outside that
horizon. These results are robust even when we knock out some control variables or endogenous factors (table S5) or replace the MID conflict data by Maoz’s dispute data (table S6) (37). The three-degree
horizon of influence can also be recovered in the more classic logistic
model (Model a2, table S7).
In addition, to theoretically complement our study, we also control
for heterogeneity in the dispute data and use only state-to-state dis-
SCIENCE ADVANCES | RESEARCH ARTICLE
Community structure of the alliance network
To better understand the three-degree peacemaking horizon of alliances and the lack of effect decay within that horizon, we examined
the community structures of the alliance networks (Fig. 4). As discussed
earlier, we seek to determine whether alliance ties have an independent
conflict-suppressing effect within three degrees of separation or whether
the observed effect is spurious, driven by the community structure of the
alliance network. We first discuss the procedure for identifying communities in the network and then perform a mediation analysis to determine whether the effect of alliance ties on conflict is strictly attributable
to the intervening variable of community structure.
We identified communities of states in the alliance network based on
the extent to which the states in the communities are better connected to
each other than to those outside the community. We define two community variables: “Within community” captures whether a pair of states
belongs to the same community, and “neighboring communities” indicates whether the pair of states is in separate, but closely linked, communities. Specifically, a pair of communities is regarded as neighboring
if they are themselves weakly connected by one or more alliances. We
expect joint membership in an alliance community to push members
toward peaceful relationships by providing members with mechanisms
for dispute settlement and mediation (39). Furthermore, states in these
close communities of allies likely have broad agreement on the security
status quo. This scenario also applies if the focal states are affiliated with
different alliance communities, but these communities are more
connected to each other than to other communities. Therefore, we also
expect alliances in the neighboring communities to have a tendency to
maintain peaceful relations. We expect that the within community and
neighboring communities together correspond with the boundary, beyond which the effectiveness of alliances drops drastically.
We present the results derived from applying the Walktrap algorithm to partition the alliance network into communities (40). Networks with a high modularity score have dense connections between
vertices within communities but sparse connections between vertices
Li et al. Sci. Adv. 2017; 3 : e1601895
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Proportion of ties
Type
Within community
Neighboring communities
Any tie
Degrees of separation
Fig. 4. Pairs of states at different degrees of separation can be categorized by
the community structure of the alliance network. Within community counts the
number of state pairs, where both states belong to the same alliance community.
Neighboring communities are composed of pairs of states that are embedded within
different communities while the two communities are themselves connected by some
ties. The “Any tie” bars denote the number of ties that belong to either within community or neighboring communities. The horizontal axis shows the proportion of alliances
that belong to a specific community structure. Because several states belong to more
than one community, some alliances fall into both within community and neighboring
communities at the same time.
in different communities. Modularity values may range in [−0.5, 1].
In our application to the alliance network, observed modularity values
range from approximately 0.55 to 0.65, which indicates obvious community clustering (fig. S7). Concern over possibly not finding the true
maximum modularity partition and the possibility of near optimal
partitions make it prudent to also consider the quality of partitions
offered by different community detection methods (41). As a check
on the robustness of our Walktrap measure, we apply several different
community detection algorithms for each system year. Among the
candidate algorithms, Walktrap usually defines a community structure
with a median modularity score (40, 42–46). Because Walktrap does
not have the most strict or loose criteria among other popular
methods, it is a fairly conservative choice. This was our motivation
for using its results in the main text.
Figure S8 presents an intuitive picture of the community structure
we find in the alliance network, where states are colored to indicate
their community membership. Because the community assignments
are based on the alliance network only, we occasionally see a few
close, politically relevant, states that are assigned different communities. An interesting example is Canada, which has been a key member
of the North Atlantic Treaty Organization (NATO) while maintaining
close relationships with the American countries. In fig. S8, Canada is
sometimes assigned to the NATO community and sometimes to the
American community. The community assignment of states that have
close ties with states in more than one community (United States and
Canada) is somewhat algorithm-dependent and may vary across years
in these special cases. For this reason, we have assigned states that
have ties with most of the states in more than one community to both
communities, as described above. Other special states, such as Japan,
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the classic definition of militarized conflict, so it is useful to differentiate
these cases from disputes that represent militarized actions authorized
by state governments. Therefore, we examine only state-directed disputes as our dependent variable in an additional test. This analysis recovers the three-degree peacemaking horizon of defensive alliances
(table S7) and strengthens our main findings reported previously.
The predicted probability of conflict between contiguous states that
are connected by an indirect alliance also drops remarkably when higherorder effects are included in the model (fig. S5). Consider, for example,
the relations between Russia and Turkey, long characterized by tension
and mistrust. They share an indirect alliance at two degrees, and we see a
sharp decline in the predicted probability of conflict when we account
for higher-order effects. Poland and Russia, third-order allies, also decrease their likelihood of conflict when we consider connections at higher orders. This effect is not significant at the fourth order.
Why is the effect of indirect alliances limited to three degrees of
separation, and why does the suppressive effect not decay? One might
look to the instability of longer path lengths in a network (for example, cutting a tie has compounding effects on the persistence of longer
paths) for an explanation of the prior. The volatility of paths through
the network increases greatly after three degrees (fig. S6), but this instability alone does not provide an explanation for the persistent
strength and lack of decay of indirect alliances’ suppression of conflict
within the first three degrees.
SCIENCE ADVANCES | RESEARCH ARTICLE
national politics can no longer be viewed as sets of bilateral relations
but that bilateral relations are embedded within international networks
that influence even states that are not formally or directly connected. This
approach and its findings step far outside of the established practice in
international relations scholarship. We have merely scratched the surface
of what indirect ties can teach us about the functioning of the international system. We have also shown that communities and neighboring
communities can exert strong effects on the behavior of their members.
Considering international networks in terms of their community
structures is also a promising area for future research (48, 49).
Our results reveal four novel and important features of the international political system. First, we have identified the network horizon
within which military alliances contribute to bilateral peace. Indirect alliances make peace more likely between states separated by as much as
three degrees. Second, we showed that the peacemaking ability of indirect alliances does not lose its potency within this horizon of influence.
The effect of (in)direct alliances on conflict is consistently strong for
states separated by up to three degrees, after which indirect alliances
quickly lose their impact. Our analysis suggests that this finding is partially driven by the influence of community structures—states cluster
into various subnetwork communities that shape state behavior beyond
the influence of formal alliances—but the horizon of three degrees of
separation also has an independent effect. Last, we found that the
impact of these alliances is strongest for contiguous states. Because
neighboring states are particularly likely to fight, any institution that
can reduce the risk of violence for contiguous states is normatively important for those seeking to engineer a safer international system.
Together, these findings suggest that policy-makers assessing threats
might have less reason to be concerned about states to whom they
are connected, even indirectly, by up to three degrees, whereas states
outside this “low-risk zone” could pose a greater threat. Therefore,
our study provides policy-makers with an additional tool to assist with
the allocation of security resources. Our findings suggest that leaders
would be wise to focus their attention on those states separated by more
than three degrees in the network of international military alliances, because these states present the greatest risk of serious conflict.
MATERIALS AND METHODS
Conflict data
Our outcome is a complete international network in which ties indicate serious interstate conflict onsets (MID levels 4 and 5) from 1965
to 2000. The original data came from version 3.0 of the MIDs data set
of the COW Project (29). A dyad in the network for a given year was
coded 1 (0 otherwise) if at least one conflict started during this time
period. MIDs were defined as cases of conflict in which the threat,
display, or use of force (including war) by one member state is explicitly directed toward the government, official representatives, official
forces, property, or territory of another state (50).
In the MIDs data set, hostility levels for each dispute are identified
by (i) no militarized action, (ii) threat to use force, (iii) display of force,
(iv) use of force, or (v) war. We included conflict onsets of levels 4 and 5
only, because these are the cases involving the deliberate use of military
force by state actors. This sometimes leads to full-scale war (defined as
MIDs involving more than 1000 battle deaths) and sometimes does not.
DISCUSSION
Our findings suggest radical new directions for the study of international politics by demonstrating that state behaviors can spread
indirectly through international networks. We have shown that interLi et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
Defensive alliances
To compute the indicators that captured the degree of separation
between two states in the defensive alliance network, we used the
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Australia, Philippines, and Pakistan, have only one defensive alliance
connection in the network (with the United States), and modularity
assignment for these states is sometimes variable. Because our analysis
incorporates both within community dyads and neighboring community dyads, these special cases do not affect our results because they
can mostly be categorized in either variable with their close friends
(for example, in 1965, the United States and Canada dyad appears
in the neighboring communities, whereas in 2000 they are assigned
to one community). Apart from these states, most of the states are
well classified into exactly one regionally based community.
Figure 4 shows that more than 90% of alliances within three degrees
of separation fall into either category of the community variables,
whereas less than 8% of alliances at four or higher degrees of separation
are contained in these community variables. Statistical analysis shows
that the within community indicator has a significantly negative effect
on conflict [main effect, 0.70; interaction with contiguity, −1.16 (−1.54
to −0.73); Model 2, Table 1], whereas neighboring communities also
significantly reduce conflict onsets [main effect, 0.63; interaction with
contiguity, −1.43 (−2.02 to −0.87)]. Thus, it appears that the sharp decline in alliance effectiveness at four degrees of separation could occur
because states at this distance no longer share membership in common
communities that represent shared interests.
However, further exploration reveals that the finding that three or
fewer degrees of separation are associated with reduced levels of violent
conflict is not epiphenomenal to the community structure. That is, both
indirect alliance ties and community structure play independent roles in
reducing the probability of conflict. In table S7, a mediation analysis
finds that the community structure only partially mediates the relationship between alliances of up to three degrees and conflict (47). The relationship between states linked through alliances of up to three degrees
and conflict (Model a2 in table S7, path c in fig. S9) is significant (P <
0.05, step 1 of Baron and Kenny’s method). Moreover, the relationships
between alliances of up to three degrees and community structure
(Models a4 and a5, P < 0.05, step 2 of Baron and Kenny’s method)
and between community structure and conflict (Model a3, P < 0.05, step
3 of Baron and Kenny’s method) are both significant. After controlling
for community structure, both community structure and the alliance
variables remain significant in predicting conflict (Model a1, P < 0.05,
step 4 of Baron and Kenny’s method), indicating that the community
structure of the alliance network partially mediates the relationship between alliances within three degrees of separation and conflict. Therefore, although the community structure exhibits a strong relationship
with alliance degrees, the three-degree peacemaking horizon appears
to be an independent feature of the alliance network, rather than an
artifact of community structure.
Our model results are robust to the choice of graph partitioning
algorithm. Models based on community variables generated by different
algorithms suggest very similar results that do not affect our substantive
inferences (for example, a partial mediation between community structure and the three-degree horizon) in any way. Because presenting those
results here would amount to reprinting the feature tables of our analysis with minor variations in the coefficient values, we do not present
tables of these results. These tables are available upon request.
SCIENCE ADVANCES | RESEARCH ARTICLE
Control variables
In addition to the key covariates and endogenous factors, we also included a set of commonly used control variables that the existing literature has shown to influence conflict. Because conflicts are more
likely between states that are geographically close to each other, we
included a measure of direct contiguity. Direct contiguity accounts
for the existence of a shared land border or a separation of no more
than 150 miles of water between the focal pair of states; the dyad was
coded 1 if the states are contiguous by this standard and 0 otherwise
(29). Because contiguity strongly affects the dynamics of both alliances
Li et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
and conflict, one could expect contiguity and alliances to be relatively
dependent. Conflicts are most likely to take place between proximate
states; thus, we expected that our alliance measures would have the
strongest effect in contiguous dyads. Therefore, we used interaction
effects between our alliance measures and contiguity to capture the
dynamics between alliances and conflict.
To examine how regime type affected the likelihood of conflict,
we included a measure of democracy in a dyad (30). As is standard in
the conflict literature, a state was regarded as a democracy if its Polity
score was 7 or higher, and a nondemocracy otherwise. A dyad in the
democracy matrix was coded 1 if both states were democracies, and 0
otherwise. According to the democratic peace literature, the probability
of conflict should be lower between two democratic states (51).
In addition, we included the dyad’s capability ratio, which is
defined as the log of state i’s capability score divided by state j’s when
i has larger capability. We obtained our data from version 4.0 of the
COW project’s National Material Capabilities data set (52) through the
EUGene data preparation and formatting tool (53). Because a weaker
state is less likely to provoke a war against a significantly more powerful
state, a large capability ratio should have a deterrent effect on conflict.
Another important exogenous factor was trade dependence. The
trade dependence of state i on state j was the volume of trade flows from
i to j, standardized by the gross domestic product (GDP) per capita of i.
We constructed this factor from version 4.3 beta of Gleditsch’s (32)
dyadic trade flow data set and version 6.0 beta of his GDP per capita
data. Because 2.5% of the values in the trade flow variable were missing,
we used single imputation to fill in these missing values based on
observations on other dyad-year variables (54). We also imputed
several missing GDP values based on the trends of surrounding data
for the same state (55). If a state’s trade depends heavily on another state,
it is unlikely that it would provoke a war and risk losing the benefits
from this trade flow. Therefore, large values of trade dependence should
inhibit conflict.
Models
Our outcome was the network of serious MIDs that involved the
deliberate use of violence, including war, by one state against a target
in another state. These data were inherently relational. Because the
conflict behavior of one pair of states may affect the conflict behavior
of other pairs of states, the data must be considered as a complex
network rather than a series of independent and identically distributed dyadic observations. Models assuming independent observations,
such as the logistic regression model, will be biased when applied to
such data (27). We used a TERGM (26, 28), a longitudinal extension
of the exponential random graph model (ERGM) first proposed by
Wasserman and Pattison (25), to model the interdependence inherent
in our data. ERGMs model networks as single draws from large
multivariate distributions and estimate parameters corresponding to
a vector of statistics computed on the observed network. An ERGM
of the network G, where we defined G as its adjacency matrix in which
Gij = 1 if state i connects to j and 0 otherwise, was specified as
PðG; qÞ ¼
expðqT hðGÞÞ
cðqÞ
ð1Þ
where q is the vector of model coefficients, h(G) is a vector of statistics
computed on G that includes both endogenous network structure
factors (for example, degree and triangles) and exogenous covariates
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ATOP database (11). This data set included all the bilateral and multilateral defensive agreements between 1815 and 2003. To match the
available date ranges for other input variables, we used the data from
1965 to 2000. We constructed the defensive alliance network for each
year, in which a dyad was coded 1 if a bilateral or multilateral defense pact existed between the states, and 0 otherwise. Because most
defensive commitments were reciprocal, we regarded the network as
undirected. Although the network was not completely undirected,
less than 1% of the defensive commitments in the ATOP data set
were directed.
The degrees of separation of alliances for a given dyad were the
length of the shortest path between them along the graph, which could
be directly calculated from the defensive alliance network. The matrices
for dyads at one degree of separation were composed of a binary coding
during the time period in which we were interested: 1 if the states of the
dyad were connected in the defensive alliance network and 0 otherwise.
The dyads in the two-degree matrices were coded as 1 if the degree of
separation between the pair was two, and 0 otherwise. Similar definitions
hold for other higher-degree variables.
To operationalize the community structure, we first used the
Walktrap algorithm to assign each state a specific community, including isolates (unallied states). We then retained those communities that
had at least four members and discarded the rest. In so doing, we obtained the major communities in the alliance network, where each
state belonged to exactly one community. However, some states had
important ties to multiple communities; for example, the United States
and Canada in 2000 had close relations with both the NATO and Organization of American States groups of states. It was not reasonable
to classify these states into one community and omit them from the
other. Therefore, we accounted for this overlapping of states by adding
a state as a member to another community if it was allied to most of
the members (more than 50% of states) there. Communities were
heavily regionally based, and in most cases, only very few major
powers, such as the United States, had memberships in more than
one community. Thus, the above step of classifying state memberships
in multiple communities has good performance in characterizing the
alliance networks. After identifying all the well-defined communities
in each system year, we defined two community variables: within community and neighboring communities. If a pair of states belonged to
the same community, then it was coded 1 in within community (0
otherwise). If two communities had one or more direct alliances
linking them, then they were regarded as neighboring communities.
Therefore, we defined the neighboring communities variable to contain all pairs of states that belonged to two different but neighboring
communities, and these pairs were coded 1 (0 otherwise). Because
there were states that were classified as belonging to more than one
community, some pairs were categorized as both within community
and neighboring community pairs.
SCIENCE ADVANCES | RESEARCH ARTICLE
pij ðqÞ ¼ PðGij ¼ 1jGij;q Þ ¼ logit
1
R
∑
!
qr dðijÞ
r ðGÞ
r¼1
ð2Þ
where G−ij refers to the entire network G except for the dyad Gij, logit−1(x) is
the inverse logistic transformation such that logit−1 (x) = 1/(1 + exp(x)),
the subscript r refers to any statistic included in h, of total number R,
anddðijÞ
r is equal to the change in hr when Nij is switched from zero to one.
1
The change statistic dðijÞ
r was computed by subtracting hr ðGij Þ from
þ1
þ1
hr ðGij Þ, where hr ðGij Þ indicates the network with the ij dyad equal
to 1, and hr ðG1
ij Þ indicates the network with the ij dyad equal to 0.
Because these change values were summed if included as an edgewise
covariate in Eq. 2, one may compute the change statistics for a given
element of the generative model and include them directly. This is
useful because it is computationally faster to calculate change statistics for a given dependency than to code new structural parameters.
When a single TERGM is applied to a pooled time series of networks, it is implicitly or explicitly assumed that all change in the
network is attributable to Eq. 1 or its equivalent Eq. 2. A bootstrap pseudolikelihood approach based on change statistics was proposed by
Desmarais and Cranmer (28). The procedure works as follows: take a
sample of s estimates of q by drawing s samples of independent (for
example, more than T − K periods removed) networks from the longitudinally observed series of networks {G1,…,GT–K} and by computing q.
The distribution of qis then used to draw unbiased confidence intervals.
For more detailed methodological treatment and applications of
TERGM, see previous studies (26, 27, 56, 57). Beyond the appropriate
selection of K, the TERGM requires no assumptions of statistical independence between states and ties, and it can obtain insight into the
underlying processes that form and maintain the network-based international system with inclusions of network statistics.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/
content/full/3/3/e1601895/DC1
Li et al. Sci. Adv. 2017; 3 : e1601895
1 March 2017
fig. S1. The number of dyads and contiguous dyads at different degrees of separation.
fig. S2. Conflict onset by degree of separation in the alliance network.
fig. S3. Interaction effect of degrees of separation and contiguity on the conditional probability
of military conflict.
fig. S4. Endogenous goodness of fit.
fig. S5. Microlevel interpretations for pairs of states connected by higher-order alliances.
fig. S6. The persistence of alliance paths.
fig. S7. Modularity score of different community detection algorithms.
fig. S8. Maps of the international system in 1965, 1990, and 2000.
fig. S9. Hypothesized mediation model.
table S1. Contiguous dyads in the defensive alliance network in 1965.
table S2. Contiguous dyads in the defensive alliance network in 1980.
table S3. Contiguous dyads in the defensive alliance network in 2000.
table S4. Summary statistics for variables used in the TERGMs.
table S5. TERGM model robustness checks.
table S6. TERGMs using Maoz MID data as robustness tests.
table S7. TERGMs using Gibler and Little’s nonprotest dispute data as robustness tests.
table S8. Logistic models for mediation analysis using Baron and Kenny’s procedure.
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Acknowledgments: We thank J. H. Fowler and Y. Lupu for helpful comments. P. Leifeld
provided technical support. Funding: Support for this research was provided by the NSF
(SES-1357622, SES-1461493, and SES-1514750) and the Alexander von Humboldt Foundation’s
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Submitted 11 August 2016
Accepted 13 January 2017
Published 1 March 2017
10.1126/sciadv.1601895
Citation: W. Li, A. E. Bradshaw, C. B. Clary, S. J. Cranmer, A three-degree horizon of peace in the
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A three-degree horizon of peace in the military alliance
network
Weihua Li, Aisha E. Bradshaw, Caitlin B. Clary and Skyler J.
Cranmer (March 1, 2017)
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doi: 10.1126/sciadv.1601895
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