UNDER SUBMISSION TO NATURE, MARCH 2016
1
A Universal Model for War and Peace
arXiv:1604.01693v1 [cs.SI] 6 Apr 2016
Weisi Guo1* , Xueke Lu2 , Samuel Johnson3
Human flourishing is often severely limited by violent
conflict, yet the root causes of war and peace are intricate
and difficult to quantify. In recent years, the increasing
availability of data has led to models which attempt to explain the complexities of warfare from various perspectives
[1]–[5]. However, a universal spatial model that can predict
areas of war and peace on a global scale remains absent.
Here, we take up the oft-cited hypothesis that land trade
routes play a key role in determining both the strategic
importance and the vulnerability to conflict of cities and
regions around the world [6], [7]. Using population data
from cities, we infer a global network by means of
the ‘gravity law’. We find that two standard network
measures, degree and betweenness, together predict with
high accuracy whether a given region will be at risk of
suffering conflict. We use these results to propose a metric
called strategic centrality, and find that it accounts for
over 80% of the variance in number of attacks among the
world’s main population zones. It is also a good predictor
of the distance to a conflict zone, and can be used to
provide an estimate of the conflict risk for a given region.
Our results demonstrate that despite the complexities underlying human violence, a pattern exists at the global scale
such that it is possible to make predictions based on simple
network measures, significantly outperforming predictions
based on established socioeconomic or geopolitical factors.
These findings suggest a particular connection between
conflict and trade routes which should be explored in more
depth. We also anticipate that the model can be used to
guide developing regions in the construction of new cities
and major transport links.
Human conflict in one guise or another has shaped our
world, enthralled historians for millennia, and continues to
represent an existential threat to humanity. The conditions
which lead to war or peace are multi-dimensional, and seem so
specific and interwoven that it is often considered simplistic or
naive to search for general patterns which might provide real
predictive power. The mathematical analysis of conflict goes
back as far as the 1940s [8] (and more recently in [2]), where
it was found that both the cumulative and non-cumulative
intensity of conflict followed power-law distributions and
exhibited fractal properties. However, the conflict data for
earlier eras are sparse, inaccurate, and often aggregated across
a whole war (i.e. the death-tolls were in the 10s to 100s of
thousands). In recent years, the greater availability of data has
1 Weisi Guo (corresponding author) is with the Warwick Institute for Science
of Cities (WISC) and the School of Engineering, University of Warwick,
United Kingdom. 2 Xueke Lu is with the School of Electronic Engineering
and Computer Science, Queen Mary University of London, United Kingdom.
3 Samuel Johnson is with the Mathematics Institute and the Centre for
Complexity Science, University of Warwick, United Kingdom.
a
j
Fi j
i
500km
No. of Cities (nodes)
7,322
No. of Trade Routes (links)
137,358
Max. Land Hop Distance
500km
Max. Water Hop Distance
100km
No. of Conflict Incidents
(2002-14)
31,731
b
c
Great Lake Cities of
USA & Canada
Top 50 High
Degree Cities
2200km to nearest
conflict zone
1400km to nearest
conflict zone
e
d
PKK
AQIM
ISIS
Taliban
AQAP
Sudan
Civil War
Boko Haram
Al Shabaab
Fig. 1. Complex network of cities and trade links. High degree cities are
far from conflict, whereas high betweenness cities are close to conflict.
Subplot a) Cities v ∈ V are represented by spots (colour indicates country
and size indicates population) and trade routes between cities are represented
by links. Links are undirected and constructed using the gravity law (weighted
by flow Fij and limited to 500 km) and represent generic flow of information,
goods, and people. Two major disconnected graphs are constructed (the
Americas, and Eurasia-Africa). Subplots b) and c) show high degree cities
(size indicates degree value D(v)) are mainly in Europe and along the Great
Lakes in North America. These cities are far away from the nearest major
conflict site. Subplots d) and e) show high betweenness cities (size indicates
betweenness value B(v)) are mainly in the Saharan Africa, the Middle East,
and South Asia. The high betweenness cities are close to major conflict sites
and the operation areas of major international terrorist groups.
led to a growing literature on building more accurate spatial
and temporal models. These include the importance of unclear
cultural boundaries [1], [9], the intensity pattern of conflicts
[2], military technology diffusion [3], the erosion of the natural
environment [4], [10], the effect of aid economy [11] and
natural resources [6], the role of political and cultural values
[12], [13], and how modern trade alliances act as a barrier
to inter-state warfare [5]. Most of the existing work focuses
on a specific region or relationships, and very few research
outputs consider the effect that a global network might have
on a region. Here we focus on post-Cold War conflict, where
UNDER SUBMISSION TO NATURE, MARCH 2016
Degree
a
2
Betweenness
d
Strategic Centrality
g
j
R2=0.82
b
e
h
R2=0.55
k
Summary Table
c
f
i
Parameter
High
Degree
High
Betweenness
High
Strategic
Centrality
Threshold
>104
>107
>104
P(A>100)
0-0.35
0.7-1.0
0.7-1.0
Distance to
Conflict
>1200km <150km
<50km
R2=0.51
Fig. 2. Scatter plot of number of attacks versus the network metrics of zones z; with smaller panel plots on the probability of major attack and
distance to nearest major conflict site. Subplots are divided in accordance to the top 250, 500, and 1000 populated zones (representing 15%, 27%, and 45%
of the total modeled population respectively). The colour gradient of the scatter plots indicate the mortality as percentage of population, showing high attack
zones suffer a disproportionately high mortality rate. The scatter plots show the data for A > 100 to highlight major conflict zones. Subplots a)-c) show
the threshold relationship between the degree of a zone and the number of attacks, whereby a high degree zone (D(z) > 104 links) experiences almost no
attacks and is on average over 1200 km away from the nearest major conflict zone. A low degree zone has a significant probability of being in major conflict
P (A(z) > 100) >0.35. Subplots d)-f) show the trend for betweenness of a zone: a high betweenness zone (B(z) > 107 shortest paths) has a high probability
of major conflict P (A(z) > 100) =0.7-1.0, and is close to existing major conflict zones (<150 km). Combining degree and betweenness, we propose strategic
centrality S(z) = B(z)/D(z) to measure the average number of shortest paths per link. Subplots g)-i) show the relationship between the strategic centrality
and the number of attacks. The results indicate a strong correlation (R2 =0.82-0.52), and a general predictor for the number of attacks has a power law form
log10 (A∗ (z)) = a log10 S(z) + b. A zone with high strategic centrality has a high probability of being in major conflict P (A(z) > 100) =0.7-1.0 and is
close to existing major conflict zones (<50 km). A summary of the findings can be found in subplot k).
conflict is taken to include international criminal violence,
terrorism, and regional warfare [10]. We apply complex network methodologies to analyse the oft-cited hypothesis that
global land trade routes play a key role in determining both
the strategic importance and the vulnerability to conflict of
regions [14], [15]. In particular, militia groups that previously
benefited from Cold War sponsorship now hunt for economic
resources in order to function as independent forces [6]. Their
mode of operation frequently involves pillaging resource rich
areas and using regions with a power vacuum as bases of
operations [7]. As a result, modern conflicts on the Eurasian
continent frequently occur in areas that are in close proximity
to high-value trade routes and in porous areas that are not
under rigorous governmental control. Using population data
for over 7,000 locations, we infer a global trade network by
means of the ‘gravity law’ (Fig. 1a). In this paper, trade
flow refers to the diffusion of goods, information, and people
between cities. We apply network metrics to identify cities of
strategic trade importance, namely their degree D (number of
neighbouring cities) and betweenness B (number of shortest
paths passing through them), combine these into a metric we
call ‘strategic centrality’, and propose an explanation for the
relationship between these network properties and conflict.
The results show that cities or zones with a high degree
are always relatively peaceful, whilst those with a high betweenness have a significant probability of experiencing severe
and prolonged violence. Geographically, the vast majority
(97%) of the top 500 high degree cities are in: i) Western,
Central, and Eastern Europe (Fig. 1b); and ii) North-East
USA along the Great Lakes (Fig. 1c). The clusters of high
degree cities form regional cohesive sub-graphs with high
topological redundancy. These regions have no reported per-
UNDER SUBMISSION TO NATURE, MARCH 2016
sistent terrorism or conflict related death tolls in the period
2002-14 and are far away from the nearest major conflict
zones (mean distance is over 1200 km)1 . On the other hand,
the top 50 high betweenness cities are almost exclusively
in the following areas: i) North and West Africa (Fig. 1d),
and ii) the Middle East, Arabian Peninsula, East Africa, and
South Asia (Fig. 1e). Most of these regions are suffering from
severe international conflicts and domestic insurgencies (e.g.
terrorism, gang warfare) and also lie on major land trade
routes. Together, they are referred to as the new silk trade route
of terrorism and organised crime [15], [16]. In particular, we
were able to identify high betweenness zones that correlated
with specific major terrorist groups and persistent conflict
zones (e.g. the Second Sudanese Civil War). This suggests
that the high betweenness cities not only attract land trade but
are also the strategic locations from a military perspective.
Conflict often occurs across a region that covers multiple
cities. We group cities into zones z with a 500 km radius
(each covering 0.5% of the global land surface area) and
examine the top 1000 populated zones. Fig. 2 shows scatter
plots of the number of attacks A(z) against the network
metrics, with panel plots on the probability of being a major
conflict zone and distance to the nearest major conflict zone.
A major conflict zone is defined as a location that experiences
persistent conflict (A(z) > 100 results in at least 78% of the
years being under attack – see SI). The results for degree
and betweenness centrality indicate a threshold behaviour,
whereby if the zone has D(z) > 104 links (high degree –
see Fig. 2a-c) or B(z) < 107 shortest paths (low betweenness
– see Fig. 2d-f), then there are very few attacks (<1/year).
Conversely, if the the zone has D(z) < 104 links (low
degree) or B(z) > 107 shortest paths (high betweenness),
then there is a high probability P (A > 100) that the zones
will experience major conflict (see Fig. 2a-f). Furthermore, a
higher number of attacks also corresponds to a higher mortality
(see Fig. 2j), which shows that the number of attacks is not
simply proportional to the population of the zone. In general,
the probability of a zone being in major conflict P (A > 100)
is shown to decay with increasing degree and decreasing
betweenness values. We also show that the average distance
from any zone to the nearest top 100 major conflict zone rises
with increasing degree. High degree zones are at least 1200 km
away from the nearest conflict zone, whereas high betweenness
zones are usually less than 150 km away.
In order to further refine the prediction of conflict, we consider the problem of a city or zone that is both high degree and
high betweenness. There is a need to combine betweenness
and degree metrics to gauge the strategic importance of cities.
We propose a new metric called strategic centrality for a zone
B(z)
, which normalises the betweenness
z, defined as S(z) = D(z)
of a city by the number of links. It is a measure of the number
of shortest paths per link connected to a city. A city with a high
betweenness will have global trade importance, while a low
degree (more isolated) will increase the relative weight of each
trade route, thereby augmenting its attractiveness to militant
1 We consider conflict to be a sustained high number of attacks, and so
isolated high profile terrorist attacks (e.g. 9/11 in New York and 7/7 in
London) do not necessarily signify major conflict zones
3
a
v
1 relay node in zone z
S1 = B1/D1
= (M-1)N/2
b
Core 1 (N Nodes)
Core 2 (N Nodes)
v
Fragmentation into K
identical connected
relay nodes in zone z
K relay nodes in zone z
SK = B1/K(D1+K-1)
≈(M-1)N/2K
Fig. 3. A graph showing K relay nodes connecting M cores, each with
N nodes. Scenario (a) shows a single relay node and Scenario (b) shows it
fragmented into K relay nodes. The solid lines indicate the shortest path and
the dashed lines indicate other paths (not shortest). Each relay node in a zone
of K relay nodes
a degree DK (v) = M N + K − 1, a betweenness
has
BK (v) = M
N 2 /K, and a strategic centrality SK ≈ M N/2K. In
2
terms of cities, fragmentation of one city into many cities will reduce
the betweenness and strategic centrality of any individual city and hence
reduce its attraction to militants (i.e., fewer trade routes). Fragmentation will
also increase the number of neighbours and its degree, which reduces its
vulnerability.
groups and vulnerability to conflict. The results indicate that
zones with a high strategic centrality suffer both a high number
of attacks and a dispassionately high mortality rate (see
Fig. 2g-i). The results indicate that the trend is similar for the
top 250 and top 1000 populated zones. The predicted number
of attacks A∗ has a power law form with respect to the strategic
centrality of the zone: log10 (A∗ (z)) = a log10 S(z) + b; or, in
linear terms, A∗ (z) = 10b S(z)a , where the best-fit parameters
are a = 4 and b = −9 for top 250 to top 1000 populated zones.
The corresponding adjusted R-squared value varies between
0.82 (top 250 populated zones) to 0.51 (top 1000 populated
zones). The results show that strategic centrality is a better
predictor of conflict than either degree or betweenness. A
low strategic centrality zone (S(z) < 104 ) will experience
no or very few attacks. On the other hand, high strategic
centrality zones (S(z) > 104 ) are on average less than 50
km away from the nearest major conflict zone. A summary
of the findings regarding degree, betweenness, and strategic
centrality, and their relationship to conflict, can be found in
the table in Fig. 2k. In order to explore the robustness of these
results with respect to the zone size and the trade flow weight
coefficient, alternative results are shown in SI. We find in
all cases very similar correlations between number of attacks
and the centrality metrics, which indicates that the proposed
method is robust to modelling parameters. We also show in SI
that the network centrality metrics presented here are not trivial
proxies for existing geopolitical or socioeconomic metrics,
which do not account for conflict in the way that our model
does.
We further expand the analysis by considering how the
changing pattern of cities would affect strategic centrality S
UNDER SUBMISSION TO NATURE, MARCH 2016
using a synthetic representation of the trade network (Fig. 3).
Each core M is a complete graph consisting of N nodes, but
the cores cannot directly connect to each other as they are too
far apart. All the core nodes are connected to each other via
K relay nodes. If the number of relay nodes increases from
1 (Fig. 3a) to K (Fig. 3b), S decreases in inverse proportion
to K, suggesting the emergence of new cities around existing
high betweenness cities would effectively reduce the S in the
zone z (Methods). Conversely, when applied to the real world
network, the prospect of merging cities could potentially make
them more vulnerable to conflict.
The model can be used to identify relatively peaceful
regions which are at particularly high risk of violence. Fig. 4
highlights cities whose strategic centrality predicts a significantly higher number of attacks than experienced (Fig. 4a).
Two major geographic areas have been identified (yellow
labels): i) Saudi Arabia, Iran, Turkey (Fig. 4b), and ii) southwestern China (Fig. 4c). Of particular interest is Saudi Arabia,
which is surrounded by several major existing conflict zones
(i.e. Yemen, Syria and Iraq, red labels). According to the
model, her major pilgrimage and capital cities are particularly
vulnerable due to their high strategic centrality. The second
region of concern is in southwestern China, where the southern
silk trade route passes from China into the Golden Triangle
of Burma and Thailand. This trend of high S and reported
growing violence suggests a potential conflict area in the
future.
“Wars are caused by undefended wealth”, claimed General
Douglas MacArthur. This is, in a nutshell, the hypothesis we
have followed here for developing a simple network metric
based on land trade routes, which we assume to be essential
for sustaining post-Cold War conflict. The fact that this method
predicts conflict with such accuracy suggests the approach is
correct. However, other potential reasons for the empirical
findings should not be dismissed out of hand. Rather, these
results highlight the need for further research to examine the
link between network structure and human struggle. Assuming
that this link is causal, as conjectured, these results may have
important policy implications. As the urban population grows
in the 21st century, the emergence of new cities and major
transport links in developing countries presents a scenario in
which to consider whether the distribution of cities and links
can have an effect on resilience to conflict [17]. Other trends
include the merging of several small cities into fewer and
larger cities and the large-scale migration that leaves many
communities depleted. These trends pose both opportunities
and risks to regional stability, and we encourage others to
investigate more deeply the role of strategic centrality in the
hope of informing development plans.
4
a
A*=10b Sa
1
5,6,7,8
2
3,4
b
Danger Zones (Key
Cities indicated)
Strategic Current Expected
Centrality Attacks Attacks
1. Mecca, Medina,
Jeddah (Saudi Arabia)
1300
12
2860
2. Abha, Jazan, Najran
(Saudi Arabia)
1000
7
1000
3. Gorgan, Mashhad
(Iran)
700
3
240
4. Ankara (Turkey)
650
2
170
5. Riyadh Province
(Saudi Arabia)
550
15
90
6. Tehran, Karaj (Iran)
520
14
70
7. Kunming, Baoshan,
Xichang (China)
500
8
65
8. Kuwait City (Kuwait) 490
4
60
4. Ankara
Adana
3. Gorgan
Aleppo
5. Tehran
Al-Raqqa
Tel Aviv
Kabul
3. Mashhad
Kermanshah
Damascus
Maymana
Rawalpindi
Baghdad
Basra
Islamabad
7. Kuwait
City
Al Khobar
4. Riyadh
1. Mecca,
Medina
Jeddah
2. Abha
Predicted Zones
of Danger
Sanaa
Existing Zones of
Major Conflict
c
Delhi
Patna
6. Baoshan
Golden
Triangle
Calcutta
Dhaka
6. Guiyang
6. Kunming
Mandalay
Fig. 4. High population zones that are either in existing conflict or are
vulnerable to future conflict. The labelled cities in red are zones which are
already in major conflict (A(z) > 100), and the zones labelled in yellow are
largely peaceful, but are experiencing rising levels of violence. In sub-plot a),
we identify outliers zones that have a low number of attacks compared with
peer zones with a similar level of strategic centrality. In the table, the current
number of attacks accumulated in each zone A(z) between 2002 to 2014 is
presented along side the predicted number of attacks A∗ (z). Two major areas
have been identified: b) Saudi Arabia and Iran, and c) southwestern China.
UNDER SUBMISSION TO NATURE, MARCH 2016
5
I. M ETHODS
this paper by considering longest inter-city highway distances
in Russia, Australia, and Canada. This way, a multi-hop road
/ railway network is created and artificial direct land links
that stretch over 500km between major cities (do not exist
in reality) are eliminated. Two further constraints are put in
place: a) eliminate sea travel (> 50km), and b) in order to
only consider major links and cities, we only consider cities
with a population over 10,000 people, which are less sensitive
to population changes (83% of all cities in the data set).
The latter 2 constraints are not strictly necessary to achieve
similar statistical results, but will create a large number of sea
routes which removes some land-based shortest paths from
strategic cities. As a result, a global multi-hop trade network
is constructed (Fig. 1a). The network comprises of 7,300+
nodes (cities weighted by population P ) and 130,000+ links
(trade routes weighted by flow F ).
1) Data Set: The paper leverages on the following open
data sets:
1) City data: 7322 cities with their latitude, longitude, and
population data from National Geospatial Intelligence
Agency 2 . The geospatial data includes cities that vary
in population from mega-cities (several millions) to
small towns. The data represents ≈25% of the world’s
total population and it includes over 2800 cities with
a population over 100,000, yielding a sufficiently high
city resolution. For the purpose of this paper, we shall
call all settlements cities. Each city is also tagged with
its country and province affiliation.
2) Conflict data: 30,000+ conflict incidents between 2002
to 2014 from the Global Terrorism Database (GTD)
[19]. The GTD contains the number of attacks and death
toll from violent conflicts, which range from small-scale
assassinations (1 death) to large-scale massacres (1000s
dead). Most of the death toll data is time stamped and
geo-tagged (longitude and latitude). The GTD data is
further processed to: (i) fill in any missing longitude and
latitude information, (ii) remove attacks with unknown
location information (less than 1%), and (iii) cluster the
number of attacks and death toll data to the nearest city
in the geospatial data set (mean distance 27km).
3) Geopolitical and socioeconomic data: the 2014 GDP per
capita and income inequality data from the World Bank
and the International Monetary Fund, and the democracy
index developed by the Economist Intelligence Unit
(EIU). We use the data sets to show that the centrality
metrics proposed are uncorrelated to existing established
geopolitical and socioeconomic metrics (see SI-D).
A. Constructing Trade Links from Gravity Law
The aim of this paper is to develop a universal model
for conflict prediction, one that avoids explicitly modeling
the complexities related to socioeconomic and geopolitical
parameters. In the post-Cold War era, the importance of land
trade routes to militant groups has been well documented [14],
[15]. In order to quantify the volume of trade flow, we use the
gravity law to infer probable trade routes. In this paper, trade
flow, denoted Fij , refers to the diffusion of goods, information,
and people between cities i and j. Existing literature has
shown that land trade is undirected and the trade flow is
proportional to the population of the two cities P and inversely
proportional to the square of the Euclidean distance d, such
P P
that: Fij = diij 2j [20], [21] (see Supplementary Information
(SI) for network model and parameter justification). Since any
finite value of population and distance will create a non-zero
value of flow, every city will be connected with every other
city. Thus, in order to consider only land routes, it is necessary
to eliminate the possibility of long distance maritime and air
routes by setting a maximum distance of a land route between
any two given cities. The maximum distance is set to 500km in
2 The dataset has been improved with population data and can be found at
[18]
B. Complex Network Metrics
As conflicts often happen over an area involving multiple
cities, we are interested in the properties of a zone z. We
define a zone as an area that has many cities, i.e., v ∈ Vz .
In the main paper, the results are presented for circular zones
of 500km in radius, each representing 0.5% of the modeled
global land surface area. Overlap of the zones is permitted and
each zone is centred on a particular city. The paper is firstly
interested in two primary measures in network science: degree
and betweenness. A city’s degree is defined as the number of
links with neighbouring cities, D(v) = deg(v). The degree is
unweighted to highlight the importance of the number of links
(neighbours), as opposed to the importance of the neighbours
or links. The total degree property of a zone z is defined as
the sum of all the degree values of each city inside
P the zone
(number of links inside a zone), i.e., D(z) = v∈Vz D(v).
Trade routes often travel the shortest multi-hop path between
a number of cities. Betweenness measures the number of
shortest paths through a node (i.e., city). The shortest path
of travelling between a city m to any other city n is defined
as the path with the least number of hops. The un-normalised
betweenness of a city Bv is defined as the number of shortest
paths that pass through this city:
X
B(v) =
σmn (v),
(1)
m6=n6=v
where σm,n (v) is the shortest path between m and n that goes
through city i. The total betweenness property of a zone z is
defined as the sum of all the betweenness values of each city
inside the zone (numberPof shortest paths that pass through
the zone), i.e., B(z) = v∈Vz B(v). Unlike the unweighted
degree, we are interested in the weight contribution of each
link. This is because multi-hop trade routes are likely to
consider a balance between: i) the shortest path, and ii) one
that also passes through major cities for increased profit (i.e.,
attracted to population P ). For the weighted network, one
can define the weight of each link between 2 arbitrary cities
i and j as inversely proportional to the expected flow value
[22]:
wij = Fij−θ ,
(2)
UNDER SUBMISSION TO NATURE, MARCH 2016
6
where Fij is the flow value and θ = [0, 1] can offer full
weighting (θ = 1) or no weighting (θ = 0). For example,
if the flow weight of a link is high, then the weight of the
link (for shortest path calculation) is negligibly small and
there is no minimum or maximum link weight. In this paper’s
main analysis and results section, we have selected to present
the results for a balanced weighted betweenness centrality
measure (θ = 0.5), such that a balance exists between the
weight of a link and the number of links. Nonetheless, we
present the results for θ = 0 (no weighting) and θ = 1
(full weighting) in the SI to demonstrate the robustness of
the methodology.
B(z)
, which
The strategic centrality for a zone z as S(z) = D(z)
normalises the betweenness of a city by the number of links. It
is a measure of the number of shortest paths per link connected
to a city. A city with a high betweenness will have global trade
importance, while a low degree (more isolated) will increase
its vulnerability to conflict. Let us consider a simple problem:
what happens to the degree and betweenness properties of the
relay nodes, when the number of relay nodes K changes. Let
us at first consider a single relay node v in a zone z (scenario
(a) in Fig. 3). The degree of the node or the zone is D1 (v) =
D1 (z) = 2N if the relay node is connected to all N nodes in
the cores. The betweenness of the node or the zone is B1 (v) =
B1 (z) = N 2 , if every shortest path passes through the relay
node. Consider now the case the single node fragmenting into
K relay nodes - see scenario (b), the degree value of each
node becomes DK (v) = D1 + (K − 1), of which there are K
in the zone (i.e., D(z) = KD(v)). As for the betweenness,
one needs to consider the average value across all relay nodes
in the zone, as each single relay node will have a different
number of shortest paths that pass through it. In total, the
same BK (z) = B1 shortest paths pass through K relay nodes
in zone z. Hence, the average betweenness in the relay nodes
is E[BK (v)] = BK (z) = BK1 . Note, no shortest paths between
relay nodes exist, as all relay nodes are directly connected to
each other. Strategic centrality is the same for each node and
B1
K]
for the whole zone: SK (v) = SK (z) = E[B
DK = K(D1 +K−1) .
More generally, if there are M cores, each with N nodes, the
degree of a relay node is:
DK (v) = M N + (K − 1),
and the betweenness of a relay node is:
2
M N
.
BK (v) =
2 K
(3)
(4)
Therefore, the strategic centrality of the K interconnecting
relay nodes is
M
N2
N
SK =
≈ (M − 1)
, (5)
2K
2 K(M N + K − 1)
for a larger number of core nodes compared to the relay nodes
(N K). Therefore, increasing the number of relay nodes K
will reduce the strategic importance of any one relay node, and
increasing the number of core nodes (M − 1)N will increase
the relay nodes’ strategic importance. The strategic centrality
analysis is equally applicable on the individual node v level
as well as the zone z level.
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Author Contributions
W.G. conceived the idea and sourced the data, W.G. and
X.L. designed, constructed, and analysed the complex network. X.L. designed the complex network metric and S.H.
provided the theoretical analysis.
Author Information
Reprints and permissions information is available at
www.nature.com/reprints
The authors have no competing financial interests to report.
Correspondence and requests for materials should be
addressed to: [email protected]
Email address of X. Lu: [email protected]
Email address of S. Johnson: [email protected]