The topology of covert conflict

0.17 MB·

by  Shishir Nagarja

The topology of covert conflict
Shishir Nagaraja, Ross Anderson
Computer Laboratory
JJ Thomson Avenue,Cambridge CB3 0FD, UK
forename.surname @ cl.cam.ac.uk
Abstract. Often an attacker tries to on disrupting the other side’s command,
6
0 disconnect a network by destroying control and communications by jamming or
0 nodes or edges, while the defender destroying his facilities. Counterterrorism
2 counters using various resilience
operations involve a similar effort but with
mechanisms. Examples include a music
n
industry body attempting to close down different tools: traffic analysis to trace
a
J a peer-to-peer file-sharing network; communications, coupled with surveillance of
medics attempting to halt the spread of
3 the flows of money, material and recruits,
2 an infectious disease by selective
followed by the arrest and interrogation of
vaccination; and a police agency trying
] to decapitate a terrorist organisation. individuals who appear to be significant
I
Albert, Jeong and Barab´asi famously nodes. Terrorists are aware of this, and take
N
analysed the static case, and showed measures to prevent their networks being
.
s that vertex
traced. Usama bin Laden described his
c -order attacks are effective against
[ strategy on the videotape captured in
scale-free networks. We extend this work

1 to the dynamic case by developing a Afghanistan as ‘Those who were trained to fly
v framework based on evolutionary game didn’t know the others. One group of people
1 theory to explore the interaction of
didn’t know the other group’ (see [14], which
0 attack and defence strategies. We show,
1 first, that naive defences don’t work describes the hijackers’ networks).
1
against vertex-order attack; second, that Connectivity matters for social dominance
0
defences based on simple redundancy
6 too,asahandfulofleadingindividualsdomuch
don’t work much better, but that
0 of the work of holding a society together. Sub-
defences based on cliques work well;
/
s third, that attacks based on centrality verting or killing these leaders is likely to be
c
work better against clique defences than the cheapest way to make an invaded coun-
:
v vertex-orderattacks do; and fourth, that
try submit. When the Norman French invaded
i defences based on complex strategies
X England in the eleventh century, they killed or
such as delegation plus clique resist
ar centrality attacks better than simple impoverished most of the indigenous landown-
clique defences. Our models thus build a ers; when the Turks, and then the Mongols, in-
bridge between network analysis and vaded India, they killed both landowners and
evolutionary game theory, and provide a
priests; when England suppressed the Scottish
framework for analysing defence and
highlands after the 1745 uprising, landowners
attack in networks where topology
matters. They suggest definitions of wereinducedtomove to Edinburghor London;
efficiencyofattackanddefence,andmay and in many of the dreadful events of the last
even explain the evolution of insurgent
century, rulers targeted the elite (Russian ku-
organisations from networks of cells to a
laks, Polish officers, Tutsi schoolteachers, ...).
more virtual leadership that facilitates
operations rather than directing them. Moving from politics to commerce, the mu-
Finally, we draw some conclusions and
sic industry spends a lot of money attempting
present possible directions for future
to disrupt peer-to-peer file-sharing networks.
research.
Techniques range from technical attacks to ag-
gressive litigation against individuals believed
1 Introduction
to have been running major nodes.
Many modern conflicts turn on connectivity. Networks of personal contacts are
In conventional war, much effort is expended important in other applications too. In public
health, for example, it often happens that a is concealed from outsiders, with the result
small number of individuals account for much that the participants appear to be ‘ordinary’
of the transmission of a disease. Thus Senegal nodes making a modest contribution in the
has been more effective at tackling the spread network, rather than important nodes that
of HIV/AIDS than other African countries, as should be targeted for close inspection and/or
they targeted prostitutes [19]. In fact, interest destruction?
in social networks has grown greatly over the
last 15 years in the humanities and social 2 Previous Work
sciences [20,9].
There has been rapid progress in recent years
Recent advances in the theory of networks
in understanding how networks can develop
have provided us with the mathematical and
organically, how their growth influences their
computational tools to understand such
topology, and how the topology in turn affects
phenomena better. One striking result is that
both their capacity and their robustness.
a network much of whose connectivity comes
There is now a substantial literature: for a
from a small number of highly-connected
book-length introduction, see Watts [21],
nodes can be very efficient, but at the cost of
while literature surveys are [1,17]
extreme vulnerability. As a simple example, if
Early work by Erd¨os and Renyi modelled
everyone in the county communicates using
networks as random graphs [11,7]; this is
one telephone exchange, and that burns down,
mathematically interesting but does not
then everyone is isolated.
model most real-world networks accurately. In
This paper starts to explore the tactical
real networks, path lengths are generally
and strategic options open to combatants in
shorter; it is well known that any two people
such conflicts. What strategies can one adopt,
are linked by a chain of maybe half a dozen
when building a network, to provide good
others who are pairwise acquainted – known
trade-offs between efficiency and resilience?
as the ‘small-world’ phenomenon. This idea
We are particularly interested in complex
was popularised by Milgram in the 60s [16].
networks, involving thousands or millions of
An explanation started to emerge in 1998
nodes, which are so complicated (or under
when Watts and Strogatz produced the alpha
such dispersed control) that the resilience
model. Alpha is a parameter that expresses
rules can only be implemented locally, rather
the tendency of nodes to introduce their
than by a central planner who deliberately
neighbours to each other; with α = 0, each
designs a network with multiple redundant
node is connected to its neighbours’
backbones.
neighbours, so the network is a set of
Is it possible, for example, to create a disconnected cliques, while with α = ∞, we
virtual high-degree node, by combining a have a random graph. They discovered that,
number of nodes which appear on external for critical values of α, a small-world network
inspection to have lower degree? For example, resulted. The alpha model is rather complex
a number of individuals might join together in to analyse, so they next introduced the beta
a ring, and use some covert communications network: this is constructed by arranging
channel to route sensitive information round nodes in a ring, each node being connected to
the ring in a manner shielded from casual its r neighbours on either side, then replacing
external inspection. There is a loose precedent existing links with random links according to
in Chaum’s ‘dining cryptographers’ a parameter β; for β = 0 no links are replaced,
construction [10], in which a number of and for β = 1 all links have been replaced, so
cryptographers pass messages round a ring in that the network has again become a random
such a way as to mask, from insiders, the graph [22]. The effect is to provide a mix of
source and destination of encrypted traffic. local and long-distance links that models
Can we build a similar construction, but in observed phenomena in social and other
which the fact of systematic message routing networks.
2
How do networks with short path lengths jamming a further third, and hoping that the
come about in the real world? The simplest remaining third would collapse under the in-
explanation involves preferential attachment. creased weight of traffic.
Baraba´si and Albert showed in 1999 how, if
new nodes in a network prefer to attach to 3 Naive Defences Don’t Work
nodes that already have many edges, this
leads to a power-law distribution of vertex Given the obvious importance of the subject,
order which in turn gives rise to a scale-free and the fact that the Albert-Jeong-Barab´asi
network [6], which turns out to be a more paper appeared in 2000, one obvious question
common type of network than the alpha or is why there has been no published work since
beta types. In a social network, for example, on how a network can defend itself against a
people who already have many friends are decapitation attack. Here is onepossibleexpla-
nation: the two obvious defences don’t work.
useful to know, so their friendship is
One of these is simply to replenish
particularly sought by newcomers. In
destroyed nodes with new nodes, and furnish
friendship terms, the rich get richer. There are
them with edges according to the same
many economic contexts in which such
scale-free rule that was used to generate the
dynamics are also of interest [13].
network initially. One might hope that some
Thekey paperforourpurposeswas written
equilibrium would be found between attack
by Albert, Jeong, and Baraba´si in 2000. They
and defence.
observedthattheconnectivity ofscale-freenet-
The other obvious defence is to replenish
works, which depends on the highly-connected
destroyed nodes, but to wire their edges
nodes,comesataprice:thedestructionofthese
according to a random graph model. In this
nodes will disconnect the network. If an at-
way, we might hope that, under attack, a
tacker removes the best-connected nodes one
network would evolve from an efficient
after another, then past some threshold point
scale-free structure into a less efficient but
the size of the largest component of the graph
more resilient random structure. In a real
collapses [2].
application, this might happen either as a
Later work by Holme, Kim, Yoon and Han result of nodes learning new behaviour, or by
in 2002 extended this from attacks on vertices selective pressure on a node population with
to attacks on edges; here, the attacker removes heterogeneous connectivity preferences: in
edges connectinghigh-degree nodes,andagain, peacetime the nodes with higher degree would
past some critical point, the network becomes become hubs, while in wartime they would be
disconnected [15]. They also suggested using early casualties.
centrality – technically, this is the‘betweenness Nice as these ideas may seem in theory,
centrality’ of Freeman [12] – as an alternative they do not work at all well in practice.
to degree for attack targeting. (A node’s cen- Figure 1 shows first (solid line) how the
trality is, roughly speaking, the proportion of vertex-order attack of Albert, Jeong and
paths on which it lies.) Computing centrality Baraba´si works against a simulated
is harder work for the attacker than observing network with no replenishment, then with
vertex degree, butit enables him to attack net- random replenishment, then with scalefree
works (such as beta networks) where there is replenishment. In the vanilla case the attack
little or no variability in vertex order. Finally, takes two rounds to disconnect the network;
in 2004, Zhao, Park and Lai modelled the cir- with random replenishment it takes three, and
cumstances in which a scale-free network can with scale-free replenishment it takes four.
suffercascadingbreakdownfrom thesuccessive It seems that, to defend against these
failure of high-connectivity nodes [23]. These kinds of decapitation attacks on networks, we
ideas find some resonance in the field of strate- will need smarter defence strategies. But how
gic studies: for example, Soviet doctrine called should these be evolved, and what sort of
for destroying a third of the enemy’s network, framework should we use to evaluate them?
3
We now formalise a model in which a game
Vertex−order attack
with naive replenishment is played with a number of rounds.Each round
consists of attack followed by recovery. Recov-
0
40 ery in turn consists of two phases: replenish-
No replenishment
Random replenishment ment and adaptation.
Scalefree replenishment
00 In the attack phase, the attacker destroys
3
a number of nodes (or, in a variant, of edges);
nt
ne this number is his budget. He selects nodes for
o 0
mp 20 destructionaccordingtosomerule,whichishis
o
C strategy. For example, he might at each round
0 destroy the ten nodes with the largest number
0
1
of edges connected to them. He executes this
strategy on the basis of information about the
0 network topology.
0 5 10 15 20 25 30 In the replenishment phase, the defend-
ing nodes recruit a number of new nodes, and
Rounds
go through a phase of establishing connections
No replenishment
– again, according to given strategies and in-
Random replenishment
Scalefree replenishment formation.
In the adaptation phase, the defending
Fig.1. Naive defences against vertex-order de- nodes may rewire links within each connected
capitation attack component of the network, in accordance with
some defensive strategy. The adaptation phase
is applied once at the start of the game,
4 A Model from Evolutionary Game
before the first round of attack; thereafter the
Theory
game proceeds attack – replenish – adapt.
An attack strategy is more efficient, for a
Previous researchers considered disruptive at- given defense strategy, if an attacker using it
tacks on networks to be a single-round game. requires a smaller budget to disrupt the net-
Such a model is suitable for applications such work. Similarly, a defense strategy is more ef-
asaconventionalwar,inwhichtheattacker has ficient if, for a given attack strategy, it com-
to expend a certain amount of effort to destroy pels the attacker to expend a higher budget
thedefender’scommand,controlandcommuni- to achieve network disruption. (We will clarify
cations, and one wishes to estimate how much; thislater oncewehave presentedanddiscussed
or a single epidemic in which a certain amount a few simulations.)
of resource must be spent to bring the disease We assume initially that the attacker has
under control. perfect information about the network topol-
However, there are many applications in ogy, and that her goal is simply to partition
which attack and defense evolve through the network – that is, divide it into two or
multiple rounds: terrorism and music-sharing more nontrivial disjoint components. We as-
are only two examples. We now develop a sume that the defender has only local informa-
framework for considering this more general tion, that it, each node shares the information
case. We apply ideas from evolutionary game available to those nodes with which it is con-
theory developed by Axelrod and others [3,4]. nected. Thus,for example, if the attacker man-
This theory studies how games of multiple ages to split the network into two components,
rounds differ from single-round games, and it there is no way for them to reconnect. We also
has turned out to have significant explanatory start off by assuming that the defence strategy
power in applications from ethology to affects only the adaptation phase, as only once
economics. nodes have connected to a network can they be
4
programmed to follow it; so the replenishment simulation, while p increases from k/(N − r)
phase is exogenous. to k/(N −1) as the replenishment proceeds. In
A further initial assumption is that the at- this strategy, the defender does nothing in the
tack anddefencebudgetsareroughlyequal.By adaptation phase.
this we will mean that for each node destroyed This models the case where new recruits to
intheattackphase,onenodewillbereplacedin a subversive network simply contact any other
the resource addition phase. Thus the network subversives they can find; no attempt is made
will neither grow or shrink in absolute size and to reshape the network in response to the cap-
we can concentrate on connectivity effects. We ture of leaders but the network is simply al-
will discuss other possible assumptions later, lowed to become more amorphous.
but the static budgets and global attack / lo-
cal defence assumptions will get us started.
5.2 Defense strategy 2 – dining
steganographers
5 Defence Evolution – First Round
Our second defensive strategy is more
To analyse the vulnerability of a network, the
sophisticated, and is inspired by the theory of
selection of network elements (nodes or edges)
anonymous communication as developed by
destroyed in each round is the attacker’s
computer scientists, most notably Chaum [10].
choice and constitutes her strategy. The
A node that acquires a high vertex order, and
attacker wishes to maximize the network
thus could be threatened by a vertex-order
damage caused per unit of work.
attack, splits itself into n nodes, arranged in a
We will start off by considering a static at-
ring. The rings have two functions. First, they
tacker, using what we know to be a reason-
provide resilience: a ring broken at one point
able attack (vertex-order), and examine how
still supports communications between all its
the defence strategy can adapt. Then we will
surviving nodes, and it is the simplest such
see what better attacks can be found against
structure. Second, nodes can route covert
the best defence we found. Then we will look
traffic between appropriate input and output
for a defence against the best attack we found
links, and use encryption and other
in the last round, and so on. There is no guar-
information-hiding mechanisms to conceal the
antee that the process converges – there may
traffic. This model was originally presented in
be a specialised attack that works well against
Chaum’s seminal ‘dining cryptographers’
each defence, and vice versa – but if evolution-
paper cited above, so we might refer to it
ary games on networks behave like more tra-
as the ‘dining steganographers’. The
ditional evolutionary games, we may expect to
collaborating nodes in each ring cannot
find some strategies that do well overall, as ‘tit
conceal the existence of communication
fortat’doesinmulti-roundprisoners’dilemma.
between them, as the cover traffic is visible to
We may also expect to gain useful insights in
the attacker. However, from the attacker’s
the process.
viewpoint it is not obvious that these n nodes
are acting as a virtual supernode.
5.1 Defense strategy 1 – random
Our focus here is on the effects of network
replenishment
topology,ratherthanonthehigher-layermech-
Our first defensive strategy is the simplest of anisms that actually implement the covertness
all, and is one of the naive defences introduced property and that provide any confidentiality
in the above section. New nodes are joined to of content or of routing data. We assume a
the graph at random. We assume that each at- worldinwhichthereissufficientencryptedtraf-
tack round removes r nodes, and the replenish- fic(SSL,SSH,DRM,...)thatencryptedtraffic
mentroundaddsexactlyrnodes,eachofwhich isnotofitselfsuspicioussolongasitiswrapped
is joined to the surviving vertices with proba- in a common ciphertext type. The attacker’s
bility p. r remains constant for each run of the input consists of traffic data collected from the
5
backbone or from ISPs, and her output con- 2. Preferential Attachment: The probability
sists of decisions to send police officers to raid that a new node connects to node i is
the premises associated with particular IP ad- Π(k ) = k /P k where k is the degree
i i j j i
dresses. Her problem is this: given an observed of node i.
pattern of communications, whom should she
investigate first? Having created the scalefree network, we
Theprecisemechanism of ringformation in then ran each of the above defensive strategies
our simulation is as follows. A vulnerable node against a vertex-order attack.
decides to create a ring and recruits for the
purpose a further n − 1 nodes from the new Results The results of the initial three simu-
nodes introduced in the most recent replenish-
lations are given in Figure 2.
ment round, or, if they are inadequate, from
among its immediate neighbours. Existing ring
members cannot be recruited, so rings may not
Vertex−order attack
overlap. Finally, recruits to a ring relinquish
with Rings and Cliques
any existing links with the rest of the network,
alinndkstuhneifroinrmg-lfyoramminogngnoaldlethsheamreesmibtseresxotefrtnhael 400 ox^x^x^x^x^^^^^^^^^^^^^^^^^^^^^^^^^^^^
x
x
ring. 0 xx
0 x
nt 3 xx
e
n x
5.3 Defense strategy 3 – revolutionary o 0
p 0 o
cells m 2
o x
C
0
Our third defensive strategy is inspired by 0
1
cells of revolutionaries, along the model o x
favoured historically by a number of insurgent 0 oooooooooooooxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxox
organisations. A node that acquires a high
0 5 10 15 20 25 30
vertex order splits itself into n nodes, all
linked with each other, with the previous
Rounds
outside connections split uniformly between
them. In graph-theoretic language, each o vertex order attack, No adaptation
supernode is a clique. + vertex order attack, Rings
As in ring formation, a node that considers ^ vertex order attack, Cliques
itself vulnerable is allowed to split itself into a
cliqueofnodes.Thenewnodesaredrawneither
fromthepoolofnewnodes,or,iftheyareinsuf-
Fig.2. Vertex order decapitation attack in
ficient,fromlow-vertex-order neighboursof the
rings, cliques and with no adaptation
clique-forming node. As before, this node’s ex-
ternal edges are distributed uniformly among
members, while other member nodes’ former The red graph in Figure 2 provides a cali-
external edges are deleted. bration baseline. As seen in the above section,
random replenishment without adaptation is
ineffective: within three rounds the size of the
Simulations – first set For our first set of
largest connected component has fallen by a
simulations, we consider a scalefree network of
half, from 400 nodes to well under 200.
N = 400 nodes. We use a Baraba´si-Albert net-
The green graph shows that rings give
work created by the following algorithm:
only a surprisingly short-term defence benefit.
1. Growth: Starting with m0 = 40 nodes, at They postpone network collapse from about
each roundwe add m = 10 new nodes,each two rounds qto about a dozen rounds.
with 3 edges. Thereafter, the network is almost completely
6
disconnected. In fact, the outcome is even
worse than with random replenishment.
Cliques, on the other hand, work well. A
few vertices are disconnected at each attack
round, but as the cyan graph shows, the net-
work itself remains robustly connected. This
may provide some insight into why, although
rings have seemed attractive to theoreticians,
those real revolutionary movements that have
left some trace in the history books have used
a cell structure instead.
6 Attack Evolution – First Round
Vertex−order and Centrality attack
with Rings and Cliques
Havingtriedanumberofdefencestrategiesand
ftohuenndetxhtatstoenpeiosfttohetmry–ocultiquaesnu–misbeerffeocftivaet-, 400 ox^x^x^x^x^x^^^^^^^^^^^^^^^^^^^^^^^^^^^
x
tack strategies to see if any of them is effective 00 xxx
against our defences, and in particular against nt 3 xx
e
n x
cliques. o 0
p 0 o
Of the attack strategies we tried against a m 2
o x
clique defence, the best performer is an attack C 0
0
based on centrality. We used the centrality 1
o
x
algorithm of Brandes [8] to select the 0 oooooooooooooxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxox
highest-centrality nodes for destruction at
each round. As before, our calibration baseline 0 5 10 15 20 25 30
is random replenishment. For this, the red and
black graphs show performance against Rounds
vertex-order and centrality attacks
o vertex order attack, No adaptation
respectively. Both are equally effective; within
+ vertex order attack, Rings
two or three rounds the size of the largest
^ vertex order attack, Cliques
connected component has been halved.
centrality attack, Rings
The green and blue graphs show that the centrality attack, Cliques
same holds for rings: the network collapses
completely after about a dozen rounds.
Fig.3. Rings and Cliques defense undervertex
Centrality attacks are very slightly more
order and centrality attacks
effective but there is not much in it.
The most interesting results from these
simulations come from the magenta and cyan
graphs, which show how cliques behave. Cyan
shows, as before, a vertex-order attack with
severity m = 10 being ineffective against a
clique defence. Magenta shows the effect on
such a network of a centrality attack. Here the
largest connected component retains about
400 nodes until the network suddenly
partitions at 14 rounds, whereafter a
largest-component size of about 200 is
maintained stably.
7
Some insight into the internal mechanics
can be gleaned from Figure 4. This shows the
average inverse geodesic length. For each
node, we find the length of the shortest path
to each other node, and take the inverse (we
take the length to be infinite, and thus the
inverse to be zero, if the nodes are in disjoint
components). We average this value over all
n(n − 1)/2 pairs of nodes. This value falls
sharply for defense without adaptation, and
falls steadily for defense with rings. These falls
reflect increasing difficulty in internode
communication. With cliques, the vertex-order
Centrality attack
attack has little effect, while the centrality
with Rings and Cliques
attack makes steadily increasing progress on a
o graph of 400 vertices, until it achieves
partition and reduces the largest component
4
eh 0. to about 200 vertices. But it makes only slow
sgt
ern 3 progress thereafter.
nvLe 0.
erage Iodesic 0.2 6.1 Clique sizes
AvGe 1 We next ran a simulation comparing how well
0.
defenseworkswhenusingdifferentsizesofrings
o
0 oooooooooooooooooooooooooooooo andcliques.Ringsizeappearstomakelittledif-
0.
ference; rings are just not an effective defence
0 5 10 15 20 25 30
other than in the very short term. However,
varying the clique size yields the results dis-
Rounds
played in Figure 5.
o Centrality attack, No adaptation
This shows that under a centrality attack,
Centrality attack, Rings
the performance of the defense increases
Centrality attack, Cliques
steadily with the size of the clique. There is
still a phase transition after about 14 rounds
or so after which the largest connected
component becomes significantly smaller, but
Fig.4. Average inverse geodesic lengths of the size of this equilibrium component
rings and clique adaption, under centrality at- increases steadily from about 150 with clique
tack size 8 to almost 300 at clique size 20.
7 Defence Evolution – Second
Round
Nowthatweknowcentralityattacksarepower-
ful,wehavetriedanumberofotherpossiblede-
fences. The most promising at present appears
tobeacompounddefencebasedoncliques and
delegation.
The idea behind delegation is fairly
simple. A node that is becoming too
well-connected selects one of its neighbours as
8
a ‘deputy’ and connects it to a second
neighbour, with which it then disconnects.
This reflects normal human behaviour even in
Vertex order attack
peacetime: busy leaders pass new recruits on
with various clique sizes
to colleagues. In wartime, and with an enemy
0 that might resort to vertex-order attacks, the
0
4 incentive to delegate is even greater. Thus a
nt terrorist leader who gets an offer from a
e 0
on 30 wealthy businessman to finance an attack
p
m might simply introduce him to a young
o 0
C 0 militant who wants to carry one out. The
st 2 leader need now maintain communications
e
g 0
ar 0 with at most one of the two.
L 1
Delegationonitsownisratherslow;ittakes
dozens of roundsfor delegation to ‘immunise’ a
0
networkagainstvertex-orderattack.Ifavanilla
0 5 10 15 20 25 30
scale-free network is going to be exposed to ei-
theravertex-orderorcentralityattackfromthe
size 5 Rounds
size 8 next round, then drastic action (such as clique
size 11 formation) is needed at once; else it will bedis-
size 14 connected within two or three rounds. Slower
size 17
defenceslikedelegationcanhoweverplayarole,
size 20
provided they are started from network forma-
tionorareasonabletimeperiod(say20rounds)
Centrality attack before the attack begins.
with various clique sizes It turns out that the delegation defence,
on its own, is rather like the rings of dining
0
40 steganographers. Network fragmentation is
nt postponed (about 14 rounds with the
e 0
n 0 parameters used here) though not ultimately
o 3
p averted.
m
o 0 What is interesting, however, is this. If we
C 0
st 2 formanetworkandimmuniseitbyrunningthe
e
rg 0 delegation strategy, then run a clique defence
a 0
L 1 as well from the initiation of hostilities, this
compound strategy works rather better than
ordinarycliques. Figure 6shows thesimulation
0 5 10 15 20 25 30 results.
Figure 7 may give some insight into the
size 5 Rounds mechanisms. Delegation results in shorter path
size 8
lengths under attack: it postpones and slows
size 11
down the growth of path length that otherwise
size 14
size 17 results from hub elimination. As a result, equi-
size 20 libriumisachieved later,andwithalargermin-
imum connected component.
Fig.5. Clique recovery with different clique
8 Conclusions and Future Work
sizes under a centrality attack
In this paper, we have built a bridge between
network science and evolutionary game theory.
9
Centrality attack Centrality attack
with Cliques and Delegation with Cliques and Delegation
0 o o
0
4
4
0 eh 0.
0 sgt
nt 3 ern 3
ne nvLe 0.
Compo 200 oo erage Iodesic 0.2
100 AvGe 0.1
ooooooooooooooooooooooooooooo 0 ooooooooooooooooooooooooooooooo
0.
0 5 10 15 20 25 30 0 5 10 15 20 25 30
Rounds Rounds
0 No replenishment
0 No replenishment Delegation
Delegation Clique
Clique Clique + Delegation
Clique + Delegation
Fig.6. Component size: clique, immunization Fig.7. Clique, immunization by delegation,
bydelegation, andcombined cliqueanddelega- and combined clique and delegation defenses
tion defenses against centrality attack against centrality attack
10