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Decapitation of networks with and without weights and direction : The economics of iterated attack and defense

Advisor : Professor Frank Y. S. LinPresented by: Tuan-Chun ChenPresentation date: Mar. 6, 2012

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AgendaIntroductionEconomic model MeasurementsEmpirical workDiscussion of resultsConclusions

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AgendaIntroductionEconomic model MeasurementsEmpirical workDiscussion of resultsConclusions

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IntroductionMany empirically observed networks

can be modeled as scale-free networks. Ex: peer-to-peer networks

Scale-free networks are more robust against random attacks than targeted attacks.

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IntroductionBetweenness centrality can be an

alternative to degree for attack targeting.

A dynamic case : each round an attacker removes a certain number of nodes, but the defenders can recruit other nodes to replace the lost ones.

The smaller the largest connected component after an attack-defense round, the more successful is the attack.

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IntroductionTwo attacks were examined:

Removal of high-degree nodes

Path centrality attack -> Combine cliques with delegation is most

effective

-> Defense strategies(i) Radom replacement > less than 3 rounds(ii) Ring replacement > 3~12 rounds (iii) Clique replacement > robust

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IntroductionContribution and plan of this paper:

Extend the former research to consider weighted and directed networks and regards the economic aspects of the attack and defense strategies.

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IntroductionReasons to consider weights and

directions :Delegation strategies and the

computation of node centralities may depend on the weight of the links.

The distance between nodes may also be relevant to assess the impact of an attack on the LCC(Largest Connected Component).

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AgendaIntroductionEconomic model MeasurementsEmpirical workDiscussion of resultsConclusions

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Economic modelAttacks and their cost:

k The number of nodes destroyed in an attack

CFA Fixed cost incurred to locate the target nodes

CX Cost of destroying one node

( ) *A FA xC k C k C

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Economic modelLocating target nodes by measuring

criticality of nodes: ◦node degree◦node path centrality◦in the case of weighted :1

1

( ) :n

ijj

R i F

R(i) The reliability of a node IFij The weight of most reliable path between i

and j

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Economic modelDefenses and their cost

CFD Fixed cost of starting a defense roundCdel(h[, q])

Cost of implementing delegation for nodes with criticality δ higher than h

h Threshold criticalityq The clique size in case delegation is based

on cliquesCcli (q) Cost of replacing each destroyed nodes by

a clique consisting of q new nodes.

( , , ) ( [, ]) * ( )D FD del cliC h k q C C h q k C q

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Economic modelDelegation strategies and their cost

(i) Before attacks start(ii) Each time new edges are added to

the network due to the defense strategy of clique formation

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Economic model(i) Before attacks start

◦ Node criticality is measured as node degree.

◦ A node i with degree δ(i) > h attempts to transfer some of its edges to its neighbors.

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Economic model

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Economic model(i) Before attacks start

◦ Node criticality is measured as node degree.

◦ A node i with degree δ(i) > h attempts to transfer some of its edges to its neighbors.

◦ :1 , ( ) ( )

( )del ji i n i h j i

C h c

Δ(i) set of edges delegated by node icj cost of delegating the jth edge in Δ(i)

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Economic model(ii) New edges are added due to clique

formation◦ Node criticality is measured using path

centrality.◦ A node i with path centrality δ(i) > h is

replaced by a clique of size q.

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Economic model

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Economic model(ii) New edges are added due to clique

formation◦ Node criticality is measured using path

centrality.◦ A node i with path centrality δ(i) > h is

replaced by a clique of size q.

:1 , ( ) ( )

( , ) ( * )del n ji i n i h j i

C h q q C c

Cn Cost of new nodes forming the cliqueΔ(i) set of edges delegated by node icj cost of delegating the jth edge in Δ(i)

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Economic modelDefenses and their cost

BD Maximum budget of defender DCFD Fixed cost of starting a defense roundq The clique size in case delegation is based

on cliquesCdel(h[, q])

Cost of implementing delegation for nodes with criticality δ higher than h

Ccli (q) Cost of replacing each destroyed nodes by a clique consisting of q new nodes.

( , , ) ( [, ]) * ( )D FD del cliC h k q C C h q k C q

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Economic modelThe cost of clique replacement

◦ A destroyed node is replaced by clique of q nodes.

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Economic model

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Economic modelThe cost of clique replacement

◦ A destroyed node is replaced by clique of q nodes.

◦ ( )

( ) *cli n jj i

C q q C c

Cn Cost of new nodes forming the cliqueΔ(i) set of edges delegated by node icj cost of delegating the jth edge in Δ(i)

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AgendaIntroductionEconomic model MeasurementsEmpirical workDiscussion of resultsConclusions

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MeasurementsLCC(Largest Connected

Component)◦ The reduction in the network connectivity makes

communication impossible between some pairs of nodes.

◦ For the attacker: smaller LCC means more successful

APL(Average Path Length)◦ APL increase means communication is still possible,

but has become more difficult.◦ For the attacker: longer APL means more successful

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AgendaIntroductionEconomic model MeasurementsEmpirical workDiscussion of resultsConclusions

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Empirical workSFNG Matlab function which was

used to synthesize undirected scale-free network.

4 synthetic networks with 400 nodes each.

1 large real network which is a snapshot of the structure of the Internet at the level of autonomous system taken on July 22, 2006. Formed by 22,963 nodes.

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Empirical workPath centrality computation

◦Unweighted networks->Freeman’s betweenness centrality

◦Weighted networks ->as path centrality the reliability

measures

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Empirical workWeighted networksUndirected

Directed

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( ) :n

ijj

R i F

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( ) :n

jij

R i F

(in-reliability)

Fij The weight of most reliable path between i and j

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Empirical work

1 1min min log(1/ )i j i jp P p P w Wpij w Wp

wF w

maxij

p

ij p P w W

F w

=> Shortest path problem with nonnegative weights

=>

Pij Set of different paths between two nodes i and j

Wp Set of edge weights in path p

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Empirical workConsider attackers with several levels of

partial knowledge on the attacked network: 100%, 80%, 60%, 40%, 20%

Conduct 2 simulations(no defense, defense) with each of 4 initial network:

Each simulation consisted of 30 attack rounds.

1. unweighted undirected

3. weighted undirected

2. unweighted directed

4. weighted directed

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Empirical workFor the larger real Internet network,

conducted two additional, longer simulation.(with defense, without defense)

Each simulation consisted of 1722 attack rounds.

Simulation with defense: After a batch of q attack rounds, the defender was allowed to perform one defense round, which replaces the most critical node destroyed in the last q rounds by a clique of q nodes.

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Empirical workParameter choice for attack and

defense

◦ Threshold criticality h was set in a network-dependent way: only 5% of nodes had criticality δ > h.

◦ Clique size q was oriented by experiments. q = 5.

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AgendaIntroductionEconomic model MeasurementsEmpirical workDiscussion of resultsConclusions

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Discussion of results1. Unweighted undirected network

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Discussion of results2. Unweighted directed network

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Discussion of results3. Weighted undirected network

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Discussion of results4. Weighted directed network

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Discussion of results5. 22,963-node Internet snapshot

network

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AgendaIntroductionEconomic model MeasurementsEmpirical workDiscussion of resultsConclusions

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ConclusionsPrevious work in this area deals only

with un weighted undirected networks and does not clearly say how much attackers and defenders can do.

Weights may represent bandwidth, trust, distance, etc.

Attacks based on “node degree” or “path centrality”(taking weights into account).

Defenses consider “delegation” and “node replacement”.

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ConclusionsEmpirical results show that there are no

significant differences in the resistance of unweighted and weighted networks.

Directed networks were more connected and had more resistance than undirected networks.

Regardless of the network type, when the attacker knows only a 20% fraction of the network topology ,her attacks are not very harmful.

Directed networks which are scale-free can successfully withstand attacks where the attacker knows as much 80% or even 100% of the network topology.

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