3rd International Conference on the Dynamics of Information Systems (DIS-2011)
Exploiting Power-Law Distribution
on Complex Network Vulnerability
Yilin Shen and My T. Thai
Department of Computer Information Science and Engineering
University of Florida
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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Power-Law Networks
Main Property:
The number of nodes having k
connections is proportional to
k-β
β is a parameter whose value is
typically in the range 2 < β < 3
Many Low Degree Nodes
Few High Degree Nodes
Internet in December 1998
http://cs.stanford.edu/people/jure/pubs/powergrowth-kdd05.ppt
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More Real Network Examples
http://www.mslima.com/mfadt/thesis/2004/08/transportation-routes.html
Continental Airlines Air Route: Most destination maps of airline
companies offer an interesting case of a power-law network.
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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Application 1: Cognitive Disruption
on Complex Networks
Koobface: A
Facebook Worm
http://www.pdastreet.com/articles/2007/11/2007-11-12-Review-Facebook-for.html
http://annarborchronicle.com/2009/01/30/geeks-gather-make-stuff/
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Application 2: The Discovery of Critical
Nodes/Commanders in Adversarial Networks
Terrorist Network by Krebs
http://vlado.fmf.uni-lj.si/pub/networks/doc/Seminar/Krebs.pdf
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 How can we fragment these adversarial
networks?
 Can we fragment the network by destroying only a
limited subset of critical nodes?
Most vulnerability related problems
will be much easier to solve when the
network has power-law distribution?
Not Really !!!
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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Formal Definitions
 Cognitive Disruption Problem (CDP)
Given a power-law graph G = (V,E) and a set of constant 0 < ci ≤1,
the problem is to minimize the size of influence nodes D ⊆ V such
that for each node vi ∈ V \ D, |D ∩ N(vi)| ≥ ci N(vi) in d rounds.
 Critical Node Disruptor Problem (CNDP)
Given an integer k and a power-law graph G=(V,E), the problem is
to find out a size bounded subset of critical nodes S ⊂ V , i.e. |S| ≤ k,
whose removal minimizes the number of connected node-pairs.
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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NP-Hardness of CDP
Theorem 1. CDP is NP-hard on power-law graphs.
Proof: The idea is to reduce CDP from vertex cover on dbounded graphs.
PowerLaw
Graph
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NP-Hardness of CNDP
Theorem 2. CNDP is NP-hard on power-law graphs.
Proof: The idea is to reduce CNDP from vertex cover on
general graphs.
Bipartite Graph
(Ferrante 08’)
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u1
u2
v1
x2
y2
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……
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……
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yn
Power-Law
Graph
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Does the power-law distribution impact
the network vulnerability?
Intuitively, the answer should be “yes” !
We prove it by showing better
inapproximability factors on CDP !
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Inapproximability Factors
Theorem 3. CDP is UG-hard to be approximated into
on power-law graphs.
The basic idea in proof is to use the reduction we
showed in the proof of NP-hardness on CDP.
-Completeness:
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-Soundness:
The inequality holds since the function
is monotonously decreasing when f (x) > 0 for any x.
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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Approximation Algorithm for CDP
 Can we devise an approximation algorithm for
CDP?
 Which algorithm will work?
 Why does the algorithm work?
…
Simple greedy algorithm will work !
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Greedy Algorithm
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v1
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Main Theorem
Theorem 4. When d=1, CDP
can be approximated into
1/(1-λ) with probability at
least
Much better than
the ratio O(log n) in
general graphs
Power-Law Network is more vulnerable !
The seed nodes to inject false information are easier to find.
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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The Behavior of Power-Law Networks
under Failures and Attacks
 Are power-law networks vulnerable in any
cases?
 If so, why do most real networks still follow
power-law distribution?
 What is the advantage?
…
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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Random Failures
Are the power-law networks also vulnerable
under the uniform failures?
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Random Failures (Cont.)
 The power-law networks are
extremely robust even when
the failure probability is
unrealistically large
 Even though PLN is
affected, the number of
node-pairs after failure is
close to original PLN
 Smaller β is better
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The Basic Idea in Proof
How to find the number of nodepairs when there is Q<0?
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The Basic Idea in Proof (Cont.)
 We use the Branching Process Method
 Use Azuma’s martingale inequality
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Outline
 Introduction
 Power-law Networks (PLN)
 Representative Applications on Network Vulnerability
 The Hardness of Vulnerability Problems on PLN
 Formal Problem Definitions (CDP & CNDP)
 NP-Hardness and Imapproximability
 Approximation Algorithm of CDP
 The Behavior of Power-Law Networks under
Failures and Attacks
 Robust under Random Failures
 Vulnerable under Men-made Attacks
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Man-made Attacks
 Preferential Attacks
 Nodes of high degrees are more likely to be attacked
 Centrality Attacks
 Intruders attack the centrality nodes intentionally
 degree-centrality, betweenness centrality …
Power-Law Networks are vulnerable
under man-made attacks !
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Preferential Attacks
 How many expected number of failure nodes
can destroy the power-law networks?
 How will the intruders attack various nodes?
A node of degree i is attacked with probability pi
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Preferential Attacks (Cont.)
 Power-Law Networks will not
be affected only when under
around expected 13% of
nodes are attacked
 Smaller β is better
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Centrality Attacks
 Nodes of high degrees are intentional attacked !
 Power-Law Networks will not
be affected only when under
5% of degree-centrality nodes
are attacked
 Smaller β is better
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Conclusions
 Vulnerability Problems on Power-Law Networks
 Most vulnerability problems remains NP-hard
 Much easier to find near-optimal solutions
(The inapproximability ratio is much lower)
 Behaviors of Power-Law Networks under
Failures and Attacks
 Robust under random failures
 Vulnerable under man-made attacks
 The smaller the exponential factor the better
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Questions?
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Thank you !
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