Exploiting the Robustness on Power-Law Networks

Yilin Shen, Nam P. Nguyen, My T. Thai

Presented by :Yilin ShenDept. Computer Information Science and Engineering University of Florida

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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

Motivation: Power-Law Networks

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Main Property:The number of nodes having kconnections is proportional to

k-β

β is a parameter whose value is typically in the range 1 < β < 4

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Internet in December 1998 http://cs.stanford.edu/people/jure/pubs/powergrowth-kdd05.ppt

Few High Degree NodesMany Low Degree Nodes

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More Real Network Examples

Many large-scale real-world networks appear to exhibit a power-law graph

Internet: β = 2.1 World Wide Web: β = 2.1 Social Networks: β = 2.3 Protein-protein interaction networks: β = 2.5

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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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() Power-law Graph

Definition (() Graph G()):

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Power-Law Random Graph Model

Form a set L containing dv disjoint copy of vertex v (mini-vertices);

Choose a random matching of the elements of L; For two vertices u and v, there is an edge between them if

and only if at least one edge of the random perfect matching was connecting copies of u to copies of v.

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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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Vulnerability Measurement

Total Pairwise Connectivity P(in residual power-law networks after the failures and attacks)

Why is Total Pairwise Connectivity an effective measurement?

It can control the balance among disconnected components while ensuring the nonexistence of giant components.

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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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Threat Taxonomy

Uniform Random Failure Each node in G() fails randomly with the same probability p

Preferential Attack Each node in G() is attacked with higher probability if it has a larger

degree Degree-Centrality Attack

The adversary only attacks the set of centrality nodes with maximum degrees in G()

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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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Two Lemmas in Literature

M. Molloy and B. Reed (1995)

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Two Lemmas in Literature (Cont.)

F. Chung et al. (2002)

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Some Fundamental Results

Relations between largest connected component and total pairwise connectivity

Robustness of power-law networks

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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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Uniform Random Failures

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The Idea of Proof

Compute the expected degree distribution of graph Gr

Use M. Molloy and B. Reed (1995) to find a threshold β0

When β β0, we use the branching process method

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Visualization

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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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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Interactive Preferential Attacks

By choosing a different parameter β′, a node of degree i in G(α, ) has probability

to be attacked Main Theorem.

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Expected Preferential Attacks

To attack the expected c nodes A node of degree i is attacked with probability

Main Theorem.

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Visualization

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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Outline

Motivation: Power-law Networks Models, Measurement and Threat Taxonomy

Power-Law Random Graph Model Vulnerability Measurement Threat Taxonomy

Preliminaries Uniform Random Failures Preferential Attacks Degree-Centrality Attacks

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Degree-Centrality Attacks

The intruders intentionally attack the “hubs”, that is, the set of nodes with highest degrees (larger than x0)

Main Theorem.

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Visualization

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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Thank you for listening!