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COVERTNESS CENTRALITY IN NETWORKS

Michael OvelgönneUMIACS

University of Marylandmov@umiacs.umd.edu

Chanhyun Kang, Anshul SawantComputer Science Dept.University of Maryland

{chanhyun, asawant}@cs.umd.edu

VS SubrahmanianUMIACS & Computer Science Dept.

University of Marylandvs@cs.umd.edu

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Motivation

Henchmen

Let’s assume there is a criminal network and we want to find a leader of this group using the henchmen.

Who is the gang leader of this network?

We may want to use centrality measures to identify important criminals in the network

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Motivation

Closeness centrality

Betweenness centrality

We can think of the vertex of a suspicious person as the leader in this network

But, if the leader is smart and understand(or know) the measures?

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MotivationIf the leader is sufficiently smart, he may- Hide in a crowd of similar actors- Have enough connections with the henchmen

The gang leader would be not like this vertex

The gang leader would be like these vertices

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MotivationTypically, if we plot centrality values and % of nodes in graph G, the distribution obeys a power law and has a long tail (closeness centrality is an exception).• A vertex that wants to stay “hidden” does not want to stick out in the

long tail.• It would prefer to be squarely near the “high percentage” part of the

distribution.

centrality value0

% of nodes

Nodes that want to stay “unnoticed” don’t want to be in this part of the

distribution.

To stay “unnoticed”, nodes want to stay here

But in order to communicate with the their own subnetwork with

lower probability of discovery, they need to be more to the right

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Motivation

Betweenness centrality

Eigenvector centrality Degree centrality

Closeness centrality

But a smart leader may know various centrality measures, so we need to consider a set C of centrality measures to identify the smart leader

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In this paper• Propose covertness centrality measure. Has two major

components: • How “common” a vertex is with regard to a set C of centrality

measures• How well the vertex can “communicate” with a user-specified set I

of vertices

• Develop algorithms to compute covertness centrality• Exact and heuristic algorithms

• Evaluate the measures and the algorithms

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Commonness• Measures how well an actor a hides in a crowd of similar

actors• CM(C, a) denotes the commonness of an actor a from the

given centrality measures C=(C1, C2, …, Ck)

Betweenness centrality Eigenvector centralityDegree centralityCloseness centrality

CMC, a The common-ness value of actor a

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( ) )(

• Instead of giving specific commonness functions, we first identify axioms that all commonness measures should satisfy

• Axioms for Commonness• Property 1. Optimal Hiding. If all vertices have the same centrality according to

all measures, then all vertices should have commonness of 1.

• Property 2. No Hiding. If the centrality of v is sufficiently different from the centrality of all other vertices according to all centrality measures, then v’s commonness is 0.

• Property 3. If the values of a centrality measure for all vertices are the same, then the commonness values for all vertices should be the same after removing the centrality measure

Commonness

( () )

( )) (

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Commonness• We suggest two measures to compute CM(C, a)

• CM1(C, a) • Compute similar actors of actor a for each centrality measure

separately

• CM2(C, a)• Compute similar actors of actor a with all centrality measures

simultaneously

CM1(C, a)

CM2(C, a) =

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Commonness : Similar actors• We consider actors similar to actor a w.r.t. one centrality measure Ci

• The probability that a randomly chosen actor excluding the actor a has a centrality Ci value within the interval Ii is

← Low Ci(a) High →

Ci centrality values

- σi : standard deviation of Ci values- α : the range of similar values

a

Ci(a) - ασi

Interval Ii

Ci(a) + ασi

Actors similar to actor a for centrality Ci

h𝑡 𝑒𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑎𝑐𝑡𝑜𝑟𝑠− 1|𝑉|−1

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• Define commonness as the sum of the squared distances separately for each centrality

CM1(C, a)

Commonness: CM1

- We compute the probability for each centrality measure

the commonness value of actor A should be larger than the other’s value if the deviation of probabilities of actor A is smaller than the other’s deviation.

Because even if the summations of the probabilities are same,

- Why not simple summation of the probabilities?

k : the number of centrality measures in C

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Commonness: CM1

• Satisfies Property 1. Optimal Hiding • If the centrality values of all actors are same, the number of the similar

actors is |V|-1. So the commonness values of all vertices is 1.

• Satisfies Property 2. No Hiding • If the centrality values of all actors are not similar to each other, the

number of similar actors is 0. So the commonness value of all vertices is 0.

• Does not satisfy Property 3- Let’s assume C={C1,C2} , the number of similar actors of actor v for C1 is r and the number of similar actors of actor v for C2 is |V|

𝐶𝑀 1 ( {𝐶 1 ,𝐶2 } ,𝑣 )=1−(1 − 𝑟 −1

|𝑉|−1)2+(1− ¿𝑉∨−1

|𝑉|− 1)2

2𝐶𝑀 1 ( {𝐶 1 } ,𝑣 )=1−(1− 𝑟 −1

|𝑉|−1)2

1

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Commonness: CM1

We compute the CM1 values using Betweenness, Closeness, Degree and Eigenvector centrality measures for the criminal network.

We can find some suspicious people who hide in a crowd. But it is not clear. There is a problem.

- α =1

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Commonness: CM1

• If the centrality measures are very different, measuring the similar actors independently for each centrality measure can lead to problems.

% of node

Normalized centrality value

The vertices will have good commonness values even if the number of similar actors for C2 is small

C1 centrality

C2 centrality

The vertices will have good commonness values even if the number of similar actors for C1 is small

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• We can also consider actors similar to a given actor a using all given centrality measures C simultaneously.

Commonness : Similar actors

Ci(a) High →Ci(a) - ασi

Interval Ii

- σi , σj: standard deviations of Ci values and Cj values

Ci centrality values

- α : the range of similar values

a

Ci(a) + ασi

Similar actors of actor a

Cj centrality values

High ↑

Cj(a)

Cj(a) + ασj

Cj(a) - ασj

Interval Ij

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Commonness: CM2

• Define commonness as the fraction of all actors that are similar to actor a in all considered dimensions

CM2(C, a)=

- The centrality values of similar actors are within all intervals generated from all centrality values of actor a

- We compute the probability that a randomly chosen actor excluding the actor a has centrality values within all the intervals from all the centrality values of a

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Commonness: CM2

• Even if the centrality measures are not correlated,

% of node

Normalized centrality value

the vertices will have small commonness values

C1 centrality

C2 centrality

a b

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Commonness: CM2

• Satisfies Property 1. Optimal Hiding• If the centrality values of all actors are the same, the number of similar

actors is |V|-1. So the commonness values of all vertices are 1.

• Satisfies Property 2. No Hiding• If the centrality values of all actors are not similar to each other, the

number of similar actors is 0. So the commonness values of all vertices are 0.

• Satisfies Property 3 - Let’s assume C={C1,C2} , the interval of actor v for C1 is I1 and the values of

C2 for all vertices are the same - The intervals of all vertices for C2 are same - So the number of similar actors for C1 and the number of similar actors for C1

and C2 are the same𝐶𝑀 2 ( {𝐶 1 } ,𝑣 )=𝐶𝑀 2 ( {𝐶1 ,𝐶 2 } ,𝑣 )

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Commonness: CM2

We compute the CM2 values using Betweenness, Closeness, Degree and Eigenvector centrality measures for the criminal network.

- α =1

Now we can find clearly some suspicious people who hide in a crowd

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Communication PotentialThe gang leader has enough connections to communicate with the henchmen for achieving their objective

For measuring the communication ability precisely, we need to use a subgraph, induced by some vertices, of the criminal network

A subgraph of G using the henchmen

G

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Communication Potential• Reflect the ability of vertex v to communicate with vertices in set I.

• : if only in-group connections are important for achieving the group’s objective • Define the communication potential based on a centrality measure D and

the group V’ • Let G’=(V’, E’) be induced subgraph of G given by V’

• : if the ability to communicate with people outside the group is important as well, the entire graph G is used

𝐶𝑃 1(𝑣)=𝐷𝐺 ′ (𝑣 )

𝐶𝑃 2(𝑣)=𝐷𝐺(𝑣 )

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Communication PotentialWe compute CP1 values using Closeness centrality

CP1

A subgraph of G using the henchmen

G

We can find some people who have good communication ability in the subgraph that contains the henchmen

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Communication Potential

CP1

We compute CP1 values using Betweenness centrality

A subgraph of G using the henchmen

Some people have better communication ability in the subgraph that contains the henchmen than others

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Communication Potential

G

We compute CP2 values using Closeness centrality

CP2

We can find some people who have good communication ability in the network

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Covertness Centrality• Covertness centrality is a combination of Commonness and

Communication potential• Let’s assume CP is normalized to the interval [0,1] like CM

- l measures the importance of Commonness vs. importance of Communication Potential

- τ is a minimum level of commonness set by the user- if CM < τ, CP is irrelevant to CC- If τ =0, CC is a classic trade-off between the CM and the CP

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Covertness Centrality

Who is the gang leader of this network?

CC

We compute CC values (λ=0.5 and τ=0) using CM2(α=1) and CP1(Closeness centrality)

The guy is the most suspicious person who leader who- Hides in a crowd of similar actors- Has enough connections to communicate with others including the henchmen

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Covertness CentralityL 0.2

l=0.2 l=0.5

l=0.8 The CC values of vertices that have a high CP value are decreased according to the increase of l

The CC values of vertices that have a high CM value are increased according to the increase of l

CC values(τ=0) varying the l (CM2(α=1) and CP2(Closeness centrality))

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CC COMPUTATION• Exact computation• Simple random sampling method

• The sample vertices are randomly chosen• Systematic sampling method

• Order all vertices by degree. Then, select k vertices by taking every n/k-th vertex starting from a start vertex randomly selected among the first n/k-th vertices

The first n/k-th vertices

High degree Low degree

A start vertex

…

n/k-th vertex n/k-th vertex

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Experimental Evaluation• We analyze the properties of the covertness centrality and

the algorithms• Dataset

• Python is used for CM1 and CM2 implementation• Evaluated on a standard desktop machine

Network #Vertices #Edges TypeURV 1133 10903 e-mail

Youtube 40k 39998 85793 friendship

Youtube 60k 59998 151481 friendship

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Evaluation : Measures• Scatter plot of the commonness scores according to CM1 and CM2 in

relation to closeness centrality• Degree, Closeness, Betweenness and Eigenvector centralities• URV dataset

• CM1 values are high because of other centrality values

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Evaluation : Measures• Distribution of CC scores depend on different λ values

• CM2 : Degree, Betweenness, Closeness and Eigenvector centrality• CP : closeness centrality• URV dataset

- Commonness is strongly negatively correlated to the base centrality measures

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Evaluation : Measures• Distribution of CC scores depend on different λ values

• CM2 : Degree, Betweenness, Closeness and Eigenvector centrality• CP : closeness centrality• URV dataset

- Covertness centrality is similar to the CP values when l is small

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Evaluation : Compute time & Accuracy• The runtime scales linearly with the number of vertices if the centrality

values are already computed.

• Comparison of the rank correlation between the exact algorithm and the sampling algorithms for the URV dataset. Very high correlation!

URV Youtube 40k Youtube 60kComputing time 0.1second 2 seconds 3 seconds

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Evaluation : Accuracy• Accuracy of sampling methods measured with Kendall’s τ

rank correlation coefficient. Very high correlation!- CM1, 100 runs for the simple sampling method

- Systematic sampling method is better than the simple sampling method

- CM2, 100 runs for the simple sampling method

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Conclusion• Defined a new concept of covertness centrality combining

• Commonness• Measures how well an actor hides in a crowd of similar actors w.r.t. a given set

of centrality measures• Proposed axioms that any good commonness function should satisfy.• Proposed two new commonness measures CM1 and CM2 and showed that CM2 satisfies

all the axioms.• Communication Potential

• Measures the ability to communicate and cooperate to achieve a common objective

• Used sampling methods for computing the covertness centrality

• Evaluated the measure and the sampling methods on YouTube and email (URV) data.

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Questions

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Related works• R. Lindelauf, P. Born, and H. Hamers, “The influence of secrecy on

the communication structure of covert networks,” Social Networks, vol. 31, no. 2, pp. 126-137, 2009• Deal with the optimal communication structure of terrorist organizations when

considering the tradeoff between secrecy and operational efficiency• Determine the optimal communication structure which a covert network

should adopt

• J. Baumes, M. Goldberg, M. Magdon-Ismail, and W. Wallace, “Discovering Hidden Groups in Communication Networks” in Intelligence and Security Informatics, 2004, vol.3073, pp. 378-389• Suggest models and e cient algorithms for detecting groups which attempt ffi

to hide their functionality – hidden groups• Use the property that hidden groups’ communications are not random

because those are planed and coordinated

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Commonness: CM1

• Define commonness as the sum of the squared distances separately for each centrality

¿𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑎𝑐𝑡𝑜𝑟𝑠 𝑓𝑜𝑟 𝑎𝑐𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑡𝑦∨−1|𝑉|−1

The probability for each centrality measure

- The commonness value of actor A should be larger than the other’s value if the deviation of probabilities of actor A is smaller than the other’s deviation.

Why not the simple summation of the probabilities?

Because even if the summations of the probabilities are same,

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Commonness: CM2

• Define commonness as the fraction of all actors that are similar to actor v in all considered dimensions

¿𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑎𝑐𝑡𝑜𝑟𝑠 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑐𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑡𝑖𝑒𝑠∨− 1|𝑉|− 1

- The similar actors’ centrality values are within the intervals generated from all centrality values of actor v