Subgroup Centrality Measures

Isabel Chen

based on: “Subgroup Centrality Measures”, Jocelyn R. Bell(Dept of Mathematical Sciences, United State Military Academy, West Point)

Network Science 2 (2): 277-297, Aug 2014

Nov 20 2014

Recall:

A network is a pair (V ,E ), with associated adjacency matrix A.

Common centrality measures:

I degree

I betweenness, closeness → based on paths

I Katz, subgraph, eigenvector → based on walks

Radial Centrality Measures

f : V × V → R

c(a) :=∑x∈V

f (a, x)

Degree

f (a, x) =

{1, if (a, x) ∈ E0, else

Closenessf (a, x) = distance between a and x

Eigenvector: Suppose Aq = λq where λ = ρ(A),

f (a, x) =

{1λqx , if (a, x) ∈ E0, else

Medial Centrality Measures

f : V × (V × V )→ R

c(a) :=1

2

∑(x ,y)∈V×V

f (a, {x , y})

Betweenness

f (a, {x , y}) =#shortest paths between x and y that contain node a

#shortest paths between x and y

Note that x 6= a and y 6= a.

Subgroup Centrality Measures

Let S ⊆ V . For any a ∈ V , define

cS(a) =∑x∈S

f (a, x) for f radial

or

cS(a) =1

2

∑(x ,y)∈S×S

f (a, {x , y}) for f medial

Local v Global

a

S ⊂ VS ′

localglobal

I Local:

cS(a) =∑x∈S

f (a, x) or1

2

∑(x ,y)∈S×S

f (a, {x , y})

I Global:

cS ′(a) =∑x∈S ′

f (a, x) or1

2

∑(x ,y)∈S ′×S ′

f (a, {x , y})

Observe thatI local w.r.t. S = global w.r.t. S ′

I global w.r.t. S = local w.r.t. S ′

Local v Global

For f radial,

cS(a) + cS ′(a) =∑x∈V

f (a, x) = c(a)

For f medial,

cS(a) + cS ′(a) + BcS(a) =1

2

∑(x ,y)∈V×V

f (a, {x , y}) = c(a)

where

BcS(a) =���1

2

∑(x ,y)∈S×S ′

f (a, {x , y})

is the ‘boundary’ measure.

Degree Example

Normalization

To account for size of S ,

I for f radial, normalize by |S | if a /∈ S , or |S | − 1 if a ∈ S .

I for f medial, normalize by(|S |2

)if a /∈ S , or

(|S |−12

)if a ∈ S .

Subgraph Centrality

Note that subgroup centrality measures are defined based on theedge structure of the underlying network (V ,E ), not the inducededge structure of (S ,E ′).

The latter is termed subgraph centrality by the author, not to beconfused with the walk-based subgraph centrality put forward byProf Estrada.

Subgroup v Subgraph

Degree

subgroup and subgraph centrality coincide.

ClosenessclS(a) =

∑x∈S

d(a, x)

Subgroup closeness adds the distance from a only to nodes in S . Itdoes not matter if a geodesic uses nodes from outside S .

Note that for ranking purposes, nodes with high clS(a) will beranked last. Alternatively, invert when normalizing, then rank asusual (with highest value ranked first).

Digression on Eigenvector Centrality

Total communicability:

TC = eβA · 1

Spectral Decomposition of f (A) = eβA:

TC = f (A) · 1 = Qf (D)QT · 1

=n∑

i=1

eβλi (qTi · 1)qi

= eβλ1

[(qT1 · 1)q1 +

n∑i=2

eβ(λi−λ1)(qTi · 1)qi

]

Therefore eigenvector centrality is the limiting case of totalcommunicability, as β →∞.

Conclusion

I Two ways to look at eigenvector centrality:I a node’s rank is proportional to the sum of the ranks of its

neighbors:

qa =∑

x∈N(a)

1

λqx

I limiting case of walk-based centrality measure

I From a walk-based point of view, it is not surprising thateigenvector subgroup measures give different results fromeigenvector (induced) subgraph measures.

Betweenness

f (a, {x , y}) =#shortest paths between x and y that contain node a

#shortest paths between x and y

Note that x 6= a and y 6= a.

Betweenness Example

1 2

3

4

5

Betweenness Example

0 3.5

1

1

0.5

Subgroup Betweenness

For any a ∈ V and any S ⊆ V ,

bS(a) =1

2

∑(x ,y)∈S×S

f (a, {x , y})

ThenbS(a) + bS ′(a) + BbS(a) = b(a)

where BbS(a) is the boundary betweenness of a relative to S :

BbS(a) =∑

(x ,y)∈S×S ′

f (a, {x , y})

Boundary Betweenness Example

I Local Betweenness BS(a) = 0

I Global Betweenness BS ′(a) = 0

I Boundary Betweenness (normalized) BbS(a) = 1 is largestpossible

Local Betweenness and Local Clustering

For a ∈ V let S = {a} ∪ N(a).

a

N(a)

bS(a) =1

2

∑(x ,y)∈S×S

B(a, {x , y}) =∑

x ,y∈N(a)(x ,y)∈E

0 +∑

x ,y∈N(a)(x ,y)/∈E

B(a, {x , y})

Local Betweenness and Local Clustering

a

x

y

z1

z2

z3

bS(a) =∑

x ,y∈N(a)(x ,y)∈E

0 +∑

x ,y∈N(a)(x ,y)/∈E

B(a, {x , y}) ≤∑

x ,y∈N(a)(x ,y)/∈E

1

Local Betweenness and Local Clustering

Therefore

1 =1(|N(a)|2

) ∑

x ,y∈N(a)(x ,y)/∈E

1 +∑

x ,y∈N(a)(x ,y)∈E

1

≥b′S(a) + Ca

where Ca is the local clustering coefficient of a:

Ca =# connected neighbors of a

# pairs of neighbors of a=

∑x ,y∈N(a)(x ,y)∈E

1

(|N(a)|2

)

The author claims:b′S(a) = 1− Ca

“thus local betweenness is a generalization of local clustering”

but in fact,b′S(a) ≤ 1− Ca

I high clustering =⇒ low local betweenness

I low local betweenness 6=⇒ high clustering

Analysis on Dolphin Network

I Lusseau et al. (2003). The bottlenose dolphin community ofDoubtful Sound features a large proportion of long lastingassociations. Behavioral Ecology and Sociobiology, 54,396-240.

I 62 dolphins: 33 males, 25 females, 4 unknown

I Edges represent ‘friendship ties’I Female dolphin SN100 (highest betweenness) splintered the

group into 2 communities:I Males split into two groupsI Females remain cohesive

Dolphin network

Table: (Overall) Rankings among the males only

Degree Closeness Betweenness Eigenvector

Beescratch 5 1 1 14Topless 1 4 10 1

Male-Dolphin subgraph

↙

n1 = 8δ1 = 0.464

n2 = 21 (18)δ2 = 0.195 (0.242)

Male-Dolphin subgraph vs subgroup betweenness

↙Highest subgraph betweenness

−→Highest subgroup bet.

Interpretation

To ‘communicate’ with the male dolphins,

I assuming communication passes through the females as well,the subgroup approach may be more effective (eg, spreadinggossip).

I if females are somehow excluded (eg communicable diseaseaffecting and carried solely by male dolphins), then thesubgraph approach is relevant.

Subgraph-Eigenvector centrality

Rankings determined by relationships withother males only

↖many male friends

Subgroup-Eigenvector centrality

Rankings determined by relationshipswith both males & females

←− not many femalefriends

↓both m & f

friends

Local v Global Subgroup-Closeness

↓

highestglobal ↗

highestlocal

Interpretation

I High global subgroup-closeness → message spreads quicklyamong the females

I High local subgroup-closeness → message spreads quicklyamong the males

Other Interesting Results

↑ladies man!

I Patchback: not in top 10 w.r.t. all other measures.

I Number 1: ranked 20th globally and 14th overall.

Divisions based on structure

Results

Much less difference between subgroup and subgraph measures.

I For both communities, very high or perfect correlationbetween subgroup v subgraph-closeness, and subgroup vsubgraph-betweenness.

I In Community 2, high correlation (0.999) for eigenvectorcentrality.

I In Community 1, correlation of 0.7342 for eigenvectorcentrality:

I Upbang is distance two away from Community 2, thereforeranks highest in the subgroup sense.

I Gallatin is further away from Community 2, therefore rankshighly only in the subgraph sense.

Conclusion and Extensions

I The different rankings obtained from different measures can(sometimes) provide information about the network.

I Proposed measures reveal properties not immediatelyapparent from the total graph nor the subgraph.

I Extends easily to weighted networks. It may be relevant toweight edges so that communications ‘prefer’ to travel withinthe subgroup if possible.

I Other centrality measures: different ’distance’ for closeness;only counting certain paths for betweenness; totalcommunicability?

I Generalize to the temporal setting.

Extension to total communicability

TC := eA1

Some approaches:I Rank is proportional to rank of immediate neighbors:

TCS(a) =∑

x∈S∩N(a)

TC (x) =∑x∈S

(a,x)∈E

eTx eA1

orSCS(a) =

∑x∈S∩N(a)

SC (x) =∑x∈S

(a,x)∈E

(eA)xx

I Walk-based: ∑x∈S

(eA)ax

I Distance-based: ∑x∈S

ξax

Francesca will tell you more...