The document discusses updates to matrix functions utilized for evaluating network centrality, focusing on how these functions change with small perturbations in the graph structure. It presents mathematical formulations of network indices based on adjacency matrix properties, addressing computational approaches and related analytical work. The findings highlight bounds on the changes in centrality measures and provide examples from network analysis in various contexts.

1. Small updates of matrix functions
used for network centrality
Francesco Tudisco
joint work with Stefano Pozza (Charles University in Prague)
Department of Mathematics and Statistics, University of Strathclyde, UK
IMA Conf. Numerical Linear Algebra and Optimization
Birmingham June 28, 2018

2. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Adjacency matrix and its powers
(A)ij = 1 if i ⇝ j, otherwise Aij = 0.
(A2)ij = n
s=1 AisAsj = #{s : i ⇝ s and s ⇝ j}
= number of paths of length 2 from i to j
(A3)ij = n
s,t=1 AisAstAtj = #{s, t : i ⇝ s and s ⇝ t and t ⇝ j}
= number of paths of length 3 from i to j
· · ·
(Ak)ij = number of paths of length k from i to j
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3. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Network indices based on f(A)
f(A) = c0I + c1A + c2A2
+ c3A3
+ · · · with ck ≥ 0
f(A)ij = weighted sum of paths of any length from i to j
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4. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Network indices based on f(A)
f(A) = c0I + c1A + c2A2
+ c3A3
+ · · · with ck ≥ 0
f(A)ij = weighted sum of paths of any length from i to j
f-Centrality: f(A)ii = importance of node i
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5. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Exponential and resolvent
Typical choices are:
ck =
1
k!
exp(A) = I + A +
1
2
A2
+
1
3!
A3
+ · · ·
and
ck = αk
rα(A) = I + αA + α2
A2
+ α3
A3
+ · · ·
Note: the longer the path length the smaller the weighting coefficient
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6. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Problem setting
We are interested in how f(A) changes when the structure of G = (V, E)
undergoes a small perturbation
Setting:
• We add remove or modify the edges in a set δE ⊆ V × V
• We obtain a graph G = (V, E) with adj(G) = A = A + δA
• S ⊆ V = set of sources of edges in δE
• T ⊆ V = set of tips of edges in δE
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7. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Why?
• Computing f(A)ii is a costly operation
• Often we are interested in the first few most important nodes
• Changes or biases are more likely to occur on nodes that are “far”
from important ones
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8. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Related work: f(A + small rank)
Important issue: Computing f(A + δA) from f(A) when δA = small rank
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9. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Related work: f(A + small rank)
Important issue: Computing f(A + δA) from f(A) when δA = small rank
• f(x) = x−1 −→ Sherman–Morrison–Woodbury formula
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10. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Related work: f(A + small rank)
Important issue: Computing f(A + δA) from f(A) when δA = small rank
• f(x) = x−1 −→ Sherman–Morrison–Woodbury formula
• f(x) is rational
Bernstein, Van Loan, SIMAX, 2010
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11. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Related work: f(A + small rank)
Important issue: Computing f(A + δA) from f(A) when δA = small rank
• f(x) = x−1 −→ Sherman–Morrison–Woodbury formula
• f(x) is rational
Bernstein, Van Loan, SIMAX, 2010
• f(x) is analytic
Beckermann, Kressner, Schweitzer, SIMAX, 2018
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12. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Related work: f(A + small rank)
Important issue: Computing f(A + δA) from f(A) when δA = small rank
• f(x) = x−1 −→ Sherman–Morrison–Woodbury formula
• f(x) is rational
Bernstein, Van Loan, SIMAX, 2010
• f(x) is analytic
Beckermann, Kressner, Schweitzer, SIMAX, 2018
We derive a-priori bounds of the type
|f(A)ii − f(A)ii| = O
1
Mδ
with δ ≈ distance i ←→ δE
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13. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Two cycles example
A = adj(G)
G = 100
99
1
2
50 200
199
198
101
→
e
No closed walk passes trough
→
e =⇒ f(A)ii is constant for all i
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14. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Two cycles example
A = adj(G) −→ A = adj(G)
G = 100
99
1
2
50 200
199
198
101
→
e
←
e = δE
T S
No closed walk passes trough
→
e =⇒ f(A)ii is constant for all i
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15. Introduction
Results
Examples
Notation: graphs and matrices
Motivations and goal
Two cycles example
A = adj(G) −→ A = adj(G)
G = 100
99
1
2
50 200
199
198
101
→
e
←
e = δE
T S
No closed walk passes trough
→
e =⇒ f(A)ii is constant for all i
• The number of paths of any length (100) → (100) increases
• Only the number of very long paths (50) → (50) increases
• We expect f(A)50,50 ≈ f(A)50,50 but f(A)100,100 ≫ f(A)100,100
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16. Introduction
Results
Examples
Bounds
Algorithm
Two key observations
δ = distG(i, S) + distG(T, i)
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17. Introduction
Results
Examples
Bounds
Algorithm
Two key observations
δ = distG(i, S) + distG(T, i)
If i /∈ S, T, then:
pk(A)ii = pk(A)ii
for every polynomial pk of degree k ≤ δ.
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18. Introduction
Results
Examples
Bounds
Algorithm
Two key observations
δ = distG(i, S) + distG(T, i)
If i /∈ S, T, then:
pk(A)ii = pk(A)ii
for every polynomial pk of degree k ≤ δ.
If f is analytic on Ω = (convex continuum), then:
f(A) = γ0I + γ1Φ1(A) + γ2Φ2(A) + γ3Φ3(A) + · · ·
Φk = k-th Faber polynomial of Ω
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19. Introduction
Results
Examples
Bounds
Algorithm
Main theorem
Theorem
Ω convex continuum containing W(A)∪W(A), ϕ : Ω → C conformal
map of Ω, f analytic on {x : |ϕ(x)| > τ} ⊇ Ω.
Then:
|f(A)ii − f(A)ii| ≤ c(τ)
1
τδ+2
where
c(τ) =
2
π
τ
τ − 1 |ϕ(x)|=τ
|f(x)| dϕ(x)
Particular choices of f and Ω lead to explicit bounds
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20. Introduction
Results
Examples
Bounds
Algorithm
Ellipse shaped Ω
W(A)
W(A)
Ω
c a
b
Bounds for exp(A) and rα(A) in terms of a, b and c
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21. Introduction
Results
Examples
Bounds
Algorithm
If δ > b − 1, then: Exponential
exp(A)ii − exp(A)ii ≤
4eℜ(c) p(δ + 1)
p(δ + 1) − a+b
δ+1
a + b
δ + 1
eq(δ+1)
p(δ + 1)
δ+1
with q(t) = 1 + a2−b2
t2+t
√
t2+a2−b2
and p(t) = 1 + 1 + (a2 − b2)/t2
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22. Introduction
Results
Examples
Bounds
Algorithm
If δ > b − 1, then: Exponential
exp(A)ii − exp(A)ii ≤
4eℜ(c) p(δ + 1)
p(δ + 1) − a+b
δ+1
a + b
δ + 1
eq(δ+1)
p(δ + 1)
δ+1
with q(t) = 1 + a2−b2
t2+t
√
t2+a2−b2
and p(t) = 1 + 1 + (a2 − b2)/t2
If 1/α /∈ Ω, then: Resolvent
rα(A)ii − rα(A)ii ≤
4
1 − a+b
(|α−1−c|−ε)pε
1
ε
a + b
|α−1 − c| − ε
1
pε
δ+1
with pε = 1 + 1 − a2−b2
(|α−1−c|−ε)2 and 0 < ε < |α−1 − c| − a
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23. Introduction
Results
Examples
Bounds
Algorithm
How to do in practice
Compute f(A)ii = eT
i Aei with non-Hermitian Lanczos method.
{v0, . . . , vt−1}, {w0, . . . , wt−1}= computed bases of Kt(A, ei), Kt(AT, ei)
Theorem
dG(m, i) = smallest k such that (vk)m ̸= 0
dG(i, m) = smallest k such that (wk)m ̸= 0
m = 1, . . . , n
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24. Introduction
Results
Examples
Bounds
Algorithm
How to do in practice
Compute f(A)ii = eT
i Aei with non-Hermitian Lanczos method.
{v0, . . . , vt−1}, {w0, . . . , wt−1}= computed bases of Kt(A, ei), Kt(AT, ei)
Theorem
dG(m, i) = smallest k such that (vk)m ̸= 0
dG(i, m) = smallest k such that (wk)m ̸= 0
m = 1, . . . , n
Elementary modification of Lanczos algorithm allows to simultaneously
compute f(A)ii, i = 1, . . . , n and the all-pair shortest-path distances
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25. Introduction
Results
Examples
Two cycles
London Transports Network
Small world networks
Two cycles
100
99
1
2
50 200
199
198
101
→
e
←
e
T S
exp(A) r1/3(A)
0 50 100 150 200
10 -20
10 -10
10 0
0 50 100 150 200
10 -20
10 -10
10 0
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26. Introduction
Results
Examples
Two cycles
London Transports Network
Small world networks
London trains
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27. Introduction
Results
Examples
Two cycles
London Transports Network
Small world networks
London trains
50 100 150 200 250 300
10-20
10-10
100
1010
50 100 150 200 250 300
10-20
10-10
100
1010
50 100 150 200 250 300
10-20
10-10
100
1010
5 nodes 15 nodes
30 nodes
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28. Introduction
Results
Examples
Two cycles
London Transports Network
Small world networks
Normalized adjacency
100 200 300 400
10 -20
10 -10
10 0
Erdös
1000 2000 3000 4000
10-20
10-10
100
Facebook
1000 2000 3000 4000 5000
10 -20
10 -10
10 0
GRQC
1000 2000 3000 4000 5000 6000
10 -20
10 -10
10 0
Gnutella
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29. Introduction
Results
Examples
Two cycles
London Transports Network
Small world networks
Reference
Details and “much more” in:
Stefano Pozza and Francesco Tudisco
Stability of network indices defined by means of matrix functions
SIAM J Matrix Anal Appl, 2018
For example: Bounds for f(A)ij, f(D−1A), f(D−1/2AD−1/2)
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30. Introduction
Results
Examples
Two cycles
London Transports Network
Small world networks
Reference
Details and “much more” in:
Stefano Pozza and Francesco Tudisco
Stability of network indices defined by means of matrix functions
SIAM J Matrix Anal Appl, 2018
For example: Bounds for f(A)ij, f(D−1A), f(D−1/2AD−1/2)
Thank you for your attention!
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