Stability of networks subgraph indexes
Stefano Pozza1
Francesco Tudisco
2
Due giorni di Algebra lineare numerica
16–17 Feb 2017, Como
1
ISTI – CNR, Pisa, E-mail: stefano.pozza@isti.cnr.it
Department of Mathematics, Università di Padova, E-mail:
francesco.tudisco@math.unipd.it.
2
Stefano Pozza, Francesco Tudisco
1 / 24
Graph G
Undirected
G = (V , E ),
Directed
V = {a, b, c, d, e, f } E = {s, t, u, v , w }
Adjacency matrix A: Ai,j = 1 if there is a edge from the node i to
the node j, otherwise Ai,j = 0.
A is symmetric if and only if the graph is undirected
Stefano Pozza, Francesco Tudisco
2 / 24
Paths and distance
distance dG (c, d) = 1 (length of the minimal path)
a path from c to b
a minimal path from c to b
dG (c, a) = 2, dG (c, {a, b}) = 1, dG (a, c) = ∞
Stefano Pozza, Francesco Tudisco
3 / 24
Paths and distance
distance dG (c, d) = 1 (length of the minimal path)
a path from c to b
a minimal path from c to b
dG (c, a) = 2, dG (c, {a, b}) = 1, dG (a, c) = ∞
Stefano Pozza, Francesco Tudisco
3 / 24
A complex network is a particular graph. For our purpose we can
think to a very big non-regular graphs obtained from real world
relationships.
Small world property: let G be a network with n nodes, then
diam(G ) ∝ log(n),
with diam(G ) the diameter, i.e, the maximal distance between two
nodes in G .
Stefano Pozza, Francesco Tudisco
4 / 24
Centrality and communicability indexes
We are interested in indexes that measure the centrality and the
communicability in terms of paths connecting the nodes.
Let A be the incidence matrix of the graph, then we have
(Ak )i,j = number of paths of length k from i to j
Hence, the matrix function defined by the series
!
∞
X
f (A)k,` =
θn An
n=0
k,`
is the f -communicability from node k to node `
(f (A)k,k is the f -centrality of k)
Stefano Pozza, Francesco Tudisco
5 / 24
Centrality and communicability indexes
We are interested in indexes that measure the centrality and the
communicability in terms of paths connecting the nodes.
Let A be the incidence matrix of the graph, then we have
(Ak )i,j = number of paths of length k from i to j
Hence, the matrix function defined by the series
!
∞
X
f (A)k,` =
θn An
n=0
k,`
is the f -communicability from node k to node `
(f (A)k,k is the f -centrality of k)
Stefano Pozza, Francesco Tudisco
5 / 24
Exponential and resolvent communicability
Our analysis focuses on this two function (but it can be extended
to other ones)
exp(A) =
X 1
An ,
n!
rα (A) =
n≥0
Stefano Pozza, Francesco Tudisco
X
αn An = (I − αA)−1 .
n≥0
6 / 24
Motivations: robustness with respect sparse noise
We derive estimates for the changes in the entries of f (A) with
respect to “small” changes in the entries of A.
Consider the graph G = (V , E ) with adjacency matrix A. Let us
add, remove or simply modify the edges in a set δE obtaining a
modify network
G̃ = (V , E ∪ δE )
with adjacency matrix A + δA.
Stefano Pozza, Francesco Tudisco
7 / 24
Motivations: robustness with respect sparse noise
We derive estimates for the changes in the entries of f (A) with
respect to “small” changes in the entries of A.
Consider the graph G = (V , E ) with adjacency matrix A. Let us
add, remove or simply modify the edges in a set δE obtaining a
modify network
G̃ = (V , E ∪ δE )
with adjacency matrix A + δA.
Stefano Pozza, Francesco Tudisco
7 / 24
Motivations
Computing the entries of f (A) is a costly operation
Often one needs to know only “who are” the first few most
important nodes
Typically the ranking of the most central nodes in G̃ does not
change with respect to those in G
The distance from important nodes and nodes having a
marginal role is typically large
Stefano Pozza, Francesco Tudisco
8 / 24
Motivations
We provide mathematical evidences in support of this observations
giving bounds for
|f (A)k,` − f (A + δA)k,` |
which enlighten the dependency on the distance that separates
either k or ` from the set of nodes touched by the new edges δE
Moreover, we provide a bound which depends on the
resolvent-communicability rα (A).
Stefano Pozza, Francesco Tudisco
9 / 24
Motivations
We provide mathematical evidences in support of this observations
giving bounds for
|f (A)k,` − f (A + δA)k,` |
which enlighten the dependency on the distance that separates
either k or ` from the set of nodes touched by the new edges δE
Moreover, we provide a bound which depends on the
resolvent-communicability rα (A).
Stefano Pozza, Francesco Tudisco
9 / 24
Example
Stefano Pozza, Francesco Tudisco
10 / 24
Example
Stefano Pozza, Francesco Tudisco
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Preliminary result
Lemma
Let G = (V , E ) be a graph with adjacency matrix A. Consider the
graph G̃ , with adjacency matrix Ã, obtained by adding, erasing, or
modifying the weights of the edges contained in δE ⊂ V × V . If
S = {s|(s, t) ∈ δE } and T = {t|(s, t) ∈ δE } are respectively the
sets of sources and tips of δE , then
(pn (Ã))k` = (pn (A))k` ,
for k ∈
/ S and ` ∈
/ T,
and for every polynomial pn of degree n ≤ dG (k, S) + dG (T , `).
Stefano Pozza, Francesco Tudisco
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Example
Stefano Pozza, Francesco Tudisco
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Faber polynomials
We use Faber polynomials approximation similarly to what done in
[P. and Simoncini, 2016].
Theorem
Let φ be the conformal mapping of E ⊃ W (Ã) ∪ W (A), ψ be its
inverse and E̊τ = {w : |φ(w )| < τ } be an open set with boundary
Γτ . If τ > 1, f is analytic in Ω̊τ and is bounded on Γτ , then
δ+1
1
τ
max |f (ψ(z))|
,
f (A) − f (Ã) ≤ 4
τ − 1 |z|=τ
τ
k`
with k ∈
/ S, ` ∈
/ T and δ = dG (k, S) + dG (T , `).
Stefano Pozza, Francesco Tudisco
14 / 24
Field of values
W (A) = {v∗ Av | v ∈ Cn , ||v|| = 1}
Notice that if A is symmetric, then W (A) = [λmin , λmax ].
Exponential bound
Let E be a set containing W (A) and W (Ã), and
δ = dG (k, S) + dG (T , `). If the boundary of E is a horizontal
ellipse with semi-axes a ≥ b > 0 and center c, and δ > b − 1 then
4e <(c) p(δ + 1)
A
e − e à ≤
k`
p(δ + 1) − a+b
δ+1
with q(δ) = 1 +
δ 2 +δ
a2 −b 2
√
,
δ 2 +a2 −b 2
Stefano Pozza, Francesco Tudisco
a + b e q(δ+1)
δ + 1 p(δ + 1)
p(δ) = 1 +
!δ+1
,
p
1 + (a2 − b 2 )/δ 2 .
15 / 24
Resolvent bound
Let E be a set symmetric with respect to the real containing
W (A) and W (Ã), δ = dG (k, S) + dG (T , `), and α ∈
/ E a positive
parameter. If the boundary of E is a horizontal ellipse with
semi-axes a ≥ b > 0 and center c, then for 0 < ε < |α − c| − a
and δ > 0
a+b
1
1 δ+1
4
,
rα (A) − rα (Ã) ≤
a+b
ε |α − c| − ε pε
k`
1 − (|α−c|−ε)p
ε
p
where pε = 1 + 1 − (a2 − b 2 )/(|α − c| − ε)2 .
Stefano Pozza, Francesco Tudisco
16 / 24
Exponential-centrality difference of the two circles example
σ(Ã) = [−2.2361, 2.2361]
Bound on the difference
10 0
10 -5
10 -10
10 -15
0
50
100
Stefano Pozza, Francesco Tudisco
150
200
17 / 24
Resolvent-centrality difference of the two circles example
Bound on the difference
10 0
10 -5
10 -10
10 -15
10 -20
0
50
100
Stefano Pozza, Francesco Tudisco
150
200
18 / 24
Exponential of the normalized Pajek/Erdos971
Erdos collaboration network (472 × 472) adding one undirected
edge.
Bound on the difference
10 0
10 -5
10 -10
10 -15
10 -20
50
100
150
200
Stefano Pozza, Francesco Tudisco
250
300
350
400
450
19 / 24
Field of values
The Hermitian part of A is HA = (A + A∗ )/2.
Theorem
Let A ≥ 0 and let à = A + em eT
n . Then
1
0 ≤ r (Ã) − r (A) ≤ 1/2 and, if H(Ã) is irreducible, then
r (Ã) − r (A) > 0.
2
Assume H(A) irreducible. For any non-negative function
f : C → R+ , such that f (r (A)) 6= 0 we have
p
f (A)mm f (A)nn
1
0 < r (Ã) − r (A) ≤
+O
f (r (A))
4
Stefano Pozza, Francesco Tudisco
20 / 24
Resolvent bound
Resolvent bound
Let à = A + em eT
n then
r (A) r (A) α
k,m α
n,` rα (A) − rα (Ã) ≤ 1 − rα (A)m,n k`
Moreover,
rα (A)k,m rα (A)n,` exp(A) − exp(Ã) ≤ α exp(α) 1 − rα (A)m,n k`
for any α > ρ(A).
Stefano Pozza, Francesco Tudisco
21 / 24
Exponential of the normalized Pajek/Erdos971
Erdos collaboration network (472 × 472) adding one directed edge
Bound on the difference
10 0
10 -5
10 -10
10 -15
10 -20
10 -25
10 -30
50
100
150
200
Stefano Pozza, Francesco Tudisco
250
300
350
400
450
22 / 24
Conclusion
We derived some bounds for the variation of the
f -communicability when some edges are modified
We showed the dependency of these bounds from the distance
between the considered nodes and the modified edges
We introduced other bounds based on the value of the original
resolvent-communicability
Preprint available soon!
Stefano Pozza, Francesco Tudisco
23 / 24
Thank you for your attention!
Stefano Pozza, Francesco Tudisco
24 / 24
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