1
Professor Ernesto Estrada
Department of Mathematics & Statistics
University of Strathclyde
Glasgow, UK
www.estradalab.org

2

3

Measure what is
measurable, and
make measurable
what is not so.
Galileo Galilei

Professor Ernesto Estrada
Department of Mathematics & Statistics
University of Strathclyde
Glasgow, UK
www.estradalab.org

  
kkp ~  kk
ekp /
~ 
 
!k
ke
kp
kk

 
 







 


2
2
2
2
1 k
kk
k
ekp



  
kkp ~ 
!k
ke
kp
kk


Poisson
Exponential
Gamma
Power-law
Log-normal
Stretched exponential
Stumpf & Ingram, Europhys. Lett. 71 (2005) 152.

   
kekp kk /
~
 
    
  

2
~
22
2//ln


k
e
kp
mk
Stumpf & Ingram, Europhys. Lett. 71 (2005) 152.

The qualitative interpretation of the skew is
complicated. For example, a zero value
indicates that the tails on both sides of the
mean balance out, which is the case for a
symmetric distribution, but is also true for an
asymmetric distribution where the asymmetries
even out, such as one tail being long but thin,
and the other being short but fat. Further, in
multimodal distributions and discrete
distributions, skewness is also difficult to
interpret. Importantly, the skewness does not
determine the relationship of mean and
median.
Wikipedia

• Collatz-Sinogowitz Index (1957)
  kGCS  1
• Bell Index (1992)
 
2
2 11






  i
i
i
i k
n
k
n
GVar
• Albertson Index (1997)
 
 


Eji
ji kkGirr
,
Unknown
upper
bounds

i
i
k
p
1










ji
ij
kk
p
11
2
1
Estrada: Phys. Rev. E 82, 066102 (2010)









ji
ij
kk
p
1
ˆ
ijijij pp ˆ

2
11
211
2



















ji
jiji
ij
kk
kkkk

 
   
 









Eij jiEij
ij
kk
G
2
11
2 

AKL 








otherwise,0
,~for1
,for
ji
jik
L
i
ij
2/12/1
2/12/1




KAKI
KLKL










otherwise,0
,~for
1
,for1
ji
kk
ji
ji
ijL

 
 
 
 .2
2
11
,
2/1
Gn
kkn
G
Eji
ji
T








L

 
2
1
n
Gn  
   
12
2~



nn
Gn
G


  0~1  G

n  210
 



n
i
j
j
j
T
j i
n 1
1
1
1
cos 


 

  
j
jj
T
nG  2
cos11

L
     






j i
jj
T
iG
2
11 

L

jjjx  cos1
jjjy  sin1
  

n
j
jx
nn
n
G
1
2
12
~
j
01 
j
jx
jy
y
x

8k 16k
pnG ,
  017.0~ G  034.0~ G

  















 

1227.2
27.0
18.1
18.1
nn
n
k
k
BA
8k 16k
  132.0~ G   108.0~ G

S. cereviciaeD. melanogaster
C. elegansH. pylori
  148.0~ G   294.0~ G
E. coli
  241.0~ G
  323.0~ G   383.0~ G Stretched exponential
Log-normal
   
kekp kk /
~
 
    
  

2
~
22
2//ln


k
e
kp
mk

Professor Ernesto Estrada
Department of Mathematics & Statistics
University of Strathclyde
Glasgow, UK
www.estradalab.org

1,305 mi.
Stanley Milgram
1933-1984

CLUSTERINGDISTANCE

5ij
d 

i
i
Ci
nodeofrelationsetransitivpossibleofnumbertotal
nodeofrelationsetransitivofnumber

   1
2
2/1 



ii
i
ii
i
i
kk
t
kk
t
C

i
iC
n
C
1

d 3.65
0.79
2.99
0.00027
1847
0.74C

d 2.65
0.28
2.25
0.05
10.5
0.69C
33

d 18.7
0.08
12.4
0.0005
926
0.3C
34

D. J. Watts
S. H. Strogatz
35

There is a high probability that the shortest path connecting
nodes p and q goes through the most connected nodes of
the network.
Sending the ‘information’ to the most connected nodes (hubs)
of the network will increase the chances of reaching the target.
The sender does not know the global structure of the network!
36

But also, there is a high probability that the most connected
nodes of the network are involved in a high number of
transitive relations.
Sending the ‘information’ to the most connected nodes
(hubs) of the network will increase the chances of getting
lost.
37

Definition: A route is a walk of length l, which is any
sequence of (not necessarily different) nodes
v1,…, vl such as for each i =1,…,l there is a link
from vl to vl+1.
Definition: The communicability between two nodes
in a network is defined as a function of the total
number of routes connecting them, giving more
importance to the shorter than to the longer ones.
38
Hypothesis: The information not only flows through the shortest
paths but by mean of any available route.

39
 0
# of walks in steps from topq l
l
G c l p q


 
The between the nodes p and q is
then mathematically defined as
where cl should:
 makes the series convergent
 gives more weight to the shorter than to the longer walks

          VpVp pfpfV
2
2
      
 Eqp
qfpAf ,
Definition: Let .
The adjacency operator is a bounded operator on
defined as .
Theorem: (Harary) The number of walks of length l
between the nodes p and q in a network is equal to .
 pq
l
A
 V2
40

The communicability function can be written as:
  qjpj
l
j
l
n
j
lpq
l
l
lpq cAcG ,,
0 10


 



41
Let be a nonincreasing ordering
of the eigenvalues of A, and let be the eigenvector
associated with .
n  21
j

j

42
    j
ee
l
A
G
n
j
qjpjpq
A
l
pq
l
pq

 



1
,,
0 !
      


 

n
j j
qjpj
pq
l
pq
ll
pq AIAG
1
,,1
0 1 


1
10

 

         1 1
1 1
2
n
n
pq n j j
j
G K p q e e p p    

  
     
 
1
1
2
1
1 1
1 .
n n
pq n j j
j
n
n
e
G K e p p
n
e
e
n ne ne
 




 
   

 pq nG K   

2cos
1
j
j
n


 
  
 
 
2
sin
1 1
j
jp
p
n n

 
 
 
2cos
1
1
2cos
1
1
2 2
sin sin
1 1 1 1
2
sin sin .
1 1 1
jn
n
pq n
j
jn
n
j
jp jq
G P e
n n n n
jp jq
e
n n n


 
 
 
 
 

 
 
 

  
          
  
   
    



      sin sin 1/ 2 cos cos         
 
   2cos 2cos
1 1
1
1 1
cos cos
1 1 1 1
j jn
n n
pq n
j
j p q j p q
G P e e
n n n n
 
    
   
    

    
    
      

1,2, ,j n  1/ nj  ,0
       2cos 2cos
0 0
1 1
cos cospq nG P p q e d p q e d
 
 
   
 
    
 / 1j n  

   xcos
0
1
cos e d I x


 


     2 2pq n p q p qG P I I  
      1 1 12 2 0n n n nG P I I    

x
xx
x
xx
x
x
xo
oo
oooo
oo
o
0 5 10 15 20
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
Subject
Secondlargestprincipalvalue
x
x
x
x
x
x
x
x
oo oo o
oo
o
0 5 10 15 20
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
x
o
o
Subject
Secondlargestprincipalvalue
  pqpq AWWG 2/12/1
exp
~ 


: Stroke : Decreased communicability
: Stroke : Increased communicability
Images courtesy of Dr. Jonathan J. Crofts.

Professor Ernesto Estrada
Department of Mathematics & Statistics
University of Strathclyde
Glasgow, UK
www.estradalab.org

 Local metrics provide a measurement of a structural
property of a single node
 Designed to characterise
 Functional role – what part does this node play in
system dynamics?
 Structural importance – how important is this node to
the structural characteristics of the system?

    j
ee
l
A
G
n
j
pjpp
A
l
pp
l
pp

 



1
2
,
0 !
1) 3.026
2) 1.641
3) 1.641
4) 3.127
5) 2.284
6) 2.284
7) 1.592
8) 1.592
1
2
3
4
5
6
7
8

1
2
3
4
56
7
8
9
 8,6,5,3,11 S
  65.451 SpGpp
 9,7,4,22 S
  70.452 SpGpp

Essential
54

55
Rank Protein Essential
?
1 YCR035C Y
2 YIL062C Y
... ... ...

Estrada: Proteomics 6 (2009) 35-40
0
10
20
30
40
50
60
70PercentageofEssentialProteinsIdentified

57
Consider that two nodes p and q are trying to communicate
with each other by sending information both ways through
the network.
ppG
Quantifies the amount of ‘information’ that is sent by p,
wanders around the network, and returns to the origin,
the node p.
pqG
Quantifies the amount of ‘information’ that is sent by p,
wanders around the network, and arrives to its destination,
the node q.
We know that:

2
def
pq pp qq pqG G G   
The goal of communication is to maximise the amount
of information that arrives to its destination by minimising
that which is lost.
58

Theorem: The function is a Euclidean distance.pq
59
   
T
pq p q p qe   Λ
φ φ φ φ
1
2
0 0
0 0
0 0 n



 
 
 
 
 
 
Λ
 
 
 
1
2
p
n
p
p
p



 
 
 
 
 
  
φ
Proof:

60
   
   
   
/2 /2
/2 /2 /2 /2
2
T
pq p q p q
T
p q p q
T
p q p q
p q
e e
e e e e
      
  
  
 
Λ Λ
Λ Λ Λ Λ
φ φ φ φ
φ φ φ φ
x x x x
x x
Proof (cont.):

61
pq
pqd
p q

62
p q
1P
p q
2P
 
 
09.5
1,
,
2/1



Pji
Eji
ij  
 
80.4
2,
,
2/1



Pji
Eji
ij
Remark. The shortest route based on the communicability
distance is the shortest path that avoids the nodes with the
highest ‘cliquishness’ in the graph.

8
1.56 10  8
3.30 10  

Fraction of nodes
removed

The routes minimizing the communicability distance are the shortest ones
that avoid the hubs of the network. Then, they can be useful to avoid
possible collapse of the networks due to the removal of the hubs.

67
Essential

 

q
qp
n
,
1

 

q
qpd
n
,
1
68

69
large ppG
Wikipedia

70
qqpp
pq
def
pq
GG
G

Theorem. The function is the cosine of the Euclidean
angle spanned by the position vectors of the nodes p and q.
pq
qqpp
pq
qp
qp
pq
GG
G
xx
xx 11
coscos 


 


Definition:

71
 







 

a
b
cR
2
2 2
4
1
11
 AT
ea 
 1
 AT
esb 
 sesc AT  

Theorem. The communicability distance induces an
embedding of a network into an (n-1)-dimensional
Euclidean sphere of radius:
Estrada et al.: Discrete Appl. Math. 176 (2014) 53-77.
Estrada & Hatano, SIAM Rev. (2016) in press.
ATT
essC 211 

 A
ediags 

Remark. The communicability distance matrix C is circum-
Euclidean:

72











2/1
2/1
2/1
1












2/1
0
2/1
2












2/1
2/1
2/1
3

jjjA 



73
0.08.0
6.0
0.0
7.0
0.0
0.1
0.1
x
y
z
2

3

1

0.0
0.1
2

3

1

0.1
0.1

74
2

1

3

x
y
z
0.1
0.0
0.1
0.1
0.1
0.0
0.1
0.2
0.0
O
1x

2x

3x


75
px

qx

O
ppp Gx 
 pq
pq
qqq Gx 

Estrada & Hatano: Arxiv (2014) 1412.7388.
Estrada: Lin. Alg. Appl. 436 (2012) 4317-4328.
qppq xxG



Professor Ernesto Estrada
Department of Mathematics & Statistics
University of Strathclyde
Glasgow, UK
www.estradalab.org

1V 2V
1 2V V V 
0
0T
B
A
B
 
  
 
1j n j    

m
m
b
fr
1
Problem:

  sinh 0Bipartivity Tr A 
Bipartivity No odd cycles

        exp sinh coshTr A Tr A Tr A 
     exp coshBipartivity Tr A Tr A 
  
  
 
  EE
EE
ATr
ATr
b even
n
j
j
n
j
j
s 




1
1
exp
cosh
exp
cosh



   even even
s s
EE EE a
b G b G e
EE EE a b

   
 
Theorem.
G G e
e
b
a

1.000sb  0.829sb  0.769sb  0.721sb 
0.692sb  0.645sb  0.645sb  0.597sb 

   
   
1 1
1 1
cosh sinh
cosh sinh
n n
j j
j j
e n n
j j
j j
b
 
 
 
 



 
 
  
  
 
 
1
1
exp
exp
exp
exp
n
j
j
e n
j
j
Tr A
b
Tr A






 



1.000sb  0.658sb  0.538sb  0.462sb 
0.383sb  0.289sb  0.289sb  0.194sb 

Hervibores
Grass
Parasitoids
Hyper-hyper-parasitoids
Hyper-parasitoids
0.766sb 
0.532eb 

j = 1 j = 2 j = 3
j = 4 j = 5 j = 6

     ˆ exp cosh sinhpq pq pq
G A A A         
 ˆ sinh 0pq pq
G A    
p
q
p
q
 ˆ cosh 0pq pq
G A   
0 and in different partitionsˆ
0 and in the same partitions
pq
p q
G
p q




1
3
7
4
6
10
8
11
9
12
5
2




















































38.919.872.673.680.636.628.777.610.705.704.773.8
19.81.1083.609.711.766.673.862.598.729.728.702.9
72.683.603.817.487.543.504.731.632.456.665.528.7
73.609.717.405.862.550.505.742.562.630.456.629.7
80.611.787.562.517.893.310.768.596.562.632.498.7
36.666.643.550.593.337.777.639.568.542.531.662.5
28.773.804.705.710.777.638.936.680.673.672.619.8
77.662.531.642.568.539.536.637.793.350.543,566.6
10.798.732.462.696.568.580.693.317.862.587.511.7
05.729.756.630.462.642.573.650.562.505.817.409.7
04.728.765.556.632.431.672.643.587.517.403.883.6
73.802.928.729.798.75.6219.866.611.709.783.61.10
ˆG








































011111000000
101111000000
110111000000
111011000000
111101000000
111110000000
000000011111
000000101111
000000110111
000000111011
000000111101
000000111110
~
G

10
12
11
8
7
9
2
3
4
1
5
6
 1 1,2,3,4,5,6C 
 2 6,7,8,9,10,11,12C 
1
3
7
4
6
10
8
11
9
12
5
2

2 3
10
4
1211
1 5 6
87 9
1
3
7
4
6
10
8
11
9
12
5
2

Professor Ernesto Estrada
Department of Mathematics & Statistics
University of Strathclyde
Glasgow, UK
www.estradalab.org

VS 
VS 2/1
 S
Let and
represents the number
of edges with exactly one
endpoint in S.
 
 
1
2
min , ,0
2n
S
S V
G S V S
S

 

     

   1 2
1 1 22
2
G
 
   

  
N  321(Alon-Milman): Let
21  

A Ramanujan graph
n = 80,  = 1/4
The Petersen graph
n = 10,  = 1

S S
  SS 
bottleneck

MySQL 30.7
93.6 56.5
St Marks St Martin

 t 
 t 
 t 
 t 

iiG
i
          

 800,628,332
1
!
1032
0
iiiiii
k
ii
k
ii
AAA
k
A
G

i i
i
1/2 1/6
1/24
i
1/40320

 iNk
     
1
1










 
n
j
kkk jNiNis
i,1


  k
jpj
n
p
n
j
ijk iN  ,
1 1
, 

     

n
p
k
jqj
n
q
n
j
pj
n
p
k pN
1
,
1 1
,
1

 


  
 
 n
p
k
jqj
n
q
n
j
pj
k
jpj
n
p
n
j
ij
k is
1
,
1 1
,
,
1 1
,



  .
1 1
,1,1
1
,1,1
1 1
1,1,1
1
1,1,1




 

 

 n
p
n
q
qp
n
p
pi
n
p
n
q
k
qp
n
p
k
pi
k is




2
1
,1
1 1
,1,1 







   
n
j
p
n
p
n
q
qp 
  in
p
p
i
k is ,1
1
,1
,1





k
Remark: The eigenvector centrality of a given node represents the
probability of selecting at random a walk of infinite length that has
started at that node.

1 2 
   

n
j
jijiiiG
2
2
,1
2
,1 expexp 
 1
2
,1 exp  iiiG 
2
ln
2
1
ln 1
,1

  iii G
i,1ln

iiGln

 
5.0
1
2
,1
,1
exp
lnln 






ii
i
i
G



  iiii GVi  1
2
,1,1 exp,0ln 

  iiii GVi  1
2
,1,1 exp,0ln 

  iiii GVi  1
2
,1,1 exp,0ln 

andVpp  ,0ln ,1 Vqq  ,0ln ,1

Chordless cycle
of length 15

MySQL
3
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
St Marks St Martin
3
1090.2 

67.1