Generalized Firefighting on the 2
dimensional infinite grid
and
Centrality measures in social
networks
Kah Loon Ng
DIMACS
Containing fires in infinite grids Ld
Fire starts at only one vertex:
d= 1: Trivial.
d = 2: Impossible to contain the fire with 1 firefighter per
time step
Containing fires in infinite grids Ld
d = 2: Two firefighters per time step needed to contain the fire.
8 time steps
18 burnt
vertices
Containing fires in infinite grids L2
• We assume that the number of firefighters available for
deployment is given by a function f (t ) that is periodic.
Justification for considering periodic functions:
t 1
1
t 2
t 3
t 4
t 5
t 6
1
2
1
1
2
Containing fires in infinite grids L2
Given a periodic function f , let
p f = period of f
pf
N f   f (i )
i 1
Nf
Rf 
pf
For example,
1

f (x )  2
3

if x  1 mod 3
if x  2 mod 3
if x  0 mod 3
(1,2,3,1,2,3,...)
We also write f  [1,2,3]
Nf 6
Rf  2
Containing fires in infinite grids L2
Given positive integers n and k , we define a periodic function
g n ,k
n 1




 [1,1,1,... 1,k ]
Given two periodic functions f and g with periods p f and p g
respectively, we say
 n f ( i )  n g (i ) 

f  g  n    

i 1
 i 1

n
n


f
(
i
)

g
(
i
)
 n  lcm( p f , p g )  


i 1
 i 1

Containing fires in infinite grids L2
If f is a periodic function and i is any positive integer, define
the translate function f i by
f i ( x )  f ( x  i )
For example, if
f  [1,1,2,2,3]  (1,1,2,2,3,1,1,2,2,3,...)
and i  3
then
f 3  [2,3,1,1,2]  ( 2,3,1,1,2,2,3,1,1,2,...)
Note that
p f i  p f
N f i  N f
R f i  R f
Containing fires in infinite grids L2
Given a positive integer n , let M n be the function with period n
defined by
n1




M n  [1,1,1,... 1, ]
where   n2   2
Note that  is chosen to be the smallest integer such
that RM n  1.5 .
Define Dn to be the periodic function of period 2n  1
n


Dn  [1,1,...,1, 2
,2
,...,


2]
n 1
Containing fires in infinite grids L2
Main Theorem:
If the number of firefighters available for deployment per time step
can be represented by a periodic function f such R f  1.5 , then
any fire that breaks out at a single vertex in L2 can be contained
after some finite time t .
Lemma 0:
Suppose f and g are two periodic functions such that f  g . If
any fire breaking out at a single vertex in L2 can be contained
“using f ” , then it can also be contained “using g ”.
Containing fires in infinite grids L2
Proof of Lemma 0:
Let s  lcm( p f , p g ) .
Case 1: If f (i )  g (i ) for all 1  i  s , we are done.
Case 2: If f (k )  g (k ) for some k  2 . Let k * be the smallest
such k .
x  f (k * )  g (k * )
k * 1
Since f  g, we have  g (i )  f (i )  x .
i 1
Time 1 to k *  1 : Deploy firefighters as in f , and
accumulate at least x “spare” firefighters.
Containing fires in infinite grids L2
Outline proof of Main Theorem:
Given any periodic function f
with R f  1.5

“Rearrange” f to get f '
that is non decreasing in its
period.
[2,3,1,4,1,2]  [1,1,2,2,3,4]
Lemma 1:
f'

g p f ,( N f  p f 1)
[1,1,2,2,3,4]  [1,1,1,1,1,8]
 Mp
[1,1,1,1,1,8]  [1,1,1,1,1,5]
g p f ,( N f  p f 1)
f
Containing fires in infinite grids L2
Outline proof of Main Theorem:

[1,1,1,1,1,8] 
Mp 
[1,1,1,1,1,5] 
g p f ,( N f  p f 1)
Lemma 4:
f
35
M pf
[1,1,1,1,1,5]
D( p f )2
[1
,1
,...,
1
,
2
,
2
,...,
2
]
 


36
37
1,1,1,1,5, 1,1,1,1,5, ………1,1,1,1,5,1,1,1,1,5,1,1,1,1,5,……
1,1,1,1,1,………………………….1,1,2,2,………………….
36
Containing fires in infinite grids L2
Outline proof of Main Theorem:
 D( p )
,1
,...,
,2
,...,
[1,1,1,1,1,5]  [1
1, 2


2]
Lemma 4:
M pf
2
f
36
37
Lemma 2: If f is a periodic function that is non decreasing
over its period, then for any positive integer i , f i  f .
f
1, 1, 2, 2, 3, 1, 1, 2, 2, 3,…
f 2
2, 2, 3, 1, 1, 2, 2, 3, 1, 1,…
Gain
Loss
Containing fires in infinite grids L2
Outline proof of Main Theorem:
 D( p )
,1
,...,
,2
,...,
[1,1,1,1,1,5]  [1
1, 2


2]
Lemma 4:
M pf
2
f
36
37
Lemma 3: If f is a periodic function that is non decreasing
over its period, then for any periodic function g with p g  p f
k
k
i 1
i 1
satisfying  g (i )   f (i )
k
k
i 1
i 1
for 1  k  p g , we have f  g .
In other words,  g (i )   f (i )
for 1  k  lcm,( p f , p g )
1,1,1,1,1,5,…..…….1,1,1,1,1,5, 1,1,1,1,1,5,………..1,5,1,1,1,..
1,1,………………………….,1, 2,2,…………………,2,2,1,1,..
36
37
Containing fires in infinite grids L2
n
n1





The strategy using Dn  [1,1,...,1, 2,2,..., 2]
End of phase 2
End of phase 1
End of phase 3
Advance firefighters
Retreat firefighters
Containing fires in infinite grids L2
The strategy using Dn :
D6  [1,1,1,1,1,1,2,2,2,2,2,2,2]
Completing phase 1:
12
12
11
2
10
10
9
4
8
8
7
6
1
3
5
7
9
11
Containing fires in infinite grids L2
The strategy using Dn :
“Delayed response” – Phase 1 can still be completed.
Containing fires in infinite grids L2
The strategy using Dn :
Completing phase 2 after phase 1 has been completed:
(0,0)
(0,0)
Containing fires in infinite grids L2
The strategy using Dn :
Completing phase 3 after phase 2 has been completed:
(0,0)
(0,0)
Containing fires in infinite grids L2
The strategy using Dn :
Finishing the job:
(0,0)
2
Retreat
firefighters
Active
vertices
(0,0)
1
3
4
Containing fires in infinite grids L2
A few points to note:
Our theorem says nothing about periodic functions f where R f  1.5 .
In fact, we can easily construct a function g where Rg  1   and
still g is sufficient to contain the outbreak.
(0,0)
Containing fires in infinite grids L2
A few points to note:
Our theorem says nothing about periodic functions f where R f  1.5 .
In fact, we can easily construct a function g where Rg  1   and
still g is sufficient to contain the outbreak.
1] is sufficient to contain the outbreak but
Obviously, [4,1,1,1,......,
we do not consider such (cheating! ) situations.
The exact time required to contain the outbreak (and thus the
number of burnt vertices) can be explicitly computed as a
function of n . However, our strategy does not guarantee that the
number of burnt vertices is minimized.
Centrality Measures in Graphs (or Social Networks)
• Centrality = Importance = Prominence?
• 3 types of centrality indices:
• degree
• betweeness
• closeness
Centrality Measures in Graphs (or Social Networks)
• Centrality indices can be computed for each vertex or a
group of vertices.
• Majority of centrality concepts are based on non-directed
and dichotomous relations
• However, for some specific purposes (for example,
measuring prestige) directed graphs or valued relations might
need to be used.
Centrality Measures in Graphs (or Social Networks)
Ca ( ni ) = centrality index for vertex ni under measure a
a = D, degree measure
a = C, closeness measure
a = B, betweeness measure
n
Ca 
 (Ca ( n )  Ca ( ni ))
*
i 1
n
max  (Ca ( n* )  Ca ( ni ))
i 1
Centrality Measures in Graphs (or Social Networks)
CD (ni ) (degree centrality)
CD (ni ) = deg( ni ) = degree of vertex ni
CD ' (ni ) =
n
CD 
1
deg ( ni )
n 1
n
 (C D ( n )  C D ( ni ))
*
i 1
n
max  (C D ( n )  C D ( ni ))
*

*
(
C
(
n
 D )  CD ( ni ))
i 1
( n  1)( n  2)
i 1
n
S D2 
2
(
C
(
n
)

C
)
D
 D i
i 1
n
Centrality Measures in Graphs (or Social Networks)
CC (ni ) (closeness centrality)
n
CC (ni )  (  d (ni , n j ))
1
j 1
CC ' ( ni ) 
n 1
n
 d ( ni , n j )
 ( n  1)CC ( ni )
j 1
n
CC 
 (CC ' ( n )  CC ' ( ni ))
n
*
i 1
n
max  (CC ' ( n )  CC ' ( ni ))
i 1
*

*
(
C
(
n
 C )  CC ( ni ))
i 1
( n  2)( n  1) /(2n  3)
Centrality Measures in Graphs (or Social Networks)
CC (ni ) (closeness centrality)
• Closeness centrality measures are related to:
• Jordan centers of a graph (subset of V (G) with
the smallest eccentricities)
• Centroid of a graph (subset of V (G) with the
smallest “weight”)
• One shortcoming of closeness measures is that it cannot be
used for disconnected graphs.
Centrality Measures in Graphs (or Social Networks)
CB ( ni ) (betweeness centrality)
Let
g jk = number of distinct shortest paths between j and k
g jk ( ni ) = number of distinct shortest paths between j and k that
passes through ni
CB ( ni )   g jk ( ni ) / g jk
j k
C B ' ( ni ) 
For each j  k , maximum
value is 1.
 g jk ( ni ) / g jk
j k
 n  1


 2 
n
CB 
*
 (C B ' ( n )  C B ' ( ni ))
i 1
n 1
Centrality Measures in Graphs (or Social Networks)
CB ( ni ) (betweeness centrality)
• We assume here that if some information (or disease) is
passed from j to k , each of the shortest paths is equally
likely to be chosen.
• If we sum g jk ( ni ) / g jk over all k , we obtain measures of
the pair-dependency of vertex j on vertex i . These values
can also be viewed as indices of how much “gate keeping” ni
does for n j .
• CB ( ni ) seems to be an improvement over CD ( ni ) and CC ( ni )
but there are still inadequacies.
Centrality Measures in Graphs (or Social Networks)
GINORI
LAMBERTESCHI
ALBIZZI
GUADAGNI
PAZZI
SALVATI
TORNABUONI
BISCHERI
MEDICI
RIDOLFI
STROZZI
ACCIAIUOLI
PERUZZI
BARBADORI
CASTELLANI
Centrality Measures in Graphs (or Social Networks)
CD ' (ni )
CC ' (ni )
CB ' ( ni )
Acciaiuoli
0.071
0.368
0.000
Albizzi
0.214
0.483
0.212
Barbadori
0.143
0.438
0.093
Bischeri
0.214
0.400
0.104
Castellani
0.214
0.389
0.055
Ginori
0.071
0.333
0.000
Guadagni
0.286
0.467
0.255
Lamberteschi
0.071
0.326
0.000
Medici
0.429
0.560
0.522
Pazzi
0.071
0.286
0.000
Peruzzi
0.214
0.368
0.022
Ridolfi
0.214
0.500
0.114
Salvati
0.143
0.389
0.143
Strozzi
0.286
0.438
0.103
Torabuoni
0.214
0.483
0.092
Centrality Measures in Graphs (or Social Networks)
CD ' (ni )
CC ' (ni )
CB ' ( ni )
Acciaiuoli
0.071
0.368
0.000
Albizzi
0.214
0.483
0.212
Barbadori
0.143
0.438
0.093
Bischeri
0.214
0.400
0.104
Castellani
0.214
0.389
0.055
Ginori
0.071
0.333
0.000
Guadagni
0.286
0.467
0.255
Lamberteschi
0.071
0.326
0.000
Medici
0.429
0.560
0.522
Pazzi
0.071
0.286
0.000
Peruzzi
0.214
0.368
0.022
Ridolfi
0.214
0.500
0.114
Salvati
0.143
0.389
0.143
Strozzi
0.286
0.438
0.103
Torabuoni
0.214
0.483
0.092
Some observations
Strozzi family
has high degree
centrality but
low closeness
centrality.
Centrality Measures in Graphs (or Social Networks)
CD ' (ni )
CC ' (ni )
CB ' ( ni )
Acciaiuoli
0.071
0.368
0.000
Albizzi
0.214
0.483
0.212
Barbadori
0.143
0.438
0.093
Bischeri
0.214
0.400
0.104
Castellani
0.214
0.389
0.055
Ginori
0.071
0.333
0.000
Guadagni
0.286
0.467
0.255
Lamberteschi
0.071
0.326
0.000
Medici
0.429
0.560
0.522
Pazzi
0.071
0.286
0.000
Peruzzi
0.214
0.368
0.022
Ridolfi
0.214
0.500
0.114
Salvati
0.143
0.389
0.143
Strozzi
0.286
0.438
0.103
Torabuoni
0.214
0.483
0.092
Some observations
Tornabouni
family has high
closeness
centrality but
low betweeness
centrality.
Centrality Measures in Graphs (or Social Networks)
CD ' (ni )
CC ' (ni )
CB ' ( ni )
Acciaiuoli
0.071
0.368
0.000
Albizzi
0.214
0.483
0.212
Barbadori
0.143
0.438
0.093
Bischeri
0.214
0.400
0.104
Castellani
0.214
0.389
0.055
CD  0.257
Ginori
0.071
0.333
0.000
CC  0.322
Guadagni
0.286
0.467
0.255
CB  0.437
Lamberteschi
0.071
0.326
0.000
Medici
0.429
0.560
0.522
Pazzi
0.071
0.286
0.000
Peruzzi
0.214
0.368
0.022
Ridolfi
0.214
0.500
0.114
Salvati
0.143
0.389
0.143
Strozzi
0.286
0.438
0.103
Torabuoni
0.214
0.483
0.092
Some observations
Centrality Measures in Graphs (or Social Networks)
Eigenvector centrality
“The centrality of a vertex does not depend on the number of
vertices it is adjacent to, but also these vertices’ centrality.”
Bonacich (1972) defines (eigenvector) centrality CE (ni )
as a positive scalar multiple of the sum of “adjacent”
centralities:
n
CE (ni )   aij CE (n j )
j 1
In matrix form, if C  (CE (n1 ),...,CE (nn ))T we have
AC  C
(Perron-Frobenius) Since A is nonnegative, there exists an
eigenvector of the maximal eigenvalue with only
nonnegative entries.
Centrality Measures in Graphs (or Social Networks)
Information centrality
Adopts the idea that information is passed along the network
along all possible paths, not necessary the shortest one.
Flow betweenes centrality
The flow betweeness of vertex i is defined as the amount of
flow through vertex i when the maximum flow is transmitted
from s to t , averaged over all s and t .
Random walk betweenes centrality
The random walk betweeness of a vertex i is the number of
times that a random walk starting at s and ending at t passes
through i along the way.
Centrality Measures in Graphs (or Social Networks)
Different measures for different types of flow
• Used goods (eg books) – consider trails in graphs?
• Money (eg a dollar note) – consider walks in graphs?
Markov process?
• Attitude/Belief – Influence process
• Gossip and infection – similarities and differences
• Package delivery – known destination? Shortest route?
Paths vs. Walks vs. Trails
Transfer vs. Duplication
Centrality Measures in Graphs (or Social Networks)
Some references:
• Freeman L.C. (1979): Centrality in social networks: Conceptual clarification.
Social Networks 1, 215-239.
•Bonacich P. (1991): Simultaneous group and individual centralities. Social
Networks 13, 155-168
•Friedkin N. E. (1991): Theoretical foundations for centrality measures. Amer.
Journal of Sociology 96, 1478-1504
•Stephenson K., Zelen M. (1989): Rethinking centrality:methods and examples.
Social Networks 11, 1-37.
•Borgatti S.P. (2005): Centrality and network flow. Social Networks 27, 55-71.
• Ruhnau B. (2000): Eigenvector centrality – a node centrality? Social Networks
22, 357-365
•Faust K. (1997): Centrality in affiliation networks. Social Networks 19, 157-191.
•Newman M. (2005): A measure of betweeness centrality based on random walks.
Social Networks 27, 39-54.
• Bell D., Atkinson J., Carlson J. (1999): Centrality measures for disease
transmission networks. Social Networks 21, 1-21.