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The block-cutpoint tree characterization of a
covering polynomial of a graph
Robert Ellis (IIT)
James Ferry, Darren Lo (Metron, Inc.)
Dhruv Mubayi (UIC)
AMS Fall Central Section Meeting
November 6, 2010
Random Intersection Model B*(n,m,p)
• Introduced: Karoński,
Scheinerman, SingerCohen `99
• Bipartite graph models
collaboration
– Activity nodes
– Participant nodes
• Random Intersection
Graph B*(n,m,p)
– Bipartite edges arise
independently with
constant probability
– Unipartite projection
onto participant nodes
m: number
of “movies”
Bipartite graph
n: number
of “actors”
Unipartite projection
Collaboration
graph: who’s
worked with
whom
Slide 2
Expected sugbraph count vs. E-R
E[XH]
 Erdős–Rényi
 n = 1000
 pER = 0.002
Subgraph H
 Random Intersection
 n = 1000
 m = 100
 p = 0.0045
 Yields pER = 0.002
60-cycle
Erdős–Rényi
Random
Intersection
1.3
1500
11
4 ´ 105
2 ´ 1015
2 ´ 1038
1.2
2 ´ 1010
3 ´ 10- 6
1700
E [X H (G( N , pER )]£
E éêëX H (B ( N , M , p)ù
ú
û
*
for p = 1- (1- pER )1 m
Slide 3
Erdős–Rényi vs. RI Model as m → ∞
• m = “number of movies”
• pER = 0.028 (edge probability)
Erdős–Rényi G(n,pER) model
1)
RC
10,000)
100)
10)
1000)
RI model (m = 2)
B*(N,M,p)
Slide 4
Erdős–Rényi vs. RI Model as m → ∞
• Theorem [Ferry, Mifflin]. For a fixed expected number of
edges pER n2  , and any graph G with n vertices, the probability
 
of G being generated by the Random Intersection model
approaches the probability of G being generated by the
Erdős–Rényi model as m → ∞.
• Formula for rate of convergence:
32
æ
ö
æ
ö
æ
ö
n
PrRC
G
(
)
RI
- 1/ 2
- 1
÷ (n - 2)e (G ) 1 + c (G ) 1 ÷
ççlog 1 ÷
÷
÷
= 1 + ççç2 çç ÷m
+
O
m
(
)
3
÷
3 ֍
÷
÷
÷
֍
çè çè3ø
PrER (G )
pER
pER
1
p
øè
ER ø
• [(Independently) Fill, Scheinerman, Singer-Cohen `00]
With m=nα, α>6, total variation distance for probability of G
goes to zero as n → ∞.
Slide 5
RI model in the constant-μ limit
Idea: Let m → ∞ and fix the expected number of movies per
actor at constant μ=pm.
This allows simplified asymptotic probabilities for random
intersection graphs on a fixed number of nodes.
• Probability formulas are from edge clique covers
• Most probable graphs have block-complete structure
• Least probable graphs have connections to Turán-type
extremal graphs
Slide 6
Edge clique covers
• Unipartite projection
corresponds to an edge clique
cover
• The projection-induced cover
encodes collaboration structure
B
B*
Unipartite projection
Hidden collaboration perspective:
• Given B*(n,m,p)=G, we can infer
which clique covers are most
likely
• This reveals the most likely
hidden collaboration structure
that produced G
G  B*
likeliest B
Slide 7
Covering polynomial of G
s ( )  X  X
“size” of clique cover S
wt (G )  min wt ( )
“weight” of S, G

wt ( )  s ( )  
s(G; x )   ai x i
ai = #least-wt covers of size i
i
Projection
wt=4, s=6
wt=4, s=7 (2 ways)
wt=5 (not least-weight)
Thus wt(G) = 4 and s(G; x) = x6 + 2x7.
Slide 8
Fixed graphs in the constant-μ limit
Theorem [Lo]. Let n be fixed and let G be a fixed graph on n vertices. In the
constant-m limit,
s (G ;  )
Pr  (n, m, p )  G   wt( G ) 1  O m 1 
m
*
• Lower weight graphs are more likely
• If G has a lower weight supergraph H, G is more likely to appear as a
subgraph of H than as an induced graph
Theorem. Let n be fixed and let G be a fixed graph on n non-isolated vertices
with j cut-vertices. In the constant-m limit,
Pr  ( n, m, p )  G  
*

n j

m
j
  
i 1 bi
rank ( G )
1  O m 
1
where the bi are block degrees of the cut-vertices of G, and f b ( x) =
is the bth Touchard polynomial.
å
ïìï bïü
í ïý
i= 0 ïïîï i ïïþ
ï
b
Slide 9
xi
Example subgraph probability
Let H be
rank(H) = 7
n(H)=8
b1= 3
2 cut-vertices; 4 blocks
b2= 2
Block-cutpoint
tree of H
3 s 1
2
   1  3  

s 1  s 
3
Pr  (8, m, p)  H  
*
1 
Stirling numbers bi 
count partitions  s 
of bi blocks into s “movies”
 6 1  3   2 1   
m7
1  Om 
1
Slide 10
Block-cutpoint tree → Least-weight supergraphs
1. Select an unvisited
cut-vertex.
2. Partition incident
blocks, merge, and
make block-complete.
3. Update block-cutpoint
tree.
4. Repeat 1 until all
original cut-vertices
are visited.
Slide 11
An extremal graph weight conjecture
Conjecture [Lo]. Let G have n vertices. Then
wt (G )  n 2 4
with equality iff there exists a bipartition V(G)= A  B
such that:
•A= n 2
•B= n 2
•The complete (A,B)-bipartite graph is a subgraph of G
•Either A or B is an independent set.
Slide 12
Related simpler questions
Conjecture. Every K4-free graph G on n vertices and
2
n
 4  m edges has at least m edge-disjoint K3’s.
Theorem [Győri]. True for G with chromatic number at
most 3.
Theorem. True when G is K4-free and
2
n
 4  m  t (n,3)  k ,
where k≤n2/84+O(1).
Slide 13