Date post: | 15-Dec-2015 |
Category: |
Documents |
Upload: | tierra-leeper |
View: | 213 times |
Download: | 0 times |
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 MeetingNovember 6, 2010
Slide 2
Random Intersection Model B*(n,m,p)• Introduced: Karoński,
Scheinerman, Singer-Cohen `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”
n: number of “actors”
Unipartite projection
Collaboration graph: who’s worked with whom
Bipartite graph
Slide 3
[ ]*
1
( ( , )
( ( , , )
1 (1 )
H ER
H
mER
X N p
X N M p
p p
G
B
E
E
for
£
é ùê úë û
= - -
Subgraph H E[XH]
Erdős–Rényi RandomIntersection
60-cycle
Expected sugbraph count vs. E-R Erdős–Rényi
n = 1000 pER = 0.002
1.3
11
´ 152 10
1.2
-´ 63 10
1500
´ 54 10
´ 102 10
1700
´ 382 10
Random Intersection n = 1000 m = 100 p = 0.0045
Yields pER = 0.002
Slide 4
RC model (m = 10,000)RC model (m = 1000)RC model (m = 100)RC model (m = 10)RC model (m = 2)RC model (m = 1)
Erdős–Rényi vs. RI Model as m → ∞
Erdős–Rényi G(n,pER) model
• m = “number of movies”• pER = 0.028 (edge probability)
RI
B*(N,M,p)
Slide 5
• Theorem [Ferry, Mifflin]. For a fixed expected number of edges pER , 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:
• [(Independently) Fill, Scheinerman, Singer-Cohen `00] With m=nα, α>6, total variation distance for probability of G goes to zero as n → ∞.
( )( )
( ) ( ) ( ) ( )3 2
1/ 2 13 3
Pr 1 1 11 2 2 log
3Pr 1RC
ER ER ER ER
nGn e G c G m O m
G p p p- -
æ öæ öæö ÷ ÷ç ÷ çç ÷ ÷= + ÷- - + +ç çç ÷ ÷÷ç çç ÷ç ÷÷çç -è ø è øè ø
Erdős–Rényi vs. RI Model as m → ∞
2
n
RI
Slide 6
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
RI model in the constant-μ limit
Slide 7
Edge clique covers• Unipartite projection
corresponds to an edge clique cover
• The projection-induced cover encodes collaboration structure
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
Unipartite projection
B
*B
*G B
likeliest B
“size” of clique cover S “weight” of S, G
ai = #least-wt covers of size i
Covering polynomial of G
X
Xs )(
)()(wt s )(wtmin)(wt
G
i
ii xaxGs );(
wt=4, s=6
Projection
wt=4, s=7 (2 ways) wt=5 (not least-weight)
Thus wt(G) = 4 and s(G; x) = x6 + 2x7. Slide 8
Slide 9
Theorem [Lo]. Let n be fixed and let G be a fixed graph on n vertices. In the constant-m limit,
• 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,
where the bi are block degrees of the cut-vertices of G, and is the bth Touchard polynomial.
0( )
b ibb i i
x xfì üï ïï ïï ïí ýï ï= ï ïï ïî þ
=å
Fixed graphs in the constant-μ limit
1)(wt
* 1);(
),,(Pr mOm
GsGpmn
G
1)(rank
1* 1),,(Pr
mO
mGpmn
G
j
i bjn
i
Slide 10
Example subgraph probabilityLet H be
rank(H) = 7 n(H)=8 2 cut-vertices; 4 blocks
31 2
1
31 3s
s s
b1= 3 b2= 2
1
Stirling numbers count partitionsof bi blocks into s “movies”
ib
s
Block-cutpointtree of H
17
26* 1
131),,8(Pr
mO
mHpm
Slide 11
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 12
An extremal graph weight conjecture
Conjecture [Lo]. Let G have n vertices. Then
with equality iff there exists a bipartition V(G)= such that:• A=• B=• The complete (A,B)-bipartite graph is a subgraph of G• Either A or B is an independent set.
4)(wt 2nG
BA
2n
2n
Slide 13
Related simpler questions
Conjecture. Every K4-free graph G on n vertices and
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
where k≤n2/84+O(1).
mn 42
,)3,(42 kntmn