Small Subgraphs in Random Graphs and the Power of Multiple Choices
The Online Case
Torsten Mütze, ETH ZürichJoint work with Reto Spöhel and Henning Thomas
Introduction
• Goal: Avoid creating a copy of some fixed graph F
• Achlioptas process (named after Dimitris Achlioptas):
• start with the empty graph on n vertices
• in each step r edges are chosen uniformly at random (among all edges never seen before)
• select one of the r edges that is inserted into the graph, the remaining r-1 edges are discarded
How long can theappearance of Fbe avoided havingthis freedom of
choice?
F = , r=2
Introduction• N0=N0(F, r, n) is a threshold:
N0=N0(F, r, n)
N = N(n) = numberof steps in the
Achlioptas process
There is a strategy that avoids
creating a copy of F withprobability 1-o(1)(as n tends to infinity)
N /N0
If F is a cycle, a clique or a complete bipartite graph with parts of equal
size, an explicit threshold function is known. (Krivelevich, Loh, Sudakov, 2007)
Every strategy will be forced
to create a copy of F withprobability 1-o(1)
N [N0
n1.2
F = , r=2
n1.286
…r=3n1.333
…r=4
n1
r=1
Our Result
• Theorem:Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is
where
• Krivelevich et al. conjecture a general threshold formula
• We disprove this conjecture and solve the problem in full generality
... we will develop an intuition for the
threshold formula in the following
r-matched graphr=2
Graph
r-matched Graphs
Random graph Random r-matched graph- generate Gn,m
- randomly partition the m edges into sets of size r
Achlioptas process after N stepsis distributed as
Threshold for the appearanceof small subgraphs in Gn,m (Bollobás, 1981):
m=m(n)
The Gluing IntuitionAnalogue of Bollobás‘ theoremfor r-matched graphs:
m=m(n)
Our key idea:relate Achlioptas processto ‘static’ objectF = , r=2
Greedy strategy:
e/v = 5/4
As long asthis subgraph does not appear,hence we do not lose.
“GluingIntuition”
J
1st Observation: Subgraph Sequences
F = , r=2
e/v = 11/8 = 1.375
Greedy strategy:
0 Maximization over a sequence of subgraphs of F
e/v = 15/10 = 1.5
Optimal strategy:
As long as this subgraph does not
appear, hence we do not lose.
2nd Observation: Edge OrderingsOrdered graph: pair
oldest edge
youngest edge2
3
7
4
6
1
5
0 Minimization over all possible edge orderings of F
1
2
2
e/v = 19/14 = 1.357...
2
F = , r=2
Optimal Strategy for ¼1:
Edge ordering ¼1:
1
12
2
2
12
e/v = 17/12 = 1.417...
Edge ordering ¼2:
12
12
2Optimal Strategy for ¼2:
F3-
F3-
F3-
F3-
F3-
F3-
F3-
F3-
4
4
4
4
F2-
F2-
F2-
F2-
3
4F1-3 F1-
r-1
4
Calculating the ThresholdMinimize over all possible edge orderings ¼ of FMaximize e(J)/v(J) over all subgraphs
J4F ¼
r2
34
F2
34
F 1
F ¼
Maximization over a sequence of subgraphs of F
5
J
H1
H2
H3
H1
H2H3
H3
H3
Calculating the Threshold (Example)
7
F6-…
F ¼
1
3 4
6
2
5
34
65
2
77
F = , r=2
Minimize over all possible edge orderings ¼ of FMaximize e(J)/v(J) over all subgraphs
J4F ¼5
3
7
4
6 5
3
7
4
65
7
7 7
7J
e(J)/v(J) = 19/14
Maximization over a sequence of subgraphs of F
1
3
7
4
6
2
5
F
(F, ¼) 23
7
4
6 5
F1-
3
7
4
6 5
F2-
Maximization over a sequence of subgraphs of F
Our Result ( explained)
• Theorem:Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is
where
Minimization over all possible edge orderings of FMaximize e(J)/v(J) over all subgraphs
J4F ¼
The F-Avoidance-Strategy (1)Measure the harmlessness of asubgraph by the parameter
21
harm
les
sdangero
us
F = , r=2
The F-Avoidance-Strategy (2)
• For each edge calculate the level of danger it entails as the most dangerous (ordered) subgraph this edge would close
• Among all edges, pick the least dangerous one
f1 f2 fr…
is more dangerous than
5
• Our strategy considers ordered subgraphs
Lower Bound ProofLemma: By our strategy, each black copy of some ordered graph is contained in a copy of some rare grey-black r-matched graph H’.
1
2
H’“History graph”
F does not appear
• Constantly many histories H’of ending up with a copy of F
• Below the threshold a.a.s. noneof the histories H’ appears in
Technical work!
F = r=2
21
12
harmless
dangerous
This might be the same edge
“Bastard”
rare: expectation at mostimplies that H’ does not appear in with probability 1-o(1)
• Goal: force a copy of F, in some fixed order regardless of the strategy
• Multiround approach (#rounds = #edges of F)
• In each round: count how many copies evolvefurther, 1st+2nd MM, small subgraphvariance calculation for r-matched graphs
• Optimize for the best upper bound
Upper Bound Proof
F = , r=2
1st round
2nd round
3rd round
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