1. Contagious sets in random graphs
Michael Krivelevich
Tel Aviv University
Joint work with: Uri Feige,
Daniel Reichman
Dedicated to Gerard Cohen
on the occasion of his 2 6 th birthday
Has probably nothing to do with his scientific (or other) activities…

2. Me and Gerard… wrote three (nice!) papers together;
didn’t learn much coding theory from the guy
(unfortunately…)
didn’t learn much French from the guy
spent quite some time in his company
in Paris and Tel Aviv
(fortunately…)
Has been big fun - so far…

3. Basic setting 𝑉 Initial seed
𝐺=(𝑉,𝐸) – graph, 𝑟≥1 –integer (=threshold) Iterative process: 𝐴 0 ⊆𝑉 – initial seed ∀𝑖≥1: 𝐴 𝑖 ≔ 𝐴 𝑖−1 ∪{𝑣∈𝑉∖ 𝐴 𝑖−1 :𝑑 𝑣, 𝐴 𝑖−1 ≥𝑟} [= adding vertices having ≥𝑟 neighbors in the current set] 𝐴 0 = 𝑖≥0 𝐴 𝑖 Def: 𝐴 0 is contagious if 𝐴 0 =𝑉 𝑉
𝐴 0
𝐴 1
𝐴 2
Initial seed

4. Extremal function, bootstrap percolation
Def.: 𝑚 𝐺,𝑟 =min{ 𝐴 0 : 𝐴 0 is contagious} - well defined: 𝑉 =𝑉 Alternative/more common terminology: - bootstrap percolation with threshold 𝑟 (more later)

5. Random vs Clever
Typical scenario for bootstrap percolation: 𝐴 0 ⊆𝑉 – a random 𝑚-subset of 𝑉 𝐺 Pr 𝐴 0 =𝑉 ? Typical size of 𝐴 0 ? min {𝑚: 𝐴 0 ⊆ 𝑅 𝑉⟹ typically 𝐴 0 =𝑉}? Here: different, allow to choose the initial seed optimally/in a sophisticated way - does make a difference (see later)

6. General bounds 𝑟=1 – not very interesting…
∀𝑣∈ 𝐴 0 – infects its connected component
⟹𝑚 𝐺,1 = # of connected components of 𝐺
Assume from now on: 𝑟≥2
mostly concentrate on: 𝑟=2
𝐴 2
𝐴 1 𝑣

7. General bounds (cont.)
Th: Reichman ′ 12 : 𝑚 𝐺,𝑟 ≤ 𝑣∈𝑉 min 1, 𝑟 𝑑 𝑣 +1 Conclusion: 𝐺- 𝑑-regular, 𝑛 vertices, 𝑟≤𝑑+1 ⟹m 𝐺,𝑟 ≤ 𝑟𝑛 𝑑+1 Tight: 𝐺:= disjoint cliques of size 𝑑+1 Need: 𝑟 vertices in every clique
𝐾 𝑑+1
𝑟 vertices

8. General bounds (cont.)
Proof: (similar to Caro-Wei’s proof of Turán’s Theorem) Notation: 𝑁 𝑣 =𝑣∪𝑁(𝑣) – closed neighborhood of 𝑣 Assume 𝑉 𝐺 ={1,…,𝑛} For a permutation 𝜎:𝑉→[𝑛], 𝑖≥1 define: 𝐿 𝑖 :={𝑣:𝑣 is #𝑖 in 𝑁 𝑣 under 𝜎} - 𝑖th layer under 𝜎

9. General bounds (cont.)
Claim: ∀ 𝜎, the set 𝐴 0 := 𝐴 0 𝜎 = 𝑖≤𝑟 𝐿 𝑖 is contagious. Proof: If not, let 𝑣 be the 1st (according to 𝜎) vertex not infected by 𝐴 0 . 𝑣∉ 𝐴 0 ⇒ 𝑣 has ≥𝑟 neighbors before it in 𝜎. Then 𝑣 gets infected – contradiction. ∎

10. General bounds (cont.)
Now: choose 𝜎 uniformly at random Recall: 𝐴 0 = 𝐴 0 𝜎 = 𝑖≤𝑟 𝐿 𝑖 ∀𝑣∈𝑉, ℙ 𝑣∈ 𝐴 0 = 𝑟 𝑑 𝑣 +1 , 𝑑 𝑣 ≥𝑟 1, otherwise ⇒ by linearity of expectation 𝔼 𝐴 0 = 𝑣∈𝑉 min 1, 𝑟 𝑑 𝑣 +1 ⇒ there exists a contagious set of at most this size. ∎

11. Can do much better for nice graphs…
From now on, fix 𝑟=2
(similar results for larger constant 𝑟)
∃ graphs 𝐺 with 𝑚 𝐺,2 =2
For “nice” graphs can expect better results:
Coja-Oghlan, Feige, K., Reichman (SODA’15):
𝐺 - 𝑑-regular on 𝑛 vertices
𝑔𝑖𝑟𝑡ℎ 𝐺 ≥2 ln ln 𝑑⟹ 𝑚 𝐺,2 =𝑂 𝑛 𝑑 2
no 𝐶 4 ⟹𝑚 𝐺,2 =𝑂 𝑛 𝑑 7/4
𝑔𝑖𝑟𝑡ℎ 𝐺 ≥7⟹ 𝑚 𝐺,2 =𝑂 𝑛 log 𝑑 𝑑 2
𝜆 𝐺 =𝑂( 𝑑 )⟹ 𝑂 𝑛 𝑑 2
no 𝐶 4 , 𝜆 𝐺 ≤ 1−𝜖 𝑑 ⟹𝑂 𝑛 log 𝑑 𝜖 2 𝑑 2
for “nice” graphs of degrees close to 𝑑
can expect 𝑚 𝐺,2
to be around 𝑂 𝑛 𝑑 2

12. Contagious sets in random graphs
Q.: 𝐺∼𝐺 𝑛,𝑝 , 𝑟=2
Typical value of 𝑚 𝐺,2 ?
Remarks:
Easy to see: 𝑝≫ 1 𝑛
⟹ whp two vertices have 𝜔(1) common neighb.+propagation
⟹ whp 𝑚 𝐺,2 =2
2. 𝐺 does not have to be connected ⟹𝑝≥ 𝐶 𝑛

13. Typical set in a random graph
Janson, Łuczak, Turova, Vallier’12:
𝐺∼𝐺 𝑛,𝑝
𝐴 0 = a fixed set of size 𝑚
Q.: How large should be 𝑚= 𝑚 2 (𝑛,𝑝) so that whp 𝐴 0 =[𝑛]?
typical scenario for bootstrap percolation
Assume: 1. 𝑝≪ 1 𝑛
2. 𝑝≥ ln 𝑛+ ln ln 𝑛+𝜔(1) 𝑛 (to ensure: whp 𝛿 𝐺 ≥2)

14. Typical set in a random graph (cont.)
JŁTV: solved the problem completely, found the threshold for having 𝐴 0 =[𝑛] Answer: 𝑎 𝑛 𝑑 2 𝑑≔𝑛𝑝 Intuition: 𝐺∼𝐺 𝑛,𝑝 𝐴 0 ⊆ 𝑛 , 𝐴 0 =𝑚 – seed Expose edges between 𝐴 0 and the rest 𝐵 1 ≔ 𝑣∉ 𝐴 0 :𝑑 𝑣, 𝐴 0 ≥2 – will be infected immediately 𝔼 𝐵 1 ≈𝑛 𝑚 2 𝑝 2 If 𝐵 1 ≫| 𝐴 0 |⟹ situation improves, snowballing… ⟹ enough to take 𝑚= 𝑐𝑛 𝑑 2

15. Our result
Th.: 𝐺∼𝐺 𝑛,𝑝 , 𝐶 𝑛 ≤𝑝 𝑛 ≤ 𝑐 log log 𝑛 1/2 𝑛 ⋅ log 𝑛 , 𝑑≔𝑛𝑝 ⟹ whp 𝑚 𝐺,2 =Θ 𝑛 𝑑 2 log 𝑑 cf.: [JŁTV] for a typical set need Θ 𝑛 𝑑 2 vertices in the seed ⟹ clever is (somewhat) better than random + similar improvement for a general 𝑟≥2.

16. Proof idea – lower bound
If 𝐴 0 ⊆[𝑛], 𝐴 0 =𝑚, satisfies: 𝐴 0 =[𝑛] Then: ∃ sequence of vertices 𝜎= 𝑣 1 ,…, 𝑣 𝑛−𝑚 , 𝑑 𝑣 𝑖 , 𝐴 0 ∪ 𝑣 1 ,…, 𝑣 𝑖−1 ≥2 (=protocol of infection spread) Then: for 𝑀≥𝑚, 𝐴 0 ∪{ 𝑣 1 ,…, 𝑣 𝑀−𝑚 } has ≥2 𝑀−𝑚 edges Find: 𝑀 for which whp in 𝐺∼𝐺 𝑛,𝑝 : no set 𝐴, 𝐴 =𝑀, 𝑒 𝐴 ≥2 𝑀−𝑚 (simple union bound)

17. Proof idea – upper bound
𝐶 0 = 𝜖 𝑑 𝑛 𝑑 2 - initial seed (part) Expose edges between 𝐶 0 and 𝑉∖ 𝐶 0 𝐵 ≔𝑁 𝐶 0 , expose 𝐺[𝐵] 𝐶 1 ≔ “large” components of 𝐺[𝐵] (size ≥ 𝑠 1 ) Observe: adding to 𝐶 0 one vertex per each component of 𝐶 1 infects all of 𝐶 1 ⟹≤ | 𝐶 1 | 𝑠 1 additional vertices to be added to the initial seed Now: all of 𝐶 1 is infected
𝐶 1
𝐶 0 𝐵

18. Proof idea – upper bound (cont.)
Now: Iterate - till reach 𝐶𝑛 𝑑 2 infected vertices ⟹apply [JŁTV] to infect most of the graph Collect few uninfected vertices (if any) (recall – possibly 𝐺 is not connected/𝛿 𝐺 <2). ∎

19. Infecting everybody with minimal resources
Q.: For which value of 𝑝(𝑛) have typically: 𝑚 𝐺,2 =2, 𝐺∼𝐺(𝑛,𝑝) ? Th.: The threshold probability 𝑝(𝑛) in 𝐺(𝑛,𝑝) for 𝑚 𝐺,2 =2 is: 𝑝 𝑛 =Θ 1 (𝑛 log 𝑛) 1/2 . + similar result for a general 𝑟≥2.

20. Open questions Sharper estimates for 𝑚 𝐺,𝑟 ?
infection time min {𝑡: 𝐴 𝑡 =𝑉} ?
Results for other models of random graphs?

21. Happy birthday Gerard! Time to start thinking of International
Draughts…
(And then off to Go…)