Distance Evolutions in
Preferential Attachment Models
Joost Jorritsma
joint with Júlia Komjáthy
Networks Training Week
April 2021
arXiv: tiny.cc/DistanceEvolutionPAM
Guiding 'real-world' example
Internet network: communicating routers and servers
~1969: 2 connected sites (UCLA, SRI)
Time
~1971: 15 connected sites
~1989: 0.5 million users
~2020: billions of connected devices
- Evolving/dynamic network:
- Vertices arrive over time.
- Connect to present vertices.
- (Nodes may be removed or replaced.)
- (Edges between 'old' vertices are removed or placed.)
- [Faloutsos, Faloutsos & Faloutsos, '99]:
- # connections per router decays as power-law: pk∼k−τ,τ>2.
- Short average hopcount between routers.
Guiding 'real-world' example
Internet network: communicating routers and servers
- Evolving network: Vertices arrive over time, connect to present vertices.
- [Faloutsos, Faloutsos & Faloutsos, '99]: Short average hopcount.
1999
dist′99(u′99,v′99)=4
2005
dist′05(u′99,v′99)=3
2021
dist′21(u′99,v′99)=2
- How does hopcount between old nodes change in growing network?
u′99
v′99
Preferential attachment models
Definition
- Start with a single vertex
- Vertices enter the network one-by-one at discrete (time-)steps t=2,3,....
- New vertex connects to old vertices according to some increasing function of the degree:
- Fixed outdegree: (m,δ):m edges/new vertex (δ>−m):
P(vt+1⟶jvi)∝degt,j(vi)+δ/m. - Variable outdegree: ∀vi∈PAt, independently, (γ,η∈(0,1))
P(vt+1⟶vi)=(γdegt(vi)+η)/t.
- Fixed outdegree: (m,δ):m edges/new vertex (δ>−m):
- Animation
Thm. [Bollobás, Riordan, Spencer, Tusnády '01; Dereich, Mörters '09]
Limiting degree distribution decays as power-law: for τm,δ=3+δ/m; τγ=1+1/γ,
pk∼k−τ.
Distances on PAMs
Thm. [Dereich, Mönch, Mörters '12].
For τ∈(2,3), Ut,Vt∼Unif(Largest cluster(t)):
distt(Ut,Vt)≍ 4∣log(τ−2)∣loglog(t)
Thm. [Dommers, v/d Hofstad, Hooghiemstra '10].
For τ>3, ∃c1,c2>0:
c1≼log(t)distt(Ut,Vt)≼c2.
Thm. [Bollobás, Riordan '04; Dereich, Mönch, Mörters '17].
For τ=3:
distt(Ut,Vt)≍loglog(t)log(t).
Thm. [Dereich, Mönch, Mörters '12; J., Komjáthy '20].
For τ∈(2,3), Ut,Vt∼Unif(Largest cluster(t)):
(distt(Ut,Vt)−4∣log(τ−2)∣loglog(t))t≥0
is a tight sequence.
Literature perspective on PAMs
-
Many static/limiting properties are well-understood
- Degree distribution.
- Typical graph distance.
- Diameter of the graph.
- Local neighborhood of a typical vertex (local weak sense).
- Component sizes.
- ...
- ...
-
Studying static properties allows for comparison to static (non-growing) models (configuration model, Norros-Reittu, ...).
-
Theorem statements do not display dynamics inherently present in PAMs (contrary to proofs).
- Exception: degree evolution (degt′(v))t′≥v of (sets of) fixed vertices.
Goal: try to understand how the graph evolves for t′≥t from perspective of graph at time t
Main Question
Consider ((distt′(Ut,Vt))t′≥t)t≥0 and define for a function f(t,t′)
Xtf:=t′≥tsup∣distt′(Ut,Vt)−f(t,t′)∣.
Q1. For τ∈(2,3), can we identify fτ(t,t′) s.t.
(Xtfτ)t≥0 is a tight sequence?
distt0(Ut0,Vt0)=7,
distt1(Ut0,Vt0)=6,
0
t2
distt2(Ut0,Vt0)=2
Ut0
Vt0
t1
t0
Q2. For τ∈(2,3), identify fτ(t,t′) s.t.
(t′≥tsup∣distt′(Ut,Vt)−fτ(t,t′)∣)t≥0 is a tight sequence.
Main Theorem
Thm. [J., Komjáthy '20+].
fτ(t,t′)=4∣log(τ−2)∣loglog(t)−log(1∨log(t′/t))∨2.
Corollary. Hydrodynamic limit [J., Komjáthy '20+].
Let t′=Tt(a):=texp(loga(t)) for a∈[0,1], then
fτ(t,Tt(a))=4(1−a)∣log(τ−2)∣loglog(t)∨2, and
a∈[0,1]suploglog(t)distTt(a)(Ut,Vt)−(1−a)∣log(τ−2)∣4⟶P0.
Main Theorem
Corollary. Hydrodynamic limit [J., Komjáthy '20+].
Let t′=Tt(a):=texp(loga(t)) for a∈[0,1], then
a∈[0,1]suploglog(t)distTt(a)(Ut,Vt)−(1−a)∣log(τ−2)∣4⟶P0.
Heuristic upper bound
Statement. [J., Komjáthy '20+].
For τ∈(2,3),
P(∀t′≥t:distt′(Ut,Vt)≤fτ(t,t′)+1/ε)≥1−ε.
Weaker statement. [J., Komjáthy '20+].
For t large, there ∃ nice t′↦εt′
P(distt′(Ut,Vt)≤fτ(t,t′)+1/ε)≥1−εt′.
"Proof" of statement by smart union bound.
- Consider (ti(t))i≥0 at integer crossings rhs (i.e., event changes).
- Distance is non-increasing.
- By weak version, the statement follows if for t large ∑iεti(t)≤ε.
- Weak statement follows from minor adaptations of [Dommers, v/d Hofstad, Hooghiemstra '10; Dereich, Mönch, Mörters '12; Caravenna, Garavaglia, v/d Hofstad '19; J., Komjáthy '20].
Unfortunately ∫t∞εt′dt′→∞.
Naive idea: Union bound.
Heuristic upper bound
Weaker statement. [J., Komjáthy '20+].
For t large, there ∃ nice t′↦εt′
P(distt′(Ut,Vt)≤fτ(t,t′)+1/ε)≥1−εt′.
Step 0, t′=t:
(t)
(t′)
(t′)
(t′)
(t′)
(t′)
(t′)
bounded-length segments
Heuristic upper bound
Weaker statement. [J., Komjáthy '20+].
For t large, there ∃ nice t′↦εt′
P(distt′(Ut,Vt)≤fτ(t,t′)+1/ε)≥1−εt′.
Step 0, t′=t:
degt(⋅)
Core(t)
- Core(t′):={v≤t′:deg(1−ε)t′(v)≥t′/log(t′)}.
- If deg(1−ε)t′(x)=s:∃y:deg(1−ε)t′(y)≈s1/(τ−2),{x↔2y}t′.
- # of iterations needed to reach core: smallest kt s.t.
(degt′(qt0))1/(τ−2)k≥t/log(t). - kt≈loglog(t)/∣log(τ−2)∣=41fτ(t,t) iterations for Ut,Vt.
Heuristic upper bound
Weaker statement. [J., Komjáthy '20+].
For t large, there ∃ nice t′↦εt′
P(distt′(Ut,Vt)≤fτ(t,t′)+1/ε)≥1−εt′.
Step 1, t′>t:
degt(⋅)
Core(t)
degt′(⋅)
Core(t′)
- Core(t′):={v≤t′:deg(1−ε)t′(v)≥t′/log(t′)}.
- If deg(1−ε)t′(x)=s:∃y:deg(1−ε)t′(y)≈s1/(τ−2),{x↔2y}t′.
- # of iterations needed: smallest kt′ s.t.
(degt′(qt0))1/(τ−2)k≥t′/log(t′). - Competing effects:
(1) Core threshold increases;
(2) degree increases. -
Strong control of degt′(qt,0):
Using Móri-martingale, Doob's maximal inequality:
degt′(qt,0)≳(t′/t)1/(τ−1) for all t′>t . - kt′≤41fτ(t,t′).
- If t′=Tt(a), # iterations is linear in a
Heuristic lower bound
Statement. [J., Komjáthy '20+]. For t sufficiently large
P(∃t′≥t:distt′(Ut,Vt)≤fτ(t,t′)−1/ε)≤ε.
2. Bound rhs using 'old' methods [Dereich, Mönch, Mörters '12].
Naive guess (similar to upper bound):
1. Find sequence (ti)i≥0 at which event changes. Then
P(∃t′≥t:distt′(Ut,Vt)≤fτ(t,t′)−1/ε)≤i∑P(distti(Ut,Vt)≤fτ(t,ti)−1/ε).
RHS not summable, new machinery needed
Heuristic lower bound
Statement. [J., Komjáthy '20+]. For t sufficiently large
P(∃t′≥t:distt′(Ut,Vt)≤fτ(t,t′)−1/ε)≤ε.
Main idea: Exploit time-dependencies in the growing graph
There is a first time of failure.
Heuristic lower bound
Statement. [J., Komjáthy '20+]. For t sufficiently large
P(∃t′≥t:distt′(Ut,Vt)≤fτ(t,t′)−1/ε)≤ε.
*In fact, we need a more advanced separation of events (using good and bad paths, inspired by [Dereich, Mönch, Mörters '12]) that makes the decomposition more interlinked.
≤∗C/(t′log3(t′))
Let* E(t,t′):={distt′(Ut,Vt)≥fτ(t,t′)−1/ε}. Recall fτ(t,t′) is non-increasing. Then
P(¬E(t,t′)∣⋂t~=tt′−1E(t,t~))≤P(there is a short path that traverses t′)
Main idea: Exploit time-dependencies in the growing graph
There is a first time of failure.
P(∃t′≥t:distt′(Ut,Vt)≤fτ(t,t′)−1/ε)=P(¬E(t,t))+t′=t+1∑∞P(¬E(t,t′)∣t~=t⋂t′−1E(t,t~))
P(∃t′≥t:distt′(Ut,Vt)≤fτ(t,t′)−1/ε)≤ε+t′=t+1∑∞C/(t′log3(t′))⟶t→∞ε
⋮
lots of computations, inductive proves, ...
Outlook and Invitation
What about (other) properties in (other/spatial) versions of the model?
Thank you!
Thank you!
Weighted version. [J., Komjáthy '20+].
Let τ∈(2,3). Equip every edge with i.i.d. Le≥0, Ut,Vt∼Unif(Largest cluster(t)). If If FL(−1) is not too flat, then we identify fτ(L)(t,t′) s.t.
(t′≥tsup∣dt′(L)(Ut,Vt)−fτ(L)(t,t′)∣)t≥0
forms a tight sequence of random variables
Def. Weighted distance: dt′(L)(x,y):=minπ from x to y at time t′∑e∈πLe
arXiv: tiny.cc/DistanceEvolutionPAM
Outlook and Invitation
What about (other) properties in (other/spatial) versions of the model?
Thank you!
Weighted version. [J., Komjáthy '20+].
Let τ∈(2,3). Equip every edge with i.i.d. Le≥0, Ut,Vt∼Unif(Largest cluster(t)). If If FL(−1) is not too flat, then we identify fτ(L)(t,t′) s.t.
(t′≥tsup∣dt′(L)(Ut,Vt)−fτ(L)(t,t′)∣)t≥0
forms a tight sequence of random variables
Def. Weighted distance: dt′(L)(x,y):=minπ from x to y at time t′∑e∈πLe
Copy of Distance evolutions
By joostjor
Copy of Distance evolutions
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