Distance Evolutions in

Preferential Attachment Models

Joost Jorritsma

joint with Júlia Komjáthy

Franco-Dutch Workshop

July 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: $p_k\sim k^{-\tau}, \tau>2$.
- 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

$\mathrm{dist}_{\color{red}{'99}}(u_{'99}, v_{'99}) = 4$

2005

$\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3$

2021

$\mathrm{dist}_{{\color{red}'21}}(u_{'99}, v_{'99}) = 2$

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: $(m,\delta): m$ edges/new vertex ($\delta>-m$):
  $$\mathbb{P}\big(v_{t+1} \overset{j}{\longrightarrow} v_i\big) \propto \mathrm{deg}_{t,j}(v_i) + \delta/m.$$
- Variable outdegree: $\forall v_i \in \mathrm{PA}_t$, independently, ($\gamma,\eta\in(0,1)$)
  $$\mathbb{P}\big(v_{t+1} \longrightarrow v_i\big) = (\gamma \mathrm{deg}_{t}(v_i)+\eta)/t.$$
Animation

Thm. [Bollobás, Riordan, Spencer, Tusnády '01; Dereich, Mörters '09]

Limiting degree distribution decays as power-law: for $\tau_{m,\delta}=3+\delta/m; \tau_\gamma=1+1/\gamma$,

$$p_k \sim k^{-\tau}.$$

Distances on PAMs

Thm. [Dereich, Mönch, Mörters '12].

For $\tau\in(2,3)$, $U_t, V_t\sim \text{Unif(Largest cluster}(t))$:

$$\mathrm{dist}_{{\color{red}t}}(U_{\color{blue}t}, V_{\color{blue}t}) \asymp 4\frac{\log\log(t)}{|\log(\tau-2)|} $$

Thm. [Dommers, v/d Hofstad, Hooghiemstra '10].

For $\tau>3$, $\exists c_1, c_2>0$:

$$c_1\preccurlyeq\frac{\mathrm{dist}_{{\color{red}t}}(U_{\color{blue}t}, V_{\color{blue}t})}{\log(t)} \preccurlyeq c_2.$$

Thm. [Bollobás, Riordan '04; Dereich, Mönch, Mörters '17].

For $\tau=3$:

$$\mathrm{dist}_{{\color{red}t}}(U_{\color{blue}t}, V_{\color{blue}t})\asymp \frac{\log(t)}{\log\log(t)}.$$

Thm. [Dereich, Mönch, Mörters '12; J., Komjáthy '20].

For $\tau\in(2,3)$, $U_t, V_t\sim \text{Unif(Largest cluster}(t))$:

$$\bigg(\left|\mathrm{dist}_{{\color{red}t}}(U_{\color{blue}t}, V_{\color{blue}t}) - 4\frac{\log\log(t)}{|\log(\tau-2)|}\right|\bigg)_{t\ge 0} $$

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 $(\mathrm{deg}_{t'}(v))_{t'\ge v}$ of (sets of) fixed vertices.

Goal: try to understand how the graph evolves for $t'\ge t$ from perspective of graph at time $t$

Main Question

Consider $\Big(\big(\mathrm{dist}_{{\color{red}t'}}(U_{\color{blue} t}, V_{\color{blue} t})\big)_{{\color{red}t'}\ge {\color{blue}t}}\Big)_{t\ge 0}$ and define for a function $f(t,t')$

$$X_t^f := \sup_{t'\ge t}\left| \mathrm{dist}_{{\color{red}t'}}(U_{\color{blue}t}, V_{\color{blue}t}) - f(t, t')\right|.$$

Q1. For $\tau\in(2,3)$, can we identify $f_\tau(t, t')$ s.t.

$(X_t^{f_\tau})_{t\ge 0}$ is a tight sequence?

$\mathrm{dist}_{t_0}(U_{t_0}, V_{t_0}) = 7,$

$\mathrm{dist}_{t_1}(U_{t_0}, V_{t_0}) = 6,$

$0$

$t_2$

$\mathrm{dist}_{t_2}(U_{t_0}, V_{t_0}) = 2$

$U_{t_0}$

$V_{t_0}$

$t_1$

$t_0$

Q2. For $\tau\in(2,3)$, identify $f_\tau(t, t')$ s.t.

$$\Big(\sup_{t'\ge t}\left| \mathrm{dist}_{{\color{red}t'}}(U_{\color{blue}t}, V_{\color{blue}t}) - f_\tau(t, t')\right|\Big)_{t\ge 0}$$ is a tight sequence.

Main Theorem

Thm. [J., Komjáthy '20+].

$$f_\tau(t,t') = 4\frac{\log\log(t) - \log(1\vee\log(t'/t))}{|\log(\tau-2)|}\vee 2.$$

Corollary. Hydrodynamic limit [J., Komjáthy '20+].
For $\varepsilon>0$, let $t'=T_t(a):=t\exp\big(\varepsilon\log^a(t)\big)$ for $a\in[0,1]$, then

$$f_\tau(t,T_t(a)) \approx 4(1-a)\frac{\log\log(t)}{|\log(\tau-2)|}\vee 2,$$ and

$$ \sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\overset{\mathbb{P}}{\longrightarrow} 0.$$

Main Theorem

Corollary. Hydrodynamic limit [J., Komjáthy '20+].
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$, then

$$ \sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\overset{\mathbb{P}}{\longrightarrow} 0.$$

Heuristic upper bound

Statement. [J., Komjáthy '20+].

For $\tau\in(2,3)$,
$$ \mathbb{P}\Big(\forall t'\ge t: \mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')+1/\varepsilon\Big) \ge 1-\varepsilon. $$

Weaker statement. [J., Komjáthy '20+].
For $t$ large, there $\exist$ nice $t'\mapsto\varepsilon_{t'}$

$$ \mathbb{P}\Big(\mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')+1/\varepsilon\Big) \ge 1-\varepsilon_{t'}. $$

"Proof" of statement by smart union bound.

Distance is non-increasing sequence of integers.
Consider $(t_i(t))_{i\ge 0}$ at integer crossings rhs (i.e., event changes).
By weak version, the statement follows if for $t$ large $\sum_i \varepsilon_{t_i(t)} \leq \varepsilon$.
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 $\int_t^\infty \varepsilon_{t'} \mathrm{d}t'\to \infty.$

Naive idea: Union bound.

Heuristic upper bound

Weaker statement. [J., Komjáthy '20+].
For $t$ large, there $\exist$ nice $t'\mapsto\varepsilon_{t'}$

$$ \mathbb{P}\Big(\mathrm{dist}_{t'}(U_t, V_t)\le f_{\tau}(t,t') + 1/\varepsilon\Big) \ge 1-\varepsilon_{t'}. $$

Step 0, $t'=t$:

$^{(t)}$

$^{(t')}$

bounded-length segments

Heuristic upper bound

Weaker statement. [J., Komjáthy '20+].
For $t$ large, there $\exist$ nice $t'\mapsto\varepsilon_{t'}$

$$ \mathbb{P}\Big(\mathrm{dist}_{t'}(U_t, V_t)\le f_{\tau}(t,t') + 1/\varepsilon\Big) \ge 1-\varepsilon_{t'}. $$

Step 0, $t'=t$:

$\mathrm{deg}_t(\cdot)$

$\mathrm{Core}(t)$

$\text{Core}(t'):=\{v\le t': \text{deg}_{(1-\varepsilon)t'}(v)\ge \sqrt{t'/\log(t')}\}$.
If $\mathrm{deg}_{(1-\varepsilon)t'}(x)=s: \exists y: \mathrm{deg}_{(1-\varepsilon)t'}(y)\approx s^{1/(\tau-2)}, \{x\overset{2}{\leftrightarrow} y\}_{t'}$.
# of iterations needed to reach core: smallest $k_t$ s.t.
$$\big(\mathrm{deg}_{t'}(q_{t_0})\big)^{1/(\tau-2)^k} \ge \sqrt{t/\log(t)}.$$
$k_t\approx\log\log(t)/|\log(\tau-2)|=\frac{1}{4}f_\tau(t,t)$ iterations for $U_t, V_t$.

Heuristic upper bound

Weaker statement. [J., Komjáthy '20+].
For $t$ large, there $\exist$ nice $t'\mapsto\varepsilon_{t'}$

$$ \mathbb{P}\Big(\mathrm{dist}_{t'}(U_t, V_t)\le f_{\tau}(t,t') + 1/\varepsilon\Big) \ge 1-\varepsilon_{t'}. $$

Step 1, $t'>t$:

$\mathrm{deg}_t(\cdot)$

$\mathrm{Core}(t)$

$\mathrm{deg}_{t'}(\cdot)$

$\mathrm{Core}(t')$

$\text{Core}(t'):=\{v\le t': \text{deg}_{(1-\varepsilon)t'}(v)\ge \sqrt{t'/\log(t')}\}$.
If $\mathrm{deg}_{(1-\varepsilon)t'}(x)=s: \exists y: \mathrm{deg}_{(1-\varepsilon)t'}(y)\approx s^{1/(\tau-2)}, \{x\overset{2}{\leftrightarrow} y\}_{t'}$.
# of iterations needed: smallest $k_{t'}$ s.t.
$$\big(\mathrm{deg}_{t'}(q_{t_0})\big)^{1/(\tau-2)^k} \ge \sqrt{t'/\log(t')}.$$
Competing effects:
(1) Core threshold increases;
(2) degree increases.
Strong control of $\text{deg}_{t'}(q_{t,0})$:
Using Móri-martingale, Doob's maximal inequality:
$\text{deg}_{t'}(q_{t,0})\gtrsim (t'/t)^{1/(\tau-1)}$ for all $t'>t$ .
$k_{t'}\le \frac{1}{4}f_\tau(t,t')$.
If $t'=T_t(a)$, # iterations is linear in $a$

Heuristic lower bound

Statement. [J., Komjáthy '20+]. For $t$ sufficiently large
$$ \mathbb{P}\Big(\exist t'\ge t: \mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')-1/\varepsilon\Big) \le \varepsilon. $$

2. Bound rhs using 'old' methods [Dereich, Mönch, Mörters '12].

Naive guess (similar to upper bound):

1. Find sequence $(t_i)_{i\ge 0}$ at which event changes. Then

$$ \mathbb{P}\Big(\exist t'\ge t: \mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')-1/\varepsilon\Big) \leq \sum_i \mathbb{P}\Big(\mathrm{dist}_{t_i}(U_t, V_t)\le f_\tau(t,t_i)-1/\varepsilon\Big).$$

RHS not summable, new machinery needed

Heuristic lower bound

Statement. [J., Komjáthy '20+]. For $t$ sufficiently large
$$ \mathbb{P}\Big(\exist t'\ge t: \mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')-1/\varepsilon\Big) \le \varepsilon. $$

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
$$ \mathbb{P}\Big(\exist t'\ge t: \mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')-1/\varepsilon\Big) \le \varepsilon. $$

*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.

$\leq^\ast C/(t'\log^3(t'))$

Let* $\mathcal{E}(t,t'):= \{\mathrm{dist}_{t'}(U_t, V_t)\ge f_\tau(t,t')-1/\varepsilon\}$. Recall $f_\tau(t,t')$ is non-increasing. Then

$\mathbb{P}\Big(\neg\mathcal{E}(t,t') \mid \bigcap_{\tilde{t}=t}^{t'-1} \mathcal{E}(t,\tilde{t}) \Big)\le\mathbb{P}\big($there is a short path that traverses $t'\big)$

Main idea: Exploit time-dependencies in the growing graph

There is a first time of failure.

$$\mathbb{P}\Big(\exist t'\ge t: \mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')-1/\varepsilon\Big) = \mathbb{P}\big(\neg \mathcal{E}(t,t)\big) + \sum_{t'=t+1}^\infty\mathbb{P}\bigg(\neg\mathcal{E}(t,t') \mid \bigcap_{\tilde{t}=t}^{t'-1} \mathcal{E}(t,\tilde{t}) \bigg)$$

$$\mathbb{P}\Big(\exist t'\ge t: \mathrm{dist}_{t'}(U_t, V_t)\le f_\tau(t,t')-1/\varepsilon\Big) \leq \varepsilon + \sum_{t'=t+1}^\infty C/(t'\log^3(t')) \overset{t\to\infty}\longrightarrow \varepsilon$$

$\vdots$

Many computations,

inductive proofs, ...

Main Theorem (extended)

Weighted-distance evolution. [J., Komjáthy '20+].
Let $\tau\in(2,3)$. Equip every edge with i.i.d. $L_e\ge 0$, $U_t, V_t\sim \text{Unif(Largest cluster}(t))$. If If $F_L^{(-1)}$ is not too flat, then we identify $f_{\tau}^{(L)}(t,t')$ s.t.

$$\Big(\sup_{t'\ge t}|d^{(L)}_{t'}(U_t, V_t) - f_{\tau}^{(L)}(t,t')|\Big)_{t\geq 0}$$

forms a tight sequence of random variables

Def. Weighted distance: $d_{t'}^{(L)}(x, y):= \min_{\pi \text{ from }x\text{ to }y \text{ at time }t'} \sum_{e\in\pi} L_e$

Weighted setting. Choose your favourite non-negative random variable $L_e$ and equip every edge upon creation with a non-negative copy of $L_e$.

arXiv: tiny.cc/DistanceEvolutionPAM

Comment. Trivial statement for most distributions with support starting at the origin: Already $d_t^{(L)}(U_t,V_t)$ is of constant order. [J., Komjáthy '20]

Distance Evolutions in

Preferential Attachment Models

Guiding 'real-world' example

Guiding 'real-world' example

Preferential attachment models

Distances on PAMs

Literature perspective on PAMs

Main Question

Main Theorem

Main Theorem

Heuristic upper bound

Heuristic upper bound

Heuristic upper bound

Heuristic upper bound

Heuristic lower bound

Heuristic lower bound

Heuristic lower bound

Main Theorem (extended)

Outlook and Invitation

Thank you!

Distance evolutions (Franco-Dutch)

Distance evolutions (Franco-Dutch)

joostjor

Distance Evolutions in

Preferential Attachment Models

Guiding 'real-world' example

Guiding 'real-world' example

Preferential attachment models

Distances on PAMs

Literature perspective on PAMs

Main Question

Main Theorem

Main Theorem

Heuristic upper bound

Heuristic upper bound

Heuristic upper bound

Heuristic upper bound

Heuristic lower bound

Heuristic lower bound

Heuristic lower bound

Main Theorem (extended)

Outlook and Invitation

Thank you!

Distance evolutions (Franco-Dutch)

More from joostjor