Distance Evolutions in

Preferential Attachment Models

Joost Jorritsma

joint with Júlia Komjáthy

Oberwolfach Workshop

January 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+].
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$, then

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

Consider $(t_i(t))_{i\ge 0}$ at integer crossings rhs (i.e., event changes).
Distance is non-increasing.
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.

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

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

$\vdots$

9 pages of computations, inductive proves, ...

Outlook and Invitation

What about (other) properties in (other/spatial) versions of the model?

Thank you!

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

Thank you!

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

forms a tight sequence of random variables

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