Preferential Attachment Random Graphs with Edge-Step Functions

Abstract

We analyze a random graph model with preferential attachment rule and edge-step functions that govern the growth rate of the vertex set, and study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function f is a monotone regularly varying function at infinity, the degree sequence of graphs associated with it obeys a (generalized) power-law distribution whose exponent belongs to (1, 2] and is related to the index of regular variation of f at infinity whenever said index is greater than $-1$. When the regular variation index is less than or equal to $-1$, we show that the empirical degree distribution vanishes for any fixed degree.

Appendices

Appendix A. Important Results on Regularly Varying Functions

In this “Appendix,” we collect some results regarding regularly varying functions that will be useful throughout the paper, as well as providing a proof for an earlier lemma.

Corollary A.1

Let $\ell$ be a slowly varying function. Then,

$$\begin{aligned} \lim _{t \rightarrow \infty } \frac{\ell (t)}{\sum _{s=1}^t\ell (s)s^{-1}} = 0 \end{aligned}$$

(A.1)

Proof

Since $s^{-1}\ell (s)$ is a RV function which is eventually monotone, we may bound the sum by the integral. Now, by Theorem 1.5.2 of [5], for a fixed small $\varepsilon$, we know that

$$\begin{aligned} \lim _{t \rightarrow \infty }\frac{\ell (xt)}{\ell (t)} =1 \end{aligned}$$

uniformly for $x \in [\varepsilon ,1]$. Therefore, for large enough t

$$\begin{aligned} \begin{aligned} \ell ^{-1}(t)\int _{1}^t\frac{\ell (s)ds}{s}&\ge \int _{\varepsilon }^1\frac{\ell (tx)dx}{\ell (t)x} \ge -(1-\delta )\log \varepsilon , \end{aligned} \end{aligned}$$

(A.2)

for some small $\delta$. This proves the desired result. $\square$

The three following results are used throughout the paper.

Corollary A.2

(Representation theorem Theorem 1.4.1 of [5]) Let f be a continuous regularly varying function with index of regular variation $\gamma$. Then, there exists a slowly varying function $\ell$ such that

$$\begin{aligned} f(t) = t^{\gamma }\ell (t), \end{aligned}$$

(A.3)

for all t in the domain of f.

Corollary A.3

Let f be a continuous regularly varying function with index of regular variation $\gamma <0$. Then,

$$\begin{aligned} f(x) \rightarrow 0, \end{aligned}$$

(A.4)

as x tends to infinity. Moreover, if $\ell$ is a slowly varying function, then for every $\varepsilon >0$

$$\begin{aligned} x^{-\varepsilon } \ell (x) \rightarrow 0 \quad \text {and}\quad x^{\varepsilon }\ell (x) \rightarrow \infty \end{aligned}$$

(A.5)

Proof

Comes as a straightforward application of Theorem 1.3.1 of [5] and Corollary A.2. $\square$

Theorem A1

(Karamata’s theorem Proposition 1.5.8 of [5]) Let $\ell$ be a continuous slowly varying function and locally bounded in $[x_0, \infty )$ for some $x_0 \ge 0$. Then

• (a) for $\alpha > -1$

  $$\begin{aligned} \int _{x_0}^{x}t^{\alpha }\ell (t)\text {d}t \sim \frac{x^{1+\alpha }\ell (x)}{1+\alpha }. \end{aligned}$$

  (A.6)

• (b) for $\alpha < -1$

  $$\begin{aligned} \int _{x}^{\infty }t^{\alpha }\ell (t)\text {d}t \sim \frac{x^{1+\alpha }\ell (x)}{1+\alpha }. \end{aligned}$$

  (A.7)

We finish this section with the proof of an earlier lemma.

Proof of Lemma 2

(i) By Potter’s theorem (Theorem 1.5.6 of [5]), if $\ell$ is slowly varying, then for every $\delta >0$ there exists $M>0$ such that

$$\begin{aligned} \frac{\ell (x)}{\ell (y)}\le 2\max \left\{ \frac{x^\delta }{y^\delta }, \frac{y^\delta }{x^\delta }\right\} \end{aligned}$$

(A.8)

for every $x,y>M$. We have

$$\begin{aligned} \int _{0}^{1} \left| \frac{\ell (ut)}{\ell (t)}-1 \right| u^{-\gamma }\mathrm {d} u = \int _{0}^{\frac{M}{t}} \left| \frac{\ell (ut)}{\ell (t)}-1 \right| u^{-\gamma }\mathrm {d} u + \int _{\frac{M}{t}}^{1} \left| \frac{\ell (ut)}{\ell (t)}-1 \right| u^{-\gamma }\mathrm {d} u. \end{aligned}$$

We then obtain

$$\begin{aligned} \int _{0}^{\frac{M}{t}} \left| \frac{\ell (ut)}{\ell (t)}-1 \right| u^{-\gamma }\mathrm {d} u\le \left( \frac{\sup _{y\in [0,M]}\ell (y)}{\ell (t)}-1 \right) \frac{M^{1-\gamma }}{t^{1-\gamma }(1-\gamma )}\xrightarrow {t\rightarrow \infty } 0, \end{aligned}$$

by Corollary A.3. Choosing $\delta <1-\gamma$ in (A.8), we see that

$$\begin{aligned} \int _{0}^{1} \left| \frac{\ell (ut)}{\ell (t)}-1 \right| u^{-\gamma }\mathbb {1}\{u\ge M/t\}\mathrm {d} u \le \int _{0}^{1} \left( 2 \max \{u^{-\delta },u^{\delta }\}-1 \right) u^{-\gamma }\mathrm {d} u<\infty , \end{aligned}$$

and therefore the LHS of the above equation tends to 0 by the dominated convergence theorem. This and another elementary application of Corollary A.3 finish the proof of item (i).

(ii) We have

$$\begin{aligned} \left| \sum _{k=1}^{t}\ell (k)k^{-\gamma } -\frac{t^{1-\gamma }\ell (t)}{1-\gamma } \right|&\le \left| \sum _{k=1}^{t}\ell (k)k^{-\gamma } -\int _{0}^{t}\ell (s)s^{-\gamma }\mathrm {d} s \right| \nonumber \\&\quad + \left| \int _{0}^{t}\ell (s)s^{-\gamma }\mathrm {d}s - \ell (t)\cdot \int _{0}^{t} s^{-\gamma }\mathrm {d}s \right| \nonumber \\&\le C + \left| \int _{0}^{t}s^{-\gamma }(\ell (s)-\ell (t))\mathrm {d}s \right| , \end{aligned}$$

(A.9)

since $\ell (s)s^{-\gamma }$ is eventually monotone decreasing. Dividing both sides by $t^{1-\gamma }\ell (t)$ and making the substitution $u=st^{-1}$ in the integral gives the result. $\square$

Appendix B. Martingales Concentration Inequalities

For the sake of completeness, we state here two useful concentration inequalities for martingales which are used throughout the paper.

Theorem B1

(Azuma-Höffeding Inequality [7]) Let $(M_n,{\mathcal {F}})_{n \ge 1}$ be a (super)martingale satisfying

$$\begin{aligned} |M_{i+1} - M_i |\le a_i \end{aligned}$$

Then, for all $\lambda > 0$ we have

$$\begin{aligned} {\mathbb {P}}\left( M_n - M_0 > \lambda \right) \le \exp \left( -\frac{\lambda ^2}{\sum _{i=1}^n a_i^2} \right) . \end{aligned}$$

Theorem B2

(Freedman’s inequality [12]) Let $(M_n, {\mathcal {F}}_n)_{n \ge 1}$ be a (super)martingale. Write

$$\begin{aligned} W_n := \sum _{k=1}^{n-1} {\mathbb {E}} \left[ (M_{k+1}-M_k)^2 \Big |{\mathcal {F}}_k \right] \end{aligned}$$

(B.1)

and suppose that $M_0 = 0$ and

$$\begin{aligned} |M_{k+1} - M_k |\le R,\quad \textit{for all }k. \end{aligned}$$

Then, for all $\lambda > 0$ we have

$$\begin{aligned} {\mathbb {P}}\left( M_n \ge \lambda , W_n \le \sigma ^2, \textit{ for some }n\right) \le \exp \left( -\frac{\lambda ^2}{2\sigma ^2 + 2R\lambda /3} \right) . \end{aligned}$$