Giant descendant trees, matchings and independent sets in the age-biased attachment graphs

TL;DR

This paper analyzes age-biased attachment graph models, revealing the limiting behavior of descendant trees and evaluating greedy algorithms for matchings and independent sets, with results on convergence and distribution.

Contribution

It introduces a detailed analysis of descendant trees and greedy algorithms in age-biased attachment graphs, providing new insights into their asymptotic properties.

Findings

Scaled descendant tree size converges to a mixture of beta-distributions for m=1.

For m>1, the descendant tree size converges to 1 almost surely.

Greedy algorithms achieve specific limiting fractions of vertices in matchings and independent sets.

Abstract

We study two models of an age-biased graph process: the $\delta$-version of the preferential attachment graph model (PAM) and the uniform attachment graph model (UAM), with $m$ attachments for each of incoming vertices. We show that almost surely the scaled size of a breadth-first (descendant) tree rooted at a fixed vertex converges, for $m=1$, to a limit whose distribution is a mixture of two beta-distributions and a single beta-distribution respectively, and that for $m>1$ the limit is $1$. We also analyze the likely performance of two greedy (online) algorithms, for a large matching set and a large independent set, and determine--for each model and each greedy algorithm--both a limiting fraction of vertices involved and an almost sure convergence rate.

Equations

$$\mathbb{P}(w=x)=\{\begin{array}{ll}\frac{d_{t,i-1}(x)+\delta }{(2m+\delta )t+2i-1+i\delta /m},& \text{ if }x\in [t],\\ & \\ \frac{d_{t,i-1}(t+1)+1+i\delta /m}{(2m+\delta )t+2i-1+i\delta /m},& \text{ if }x=t+1.\end{array}$$

$$\mathbb{P}\left(w=x\right)=\{\begin{array}{ll}\frac{1+\delta }{(2+\delta )t+(1+\delta )},& \text{ if }x=t+1,\\ & \\ \frac{d_t(x)+\delta }{(2+\delta )t+(1+\delta )},& \text{ if }x\in [t].\end{array}$$

$$h(z)=h_{m,\delta }(z):=2{\left[1-\left(\frac{m+\delta }{2m+\delta }\right)z\right]}^m-z-1,$$

$$\underset{t\to \infty }{\lim }{t}^{\alpha }\max \{0,x(t)-{\rho }_{m,\delta }\}=0.$$

$$r_1=0.5000,$$

$$r_{10}=0.8863,$$

$$\lim_{t\to\infty}t^{\alpha}\biggl{|}\frac{2|M(t)|}{t}-r_{m}\biggr{|}=0.$$

$$r_1=0.6667,$$

$$r_{10}=0.9386,$$

$$\lim_{t\to\infty}t^{\chi}\biggl{|}\frac{|I(t)|}{t}-w_{m}\biggr{|}=0.$$

$$\lim_{t\to\infty}t^{\alpha}\Big{|}\frac{|I(t)|}{t}-w_{m}\Big{|}=0.$$

$$w_{m}=\frac{\log m}{m}+O\bigl{(}m^{-1}\log\log m\bigr{)}.$$

$$Y(t)=\{\begin{array}{rlrl}& 2X(t),& & \text{if }r\text{ looped on itself},\\ & 2X(t)-1,& & \text{if }r\text{ selected a vertex in }[r-1].\end{array}$$

$$\mathbb{P}(X(t+1)$$

$$=\{\begin{array}{rlrl}& \frac{(2+\delta )X(t)}{(2+\delta )t+(1+\delta )},& & \text{if }r\text{ looped on itself},\\ & \frac{(2+\delta )X(t)-1}{(2+\delta )t+(1+\delta )},& & \text{if }r\text{ selected a vertex in }[r-1].\end{array}$$

$$\mathbb{P}(X(t+1)=X(t)+1|\circ )$$

$$\mathbb{P}(X(t+1)=X(t)|\circ )$$

$$\mathbb{P}(X(t+1)=X(t)+1|\circ )$$

$$\mathbb{P}(X(t+1)=X(t)|\circ )$$

$$\mathbb{E}\left[{Z(t+1)^{(\ell)}\big{|}\circ}\right]=\left(\frac{t+\beta+\ell}{t+\beta}\right)Z(t)^{(\ell)}.$$

$$\mathbb{E}[Z^k(t+1)|\circ ]$$

$$=\frac{Z(t)}{t+\beta }\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{k}{j}\right)Z^j(t)+Z^k(t)\left(1-\frac{Z(t)}{t+\beta }\right)$$

$$=Z^k(t)+\frac{Z(t)}{t+\beta }\sum _{j=0}^{k-1}\left(\genfrac{}{}{0ex}{}{k}{j}\right)Z^j(t)$$

$$=Z^k(t)\frac{t+\beta +k}{t+\beta }+\frac{1}{t+\beta }\sum _{j=1}^{k-1}\left(\genfrac{}{}{0ex}{}{k}{j-1}\right)Z^j(t).$$

$$z^{(\ell )}=\sum _{k=1}^{\ell }{z}^{k}s(\ell ,k),$$

$$\sum _{\ell \ge 1}{\eta }^{\ell }\frac{s(\ell ,k)}{\ell !}=\frac{1}{k!}{\log }^k\frac{1}{1-\eta },|\eta |<1,$$

$$\displaystyle\mathbb{E}\bigl{[}Z^{(\ell)}(t+1)|\circ\bigr{]}=\sum_{k=1}^{\ell}s(\ell,k)\mathbb{E}\bigl{[}Z^{k}(t+1)|\circ\bigr{]}$$

$$\displaystyle=(t+\beta)^{-1}\sum_{k=1}^{\ell}s(\ell,k)\cdot\Biggl{(}\!(t+\beta+k)Z^{k}(t)+\sum_{j=0}^{k-1}\binom{k}{j-1}Z^{j}(t)\Biggr{)}$$

$$=:(t+\beta {)}^{-1}\sum _{i=1}^{\ell }\sigma (\ell ,i)Z^i(t),$$

$$\sigma (\ell ,i)$$

$$\ell s(\ell ,i)=\sum _{k=i}^{\ell }s(\ell ,k)\left(\genfrac{}{}{0ex}{}{k}{i-1}\right).$$

$$\sum _{\ell \ge 1}\frac{{\eta }^{\ell }}{\ell !}\sum _{k=i}^{\ell }s(\ell ,k)\left(\genfrac{}{}{0ex}{}{k}{i-1}\right)=\sum _{k\ge i}\left(\genfrac{}{}{0ex}{}{k}{i-1}\right)\sum _{\ell \ge k}\frac{{\eta }^{\ell }}{\ell !}s(\ell ,k)$$

$$=\sum _{k\ge i}\left(\genfrac{}{}{0ex}{}{k}{i-1}\right)\frac{1}{k!}{\log }^k\frac{1}{1-\eta }=\frac{1}{(i-1)!}\left({\log }^{-1}\frac{1}{1-\eta }\right)\sum _{s\ge 1}\frac{1}{s!}{\log }^s\frac{1}{1-\eta }$$

$$=\frac{1}{(i-1)!}\left({\log }^{i-1}\frac{1}{1-\eta }\right)\left(\frac{1}{1-\eta }-1\right)=\frac{1}{(i-1)!}\left({\log }^{i-1}\frac{1}{1-\eta }\right)\frac{\eta }{1-\eta }.$$

$$\sum _{\ell \ge 1}\frac{{\eta }^{\ell }}{\ell !}\ell s(\ell ,i)=\eta \sum _{\ell \ge 1}\frac{\ell {\eta }^{\ell -1}}{\ell !}s(\ell ,i)$$

$$\displaystyle=\eta\frac{d}{d\eta}\Biggl{(}\frac{1}{i!}\log^{i}\frac{1}{1-\eta}\Biggr{)}=\frac{1}{(i-1)!}\left(\log^{i-1}\frac{1}{1-\eta}\right)\frac{\eta}{1-\eta}.\qed$$

$$f(x;a,b)=\frac{\Gamma (a+b)}{\Gamma (a)\Gamma (b)}{x}^{a-1}(1-x{)}^{b-1},x\in (0,1),$$

Full text

During the time of this research, the first coauthor was supported by NSF Fellowship (Award No. 1502650).

Research of the second author supported in part by NSF Grant DMS 1661063

1. Introduction

It is widely accepted that graphs/networks are an inherent feature of life today. The classical models $G_{n,m}$ and $G_{n,p}$ of Erdős and Rényi [17] and Gilbert [22], respectively, lacked some salient features of observed networks. In particular, they failed to have a degree distribution that decays polynomially. Barabási and Albert [3] suggested the Preferential Attachment Model (PAM) as a more realistic model of a “real world” network. There was a certain lack of rigour in [3], and later Bollobás, Riordan, Spencer and Tusnády [6] gave a rigorous definition.

Many properties of this model have been studied. Bollobás and Riordan [7] studied the diameter and proved that with high probability (whp) PAM with $n$ vertices and $m>1$ attachments for every incoming vertex has diameter $\approx\log n/\log\log n$. Earlier result by Pittel [33] implied that for $m=1$ whp the diameter of PAM is of exact order $\log n$. Bollobás and Riordan [9, 10] studied the effect on component size from deleting random edges from PAM and showed that it is quite robust whp. The degree distribution was studied in Mori [29, 30], Flaxman, Frieze and Fenner [18], Berger, Borgs, Chayes and Saberi [4]. Peköz, Röllin and Ross [31] established convergence, with rate, of the joint distribution of the degrees of finitely many vertices. Acan and Hitczenko [2] found an alternative proof, without rate, via a memory game. Pittel [35] used the Bollobás-Riordan pairing model to approximate, with explicit error estimate, the degree sequence of the first $n^{m/(m+2)}$ vertices, $m\geq 1$, and proved that, for $m>1$, PAM is connected with probability $\approx 1-O((\log n)^{-(m-1)/3})$. Random walks on PAM have been considered in the work of Cooper and Frieze [14, 15]. In the first paper there are results on the proportion of vertices seen by a random walk on an evolving PAM and the second paper determines the asymptotic cover time of a fully evolved PAM. Frieze and Pegden [21] used random walk in a “local algorithm” to find vertex 1, improving results of Borgs, Brautbar, Chayes, Khanna and Lucier [12]. The mixing time of such a walk was analyzed in Mihail, Papadimitriou and Saberi [23] who showed rapid mixing. Interpolating between Erdős-Rényi and preferential attachment, Pittel [34] considered birth of a giant component in a graph process $G_{M}$ on a fixed vertex set, when $G_{M+1}$ is obtained by inserting a new edge between vertices $i$ and $j$ with probability proportional to $[\text{deg}(i)+\delta]\cdot[\text{deg}(j)+\delta]$, with $\delta>0$ being fixed. Confirming a conjecture of Pittel [34], Janson and Warnke [26] recently determined the asymptotic size of the giant component in the supercritical phase in this graph model.

The previous paragraph gives a small sample of results on PAM that can be related to its role as a model of a real world network. It is safe to say that PAM has now been accepted into the pantheon of random graph models that can be studied purely from a combinatorial aspect. For example, Cooper, Klasing and Zito [16] studied the size of the smallest dominating set and Frieze, Pérez-Giménez, Prałat and Reiniger [19] studied the existence of perfect matchings and Hamilton cycles.

One source of our inspiration was the work of Mori [29, 30], see also Katona and Móri [27], Hofstad [28]. They were able to construct a family of martingales in a form of factorial products, with arguments being the degrees of individual vertices. This allowed them to analyze the limiting behavior of vertex degrees in a $\delta$-version of PAM. In this paper we construct a new factorial-type martingale with an argument being the total size of a “descendants” subtree. This is a generalization of the martingale for $\delta=0$, found by Pittel [35].

2. Our Results

For each of the two models, PAM and UAM, we study the descendants tree of a given vertex $v$; it is a maximal subtree rooted at $v$ and formed by increasing paths starting at $v$. The number of vertices in this subtree is a natural influence measure of vertex $v$. We also analyze the performance of two on-line greedy algorithms, for finding a large matching set and for a large independent set. We carry out this analysis in the context of the PAM graph process described in Bollobás [6] and its extension taken from Hofstad [28, Ch. 8], and the UAM graph process, see [19] and [1].

The PAM graph process, $\delta$-extension:

Vertex $1$ has $m$ loops, so its degree is $2m$ initially. Recursively, vertex $t+1$ has $m$ edges, and it uses them one at a time either to connect to a vertex $x\in[t]$ or to loop back on itself.

To be more precise, let us denote by $d_{t,i-1}(x)$ the degree of vertex $x$ just before the $i$-th edge of vertex $t+1$ arrives. Denoting by $w$ the random receiving end of the $i$-th edge emanating from vertex $t+1$, we have

[TABLE]

We will use the notation $\{G_{m,\delta}(t)\}$ for the resulting graph process.

For $m=1$, writing $d_{t}(x)$ for $d_{t,0}(x)$, the probabilities above can be written a little more simply

[TABLE]

Bollobás and Riordan [7] discovered the following coupling between $\{G_{m,0}(t)\}_{t}$ for $m>1$ and $\{G_{1,0}(mt)\}_{t}$. Start with the $\{G_{1,0}(t)\}$ random process and let the vertices be $v_{1},v_{2},\dots$. To obtain $\{G_{m,0}(t)\}$ from $\{G_{1,0}(mt)\}$,

1. (1)

collapse the first $m$ vertices $v_{1},\dots,v_{m}$ into the first vertex $w_{1}$ of $G_{m,0}(t)$, the next $m$ vertices $v_{m+1},\dots,v_{2m}$ into the second vertex $w_{2}$ of $G_{m,0}(t)$, and so on; 2. (2)

keep the full record of the multiple edges and loops formed by collapsing the blocks $\{v_{(i-1)m+1},\dots,v_{im}\}$ for each $i$.

Doing this collapsing indefinitely we get the jointly defined Bollobás-Riordan graph processes $\{G_{m,0}(t)\}$ and $\{G_{1,0}(mt)\}$. The beauty of the $\delta$-extended Bollobás-Riordan model is that similarly this collapsing operation applied to the process $\{G_{1,\delta/m}(mt)\}$ delivers the process $\{G_{m,\delta}(t)\}$, Hofstad [28].

(For the reader’s convenience we present the explanation in Appendix.)

Remark 2.1*.*

Note that the process is well defined for $\delta\geq-m$ since for such $\delta$, all the probabilities defined in (2.1) are nonnegative and add up to 1. For $m=1$ and $\delta=-1$, it is easy to see from (2.2) that there is no loop in the graph except the loop on the first vertex. Hence, a vertex $u>1$ starts with degree 1 and then its degree does not change since as long as $d_{t}(u)+\delta=0$, the vertex cannot attract any neighbors (again from (2.2)). As a result, in this case the graph is a star centered at vertex 1. It follows from the above coupling that $G_{m,-m}(t)$ is also a star centered at vertex $1$ and the key problems we want to solve have trivial solutions in this extreme case.

The UAM graph process

Conceptually close to the preferential attachment model is the uniform attachment model (UAM). In this model, vertex $t+1$ selects uniformly at random (repetitions allowed) $m$ vertices from the set $[t]$ and attaches itself to these vertices. (See Acan and Pittel [1] for connectivity and bootstrap percolation results.) This model can be thought of the limit of the PAM model as $\delta\to\infty$ except that loops are not allowed in this case.

2.1. Number of Descendants

Fix a positive integer $r$ and let $X(t)$ denote the number of descendants of $r$ at time $t$. Here $r$ is a descendant of $r$ and $x$ is a descendant of $r=O(1)$ if and only if $x$ chooses to attach itself to at least one descendant of $r$ in Step $x$. In other words, if we think of the graph as a directed graph with edges oriented towards the smaller vertices, vertex $x$ is a descendant of $r$ if and only if there is a directed decreasing path from $x$ to $r$. In Pittel [35] $X(t)$ was proposed as an influence measure of vertex $r$ at time $t$.

We prove two theorems.

Theorem 2.2**.**

Suppose that $m=1$ and $\delta>-1$ and set $p(t):=X(t)/t$. Then almost surely (i.e. with probability $1$), $\lim p(t)$ exists, and its distribution is the mixture of two beta-distributions, with parameters $a=1$, $b=r-\frac{1}{2+\delta}$ and $a=\frac{1+\delta}{2+\delta}$, $b=r$, weighted by $\frac{1+\delta}{(2+\delta)r-1}$ and $\frac{(2+\delta)(r-1)}{(2+\delta)r-1}$ respectively. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$.

Note. (i) The proof is based on a new family of martingales $M_{\ell}(t):=\frac{\left(X(t)+\frac{\gamma}{2+\delta}\right)^{(\ell)}}{(t+\beta)^{(\ell)}}$, $(z)^{(\ell)}$ standing for the rising factorial. This family definitely resembles the martingales Mori [29, 30] used for the individual vertices’ degrees. For instance, if $D_{j}(t)$ is the degree of vertex $j$ at time $t\geq j$, then for some deterministic $\gamma_{k}(t)$, $Z_{j,k}(t):=\gamma_{k}(t)(D_{j}(t)+\delta)^{(k)}$ is a martingale, Hofstad [28]. However, $M_{\ell}(t)$ depends on $X(t)$, a global parameter of the PAM graph. Unsurprisingly, the proof that each $M_{\ell}(t)$ is indeed a martingale requires a very different method. (ii) Whp $G_{1,\delta}$ is a forest of $\Theta(\log t)$ trees rooted at vertices with loops. For the preferential attachment tree (no loops), Janson [25] recently proved that the scaled sizes of the principal subtrees, those rooted at the root’s children and ordered chronologically, converge a.s. to the GEM distributed random variables. His techniques differ significantly.

For $m>1$, we use Theorem 2.2 to prove a somewhat surprising result that, for $r=O(1)$, almost surely all but a vanishingly small fraction of vertices are descendants of the vertex $r$, (cf. [25]).

Theorem 2.3**.**

Let $m>1$ and $\delta>-m$ and let $p_{X}(t)=X(t)/t$, $p_{Y}(t)=Y(t)/(2mt)$, where $Y(t)$ is the total degree of the descendants of $r$ at time $t$. Then almost surely $\lim_{t\to\infty}p_{X}(t)=\lim_{t\to\infty}p_{Y}(t)=1$.

For the case of UAM, we have the following result.

Theorem 2.4**.**

Consider the UAM graph process $G_{t,m}$. Given $r>1$, let $X(t)$ be the cardinality of the descendant tree rooted at vertex $r$, and let $p(t):=X(t)/t$.

1. (i)

For $m=1$, almost surely, $\lim p(t)$ exists and it has the same distribution as the minimum of $(r-1)$ independent $[0,1]$-Uniforms. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$. 2. (ii)

For $m>1$, almost surely $\lim_{t\to\infty}p(t)=1$.

2.2. Greedy Matching Algorithm

We analyze a greedy matching algorithm; a.s. it delivers a surprisingly large matching set even for relatively small $m$. This algorithm generates the increasing sequence $\{M(t)\}$ of partial matchings on the sets $[t]$, with $M(1)=\emptyset$. Suppose that $X(t)$ is the set of unmatched vertices in $[t]$ at time $t$. If $t+1$ attaches itself to a vertex $u\in X(t)$, then $M(t+1)=M(t)\cup\{\{u,t+1\}\}$, otherwise $M(t+1)=M(t)$. (If $t+1$ chooses multiple vertices from $X(t)$, then we pick one of those as $u$ arbitrarily.)

Consider first the PAM graph. Let

[TABLE]

and let $\rho=\rho_{m,\delta}$ be the unique root $\rho=\rho_{m,\delta}$ in the interval $[0,1]$ of $h(z)=0$: $\rho_{m,\delta}\in(0,1)$ if $\delta>-m$.

Theorem 2.5**.**

Let $M(t)$ and $X(t)$ be the set of greedy matching and the set of uncovered vertices at time $t$, and let $x(t)=X(t)/t$. For any $\delta>-m$ and $\alpha<1/3$, almost surely,

[TABLE]

In consequence, the Greedy Matching Algorithm a.s. finds a sequence of nested matchings $\{M(t)\}$, with $M(t)$ of size $(1-o(1))(1-\rho_{m,\delta})t/2$, at least.

Remark 2.6*.*

Observe that $\rho_{m,-m}=1$, which makes it plausible that the maximum matching size is minuscule compared to $t$. In fact, by Remark 2.1, $G_{m,-m}(t)$ is the star centered at vertex $1$ and hence the maximum matching size is 1.

Remark 2.7*.*

Consider the case $\delta=0$. Let $r_{m}:=1-\rho_{m,0}$; some values of $r_{m}$ are:

[TABLE]

With a bit of calculus, we obtain that $r_{m}=1-2m^{-1}\log 2+O(m^{-2})$.

Theorem 2.8**.**

Let $M(t)$ denote the greedy matching set after $t$ steps of the UAM process. Let $r_{m}$ denote a unique positive root of $2(1-z^{m})-z=0$: $r_{m}=1-m^{-1}\log 2+O(m^{-2})$. Then, for any $\alpha<1/3$, almost surely

[TABLE]

Some values of $r_{m}$ in this case are:

[TABLE]

2.3. Greedy Independent Set Algorithm

The algorithm generates an increasing sequence of independent sets $\{I(t)\}$ on vertex sets $[t]$. Namely, $I(1)=\{1\}$, and $I(t+1)=I(t)\cup\{t+1\}$ if $t+1$ does not select any of the vertices in $I(t)$; if it does, then $I(t+1)=I(t)$. $I(t)$ is also a dominating set for the PAM/UAM graph with vertex set $[t]$; indeed if a vertex $\tau\in[t]\setminus I(t)$ did not have any neighbor in $I(t)$, then vertex $\tau$ would have been added to $I(\tau-1)$ at step $\tau$. (Pittel [32] analyzed performance of this algorithm applied to Erdős–Rényi random graph with a large, but fixed vertex set.)

For the PAM case we prove

Theorem 2.9**.**

Let $w_{m}$ denote the unique root of $-w+(1-w)^{m}$ in $(0,1)$. For any \chi\in\Bigl{(}0,\min\Bigl{\{}\frac{1}{3},\frac{2m+2\delta}{3(2m+\delta)}\Bigr{\}}\Bigr{)}, almost surely

[TABLE]

Remark 2.10*.*

Thus the limiting scaled size of the greedy independent set does not depend on $\delta$, but the convergence rate does.

For the UAM case we prove an almost identical

Theorem 2.11**.**

Let $w_{m}$ be the unique positive root of $-w+(1-w)^{m}$ in $(0,1)$. Then, for any $\alpha<1/3$, almost surely

[TABLE]

Remark 2.12*.*

Let $w_{m}$ the unique positive root of $(1-w)^{m}-w$ in $(0,1)$. As $m\to\infty$,

[TABLE]

So, for $m$ large and both models, a.s. for all large enough $t$ the greedy algorithm delivers an independent set containing a fraction $\sim\frac{\log m}{m}$ of all vertices in $[t]$. It was proved in [19] that for each large $t$ with probability $1-o(1))$ the fraction of vertices in the largest independent set in the PAM graph process and in the UAM graph process is at most $(4+o(1))\frac{\log m}{m}$ and $(2+o(1))\frac{\log m}{m}$, respectively. We conjecture that for each of the processes there exists a corresponding constant $c$ such that, for $m\to\infty$, a.s. for all large $t$ the largest independent set contains a fraction $\sim c\frac{\log m}{m}$ of all $t$ vertices.

Since $I(t)$ is dominating, our results prove, for $m$ large, a.s. existence for all large $t$ of relatively small dominating sets, of cardinality $\sim t\frac{\log m}{m}$.

3. Descendant trees

Instead of referring the reader back to “Our results” section, we start this, and other proof sections, with formulating in full the claim in question.

Proof of Theorem 2.2

Theorem 2.2**.**

Suppose that $m=1$ and $\delta>-1$ and set $p(t):=X(t)/t$. Then almost surely (i.e. with probability $1$), $\lim p(t)$ exists, and its distribution is the mixture of two beta-distributions, with parameters $a=1$, $b=r-\frac{1}{2+\delta}$ and $a=\frac{1+\delta}{2+\delta}$, $b=r$, weighted by $\frac{1+\delta}{(2+\delta)r-1}$ and $\frac{(2+\delta)(r-1)}{(2+\delta)r-1}$ respectively. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$.

Proof.

For $t\geq r>1$, let $X(t)=X_{m,\delta}(t)=X_{m,\delta}(t,r)$ and $Y(t)=Y_{m,\delta}(t)=Y_{m,\delta}(t,r)$ denote the size and the total degree of the vertices in the vertex set of the subtree $T(t)=T_{m,\delta}(t,r)$ rooted at $r$; so $X(r)=1$ and $Y(r)\in[m,2m]$, where $m$ ($2m$ resp.) is attained when vertex $r$ forms no loops (forms $m$ loops resp.) at itself. Introduce $p(t)=p_{Y}(t)=\frac{Y(t)}{2mt}$ and $p_{X}(t)=\frac{X(t)}{t}$. This notation will be used in the proof of Theorem 2.3 as well, but of course $m=1$ Theorem 2.2.

Here

[TABLE]

(In particular, $p_{X}(t)=p(t)+O(t^{-1})$.) So, by (2.2),

[TABLE]

Thus we are led to consider the process $X(t)$ such that

[TABLE]

$\gamma=0$ if $r$ looped on itself, $\gamma=-1$ if $r$ selected a vertex in $[r-1]$. Letting $\beta=\frac{1+\delta}{2+\delta}$, the above equation can be written as

[TABLE]

For $\delta=0$ the following claim was proved in Pittel [35]. We denote by $z^{(\ell)}$ the rising factorial $\prod_{j=0}^{\ell-1}(z+j)$.

Lemma 3.1**.**

Let $\beta=\frac{1+\delta}{2+\delta}$ and $Z(t)=X(t)+\frac{\gamma}{2+\delta}$. Then, conditioned on the attachment record during the time interval $[r,t]$, i.e. starting with attachment decision by vertex $r$, we have

[TABLE]

Consequently $M(t):=\frac{Z(t)^{(\ell)}}{(t+\beta)^{(\ell)}}$ is a martingale.

Proof.

By (3.1), we have: for $k\geq 1$, and $t\geq r$,

[TABLE]

Next recall that

[TABLE]

where $s(\ell,k)$ is the signless, first-kind, Stirling number, i.e. the number of permutations of the set $[\ell]$ with $k$ cycles. In particular,

[TABLE]

Comtet [13, Section 5.5]. Using (3.2) and (3.3), we have

[TABLE]

We need to show that $\sigma(\ell,i)=(t+\beta+\ell)s(\ell,i)$ for $k<\ell$, which is equivalent to

[TABLE]

To prove the latter identity, it suffices to show that, for a fixed $i$, the exponential generating functions of the two sides coincide. By (3.4),

[TABLE]

And, using (3.4) again,

[TABLE]

To identify the $\lim_{t\to\infty}p(t)$, recall that the classic beta probability distribution has density

[TABLE]

parametrized by two parameters $a>0$, $b>0$, and moments

[TABLE]

We can now complete the proof of Theorem 2.2. By Lemma 3.1, we have $\mathbb{E}[M(t)|\gamma]=M(r)$, i.e.

[TABLE]

For every $\ell\geq 1$, by martingale convergence theorem, conditioned on $\gamma$, there exists an integrably finite $\mathcal{M}_{\gamma,\ell}$ such that a.s.

[TABLE]

and

[TABLE]

So, using the notation $p_{X}(t)=X(t)/t$, we have: a.s.

[TABLE]

and

[TABLE]

This means that $\mathcal{M}_{\gamma,1}$ is beta-distributed with parameters $1+\frac{\gamma}{2+\delta}$ and $r+\beta-1-\frac{\gamma}{2+\delta}$. By the definition of $\gamma$ and (2.2), we have

[TABLE]

We conclude that $\lim_{t\to\infty}p(t)$ has the distribution which is the mixture of the two beta distributions, with parameters $a=1$, $b=r-\frac{1}{2+\delta}$, and $a=\frac{1+\delta}{2+\delta}$, $b=r$, weighted by $\frac{1+\delta}{(2+\delta)r-1}$ and $\frac{(2+\delta)(r-1)}{(2+\delta)r-1}$ respectively. This completes the proof of Theorem 2.2. ∎

3.1. Proof of Theorem 2.3

Theorem 2.3**.**

Let $m>1$ and $\delta>-m$ and let $p_{X}(t)=X(t)/t$, $p_{Y}(t)=Y(t)/(2mt)$, where $Y(t)$ is the total degree of the descendants of $r$ at time $t$. Then almost surely $\lim_{t\to\infty}p_{X}(t)=\lim_{t\to\infty}p_{Y}(t)=1$.

Proof.

We need to derive tractable formulas/bounds for the conditional distribution of $Y(t+1)-Y(t)$. First, let us evaluate the conditional probability that selecting the second endpoints of the $m$ edges incident to vertex $t+1$ no loops will be formed. Suppose there has been no loop in the first $i-1$ steps, $i\in[m]$; call this event $\mathcal{E}_{i-1}$. On event $\mathcal{E}_{i-1}$, as the $i$-th edge incident to $t+1$ is about to attach its second end to a vertex in $[t]\cup\{t+1\}$, the total degree of all these vertices is $2mt+i-1$ ($1\leq i\leq m$). So, by the definition of the transition probabilities (items (a), (b), (c)) we have

[TABLE]

“$\circ$” indicating conditioning on the full record of $i-1$ preceding attachments such that the event $\mathcal{E}_{i-1}$ holds. Crucially this conditional probability depends on $i$ only. Therefore the probability of a given full loops-free record of the $m$ attachments is equal to the corresponding probability for the “no loops in $m$ attachments process”, multiplied by

[TABLE]

Lemma 3.2**.**

If no loops are allowed in the transition from $t$ to $t+1$, then for $a\in[m]$,

[TABLE]

and

[TABLE]

Proof.

Vertex $t+1$ selects, in $m$ steps, a sequence $\{v_{1},\dots,v_{m}\}$ of $m$ vertices from $[t]$, with $t$ choices for every selection. Introduce $\mathbb{\boldsymbol{I}}=\{\mathbb{I}_{1},\dots,\mathbb{I}_{m}\}$, where $\mathbb{I}_{i}$ is the indicator of the event $\{v_{i}\in V(T(t))\}$. The total vertex degree of $[t]$ (of $V(T(t))$ respectively) right before step $i$ is $2mt+i-1$ \bigl{(}Y(t)+\mu_{i}\text{ respectively},\mu_{i}:=|\{j<i:\mathbb{I}_{j}=1\}|\bigr{)}. Conditioned on this prehistory,

[TABLE]

Therefore a sequence $\mathbb{\boldsymbol{I}}$ will be the outcome of the $m$-step selection with probability

[TABLE]

Furthermore, for $a\in[m]$, on the event $\{Y(t+1)=Y(t)+m+a\}$ for each admissible $\mathbb{\boldsymbol{I}}$ we have

[TABLE]

so that

[TABLE]

Since the total number of admissible sequences $\mathbb{\boldsymbol{I}}$ is $\binom{m}{a}$, we obtain the first formula in Lemma 3.2. The second formula is the case of $\mathbb{P}(\mathbb{\boldsymbol{I}})$ with ${a=0}.\qed$

It is clear from the proof of Lemma 3.2 that $\{\mathbb{P}_{m}(a)\}_{0\leq a\leq m}$,

[TABLE]

is a probability distribution of a random variable $D$, a “rising-factorial” counterpart of the binomial $\mathcal{D}=\text{Bin}(m,p=Y(t)/2mt)$. Define the falling factorial $(x)_{\ell}=x(x-1)\cdots(x-\ell+1)$. It is well known that $\mathbb{E}[(\mathcal{D})_{\mu}]=(m)_{\mu}p^{\mu}$, $(\mu\leq m)$. For $D$ we have

[TABLE]

since the sum over $a\geq\mu$ is $\sum_{\nu\geq 0}\mathbb{P}_{m-\mu}(\nu)=1$.

From Lemma 3.2 and (3.8) we have: if no loops during the transition from $t$ to $t+1$ are allowed, then

[TABLE]

and

[TABLE]

What if the ban on loops at the vertex $t+1$ is lifted? From the discussion right before Lemma 3.2, we see that both \mathbb{E}[\bigl{(}Y(t+1)-Y(t)\bigr{)}\mathbb{I}(\text{no loops})|\circ] and \mathbb{E}[\bigl{(}X(t+1)-X(t)\bigr{)}\mathbb{I}(\text{no loops})|\circ] are equal to the respective RHS’s in (3.1) and (3.10) times $\Pi_{m}(t)=1-O(t^{-1})$. Consequently, adding the terms $O(t^{-1})$ to the RHS of (3.1) and to the RHS of (3.10) we obtain the sharp asymptotic formulas for $\mathbb{E}[Y(t+1)-Y(t)|\circ]$ and $\mathbb{E}[X(t+1)-X(t)|\circ]$ in the case of the loops-allowed model. We will also need

[TABLE]

where $p_{Y}(t)=\frac{Y(t)}{2mt}$ and $p_{X}(t)=\frac{X(t)}{t}$ as defined in the beginning of the section.

To continue the proof of Theorem 2.3, we note first that $mX(t)\leq Y(t)\leq 2mX(t)$. The lower bound is obvious. The upper bound follows from induction on $t$: Suppose $Y(t)\leq 2mX(t)$. If $X(t+1)=X(t)$, then $Y(t+1)=Y(t)\leq 2mX(t)=2mX(t+1)$. If $X(t+1)=X(t)+1$, then $Y(t+1)\leq Y(t)+2m\leq 2mX(t)+2m=2mX(t+1)$. Therefore, by the definition of $p(t)$, we have

[TABLE]

in particular, $p(t)\in[0,1]$ since $\delta\geq-m$. We will also need

[TABLE]

So, using (3.1), we compute

[TABLE]

Likewise

[TABLE]

Multiplying the equation (3.12) by $\frac{2m}{2m+\delta}$, the equation (3.13) by $\frac{\delta}{2m+\delta}$, and adding them, we obtain

[TABLE]

From the first line in (3.14) it follows that

[TABLE]

implying that

[TABLE]

Since $1-z-(1-z)^{m}\geq 0$ on $[0,1]$, the second line in (3.14) implies that $|q(t)|\leq q(t)+O(t^{-2})$. Since $\sum_{t}t^{-2}<\infty$, we see that $\sum_{t}\mathbb{E}[|q(t)|]<\infty$.

So a.s. there exists $Q:=\lim_{\tau\to\infty}\sum_{1\leq t\leq\tau}q(t)$, with $\mathbb{E}[|Q|]\leq\sum_{t}\mathbb{E}[|q(t)|]<\infty$, i.e. a.s. $|Q|<\infty$. Introducing $Q(t+1)=\sum_{\tau\leq t}q(\tau)$, we see from (3.14) that $\{p(t+1)-Q(t+1)\}_{t\geq 1}$ is a martingale with $\sup_{t}|p(t+1)-Q(t+1)|\leq 1+\sum_{\tau\geq 1}|q(\tau)|$. By the martingale convergence theorem we obtain that there exists an integrable $\lim_{t\to\infty}(p(t)-Q(t))$, implying that a.s. there exists a random $p(\infty)=\lim_{t\to\infty}p(t)$. The (3.14) also implies that

[TABLE]

Since $m+\delta>0$ and

[TABLE]

and the series $\sum_{t\geq 1}t^{-1}$ diverges, we obtain that $\mathbb{P}(p(\infty)\in\{0,1\})=1$.

Recall that $p(t)\geq\frac{m+\delta}{2m+\delta}\,p_{X}(t)$. If we show that a.s. $\liminf_{t\to\infty}p_{X}(t)>0$, it will follow that a.s. $p(\infty)>0$, whence a.s. $p(\infty)=1$, implying (by $p(t)\leq p_{X}(t)$) that a.s. $p_{X}(\infty)$ exists, and is $1$, and consequently (by the formula for $p(t)$) a.s. $p_{Y}(\infty)$ exists, and is $1$.

So let’s prove that a.s. $\liminf_{t\to\infty}p_{X}(t)>0$. Recall that we did prove the latter for $m=1$. To transfer this earlier result to $m>1$, we need to establish some kind of monotonicity with respect to $m$. The coupling described in Section 2 to the rescue!

Lemma 3.3**.**

For the coupled processes $\{G_{m,\delta}(t)\}$ and $\{G_{1,\delta/m}(mt)\}$, we have $X_{m,\delta}(t,r)\geq m^{-1}X_{1,\delta/m}(mt,mr)$.

Proof.

Let us simply write $G_{1}$ and $G_{m}$ for the two graphs $G_{1,\delta/m}(mt)$ and $G_{m,\delta}(t)$, respectively. Similarly, write $T_{1}$ and $T_{m}$, respectively, for the descendant tree in $G_{1,\delta/m}(mt)$ rooted at $mr$ and the descendant tree in $G_{m,\delta}(t)$ rooted at $r$. If $v_{a}\in T_{1}$, i.e. $v_{a}$ is a descendant of $mr$, then for $b=\left\lceil a/m\right\rceil$ we have $w_{b}=\{v_{m(b-1)+i}\}_{i\in[m]}\ni v_{a}$, implying that $w_{b}$ is a descendant of $r$ in $G_{m}$, i.e. $w_{b}\in T_{m}$. (The converse is generally false: if $w_{b}$ is a descendant of $r$, it does not mean that every $v_{m(b-1)+i}$, ($i\in[m]$), is a descendant of $mr$.) Therefore

[TABLE]

Thus, to complete the proof of the theorem, i.e. for $\delta>-m$, we (a) use Theorem 2.2, to assert that for the process $\{G_{1,\delta/m}(t)\}$, a.s. $\lim_{t\to\infty}p_{X}(t)>0$; (b) use Lemma 3.3, to assert that a.s. $\liminf_{t\to\infty}p_{X}(t)>0$ for $\{G_{m,\delta}(t)\}$ as well. The proof of Theorem 2.3 is complete.

∎

3.2. Proof of Theorem 2.4

Theorem 2.4**.**

Consider the UAM graph process $G_{t,m}$. Given $r>1$, let $X(t)$ be the cardinality of the descendant tree rooted at vertex $r$, and let $p(t):=X(t)/t$.

1. (i)

For $m=1$, almost surely, $\lim p(t)$ exists and it has the same distribution as the minimum of $(r-1)$ independent $[0,1]$-Uniforms. Consequently a.s. $\liminf_{t\to\infty}p(t)>0$. 2. (ii)

For $m>1$, almost surely $\lim_{t\to\infty}p(t)=1$.

Proof.

By the definition of the UAM process, we have

[TABLE]

(i) Consider $m=1$. For $r=1$, we have $p(t)\equiv 1$. Consider $r\geq 2$. The equation (3.15) is the case $\delta=-1$, $\gamma=0$ of (3.1). By Lemma 3.1, we claim that $M(t):=\frac{(X(t))^{(\ell)}}{t^{(\ell)}}$ is a martingale. So arguing as in the proof of Theorem 2.2, we obtain that almost surely (a.s.) $\lim p(t)=p(\infty)$ exists, and the limiting distribution of $p(\infty)$ is a beta-distribution with parameters $a=1$ and $b=r-1$. That is, the limiting density is $(r-1)(1-x)^{r-2}$, $x\in[0,1]$. Therefore a.s. $\lim p(t)>0$.

(ii) Consider $m>1$. Clearly $G_{t,1}\subset G_{t,m}$. Therefore a.s. $\liminf p(t)>0$ as well. Furthermore, it follows from (3.15) that

[TABLE]

which is a special case of (3.14), with $O(t^{-2})$ dropped. So we obtain that \mathbb{P}\bigl{(}p(\infty)\in\{0,1\}\bigr{)}=1, which in combination with $\mathbb{P}(p(\infty)>0)=1$ imply that $\mathbb{P}(p(\infty)=1)=1$. ∎

4. A technical lemma

In this section, we will prove Lemma 4.2. We need the following Chernoff bound for its proof. (See e.g. [24, Theorem 2.8].)

Theorem 4.1**.**

If $X_{1},\dots,X_{n}$ are independent Bernoulli random variables, $X=\sum_{i=1}^{n}X_{i}$, and $\lambda=\mathbb{E}[X]$, then

[TABLE]

Lemma 4.2**.**

Let $\{X(t)\}_{t\geq 0}$ be a sequence of random variables such that $X(0)=0$ and $X(t+1)-X(t)\in\{0,1\}$. Let $x(t)=X(t)/t$ and assume

[TABLE]

where $h$ is a continuous, strictly decreasing function with $h(0)>0$ and $h(1)<0$, so that $h(x)$ has a unique root $\rho\in(0,1)$. Assume also that $h^{\prime}(x)<-1$ in $(0,1)$. Then, for any $\gamma<1/3$, almost surely

[TABLE]

Lemma 4.3** (Extensions of Lemma 4.2).**

Lemma 4.2 can be extended in a couple of ways as follows.

1. (a)

If the hypothesis $X(t+1)-X(t)\in\{0,1\}$ in Lemma 4.2 is replaced with $X(t+1)-X(t)\in\{-1,1\}$, then the conclusion of Lemma 4.2 still holds. This follows from minor modifications in the proof of Lemma 4.2. 2. (b)

If the inequality sign in 4.1 is replaced with an equality sign, .i.e. under the condition $\mathbb{E}\left[{x(t+1)-x(t)\mid\circ}\right]=\frac{h(x(t))}{t}+O(t^{-2})$, we have the following conclusion: for any $\gamma<1/3$, almost surely

[TABLE]

Proof of (b).

First of all, by Lemma 4.2, we have $\lim_{t\to\infty}t^{\gamma}\max\{0,x(t)-\rho\}=0$ almost surely. Second, let $g(z)=-h(1-z)$, so that $g(0)>0$ and $g(1)<0$, and in $(0,1)$, we have $g^{\prime}(z)=h^{\prime}(1-z)<-1$. Letting $y(t)=1-x(t)$,

[TABLE]

Applying Lemma 4.2 with $X_{1}(t)=t-X(t)$, and then switching back to $X(t)$, we see that $\lim_{t\to\infty}t^{\gamma}\max\{0,\rho-x(t)\}=0$ almost surely, as well. ∎

Proof of Lemma 4.2.

Let $\varepsilon=\varepsilon_{t}:=t^{-1/3}\log t$. We will show

[TABLE]

Once we show (4.2), the Borel-Cantelli lemma gives

[TABLE]

which proves what we want. Let us prove (4.2).

For $T\in[0,t)$, let $\mathcal{E}_{T}$ be the event that {$x(t)>\rho+\varepsilon$ and $T$ is the last time such that $X(\tau)\leq(\rho+\varepsilon/2)\tau$}, that is,

[TABLE]

Since $X(t+1)-X(t)\in\{0,1\}$, we have

[TABLE]

Using $X(t)=tx(t)$ above , we get

[TABLE]

implying

[TABLE]

We conclude that

[TABLE]

Now let us fix a $T\in[0,s]$ and bound $\mathbb{P}(\mathcal{E}_{T})$. The main idea of the proof is that, as long as $x(\tau)>\rho$, by Equation (4.1), the process $\{x(\tau)\}$ has a negative drift.

Let $\xi_{\tau}$ denote the indicator of the event $\{x(\tau-1)>\rho+\varepsilon/2\text{ and }X(\tau)=X(\tau-1)+1\}$ and let $\mathcal{Z}_{T}:=\xi_{T+2}+\cdots+\xi_{t}$. On the event $\mathcal{E}_{T}$, the sum $\mathcal{Z}_{T}$ counts the total number of upward unit jumps ($X(\tau)-X(\tau-1)=1$, $\tau\in[T+2,t]$) and therefore

[TABLE]

Since $X(T+1)\leq X(T)+1\leq T(\rho+\varepsilon/2)+1$, we must have

[TABLE]

Writing $p(x(\tau)):=\mathbb{P}\left(X(\tau+1)=X(\tau)+1\mid\circ\right)$,

[TABLE]

so that $p(x(\tau))=h(x(\tau))+x(\tau)+O(\tau^{-1})$.

Recall that, for $\tau\geq T+1$, we have $x(\tau)>\rho+\varepsilon/2$. Since $h^{\prime}(x)<-1$ in $(0,1)$, the sum $h(x)+x$ is decreasing in $(0,1)$. Hence, conditioning on the full record (up to and including time $\tau$),

[TABLE]

Hence, the sequence $\{\xi_{\tau}\}$ is stochastically dominated by the sequence of independent Bernoulli random variables $B_{\tau}$ with parameters \min\bigl{(}\rho+O(\tau^{-1}),1\bigr{)}. Consequently, $Z_{T}$ is stochastically dominated by $1+\sum_{j=T+2}^{t}B_{j}$, and

[TABLE]

For the choice of $\varepsilon$ we have, (4.3) gives

[TABLE]

Thus, by the Chernoff bound in Theorem 4.1 and using (4.3),

[TABLE]

Using the union bound on $T$ we complete the proof of (4.2) and of the lemma. ∎

5. Greedy Matching Algorithm

Recall that the greedy matching algorithm (for either of two graph models) generates the increasing sequence $\{M(t)\}$ of partial matchings on the sets $[t]$, with $M(1)=\emptyset$. Given $M(t)$, let

[TABLE]

5.1. The PAM case

Theorem 2.5**.**

Let $X(t)$ be the number of unmatched vertices at time $t$ in the greedy matching algorithm. For $\delta>-m$, let $\rho_{m,\delta}$ be the unique root in $(0,1)$ of

[TABLE]

Then, for any $\alpha<1/3$, almost surely,

[TABLE]

In consequence, the Greedy Matching Algorithm a.s. finds a sequence of nested matchings $\{M(t)\}$, where the number of vertices in $M(t)$ is asymptotically at least $(1-\rho_{m,\delta})t$.

Proof.

Notice first that, for $\delta>-m$, the function $h(z)$ is decreasing on $(0,1)$ and $h(z)=0$ does have a unique solution in the same interval.

We will prove our claim first for a slightly different model that does not allow any loops other than at the first vertex. In this model, vertex $1$ has $m$ loops, and the $i$-th edge of vertex $t+1$ attaches to $u\in[t]$ with probability

[TABLE]

Loops not allowed except at vertex 1. In this case, since each degree is at least $m$, we have $Y(t)\geq mX(t)$ and hence $y(t)\geq x(t)/2$. Also, since

[TABLE]

we have

[TABLE]

Since \mathbb{P}(\text{vertex t+1 has some loop})=O(t^{-1}), using $Y(t)\geq mX(t)$ in the last step below, by Lemma 3.2 we get

[TABLE]

Using (5.3) and (5.1) gives

[TABLE]

where $h(z)$ is as defined in (5.1). Note that $X(0)=0$ and $h(z)$ satisfies the conditions given in Lemma 4.2 and Lemma 4.3, namely, $h(0)>0$, $h(1)<0$, and $h^{\prime}(z)<-1$ for $z\in(0,1)$. The conclusion of the theorem follows from the first part of Lemma 4.3 in this case.

Loops allowed everywhere. The above analysis is carried over to this more complicated case via an argument similar to the one for the descendant trees in the subsections 1.1. Here is a proof sketch. First, the counterpart of (5.1) is:

[TABLE]

see (3.7) for $\Pi_{m}(t)$. Therefore we obtain again (5.1). The rest of the proof remains the same. ∎

Remark 5.1*.*

Let $r=r_{m,\delta}:=1-\rho_{m,\delta}$, where $\rho_{m,\delta}$ is the unique root in $(0,1)$ of

[TABLE]

Then, $r$ is the unique root in $(0,1)$ of

[TABLE]

Thus, by Theorem 2.5, we have

[TABLE]

almost surely, where $1-x(t)$ is the fraction of the vertices in $M(t)$. See (2.3) for various $r$ values when $\delta=0$.

Remark 5.2*.*

When $\delta\to\infty$, the function $f_{m,\delta}(z)$ as defined above converges to $2-z-2z^{m}$ in $(0,1)$. So it is plausible that for the case of uniform attachment model, the number of vertices in $M(t)$ is asymptotically $rt$, where $r$ is the unique root of $2-z-2z^{m}$. This is in fact the case as shown in the next theorem.

5.2. The UAM case

Theorem 2.8**.**

Let $M(t)$ denote the greedy matching set after $t$ steps of the UAM process. Let $r_{m}$ denote a unique positive root of $2(1-z^{m})-z=0$: $r_{m}=1-m^{-1}\log 2+O(m^{-2})$. Then, for any $\alpha<1/3$, almost surely

[TABLE]

Proof.

Let $X(t)=t-2|M(t)|$ as before. In particular, we have $X(0)=0$ and $X(1)=1$. At each step $t\geq 2$, we check the edges incident to vertex $t$. If some of the edges end at vertices that do not belong to $M(t)$, then we choose the largest (youngest) of those vertices, say $w$, and set $M(t):=M(t-1)\cup\{(t,w)\}$ and $X(t)=X(t-1)-1$. Otherwise, $M(t)=M(t-1)$ and $X(t)=X(t-1)+1$. Let $x(t)=X(t)/t$ be the fraction of unmatched vertices after step $t$. Then $X(t)$ is a Markov chain with

[TABLE]

since for $X(t+1)-X(t)=1$ to happen, each of the $m$ choices made by vertex $t+1$ must lie outside of $M(t)$, the probability of each such choice is $1-x(t)$, and the choices are independent of each other. With the remaining probability vertex $t+1$ chooses at least one of $m$ vertices from $X(t)$, in which case $X(t)$ decreases by 1. Consequently,

[TABLE]

Dividing both sides with $t+1$, we obtain

[TABLE]

where $h(z)=2(1-z)^{m}-z-1$. The function $h$ meets the conditions of the second part of Lemma 4.3. Hence $\lim_{t\to\infty}t^{\gamma}|x(t)-\rho_{m}|=0$ a.s. On the other hand, if $\rho_{m}$ is the unique root of $h$ in $(0,1)$, then $r_{m}:=1-\rho_{m}$ is the unique root of $2(1-z^{m})-z$ in $(0,1)$. Since $x(t)-\rho_{m}=(1-2|M(t)|/t)-(1-\rho_{m})=\rho_{m}-2|M(t)|/t$, we also have, a.s.

[TABLE]

This completes the proof. ∎

6. Analysis of Greedy Independent Set Algorithm

The algorithm, for both PAM and UAM cases, generates the increasing sequence of independent sets $\{I(t)\}$ on the sets $[t]$, with $I(1):=\{1\}$. If vertex $t+1$ does not select a single vertex from $t$, we set $I(t+1)=I(t)\cup\{t+1\}$; otherwise $I(t+1):=I(t)$. Given $I(t)$, let

[TABLE]

Since each insider selects only among outsiders, the total degree of insiders is $m|I(t)|+Z(t)$ and

[TABLE]

6.1. The PAM case

Theorem 2.9**.**

Let $w_{m}$ denote the unique root of $-w+(1-w)^{m}$ in $(0,1)$. For any \chi\in\Bigl{(}0,\min\Bigl{\{}\frac{1}{3},\frac{2m+2\delta}{3(2m+\delta)}\Bigr{\}}\Bigr{)}, almost surely

[TABLE]

Proof.

By the definition of the algorithm, we have

[TABLE]

Now $U(t)=0$ means that vertex $t+1$ selects all $m$ vertices from the outsiders set, so that

[TABLE]

The leading term in the first equality is the exact (conditional) probability of $\{U(t)=0\}$ when no loops at vertices other than the first vertex are allowed, and the extra $O(t^{-1})$ is for our, more general, case when the loops are admissible. So using $i(t)=|I(t)|/t$, we have

[TABLE]

Next $Z(t+1)=Z(t)+U(t)$, so that

[TABLE]

and using $z(t)=Z(t)/mt$, we have

[TABLE]

Introduce $w(t)=i(t)\frac{m+\delta}{2m+\delta}+z(t)\frac{m}{2m+\delta}$. Multiplying the equations (6.2) and (6.3) by $\frac{m+\delta}{2m+\delta}$ and by $\frac{m}{2m+\delta}$ and adding the products, we obtain

[TABLE]

The function $f$ is qualitatively similar to the function $h$ in the proof of Theorem LABEL:th3repeat. Indeed $f(w)$ is strictly decreasing, with $f(0)=1$ and $f(1)=-1$. Therefore $f(w)$ has a unique root $w_{m}\in(0,1)$; it is not difficult to see that

[TABLE]

Let us prove that, for any \alpha<c(m,\delta):=\min\Bigl{\{}1,\frac{2m+2\delta}{2m+\delta}\Bigr{\}}(\subseteq(0,1]), and $A\geq A(\alpha)$, we have

[TABLE]

First

[TABLE]

and similarily

[TABLE]

Therefore $|w(t+1)-w(t)|\leq 2/(t+1)$, and consequently

[TABLE]

So, conditioning on prehistory, we have

[TABLE]

By (6.4), $f^{\prime}(w)\leq-\frac{m+\delta}{2m+\delta}$; therefore, with $c(m,\delta)=\frac{2m+2\delta}{2m+\delta}$, the last inequality gives

[TABLE]

leading to a recursive inequality

[TABLE]

The bound (6.5) follows from (6.7) by a straightforward induction on $t$. Next we use (6.5) to prove that, for $A_{1}$ large enough,

[TABLE]

Using (6.6), we have

[TABLE]

Consequently

[TABLE]

Taking expectations of both sides, and using Cauchy’s inequality and (6.5), we obtain

[TABLE]

since $w_{m}=(1-w_{m})^{m}$, and

[TABLE]

So it suffices to show existence of $A_{1}$ such that

[TABLE]

holds for $t>t_{0}$ where $t_{0}$ depends only on $A$ and $\alpha$. For large $t$, the above inequality becomes

[TABLE]

and A_{1}>A\Bigl{(}\frac{2m}{2-\alpha}\Bigr{)}^{2} does the job. So (6.8) is proved. Therefore, by Markov inequality,

[TABLE]

This inequality already means that $i(t)\to w_{m}$ in probability. Let us show that, considerably stronger, $i(t)\to w_{m}$ with probability $1$, at least as fast as $t^{-\chi}$, for any given $\chi<1/3$. Pick $\beta>1$ and introduce a sequence $\{t_{\nu}\}$, $t_{\nu}=\lfloor\nu^{\beta}\rfloor$. By (6.9), we have

[TABLE]

provided that $\beta>(\alpha-2\chi)^{-1}$, which we assume from now. For such choice of $\beta$, by Borel-Cantelli lemma with probability $1$ for all but finitely many $\nu$ we have $|i(t_{\nu})-\rho|\leq t_{\nu}^{-\chi}$. Let $t\in[t_{\nu},t_{\nu+1}]$. By (6.6), we have

[TABLE]

uniformly for all $\nu$. So if $|i(t_{\nu})-\rho|\leq t_{\nu}^{-\chi}$, then (using $t_{\nu}=\Theta(\nu^{\beta})$), we have: for $t\in[t_{\nu},t_{\nu+1}]$,

[TABLE]

Since $|i(t_{\nu})-w_{m}|\leq t_{\nu}^{-\chi}$ holds almost surely (a.s.) for all but finitely many $\nu$’s, we see then that a.s. so does the bound |i(t)-w_{m}|=O\bigl{(}t^{-\min(\chi,\beta^{-1})}\bigr{)} for all but finitely many $t$’s. Now by taking $\beta$ sufficiently close to $(\alpha-2\chi)^{-1}$ from above, we can make $\min(\chi,\beta^{-1})$ arbitrarily close to $\min(\chi,\alpha-2\chi)$ from below. It remains to notice that $\min(\chi,\alpha-2\chi)$ attains its maximum $\alpha/3$ at $\chi=\alpha/3$. The proof of Theorem 2.9 is complete. ∎

6.2. The UAM case

Theorem 2.11**.**

Let $w_{m}$ be the unique root of $-w+(1-w)^{m}$ in $(0,1)$. Then, for any $\alpha<1/3$, almost surely

[TABLE]

This following remark is already stated as Remark 2.12.

Remark 6.1*.*

Thus, the convergence rate aside, almost surely the greedy independent algorithm delivers a sequence of independent sets of asymptotically the same size as for the PAM case.

Proof.

Let $i(t)=|I(t)|/t$. From the definition of the UAM process and the greedy independent set algorithm, we obtain

[TABLE]

Therefore

[TABLE]

or equivalently

[TABLE]

The function $-x+(1-x)^{m}$ differs by a constant positive factor from the function $f$ in (6.4). The function $f$ meets the conditions of Lemma 4.2, and $r_{m}$ is a unique root of $f$. Therefore, for any $\alpha<1/3$, a.s. $\lim_{t\to\infty}t^{\alpha}|i(t)-r_{m}|=0$. ∎

Appendix

In order to show that the coupling described in Section 3 really works, we can compute the probability that the $i$-th edge of vertex $w_{t+1}$ connects to vertex $w_{x}$ in the coupling and compare it with the probability in (2.1). Let us denote by $\{G^{\prime}_{m,\delta}(t)\}$ the process obtained by collapsing the vertices of $\{G_{1,\delta/m}(mt)\}$. Note that $(mt+i)$-th edge of the $\{G_{1,\delta/m}(mt)\}$-process becomes the $i$-th edge of $w_{t+1}$ after the collapsing. Hence the $i$-th edge of vertex $w_{t+1}$ connects to $w_{x}$ ($x\leq t$) if and only if the $(mt+i)$-th edge of $\{G_{1,\delta/m}(mt)\}$-process connects $v_{mt+i}$ with one of the vertices $v_{m(x-1)+1},\dots,v_{mx}$. Let us denote by $d_{mt+i-1}(v_{y})$ the degree of $v_{y}$ ($y\leq mt+i$) just before the $(mt+i)$-th edge of $\{G_{1,\delta/m}\}$-process is drawn. Also, let $D_{t,i-1}(w_{x})$ denote the degree of $w_{x}$ at the exact same time. Hence, by definition,

[TABLE]

By (2.2), for $x\leq t$, the probability that $v_{mt+i}$ connects to one of the vertices $v_{m(x-1)+1},\dots,v_{mx}$ (equivalently, the probability that the $i$-th edge of $w_{t+1}$ connects to $w_{x}$) is

[TABLE]

Similarly, the probability that $v_{mt+i}$ selects one of $v_{mt+1},\dots,v_{mt+i}$ (equivalently, the probability that the $i$-th edge of $w_{t+1}$ is a loop) is

[TABLE]

Note that the two probabilities above are the same as those in (2.1) if we replace $D_{t,i-1}(w_{x})$ with $d_{t,i-1}(x)$. Moreover, the two processes, $\{G^{\prime}_{m,\delta}(t)\}$ and $\{G_{m,\delta}(t)\}$ as defined by (2.1), both start with $m$ loops on the first vertex, which implies $d_{1,0}(\cdot)=D_{1,0}(\cdot)$. This gives us that $\{G_{m,\delta}(t)\}$ and $\{G^{\prime}_{m,\delta}(t)\}$ are equivalent processes, that is, at every stage, they produce the same random graph.