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Network Science (2025), 13, e5, 1–20
doi:10.1017/nws.2025.3
RESEARCH ARTICLE
Exponentially preferential trees
Rafik Aguech1
, Hosam Mahmoud2, Hanene Mohamed3and Zhou Yang2
1Laboratory AGA, University of Monastir, Monastir, Tunisia, 2Department of Statistics, The George Washington
University, Washington, USA, and 3Modal’X, UPL, Université Paris Nanterre, Nanterre, France
Corresponding author: Hosam Mahmoud; Email: hosam@gwu.edu
Abstract
We introduce the exponentially preferential recursive tree and study some properties related to the degree
profile of nodes in the tree. The definition of the tree involves a radix a>0. In a tree of size n(nodes), the
nodes are labeled with the numbers 1, 2, ...,n. The node labeled iattracts the future entrant n+1 with
probability proportional to ai.
We dedicate an early section for algorithms to generate and visualize the trees in different regimes. We
study the asymptotic distribution of the outdegree of node i,asn→∞, and find three regimes according to
whether 0 <a<1 (subcritical regime), a=1 (critical regime), or a>1 (supercritical regime). Within any
regime, there are also phases depending on a delicate interplay between iand n, ramifying the asymptotic
distribution within the regime into “early,” “intermediate” and “late” phases. In certain phases of certain
regimes, we find asymptotic Gaussian laws. In certain phases of some other regimes, small oscillations in
the asymototic laws are detected by the Poisson approximation techniques.
Keywords: Random structure; recursive tree; affinity; Gaussian law; Poisson approximation
AMS subject classifications: 60F05; 60G20; 60F05
1. Scope
The recursive tree is a classic hierarchical structure. Several models of randomness are used in a
variety of applications. Dozens of research papers have been devoted to the uniform model alone,
many of them are surveyed in Smythe and Mahmoud (1995). Today, the uniform recursive tree
is a standard entry in books on random structures (Drmota, 2009; Hofri and Mahmoud, 2018;
Frieze and Karo´
nski, 2016). The uniform recursive tree is used as a model in many applications.
Some of the classic applications are in pyramid schemes (Gastwirth and Bhattacharya, 1984)and
Philology (Najock and Heyde, 1982).
Driven by other applications, interest was developed into nonuniform models, wherein the
attachment of new nodes is done preferentially according to some criterion. The earliest of these
preferential models is a probability scheme in which nodes of higher outdegrees are favored
(Szyma´
nski, 1987). Other preferential ideas are based on node fitness (Dereich and Ortgiese,
2014), old age of nodes (Hofri and Mahmoud, 2018), node Youthfulness (Lyon and Mahmoud,
2020), and power-weights on the nodes (Lyon and Mahmoud, 2022).
1.1 A proposed preferential model
In the present investigation, we propose a new preferential model parameterized by a real positive
radix, which we call a. In this exponential model, the affinity of a node is the radix raised to the
node label. A precise definition is given in Subsection 1.3.
C
The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the
Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and
reproduction, provided the original article is properly cited.
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2R.Aguechet al.
For instance, if the radix is less than 1, nodes that appear first (older in the tree) have
an attraction power that is larger than newer nodes. One sees such a phenomenon in the
growth of networks, where older nodes have a bigger chance of growth than younger ones.
For instance, in a graph representing the growth of technology companies, nodes representing
giants like MicrosoftC
and AppleC
have a higher chance of attracting new subscribers than a node
representing a small start-up company.
We assume that the reader is familiar with the jargon of trees, such as “node,” “vertex,” “edge,”
“root,” “ancestors,” “descendants,” “children,” “parents,” “recruiting,” “affinity,” etc.
1.2 The building algorithm
A recursive tree is grown by an attachment algorithm that operates in the following way. Initially
(at time n=0), there is a node labeled 1. At each subsequent epoch n≥1 of discrete time, a node
labeled n+1 joins the tree by choosing one of the existing nodes as a parent and attaching itself
to it via a new edge. The parent selection is determined according to some probability model on
the set {1, ...,n}. Note that the labels on any root-to-leaf path are in an increasing sequence. For
this reason, some authors call these structures “increasing trees” (Bergeron et al., 1992).
These trees have been studied under several probability models, notably including the natu-
ral uniform model and preferential models based on favoring certain nodes according to some
criterion. The first preferential criterion in the literature is to select a node with probability pro-
portional to 1 plus its outdegree (Szyma´
nski, 1987; Mahmoud et al., 1993). This model gained
popularity, as it offers scalability properties and power laws that are met in certain trees in nature
(Barabási and Albert, 1999). For over a decade, the terminology “preferential attachment” stood
solely for preference by node outdegrees.
More recently, authors broke away from this narrower definition of “preference” to tree models
with other types of preference (Hofri and Mahmoud, 2018; Lyon and Mahmoud, 2020;Lyonand
Mahmoud, 2022).
1.3 Exponentially preferential trees
In this investigation, we consider a new exponentially preferential attachment algorithm, wherein
node iat time n−1 recruits with probability proportional to ai, for some positive constant a.
Specifically, if we define An,ias the event that node irecruits (the node labeled n+1) when the
tree has nnodes in it (that is, when the tree is of age n−1), we would have
P(An,i)=ai
n
j=1aj=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1
n,ifa=1;
(a−1) ai−1
an−1, otherwise.
(1)
In the sequel, we observe a trichotomy of the positive real line into three regimes for a,and
in each regime we have a different behavior. We call the regime 0 <a<1subcritical, the regime
a=1critical,andtheregimea>1supercritical.
We call a tree grown according to this distribution for the choice of parent an exponentially
preferential tree with radix a. Since this is the only kind we study in this manuscript, we refer to it
simply as the “tree” most of the time. When a=1, we have the special case of uniform recruiting,
which is extensively studied (Bergeron et al., 1992;Drmota,2009; Hofri and Mahmoud, 2018;
Frieze and Karo´
nski, 2016; Smythe and Mahmoud, 1995).
Figure 1displays the six exponentially preferential trees of size 4 with radix a=1/2. The num-
bers above the trees are their probabilities. Note the high probability assigned to the bushiest tree
at the far right. In the uniform model, this tree only has probability 1/6. To discern the these
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Network Science 3
1
21
1
2
3
4
2
21
1
2
34
4
21
1
2
3
4
4
21
1
2
4
3
2
21
1
23
4
8
21
1
234
Figure 1. The exponentially preferential attachment recursive trees of size 4 with radix a=1/2 and their probabilities.
probabilities, we illustrate the computation for one tree. We choose the fourth one from the left
(with probability 4/21), as it has multiple nodes recruiting and a node (the root) recruiting twice.
Initially, we have a root node labeled with 1. With probability 1, this root node recruits the node
labeled with 2. So, the tree
1
2
appears with probability 1. The nodes 1 and 2 are now in a competition to attract node 3, with
probabilities (1/2)1
(1/2)1+(1/2)2=2/3and (1/2)2
(1/2)1+(1/2)2=1/3. The tree
1
23
emerges with probability 1 ×2/3. The nodes labeled with 1, 2, and 3 are now in competi-
tion to attract the node labeled with 4, with respective probabilities, (1/2)1
(1/2)1+(1/2)2+(1/2)3=4/7,
(1/2)2
(1/2)1+(1/2)2+(1/2)3=2/7, and (1/2)3
(1/2)1+(1/2)2+(1/2)3=1/7. Whence, if the node labeled with 2 is the
one that recruits, we get the third tree on the right in Figure 1with probability 1 ×2/3×2/7=
4/21.
This example illustrates the dynamic nature of the attraction probability at node i. While aiis
a fixed number, the scaling used is changing with time.
2. Generation and visualization
Before we present any theoretical results, it may help the reader grasp the gist of the varied
behavior of the random exponentially preferential trees in the three regimes with diagrams.
To produce images, we first need a generating algorithm to provide the data. We present one
such algorithm that sequentially cranks out the edges that join the tree. The edges appear in the
form (n+1, r), where ris the recruiter, when the tree size is n. For instance, the pair (78, 50) stands
for an edge joining the vertex labeled 78 to a tree of size 77, in which node 50 is the recruiter.
As an illustration of the action of the algorithm, we trace through the evolution of the third
tree on the right in figure 1. The algorithm builds the tree edge by edge, first constructing the edge
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4R.Aguechet al.
(2,1), then adding the edge (3,2) and finally completing the description of this tree by adding the
edge (4,2) to the list of edges.
Once the tree description is obtained in the form of a list of edges, we can visualize the tree by
a drawing obtained with the aid of a tree-graphing package.
The algorithm assumes it can access the function
F(s,i)←
i
r=1
(a−1) ar−1
as−1=ai−1
as−1,
which accumulates the probabilities P(As,1)+···+P(As,i) for the purpose of generating the
recruiting index. With F(s,i) having a closed form, it can be evaluated in O(1) time.
The building algorithm assumes the existence of the primitive function random,which
generates a random number uniformly distributed over the interval (0, 1).
The core of the algorithm repeats the calculation of an index when the tree is of size “ size,” for
size =1, ...,n−1. At each size between 1 and n−1, a random variable Udistributed uniformly
between 0 and 1 is generated. If the value of Ufalls between F(size,r−1) and F(size,r), we take
the recruiter to be r. This recruiter is receiving the node size +1, and we store the pair (edge)
(size +1, r) in the array Rof recruiters. At the end of the execution of the algorithm, the array R
holds a complete description of a tree of size n.
Here is a possible version in pseudo code:
for size from 1to n−1do
U←random
r←1
while U>F(size,r)and r<size do
r←r+1
R[size]←(size +1, r)
The inner while loop may run an order of size in the worst case, and the outer for loop is driving
size through n−1 iterations. The overall execution performs in O(n2) time. By this algorithm, we
obtained three trees of size n=100 each, under the settings a=1/2, a=1, a=2, respectively.
The data (edges) were then fed into the tree-drawing package “Pyvis,” which produced the three
images in Figure 2. The root of each tree is shown as a red star. The figure shows a random tree
in the subcritical regime with radix a=1/2 (top left), a random uniform (standard) recursive
tree, with radix a=1 (top right), and a random tree in the supercritical regime with radix a=2
(bottom).
In Figure 2, we chose a drawing style to fill the space, rather than one going down vertically (as
in the more traditional vertical drawing as in Figure 1). The vertical drawing would use the space
sparsely.
The reader will notice right away that in the subcritical tree (the one at the top left in Figure 2),
the nodes cluster near the root, making it a shrubby structure. In the uniform tree (the one at the
top right in Figure 2), the nodes are all over the place, whereas in the supercritical tree (the one
at the bottom in Figure 2), many nodes drag the tree toward higher altitudes, making it a tall and
scrawny tree with short branches sprouting out of a main thin trunk.
2.1 Notation
The indicator of event Eis IE. The following presentation involves H(r)
p=p
k=11/kr,thepth
harmonic number of order r.1The harmonic numbers of the first two orders have well-known
asymptotic equivalents (as n→∞):2
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Network Science 5
Figure 2. Randomly generated trees of size 100: subcritical (top left) with radix 1/2, uniform (top right) with radix a=1,
supercritical (bottom) with radix 2.
Hn=ln n+γ+O1
n;(2)
H(2)
n=π2
6+O1
n.(3)
Ceils and floors appear in the calculations. The floor of a real number xcan be represented by
removing the floor and compensating for the fractional part of x, sometimes denoted by {x}.That
is, we have
x=x−{x}.
For i≥1, and a>1, in the sequel, we use the numbers
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6R.Aguechet al.
c(1)
i(a)=∞
k=i
1
ak−1,
c(2)
i(a)=∞
k=i
1
(ak−1)2.
We denote the normally distributed random variable with mean μand variance σ2>0by
N(μ,σ2) and denote the Poisson random variable with parameter λ>0by Poi(λ). The notation
L(X) stands for the law (probability distribution) of a random variable X.
The total variation distance (dTV ) between the laws of the nonnegative integer-valued random
variables Xand Yis defined as
dTVL(X), L(Y)=1
2
∞
j=0P(X=j)−P(Y=j).
As some authors do, we simplify the notation of the total variation distance to dTV (X,Y), but it
should be understood that it is the distance between the laws of these variables.
The following theorem by Barbour and Holst (Barbour and Hall, 1984) is beneficial in
obtaining Poisson approximations for node degrees.
Theorem 2.1. Let X1,...,Xnbe independent Bernoulli random variables, such that P(Xk=1) =
pk,for i =1, ...,n,and let Sn=n
k=1Xkbe the sum of these variables. Define
λn,1 =
n
k=1
E[Xk]=
n
k=1
pk,and λn,2 =
n
k=1
p2
k.
We have
dTV Sn,Poi
λn,1≤(1 −e−λn,1 )λn,2
λn,1
.
The theorem is one of several versions fitting in the machinery of Poisson approximation (and the
more general framework of Chen−Stein methods) (Barbour et al., 1992).
3. Node outdegrees
Let n,ibe the outdegree of node iin a tree of size n. It is related to the degree of node i.Exceptfor
the root, any node degree is 1 plus its outdegree. As the root is the only node that does not have a
parent, its degree is the same as its outdegree.
Remark 3.1. In a tree of size n,the sum of the outdegrees is n −1.
The outdegree of node iincreases, when it recruits, and we have the representation
n,i=
n−1
k=i
IAk,i.(4)
3.1 Mean
Take expectations of (4). We then get, for n≥2, a mean value for the ith degree in the form
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Network Science 7
E[n,i]=
n−1
k=i
P(IAk,i=1) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
n−1
k=i
1
k,ifa=1;
n−1
k=i
(a−1) ai−1
ak−1, otherwise.
(5)
Proposition 3.1. Let n,ibe the outdegree of node i in an exponentially preferential tree of size n
and radix 0<a<1. As n →∞,we have
E[n,i]=(1 −a)ai−1(n−i)+O(ai), if a <1;
ln n−ln i+O1
i,if a =1.
Otherwise, a is greater than 1, and the case is ramified according to the relationship between i and n.
As n →∞,we have the phases:
E[n,i]=(a−1) ai−1c(1)
i(a)+O1
an−i,ifixed;
1−1
an−i+O1
ai,n≥i→∞.
Proof. We need to cover three different regimes:
•The subcritical regime (0 <a<1): Start with the lower display in (5) in the form
E[n,i]=(1 −a)ai−1
n−1
k=i
1
1−ak.
A Taylor series expansion of the summand yields:
E[n,i]=(1 −a)ai−1
n−1
k=i1+O(ak)
=(1 −a)ai−1n−i+On−1
k=i
ak
=(1 −a)ai−1(n−i)+O(ai).
•The critical regime (a=1): Here we use the upper display in (5), yielding
E[n,i]=
n−1
k=i
1
k=Hn−1−Hi−1.
Using the asymptotic equivalent in (2), as both nand iapproach ∞,weget
E[n,i]=ln n−ln i+O1
i,
a known result (Javanian and Asl, 2003;Mahmoud,2019).
•The supercritical regime a>1: Again, we start with the lower display in (5). We have two
phases in the lives of the various nodes:
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8R.Aguechet al.
(i) The index iisaconstant.Inthisphase,wehave
E[n,i]=(a−1) ai−1
n−1
k=i
1
ak−1=(a−1) ai−1c(1)
i(a)+O1
an−i.
(ii) The index iis increasing with n. We can approximate the series in the lower display in (5)
with a geometric series. To this end, we quantify the absolute error En,ifrom the bound
En,i:=
n−1
k=i
1
ak−1−
n−1
k=i
1
ak=
n−1
k=i
1
ak−1−
n−1
k=i
1
ak=
n−1
k=i
1
ak(ak−1).
From a Taylor series expansion of the summand, we get
En,i:=
n−1
k=i
1
a2k1+O1
ak
=1
a2i1−1/a2(n−i)
1−1/a2+On−1
k=i
1
a3k
=1
a2(i−1) 1−1/a2(n−i)
a2−1+O1
a3(i−1) (1 −1/a3(n−i)
a3−1
=O1
a2i.
In this phase, we have
E[n,i]=(a−1) ai−1n−1
k=i
1
ak+En,i
=(a−1) ai−1n−1
k=i
1
ak+O1
a2i
=(a−1) ai−11−(1/a)n−i
ai(1 −1/a)+O1
ai
=1−1
an−i+O1
ai.
3.2 Variance
In (4), the indicators IAn,i,forn≥1, are independent. It follows that
Var[n,i]=Varn−1
k=i
IAk,i=
n−1
k=i
Var[IAk,i].
That is, we have
Var[n,i]=(a−1) ai−1
n−1
k=i
1
ak−1−(a−1)2a2i−2
n−1
k=i1
ak−12.(6)
This combinatorial form is reducible in the uniform case (a=1), where we get
Var[n,i]=
n−1
k=i
1
k1−1
k=Hn−1−Hi−1−H(2)
n−1−H(2)
i−1.(7)
Again we have an asymptotic trichotomy.
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Network Science 9
Proposition 3.2. Let n,ibe the outegree of node i in an exponential preferential tree of size n and
radix 0<a<1. We then have
Var[n,i]=(1 −a)ai−11−(1 −a)ai−1(n−i)+O(ai), if a <1;
ln n−ln i+O(1
i), if a =1.
Otherwise, a is greater than 1, and the case is ramified according to the relationship between i and n.
As n →∞,we have the phases:
Var[n,i]=⎧
⎪
⎨
⎪
⎩
(a−1) ai−1c(1)
i(a)
−(a−1)2a2i−2c(2)
i(a)+O1
an−i,ifixed;
2
a+1−1
an−i+a−1
(a+1)a2n−2i+O1
ai,n≥i→∞.
Proof. We need to cover three different regimes:
•The subcritical regime (0<a<1): We first write the exact variance in (6) in the form
Var[n,i]=(1 −a)ai−1
n−1
k=i
1
1−ak−(1 −a)2a2i−2
n−1
k=i1
1−ak2.
As we did in the proof of the mean, a Taylor series expansion for each series yields
Var[n,i]=(1 −a)ai−1(n−i)+O(ai)
−(1 −a)2a2i−2(n−i)+O(a2i)
=(1 −a)ai−11−(1 −a)ai−1(n−i)+O(ai).
•The critical regime (a=1): Asymptotically, as both nand i≤napproach ∞,from(7)and
the asymptotics in (2)–(3), we get
Var[n,i]=ln n+γ+O1
n−ln i+γ+O1
i
−π2
6+O1
n−π2
6+O1
i
=ln n−ln i+O1
i,
a known result (Javanian and Asl, 2003;Mahmoud,2019).
•The supercritical regime (a>1): In (6), we have phases:
(i) The index iis a constant. In this case, we have
Var[n,i]=(a−1) ai−1c(1)
i(a)+O1
an−i
−(a−1)2a2i−2c(2)
i(a)+O1
a2n−2i
=(a−1) ai−1c(1)
i(a)−(a−1)2a2i−2c(2)
i(a)+O1
an−i.
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10 R. Aguech et al.
(ii) The index iis increasing with n. We resort again to the approximation of the two sums
by geometric series. In this phase, we have
Var[n,i]=(a−1) ai−1n−1
k=i
1
ak+O1
a2i
−(a−1)2a2i−2n−1
k=i
1
a2k+O1
a4i
=(a−1) ai−11−(1/a)n−i
ai(1 −1/a)
−(a−1)2a2i−21−(1/a2)n−i
a2i(1 −1/a2)+O1
ai
=1−1
an−i−a−1
a+11−1
a2n−2i+O1
ai
=2
a+1−1
an−i+a−1
(a+1)a2n−2i+O1
ai.
Corollary 3.1. In the subcritical regime (0<a<1), when (n−i)ai→∞,we have
n,i
(n−i)ai
P
−→ 1−a
a,
and in the critical regime (a =1), when n/i→∞,we have
n,i
ln (n/i)
P
−→ 1,
3.3 Distributions
In view of the trichotomy, we observed in the mean and variance, it should be anticipated that the
asymptotic distribution of the outdegree would have three regimes, too, according as where ais
on the real line.
Theorem 3.1. Let n,ibe the outegree of node i in an exponentially preferential tree of size n and
radix a >0. We then have:
(i) Let g(n)be a positive integer-valued function increasing to infinity, such that
g(n)=o(lnn). In the subcritical regime (0 <a<1), we have phases:
(a) In the early phase (1 ≤i≤log 1
an−g(n)),3as n →∞,wehave
4
n,i−1−a
aain
√ain
L
−→ N0, 1−a
a.
(b) In the intermediate phase (i=log 1
an+c,and c ∈Z), we have
dTV n,i,Poi
1−a
aac−{log 1
an}→0.
(c) Let h(n)be a positive integer-valued function increasing to infinity, that grows faster than
a constant, but remains at most n −log1
an.In the late phase (i=log 1
an+h(n)), we
have
n,i
P
−→ 0.
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Network Science 11
(ii) In the critical regime (a=1), we have phases:
(a) In the early phase n/i→∞,we have
n,i−ln (n/i)
ln (n/i)
L
−→ N(0, 1).
(b) In the intermediate phase i ∼cn (and c ∈(0, 1)), we have
n,i
P
−→ Poi ln 1
c.
(c) In the late phase i ∼n,we have
n,i
P
−→ 0.
(iii) In the supercritical regime (a>1), we have phases:
(a) In the early phase (ifixed),asn→∞,we have
n,i
a.s.
−→ ∗
i,
and, for k ≥1, the limiting random variable has the distribution
P(∗
i=k)=(a−1)ka(i−1)klim
n→∞ n−1
=i
1
a−1
i≤j1<j2<···<jk≤n−1
×
i≤m≤n−1
m∈{j1,...,jk}(am−1) −(a−1)ai−1.
(b) Let b(n)be a function growing to infinity, in such a way that n −b(n)also grows to
infinity. In the intermediate phase (1 ≤i=n−b(n)), we have5
P(n,n−b(n)=k)=(a−1)ka(n−b(n)−1)kn−1
=i
1
a−1
×
n−b(n)≤j1<j2<···<jk≤n−1
i≤m≤n−1
m∈{j1,...,jk}(am−1) −(a−1)an−b(n)−1.
(c) In the late phase (1 ≤i=n−c,with c ∈N), we have n,n−c
a.s.
−→
c,and
chas the
distribution
P(
c=k)=(a−1)k
ac(c+1)/2
1≤r1<r2<···<rk≤c
1≤s≤c
s∈{r1,...,rk}
(as−a+1).
Proof. We need to cover three different regimes:
(i) The subcritical regime: Within the regime 0 <a<1, we recognize phases:
(a) The early phase (1 ≤i=log 1
an−g(n)): Let
s2
n,i=Var[n,i]=
n−1
k=i
Var[IAk,i].
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12 R. Aguech et al.
From the calculation of the variance in this regime (cf. Proposition 3.2), we have s2
n,i∼
(1 −a)ai−1(n−i), as ai(n−i)→∞. The case is amenable to normality via Lindeberg’s
central limit theorem, if n−i→∞, in which case we have a sum of a large number of
independent indicators (Bernoulli random variables).
For ito be in this phase, we must have log1
a(ai(n−i)) →∞,asn→∞.Thatistosay,
−i+log1
an1−i
n=−i+log 1
an+log1
a1−i
n=log1
an−i+O(i/n)
must increase to ∞.Iflog 1
an−i→∞, such an asymptotic relation holds, when i
increases up to log 1
an−g(n), for any positive integer function g(n)thatiso(lnn).
Fix ε>0, and define Lindeberg’s quantity
Ln,i(ε)=1
s2
n,i
n−1
k=iIAk,i−E[IAk,i]>εsn,i
I2
Ak,idP,
where Pis the underlying probability measure. The indicators are Bernoulli random
variables bounded by 1. Hence, we have
IAk,i−E[IAk,i]≤IAk,i+E[IAk,i]≤2,
whereas εsn,igrows to infinity, no matter how small εis, or what the value of iis within
the specified phase. In other words, the sets in the integration are all empty for large
enough n(greater than some n0=n0(ε,i)). We can now read the Lindeberg quantity as
Ln,i(ε)=1
s2
n,i
n
k=iφ
I2
Ak,idP=0.
We have verified that, within the phase i=log 1
an−g(n), the quantity Ln,i(ε)→0,
for all ε>0. Thus, we have the Gaussian law
n,i−(1 −a)ai−1(n−i)+O(ai)
(1 −a)ai−11−(1 −a)ai−1(n−i)+O(ai)
L
−→ N(0, 1).
Toward simpler appearance, we use Slutsky’s theorem (Karr, 1993), pp. 146–147 to
remove some factors:
n,i−1−a
aain
√ain
L
−→ N0, 1−a
a.
(b) The intermediate phase (i=log 1
an+c):
Let i=log1
an+c=log1
an−rn+c, where rn={log 1
an}∈[0, 1), and c∈Z.Inthis
phase, we have
ai(n−i)=alog 1
an−rn+c(n−log1
an+rn−c)∼ac−rn
n×n=ac−rn.
In the notation of Theorem 2.1,wehave
λn,1 =
n
k=i
P(IAk,i=1) =
n
k=i
(1 −a)ai−1
1−ak∼(1 −a)ai−1(n−i)∼1−a
aac−rn,
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Network Science 13
and
λn,2 =
n−1
k=i
P2(IAk,i=1)
=
n−i
k=i
(1 −a)2a2i−2
(1 −ak)2
∼(1 −a)2a2i−2n−i+O(ai)
∼1−a
a2a2( log 1
an−rn+c)×n
∼1−a
a2a2c−2rn
n2×n
→0.
By that theorem, we conclude
dTV n,i,Poi
1−a
aac−rn→0.
(c) The late phase (i=log 1
an+h(n)≤n,with h(n)→∞): In this late phase,
Var[n,i]∼(1 −a)ai−1(n−i)→0. As is well known, convergence of the variance of
a sequence of random variables to 0 implies that the sequence converges weakly to a
constant, a consequence of Chebyshev’s inequality. The limiting constant must be the
constant obtained from the L1converges n,i
L
−→ 0.
(ii) The critical regime: In this uniform attachment case, the distribution as stated is known.
We refer the reader to two different proofs in Javanian and Asl (2003); Mahmoud (2019).
(iii) The supercritical regime: In this regime we, work from the exact distribution to produce
local limit theorems in the different phases. For n,ito be equal to k, node imust recruit
ktimes and fail to recruit n−i−ktimes. We can partition the event n,i=kinto disjoint
sets according to the the size of the tree at the times of recruiting. Suppose the ksuccesses
in recruiting occur when the tree sizes are i≤j1<j2<...jk≤n−1. The probability of
this event is
∈{j1,...,jk}
P(IA,i=1)
i≤m≤n−1
m∈{j1,...,jk}
P(IAm,i=0).
Using the probabilities in the lower display in (1), we obtain
P(n,i=k)=
i≤j1<j2<···jk≤n−1
∈{j1,...,jk}
P(IA,i=1)
×
i≤m≤n−1
m∈{j1,...,jk}
P(IAm,i=0)
=
i≤j1<j2<···<jk≤n−1
∈{j1,...,jk}
(a−1)ai−1
a−1
×
i≤m≤n−1
m∈{j1,...,jk}1−(a−1)ai−1
am−1(8)
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14 R. Aguech et al.
=(a−1)ka(i−1)kn−1
=i
1
a−1
i≤j1<j2<···<jk≤n−1
×
i≤m≤n−1
m∈{j1,...,jk}(am−1) −(a−1)ai−1.(9)
According to Kolmogorov’s criterion (Theorem 22.3 in Billingsley (2012)), for indepen-
dent zero-mean random variables X1,X2,X3,...,when∞
k=1Var[Xk]<∞, the sum
n
k=1Xkconverges almost surely to a limit.
In the supercritical regime, when n−i→∞,wehave
n
k=i
VarIAk,i−E[IAk,i]≤∞
k=1
(a−1) ai−1
ak−11−(a−1) ai−1
ak−1<∞.
In view of Kolmogorov’s criterion, when n−i=g(n)→∞,
n,i−E[n,i]=
n
k=i
IAk,i−
n
k=i
E[IAk,i]
converges to a limit.
Having shown that in the supercritical phase we have n
k=iE[IAk,i]=E[n,i] is convergent
(cf. Proposition 3.1), we see right away that, if n−i→∞,wewouldhaven,i=n
k=iIAk,i
converging to a limit.
(a) The early phase (ifixed): Certainly, in this phase n−i→∞,asn→∞.By
Kolmogorov’s criterion, n,iconverges to a limit, which we call
i. We can determine
the distribution of
i:=limn→∞ n,iby the following argument.
Since n,iconverges almost surly, it also converges in distribution. The limit of the latter
probabilities exists (and must be the distribution of the almost-sure limit, too). Indeed,
n,iconverges almost surely to a limit ∗
iwith a distribution determined as the limit of
the probabilities in (9).
(b) The intermediate phase (igrows faster than a constant, but slower than n−c,for any
c∈N). In this phase, iis n−d(n), for an integer-valued function d(n)→∞, in such a
way that n−d(n) also tends to infinity. In this case, for any fixed k,(8) takes the form:6
P(n,n−d(n)=k)=
n−d(n)≤j1<j2<···<jk≤n−1
∈{j1,...,jk}
(a−1)an−d(n)−1
a−1
×
n−d(n)≤m≤n−1
m∈{j1,...,jk}1−(a−1)an−d(n)−1
am−1
=(a−1)kak(n−d(n)−1)
n−1
r=n−d(n)(ar−1)
n−d(n)≤j1<j2<···<jk≤n−1
×
n−d(n)≤m≤n−1
m∈{j1,...,jk}am−1−(a−1)an−d(n)−1.
(c) The late phase (1≤i=n−c,andc∈N): Starting with the probabilities in the form (8),
we write an asymptotic equivalent.
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Network Science 15
Note that while the indices in the sum are large numbers there, is only a finite number
of them (cof them to be exact). It is therefore legitimate to take the asymptotic terms
individually:
P(n,n−c=k)∼
n−c≤j1<j2<···<jk≤n−1
∈{j1,...,jk}
a−1
a−(n−c)+1
×
n−c≤m≤n−1
m∈{j1,...,jk}1−a−1
am−(n−c)+1
∼(a−1)k
n−1
r=n−car−(n−c)+1
×
n−c≤j1<j2<···<jk<n−1
n−c≤m≤n−1
m∈{j1,...,jk}
(am−(n−c)+1−a+1)
→(a−1)k
aa2...ac
1≤r1<r2<···<rk≤c
1≤s≤c
s∈{r1,...,rk}
(as−a+1).
3.4 Illustrative examples
In any of the three regimes of a, there is an intriguing interplay between nand i. We only dis-
cuss interpretations and examples from the subcritical and supercritical regimes, since the critical
phase is well studied, and illustrative examples of it can be found elsewhere. We refer a reader
interested in a discussion of the critical regime to Javanian and Asl (2003); Mahmoud (2019).
3.5 The subcritical regime
When a<1, the term ai−1in E[n,i] decreases exponentially fast in i.Takea=1/2, for instance.
With this radix, for iin the subcritical phase, we have the convergence
n,i−n
2i
n
2i
L
−→ N(0, 1).
A Gaussian law holds so long as iis well below log 1
an(differing by an increasing function
from that critical level). When iapproaches the critical phase, Poisson approximations kick in to
replace the normal distribution. When iis tied to log 1
anby a constant, it is related to log 1
an
via corrections obtained by removing the floors. These corrections are oscillating functions in n
and are uniformly dense on the real line (Kuipers and Niederreiter, 1974). So, there is not really
convergence to a Poisson limit, but rather approximations to a family of Poisson distributions,
with parameters lying in the range [ 1
2c,2
2c).
Table 1shows the behavior in the subcritical regime for a=1/2 and some selected phases. Note
the fourth and fifth entries (from the top) which lie in the intermediate phase, where there is no
limit per se, but rather good approximations by various Poisson distributions. For instance, in the
phase i=log2n−5, Poi (18.4136)is a good approximation for the distribution of 20000,10 at
n=20000, Poi (18.4182)is a good approximation for the distribution of 20001,10 at n=20001,
and Poi (18.4228)is a good approximation for the distribution of 20002,10 at n=20002.
For the last two entries in Table 1, the variance is diminishing at a fast rate, and the sum of
the variances converges. By Kolmogorov’s theorem, we have an almost-sure convergence in both
cases.
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16 R. Aguech et al.
Tab le 1 . The asymptotic mean, variance, and distribution of the outdegree of an exponentially preferential tree
with radix 1/2 in some selected phases
iMean Variance Distribution
1n
2
n
4
n,1−1
2n
√n
L
−→ N0, 1
4
...............................................................................................................................................................................................................................................
5n
32
n
32
n,5−1
32 n
√n
L
−→ N0, 1
32
...............................................................................................................................................................................................................................................
ln ln n−38n
2ln ln n
8n
2ln ln n
n,ln ln n−3−8n
2ln ln n
8n
2ln ln n
L
−→ N(0, 1)
...............................................................................................................................................................................................................................................
log2n−532×2{log2n}32 ×2{log2n}dTV n,log2n−5,Poi 32 ×2{log2n}→0
...............................................................................................................................................................................................................................................
log2n+52{log2n}
32
2{log2n}
32 dTV n,log2n+5,Poi 2{log2n}
32 →0
...............................................................................................................................................................................................................................................
√n+πn
2√n+π
n
2√n+πn,√n+π
a.s.
−→ 0
...............................................................................................................................................................................................................................................
n−532
2n32
2nn,n−5
a.s.
−→ 0
Tab le 2. The limiting value of the outde-
gree of the first few entries in an expo-
nentially preferential tree with radix 2
ilimn→∞ E[n,i]
1 1.607
.......................................................................................
2 1.213
.......................................................................................
3 1.093
.......................................................................................
4 1.044
.......................................................................................
5 1.021
.......................................................................................
6 1.010
.......................................................................................
7 1.005
.......................................................................................
8 1.002
.......................................................................................
9 1.001
.......................................................................................
10 1.000
.......................................................................................
11 1.000
.......................................................................................
12 1.000
3.6 The supercritical regime
As an instance, take a=2. Over an extended period of time, the average root outdegree converges:
E[n,1]→c(1)
1(2) ≈1.607. Table 2shows the asymptotic outdegree for the first 12 entries in the
tree, approximated to three decimal places.
A late entry in the tree, such as i=n−n1/4, has an average outdegree
E[n,i]=1+O1
2n1/4
A much later entrant, such as a node with with the index n−4 has an average outdegree
E[n,i]=1−1
24+O1
2n=15
16 +O1
2n.
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Network Science 17
We discern a “thinning” occurring in the tree. At a node with a high index i, the subtree of
descendants is tapered almost into a path.
At a=2, cancellations occur in the formula in the expression in Theorem 3.1 (iiia), greatly
simplifying the calculation for the root outdegree (also its degree, the case i=1) and giving
transparency into the limiting distribution:
P(n,1 =k)→P(∗
i=k)
=lim
n→∞ n−1
=1
1
2−1
1≤j1<j2<···jk≤n−1
m∈{j1,...,jk}
(2m−2).
Note that if j1= 1, the product retains m=1 as one of its indices and 2m−2=0 for this value
of m, annihilating the entire part of the expression with j1=1 in the sum, simplifying it further to
P(∗
1=k)=lim
n→∞ n−1
=1
1
2−1
1<j2<···<jk≤n−1
m∈{1,j2,...,jk}
2(2m−1−1)
=lim
n→∞ n−1
=1
1
2−1
1<j2<···<jk≤n−1
2n−k−1n−1
m=2(2m−1−1)
(2j2−1−1) ···(2jk−1−1)
=lim
n→∞ 2n−k−1
2n−1−1
1<j2<···<jk≤n−1
1
(2j2−1−1) ···(2jk−1−1)
=1
2k
1<j2<···<jk≤∞
1
(2j2−1−1) ···(2jk−1−1).
The first few values in the sequence P(∗
1=k)are
7
P(∗
1=1) =1
2;
P(∗
1=2) =1
4
∞
=2
1
2−1≈0.1516737881 ...
P(∗
1=3) =1
8
∞
=2
∞
m=+1
1
(2−1)(2m−1) ≈0.01442126698 ....
The first three values alone contain about 0.6661 of the mass of the limiting distribution. The
root comes early and stays the longest in the tree. So, it has repeated chances for recruiting.
However, the probabilities are very quickly diminished by the appearance of nodes of higher
indices, with higher chances of recruiting. While the limit distribution of the root outdegree
(which is also the degree) is supported on N, there is a high probability of remaining small
(confined to the values 1,2,3).
The probability formula in the intermediate phase of the supercritical regime is unwieldy
(Theorem 3.1 (iiib)), yet it can be used to tell us something about the asymptotic structure of
the tree. With a=2andi=n−ln n, we can compute the probability of the intermediate node
iin the following way. Here, b(n)isln n.Fork=0, the set {j1,...,j0}is empty, and the only
product that stands is the one on the full set {n−h(n), ...,n−1}.Wehave
P(n,n−ln n=0) =1
n−1
r=n−ln n(2r−1)
n−1
m=n−ln n2m−1−2n−ln n−1.
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18 R. Aguech et al.
At n=1000, this probability is about 0.2933.
The very late nodes have the lion’s share. Consider a=2, and the tree when the size is some
large n. The outdegree of node nis 0, as it has not recruited yet (which is consistent with zero mean
and zero variance as given by the exact and asymptotic formulas). Being of the second to highest
index in the tree, node n−1, has a chance of recruiting node nand has a chance of missing. The
probability of node n−1 recruiting node nis 2n−1(2 −1)/(2n−1) →1/2. Thus, the outdegree
of the penultimate node is asymptotically distributed like a Bernoulli(1/2) random variable.
Node n−2 has two chances at recruiting by time nand has an asymptotic distribution on
{0, 1, 2}with mean 1/4, and so on. According to Theorem 3.1 (iiic), we have
P(
n,n−2=k)→P(
2=k)→1
23
1≤r1<r2<···<rk≤2
1≤s≤2
s∈{r1,...,rk}
(2s−1),
for k=0, 1, 2. For k=0, the set r1,r2,...,r0is empty, and we compute
P(
2=0) →1
8
1≤s≤2
s∈φ
(2s−1) =1×3
8=3
8.
Further, we have
P(
2=1) →1
8
1≤k1≤2
1≤s≤2
s∈{k1}
(2s−1) =3+1
8=4
8.
We can find P(
2=2) from 1 −P(
2=0) −P(
2=1) =1/8. In summary, we have
2=⎧
⎪
⎨
⎪
⎩
0, with probability 3/8;
1, with probability 4/8;
2, with probability 1/8.
Remark 3.2. We discussed phases, where the growth of i is systematically increasing toward n.
However, there is no limit to how bizarre the sequence i =i(n)can be. For example, i(n)might
be a sequence alternating between two (or more values), such as the sequence i(n)=5+(−1)n,in
which case the degree of node i does not converge to a limit. Or, i(n)might alternate between low and
high values, such as
i(n)=1n,even;
ln n+0.76884,nodd.
Even worse, i(n)may not have any structure at all. We reckon that such sequences are not interesting
and do not appear in practice.
4. Concluding remarks
We discussed an exponentially preferential model of recursive trees, wherein node irecruits with
probability proportional to ai. The proportionality constant is time dependent, which captures
the reality of dynamic change in networks. The radix agoverns the behavior of the tree. The case
a=1 is critical and corresponds to the standard well-studied uniform model of random recursive
trees. This criticality creates three regimes, the subcritical (0 <a<1), the critical (a=1), and the
supercritical (a>1). Each regime has its own early, intermediate, and late phases. They are not
the same. For instance, the early phase in the subcritical regime extends to O(lnn), whereas in the
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Network Science 19
critical regime, the early phase stops at n/i→∞and in the supercritical regime the early phase is
restricted to fixed i.
Gaussian behavior appears only in the early phases of the subcritical and critical regimes. Other
behavior is detected, too. For instance, Poisson approximations are the appropriate asymptotic
behavior in the intermediate phases of both the subcritical and critical regimes, with oscillations
in the case of the subcritical regime.
Consistently, in the late phase of all three regimes (noting they start differently), n,iconverges
to a constant, with constant being 0 in the subcritical and critical regimes, while it is a positive
constant in the supercritical regime. The intuition behind these constants is that in the subcritical
and critical regimes, the nodes in the early and intermediate phases gobble up the largest share of
recruits, but in the supercritical regime, the highest labeled nodes in the late stage are the most
attractive.
The dependence on the radix ahosts a broad range of applicability. For example, with a=1
alone, the entire class of uniform recursive trees comes in with its plethora of known applications.
The range 0 <a<1 corresponds to applications where early nodes are the most attractive, such as
the growth of companies and subsidiary branches, where the headquarters and its closest branches
acquire influence and wealth over time. The case a>1 corresponds to applications where late
nodes are the most attractive, such as some social networks in which newcomers are the most
anxious to expand their circles.
Future research may extend to deal with other tree properties, such as the maximal degree,
the depth of nodes, and the total path length (among many others). We may also consider other
weights and generalizations.
Supplementary material. The supplementary material for this article can be found at https://doi.org/
10.1017/nws.2025.3.
Notes
1The superscript ris often dropped, when it is 1.
2The number γ≈0.5772 is Euler–Mascheroni constant.
3The reader should be alerted to that the words “early,” “intermediate,” and “late” mean different things in the different
regimes.
4Oneshouldtakenotethatiis a node index and is always an integer.
5The ultimate formula is unwieldy, but can be used to discover the probability for small k, such as, for example that an
intermediate node is a leaf (k=0).
6The ultimate formula is unwieldy, but can be used to discover the probability for small k, such as, for example that an
intermediate node is a leaf (k=0).
7For k=1, the sum is for a product that does not exist, to be interpreted as 1.
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Cite this article: Aguech R., Mahmoud H., Mohamed H. and Yang Z. (2025). Exponentially preferential trees. Network Science
13, 1–20. https://doi.org/10.1017/nws.2025.3
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