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On the Skewed Degree Distribution

of Hierarchical Networks

Bijan Ranjbar-Sahraei

Maastricht University, The Netherlands

Email: b.ranjbar@ieee.org

Karl Tuyls

University of Liverpool, United Kingdom

Email: k.tuyls@liverpool.ac.uk

Haitham Bou Ammar

University of Pennsylvania, United States

Email: haithamb@seas.upenn.edu

Gerhard Weiss

Maastricht University, The Netherlands

Email: gerhard.weiss@maastrichtuniversity.nl

Abstract—In this paper, a prestige-based evolution process

is introduced, which provides a formal framework for the

study of linear hierarchies seen in human societies. Due to

the deterministic characteristics of the proposed model, we are

capable of determining equilibria in closed form. Surprisingly,

these stationary points recover the power-law degree distribution

as the shared property of the resulting hierarchal networks,

explaining the prevalence of hierarchies in societies. This result

sheds light on the evolutionary advantages of hierarchies.

I. INTRODUCTION

To analyze the emergence of social networks, a variety of

mathematical models have been proposed. The earliest dates

back to the 1900’s, where Yule [1] studied the biological

evolution of species based on age and population data. Others,

e.g., Lotka [2] provided rules required for describing and

analyzing scientiﬁc publications. Resulting from these and

other studies, was the emergence of the power-law degree

distribution [3] as a shared common characteristic for a wide-

range of networks including but not limited to, the world wide

web, protein-protein interaction, airlines and social networks.

Given such a widely-shared characteristic, Barab´

asi and

Albert suggested a preferential attachment model for the

generation of scale-free graphs exhibiting a power-law degree

distribution [4]. As noted by Durret [5], the deﬁnition of

their process was rather informal. Since then, different precise

forms of the Barab´

asi-Albert model have been studied in

literature [6]. Though successful at recovering the power-law

degree distribution, these studies impose several restricting

assumptions on the underlying graph generating process. For

instance, such techniques typically adopt a binary attachment

model, in which two nodes are either connected or not [4], [7].

On the other hand, the existence of hierarchical relation-

ships is another shared common characteristic for a wide-range

of networks. Research has shown that human physique (e.g.,

stature) and body hormones play a crucial role in enabling

dominance in the society. Contrary to animal societies which

base hierarchies on dominance, however, human societies

replace dominance by “prestige” to construct reciprocal rela-

tionships between leaders and followers [8]. Thus, evolutionary

considerations of real-world networks suggests the emergence

of scale-free behavior (i.e., networks exhibiting a power-law

degree distribution) in networks as a resultant of hierarchal

attachment processes which are not reﬂected through current

preferential attachment models. Apart from this modeling

restriction, another problem inherent to existing binary models

lies in their explanatory capabilities. For instance, they fail to

manifest connection strengths between individuals; a property

being at the core of behavioral emergence in real networks [9].

To provide more realistic modeling outcomes, in this paper,

we contribute by proposing a deterministic hierarchal graph at-

tachment process for prestige-based human societies. Contrary

to preferential attachment models, our approach only assumes

hierarchal connections between individuals, thus bridging the

modeling gap to real-world evolutionary networks. Among

many advantages, our deterministic setting enables the deriva-

tion of the degree distribution in closed-form. Performing

this derivation recovers, surprisingly, the power-law degree

distribution as the main property of the resultant hierarchal

networks, which explains the prevalence of such hierarchies

in societies.

II. NOTATI ON S

A network is described via a graph, G= (V,W), con-

sisting of a set of Nnodes (or vertices) V={v1,...,vN}

and an N×Nadjacency matrix W= [wij ]where non-

zero entries wij indicate the weighted connection from vj

to vi. In this paper we consider the undirected graphs which

have symmetrical W. The neighborhood Nof a node vi

is deﬁned as the set formed by its connected vertices, i.e.,

N(vi) = ∪jvj:wij >0.

The node’s degree, deg(vi), is given by the cardinality of its

neighborhood. The strength of a node is of major importance

in hierarchical networks. Next, we deﬁne three concepts: (1)

relative strength (2) strength observation and (3) absolute

strength.

Relative Node Strength: The relative strength of jth node

with respect to ith node with i>jis denoted by Ψi(vj)and

represents the sum over all edge weights between jth node

and every kth node where k < i. Namely,

Ψi(vj) =

i−1

X

k=1

wjk , i > j. (1)

In other words, when node iis monitoring node jwith

j < i, it just observes those connections from other ks to j

which k < i.

Strength Observation The Strength observation of the ith

node is denoted by the vector

~

Ψi= [Ψi(v1),Ψi(v2),...,Ψi(vi−1)]T

with cardinality i−1.

Absolute Node Strength: The absolute strength of ith

node is deﬁned as

Ψ(vi) =

N

X

k=1

wik.(2)

III. NET WORK DYNAM IC S

Here, we propose a dynamical process which captures the

edge dynamics of a complete network. Let ω={wij |∀i, j =

1,2, ..., N, i > j}denote the state vector of the process, where

each state variable wij corresponds to the weight of the link

between the jth and ith node. In the very general case, one

considers the rate of changes in wij as a function of all state

variables

˙wij =f(ω)(3)

In this paper, however, the focus is on hierarchical networks

in which for any i>j,˙wij is a function of wij itself and the

strength observation of the ith node ~

Ψi

˙wij =fΨ(wij ,~

Ψi), i > j (4)

In other words, the dynamics of the linking strength between

iand jis independent of any other node lwhere lis higher

than ior jin the hierarchy.

Using fΨfrom Equation 4, and sorting the state variables

wij increasingly (based on Ni+j), the overall dynamic process

can be written as

˙ω=d

dtw21, . . . , wN(N−1) T

=hfΨ(w21,~

Ψ2), . . . , fΨ(wN(N−1),~

ΨN)iT

Next, we introduce a possible strategy for fΨ(.), namely

f(P)

Ψ(·)which represents the Prestige-based attachment (PA)

models.

IV. PRESTIGE-BA SED ATTACHMENT MODEL

The overall strength of node iin establishing connection

with every other jth node is assumed to be limited and sums-

up to 1. The prestige-based attachment model can be formally

derived as follows. Let

˙wij =f(P)

Ψ(wij ,Ψi(vj),

~

Ψi

)

and

f(P)

Ψ(wij ,Ψi(vj),

~

Ψi

) = Ψi(vj)

~

Ψi

−wij , j < i (5)

where ||.|| denotes the ﬁrst norm.

By studying the dynamic process proposed in Equation 5,

it can be easily seen that ˙wij , i > j is a function of every wkl

for k, l < i. Without loss of generality we assume w(P)

11 = 1,

such that

Ψ2(v1)=1 (6)

and w(P)

ii = 0 for every i > 1. It is straightforward to compute

the equilibrium point of this system

w(P)

ij =Ψi(vj)

~

Ψi

(7)

The equilibrium point (7) explains that the connection

strength between node iand node jdepends on the connection

strength among nodes iand jand every kth node with

k < max(i, j). To illustrate, imagine Nagents who are all

connected to each other, and continuously the agents with

higher order share their available resources with agents with

lower order. The strength of link between iand jshows

the amount of resources which are transmitted from ito j.

According to (7) the 2nd individual shares all of her resources

with 1st individual (i.e., w21 =1

1= 1). The 3rd individual

shares one third of her resources with the 2nd individual and

two third of it with the 1st individual (i.e., w32 =1

1+2 =1

3

and w31 =2

1+2 =2

3). With the same respect, ith individual

shares portions of her resources with each of the jindividuals

where j < i, while those with lower order receive more. We

call this a prestige-based model as the lower orders reﬂect a

kind of prestige in the group and high prestige agents receive

more than agents with lower prestige.

An immediate result of (7) is that

i

X

j=1

w(P)

ij =

i−1

X

j=1

w(P)

ij =

i−1

X

j=1

Ψi(vj)

~

Ψi

=

~

Ψi

~

Ψi

= 1.(8)

Next, we study the amount of resources each individual

receives in such prestige-based network (captured by node’s

strengths), and also compute the distribution of node strengths.

A. Analysis of Node’s Strength

Here we determine a closed form solution for the sum over

the strength of every jth node from the perspective of the ith

node, where j < i.

Lemma 1: In the prestige-based attachment model, the

sum of the relative node strengths of every jth node from

perspective of the ith node, where j < i is as following

K(i) : k~

Ψik= 2i−3.

Proof: The above lemma can be proved by using induc-

tion:

Initial Step: According to Equation (6) we have k~

Ψ2k=

Ψ2(v1)=1. Therefore, K(i)holds for i= 2.

Inductive Step: Let

K(i−1) : k~

Ψi−1k= 2i−5,

and also note that Ψi(vj) = Ψi−1(vj) + w(P)

(i−1)j. Therefore

we can write

k~

Ψik=

i−1

X

j=1

Ψi(vj)

=Ψi(vi−1) +

i−2

X

j=1 Ψi−1(vj) + w(P)

(i−1)j

=

i−1

X

j=1

w(P)

(i−1)j+k~

Ψi−1k+

i−2

X

j=1

w(P)

(i−1)j

By using K(i−1) and Equation 8, we’ll get

k~

Ψik=

i−1

X

j=1

Ψi(vj) = 1 + 2i−5 + 1 = 2i−3(9)

Therefore, K(i)holds for every i, concluding the proof.

B. Analysis of Edge Weights

We can compute the edge weight between ith node and

jth node as follows.

Lemma 2 (Edge Weight): For the weighted graph G,

evolved with PA model, ith node is connected to jth node

with an edge of weight

K(i) : w(P)

ij =1

2i−2

i−j

Y

k=1

2i−2k

2i−2k−1,∀j < i. (10)

Proof:

The validity of Equation 10 can be proved for each iand

for every j < i using induction.

Initial Step: The second node is connected to the ﬁrst node

with w(P)

21 = 1, meaning that K(2) holds.

Inductive Step: Now assume that

K(i−1) : w(P)

(i−1)j=1

2i−4

i−j−1

Y

k=1

2i−2k−2

2i−2k−3

holds for every j < i −1. For computing the edge weight

between ith and jth node, recall that Ψi(vj) = Ψi−1(vj) +

w(P)

(i−1)j. By using (7) and Lemma 1, it can be seen that:

Ψi(vj) = Ψi−1(vj) + w(P)

(i−1)j

= (2i−5)w(i−1)j+w(P)

(i−1)j

= (2i−4)w(i−1)j(11)

Using Equations 7, 11 and Lemma 1, the edge weight

between ith and jth nodes can be written as

w(P)

ij =Ψi(vj)

Pi−1

k=1 Ψi(vk)=1

2i−2

i−j

Y

k=1

2i−2k

2i−2k−1

for j < i −1. Therefore, K(i)holds for every i, concluding

the proof.

Before, computing the distribution of strengths for the PA

model, we present the following proposition providing the

relative strength of the jth node from the perspective of the

ith for every i>j (i.e., Ψi(vj)) in closed form.

Proposition 1 (Relative Node Strength): For the weighted

graph G, evolved according to the PA model, the strength of

the jth node from perspective of the ith node is given by

K(i) : Ψi(vj) = Qi

k=j+2 2k−4

2k−5for j < i −1

Ψi(vj) = 1 for j=i−1.(12)

Proof: Again, induction can be used to prove the validity

of Equation 12. Starting with the initial step we get

Initial Step: From Equation 6, the strength of the ﬁrst

node from the perspective of the second node is Ψ2(v1)=1.

Besides, using Lemma 2 we can deduce that

Ψ3(v1) = w(P)

11 +w(P)

21

3=2

3.

Therefore, K(2) holds. For the inductive step we proceed as

follows

Inductive Step: Assume that following holds.

K(i−1) : Ψi−1(vj) = Qi−1

k=j+2 2k−4

2k−5for j < i −2

Ψi−1(vj)=1 for j=i−2.

For computing Ψi(vj), consider Ψi(vj) = Ψi−1(vj) +

w(P)

(i−1)j. Using Equation 7 and Lemma 1, we can show that

for every j < i −1

Ψi(vj) = Ψi−1(vj) + w(P)

(i−1)j=

i

Y

k=j+2

2k−4

2k−5

Besides using Equation (8), Ψi(vj)= 1 for j=i−1.

Therefore, K(i)holds for every iand the proof is concluded.

Lemma 3 (Global Strength): For the weighted graph G,

evolved with the PA model, the global strength of the ith node

is Ψ(vi) = QN+1

k=i+2 2k−4

2k−5for i<N

Ψ(vi)=1 for i=N. (13)

Proof: We know that Ψ(vi) = ΨN(vi) + w(P)

iN for every

i<N. Using Equation 7 and Proposition 1, we have

Ψ(vi) = ΨN(vi) + ΨN(vi)

2N−3

=2N−2

2N−3

N

Y

k=i+2

2k−4

2k−5

=

N+1

Y

k=i+2

2k−4

2k−5

for every i < N. Based on Equation (8) we have

Ψ(vN) =

N−1

X

i=1

w(P)

Ni = 1.

This concludes the proof.

Finally, we can compute the strength distribution in a

closed form. In analysis of weighted networks, typically the

Distribution Function (DF) of node strengths is deﬁned as

P(k) =

vi|∀i, k ≤Ψ(vi)< k + 1

(14)

where Ψ(vi)deﬁned in (2) denotes the strength of node vi.

To ease the analysis, in this work we make use of the Com-

plementary Cumulative Distribution Function (CCDF) deﬁned

as:

Pc(k) =

vi|∀i, Ψ(vi)≥k

(15)

The following lemma clariﬁes the relation between the DF and

CCDF for a network with power-law Distribution.

Lemma 4 (Power-law Distribution): Consider a power-

law distribution in form of P(k) = ck−α, where αis the

power-law exponent. The CCDF Pc(k)also follows a power-

law but with an exponent α−1.

Proof: Can be easily seen by simple integration.

The following theorem provides the strength distribution

of a PA model.

Theorem 1 (Strength Distribution): For the weighted

graph Gevolved with the PA model, the distribution of the

global strength kfollows a power-law with exponent −3

P(k)∝k−3.

For proving Theorem 1, we need the following lemma, ﬁrst.

Lemma 5 (Fraction Product Series): Consider the follow-

ing product of fractions

L(i, N ) =

N+1

Y

k=i+2

2k−4

2k−5,

then γi−1

2< L(i, N )< γ(i−1)−1

2, with γ=√N−1

Proof: We use the comparison test to compute the lower

and upper bounds of L(i, N ). Firstly, consider

Q(i, N ) =

N+1

Y

k=i+2

2k−5

2k−6(16)

Clearly, L(i, N)< Q(i, N )and L(i, N )Q(i, N) = 2N−2

2i−2.

Therefore, L(i, N )<q2N−2

2i−2≤γ(i−1)−1

2concluding the

upper-bound. To determine the lower bound, deﬁne

Q0(i, N ) =

N+1

Y

k=i+2

2k−3

2k−4(17)

It can be shown that L(i, N)> Q0(i, N )and

L(i, N )Q0(i, N) = 2N−1

2i−1. Therefore,

L(i, N )>r2N+ 1

2i−1>r2N−2

2i≥γi−1

2.(18)

Using results of Lemmas 3, 4 and 5 we give the following

proof for Theorem 1.

Proof: The following lower and upper bounds can be

computed for the strength of the ith node

γi−1

2<Ψ(vi)< γ(i−1)−1

2(19)

where γ=√N−1. From Equation (19), we have

Ψ(vi)≥k, for i∈ {1,2,3,...,γ2

k2}(20)

Pc(k) =

1,2,3,..., γ2

k2

'γ2k−2(21)

Therefore,

Pc(k)∝k−2(22)

Using Lemma 4, we have

P(k)∝k−3,(23)

thus proof is concluded.

V. CONCLUSION

In this paper we proposed a dynamical model for prestige-

based hierarchical systems. Although, the dynamical system

was described using simple hierarchical rules, the derived sta-

tionary points had been shown to recover a power-law degree

distribution with exponent -3 (Theorem 1). Emergence of this

degree distribution despite the simple hierarchical structure

explains how hierarchical social structures have survived in

human societies. There are various interesting future directions

of this work. We plan to validate the attained results through

data gathered from real-world networks. Moreover, our model

is proposed in form of a dynamical process, which makes it

possible for the development of control strategies. The overall

idea would be the control of the evolution of the network to

arrive at speciﬁc network forms.

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