A Pyramid Scheme Model Based on" Consumption Rebate" Frauds

arXiv:1904.08136v3 [physics.soc-ph] 26 Jun 2019

A Pyramid Scheme Model Based on “Consumer

Rebate” Frauds

Yong Shi

∗

, Bo Li

†

and Wen Long

‡

School of Economics and Management,

University of Chinese Academy of Sciences, Beijing, 100190, China

Research Center on Fictitious Economy and Data Science,

Key Laboratory of Big Data Mining and Knowledge Management,

Chinese Academy of Sciences, Beijing, 100190, China

June 27, 2019

Abstract

There are various types of pyramid schemes which have inﬂicted or are in ﬂicting

losses on many people in the world. We propose a pyramid scheme model which

has the principal characters of m any pyramid schemes appeared in recent years:

promising high returns, rewarding the participants recruiting the next generation

of participants, and the organizer will take all the money away when he ﬁnds

the money from the new participants is not enough to p ay the previous partici-

pants interest and rewards. We assume the pyramid scheme carries on in th e tree

network , ER random network, SW small-world network or BA scale-free network

respectively, then give the analytical results of how many generations the pyramid

scheme can last in these cases. We also use our model to analyse a pyramid scheme

in the real world and we ﬁnd the connections between participants in th e pyramid

scheme may constitute a SW small-world network.

∗

Email: y s[email protected]

†

Email:libo312@mails.ucas.edu.cn

‡

Email: lo[email protected], corresponding author

1 Introduction

The column Topics in Focus of China Centr al Television (CCTV), one of the most-

watched shows on CCTV, exposed a so-called “ consumption rebate” platform named

“RenRenGongYi” on June 18, 2017 which actually was a pyramid scheme

. The alleged

operation pattern of “RenRenGongYi” was to let franchisees give the ﬁxed proportion

of customers’ consumption to the platform to form a fund pool, such as 24%, 12% or

6%, then the platform return the money t o customers and franchisees by instalments.

But most transactions are fa br icat ed in “RenRenGongYi”, and the platform in fact

lured the participants by promising high returns to invest in the f und pool and rewarded

the participants who attra cted the next generation of participants. Franchisees may be

mostly ﬁctitious because most transactions are fabricat ed, and the participants of t he

pyramid scheme are mainly consumers. For example, If a participant fa bricat es a 100

yuan consumption (the pa r ticipant was both the f r anchisee and the consumer) and gave

24 (or 12 or 6) yuan to the platform, he/she could gradually get a consumption rebate

of nearly 1 00 yuan, which was more than 4 (or 8 or 16) times the principal. In less than

a month, 5,267 franchisees and 48,505 consumers were involved in the platform. From

the pr oject oﬃcially opened on December 1, 2016 to the crash at the end of the month,

the amount absorbed in just one month reached 1 billion RMB.

Besides “RenRenGongYi”, there are many more platforms of this form and Chinese

government have warned the risk of “consumption rebate” pla tforms

. There are many

other forms of pyramid schemes, such as IGOFX

originated from Malaysia and MMM

originated from Russia.

Pyramid schemes are diﬀerent from ordinary Ponzi schemes named after the epony-

mous fraudster Charles Ponzi(1882-1949), though in both Ponzi and pyramid schemes,

existing investors are paid by the money of new investors

. In a Ponzi scheme pa r-

ticipants believe they are actually earning returns from their investment. While in a

pyramid scheme, participants are aware that they are earning money by ﬁnding new

participants. They become part of the scheme.

Pyramid schemes and Ponzi schemes have been researched fro m some diﬀerent per-

spectives. Joseph Gastwirth proposed a probability model of a pyramid scheme and

concluded that the vast majority of participants have less than a ten percent chance of

recouping their initial investment [

1]. St imulated by the Madoﬀ investment scandal in

Topics in Focus on June 18, 2017 can be viewed at http://tv.cctv.com/2017/06/18/VIDE8gtfpFkpiB

v2QYnNuGIF170618.shtml.

http://www.mps.gov.cn/n2253534/n2253543/c6108362/content.html

https://news.china.com/news100/11038989/20170707/30931598

all.html

https://en.wikipedia.org/wiki/MMM

(Ponzi scheme company)

http://www.51voa.com/VOA Special English/pyramid-vs-ponzi-69196.html

2008, Marc Artzrouni put forward a ﬁrst order linear diﬀerential equation to describe the

Ponzi schemes [

2]. The model of Marc Artzrouni depends on the following parameters:

a promised but unrealistic interest rate, the actual realized nominal interest rate, the

rate at which new deposits are accumulated and the withdrawal rate. Marc Artzrouni

gave the conditions on these parameters for the Ponzi scheme to be solvent or to col-

lapse. The model was ﬁtted to data available on Charles Ponzi’s 1920 eponymous scheme

and illustrated with a philanthropic version of the scheme. Tyler Moore et al. make an

empirical ana lysis of these High Yield Investment Programs but not put forward a math-

ematical model [3]. A High Yield Investment Program (HYIP) is considered to be an

online Ponzi scheme, because it pays outrageous levels of interest using money from new

investors. Diﬀerent from the traditional Ponzi schemes, there are many sophisticated

investors understanding the fraud, but hope to proﬁt by joining early, and investors can

not withdraw their mo ney at any time. Anding Zhu, Peihua Fu et al. researched some

problems when Po nzi scheme diﬀuses in complex networks [

4, 5].

Since the introduction of random networks, small-world networks and scale-free net-

works, complex networks have attracted great attention f r om researchers in various ﬁelds

such as management and statistical physics. Researches show that many natural and

social phenomena have small-world or scale-free char acteristics. At present, complex net-

works have been successfully applied to improve transportation networks [

6, 7], analyze

innovative networks [8], research the spread of infectious diseases and rumors [9,10].

Most of the existing literature focuses on the resear ch of Ponzi schemes, including the

spread of Ponzi schemes in complex networks, while the research on pyramid scheme is

relatively few. To the best of our knowledge, for the pyramid schemes of “consumption

rebate” type, no scholar has put forward a model based on it at present. In order to

understand and explain the operation mechanism and characteristics of the pyramid

schemes of “consumption rebate type, then provide ideas for monitoring this kind of

pyramid schemes, and oﬀ er the basis for further research, we propose a pyramid scheme

model which has the pr incipal characters of many pyramid schemes appeared in recent

years: pro mising hig h returns, rewarding the participants recruiting the next generation

of participants, and the organizer will take all the money away when he ﬁnds the money

from the new pa r ticipants is not enough to pay the previous participants interest and

rewards. We assume the pyramid scheme carries on in the tr ee network, ER rando m

network, SW small-world network or BA scale-free network respectively, then give the

analytical results of how many generations the pyramid scheme can last in these cases.

We also use our model to analyse a pyramid scheme in the real world and we ﬁnd the

connections between par t icipants in the pyramid scheme may constitute a SW small-

world network.

This paper is organized as follows. In Sec. 2, we brieﬂy introduce the tree networ k,

the random network, the small-world network and the scale-free network. In Sec. 3, we

propose our pyramid scheme model. In Sec. 4, we analyse a pyramid scheme in real

world. Some discussions and conclusions are given in Sec. 5.

2 Networks

2.1 Tree network

Tree networks are connected acyclic graph. The word “tree” suggests branching out

from a roo t and never completing a cycle. Tree networks a r e hierarchical, and each node

can have an arbitrary number of child nodes. Trees as graphs have many applications,

esp ecially in data storage, searching, and communication [

11].

2.2 Random network

Random network, also known as stochastic network or stochastic gra ph, refer to

complex network created by stochastic process. The most typical random network is

the ER model proposed by Paul Erd¨os a nd Alfred R´eney [12]. ER model is based

on a “natural” construction method: suppose there are n nodes, and a ssume that the

possibility of connection between each pair of nodes is constant 0 < p < 1. The network

constructed in this way is ER model network. Scientists ﬁrst used this model to explain

real-life networks.

2.3 Small-world network

The original model of small-wor ld was ﬁrst proposed by Watts and Strogatz, and

it is the most classical model of small-world network which called SW small-world net-

work [

13]. The WS small-world networ k model can be constructed as follows: take a

one-dimensional lattice of L vertices with connections o r bonds between nearest neigh-

bors and periodic boundary conditions (the lattice is a ring), then go through each of the

bonds in turn and independently with some probability φ “rewiring” it. Rewiring in this

context means shifting one end of the bond to a new vertex chosen uniformly a t random

from the whole la t tice, with the exception that no two vertices can have more than one

bond running between them, and no vertex can be connected by a bond to itself. The

most striking feature of small-world networks is that most nodes are not neighbors of

one another, but the neighbors of any given node are likely to be neighbors of each other

and most nodes can be reached from every other node by a small number of ho ps or

steps. It has been found that many networks in r eal life small-world property, such as

social networks [

14], the connections of neural networks [13], and the bond structure of

long macromolecules in the chemical [1 5].

2.4 Scale-free network

A scale-free network is a network whose degree distribution follows a power law, at

least asymptotically. The ﬁrst model of scale-free network is proposed by Barabasi and

Albert, which is called BA scale-free network [16]. BA model describes a growing o pen

system starting from a group of core nodes, new nodes are constantly added to the

system. The two basic assumptions of BA scale-free network model are: (1) from m

nodes, a new node is added to each t ime step, and m nodes are selected to be connected

to the new node in m

nodes(m ≤ m

); (2) The probability Π

that the new node is

connected to an existing node i satisﬁes Π

= k

N−1

j=1

, where k

denotes the degree

of the node i and N denotes the number of nodes. In this way, when added enough new

nodes, the network generated by the model will reach a stable evolutio n state, and then

the degree distribution follows the power law distribution. Ref [17] reported that the

degree distribution of many networks in real world is a pproximate or exact obedience to

power law distribution.

3 The model

3.1 Assumptions

We consider a simple pyramid scheme while it meets the basic features of many

pyramid schemes in the real world, especially the “consumption rebate” platforms. First,

it has an organizer that a ttracts par t icipants through promising high r ate of return

compared t o normal interest rate. Besides the promising return, any participant will be

rewarded by the organizer with a propor t ion of the total investment of the participants

he or she directly attracted, thus the early participants will be motivated enough to

recruit the next-generation participants and the next-generation participants will do the

same thing in order to get more returns. Secondly, we assume all the participants at

current generation are r ecruited by the participants at the upper generation, and the

organizer pays the participants at the previous generations the interests and rewards

when all possible participants at current generation have jo ined in the scheme. The

third assumption is that the organizer will take all the money away when he ﬁnds the

money from the new participants is not enough to pay the previous participants interest

and rewards. To simplify the model, we also assume all the part icipants invest the same

amount o f money and invest only once. Figure 1 is a schematic diagram of pyra mid

Figure 1: A schematic diagram of pyramid scheme. From top to bottom are the organizer,

the ﬁrst genera tion participants and the second generation participants.

scheme, it has one organizer and two generations of participants.

Based on these assumptions, we discuss the pyramid scheme spreads in tree networ k,

random network, small world network, and scale-free network below.

3.2 Tree network case

If the pyramid scheme expands in the form o f tree network that has a constant

branching coeﬃcient α and the root node of the tr ee network r epresents the organiser,

we can simply write the number of participants at the g-th generation as n

g−1

and the

total amount of money ent ering the pyramid scheme at the g-th generation as mn

g−1

where n

is the number of participants at the ﬁrst generation and m is the money

amount of every participant invests. For simpliﬁcation, we assume n

= α and m = 1.

We suppose the number of all potential participants is N in this case. Removing the

interest and rewards, the relationship between the net inﬂow of money M of the pyramid

scheme and the generation g when all po ssible participants at g-th generation have joined

in the scheme can be given by

M(g) = α

− r

g−1

i=1

− r

, (1)

where r

is the promised rate of return of the organizer, r

is the ratio of the money

rewarded with a participant to the tota l investment of the participants he or she directly

recruited. Normally in real pyramid scheme cases, r

and r

are between 0% and 5 0%.

The ﬁrst term of Eq. (

1) represents the investment of all the participants, the second term

represents the interest paid to the participants before the generation g, and the third

term represents the rewards paid to the recruiters of participants at the g-th generatio n.

Notice in our pyramid scheme, the participants at the g-t h generation are all recruited

by the participants at g − 1-th g eneratio n.

The second term of Eq. (

1) is the sum of geometric sequences, after summing them

up Eq. (

1) can be rewritten as

M(g) =

α − 1

[(1 − r

)α

− (1 + r

− r

)α

g−1

+ r

]. (2)

Through Eq. (

2) we can ﬁnd that if the branching coeﬃcient α satisfy the condition

α ≥

1 − r

+ r

1 − r

, (3)

the inﬂow of money M(g) of the pyramid scheme is always positive, so the pyramid

scheme will continue forever under the circumstances.

However, the potential participants are limited to N and the pyramid scheme will

stop eventually. The maximum generation G of the pyramid scheme is given by

T R

= ⌊log

(

Nα − N + α

)⌋ + 1, (4)

where ⌊x⌋ is the int eger part of x. At the G-th generation all the potential participants

have joined the pyramid scheme, and the organizer will take away all the money and

don’t pay the interest and rewards any more. We can write the ﬁnal income of the

pyramid as

= N − r

G−2

i=1

(G − i − 1)α

− r

G−1

i=2

, (5)

and the income of the participants at i-th generation is

(

(G − i − 1)α

+ r

i+1

− α

, for 1 ≤ i ≤ G − 2.

−α

, for G − 2 < i ≤ G.

(6)

Figure

2(a) shows the analytical result and the simula t ive result of maximum gener-

ation G

when the branching coeﬃcient α changes, and we take the parameters value

N = 10000, r

= 0.1, r

= 0.1. Figure

2(b) shows the analytical result and the simulative

result of maximum generatio n G

when the possible participants N changes, and we

take the parameters value α = 4, r

= 0.1, r

= 0.1. Figur e 2 illustrates intuitively that

in the tree network case, if other conditions of the pyramid scheme remain unchanged,

the larger the branch coeﬃcient , that is, the mo re new participants each person re-

(a) (b)

Figure 2: (a) The analytical result and the simulative result of ma ximum generation G

when the branching coeﬃcient α changes. We take the parameters value N = 10000,

= 0.1, r

= 0.1; (b) The analytical result and the simulative result of maximum

generation G

when the possible participants N chang es. We take the parameters

value α = 4, r

= 0.1, r

= 0.1.

cruits, the fewer generations the pyramid scheme can last. On t he other hand, when

other conditions remain unchanged, the larger the number of potential participants, the

more generations the pyramid scheme can sustain, but every new generation needs more

participants, and this growth of new pa r t icipants is exponential.

3.3 Random network case

If the pyramid scheme takes place in a ER random network has an average degree k

and N nodes, we assume the organizer is a random node in t he network and other nodes

represent the potential participants. The organizer r ecruits the potential participants

nearest to him as the ﬁrst generation participants, and the ﬁrst generation participants

recruits the potential participants nearest to them as the second generation participants,

and so on. So the value of the generation of any participant in the pyramid scheme

is the shortest path length from the node represents t he organizer. Ref. [

18] gives the

approximate analytical results for the distribution of shortest path lengths in ER random

networks, the number of nodes at the i-th generation is about k

if i ∗ log

k < 1 and all

the nodes are included in the pyramid scheme if i ∗ log

k > 1. Therefore, the pyramid

scheme in ER random network is approximat e to the case in the tree network above and

the diﬀerence is the branching coeﬃcient α should be replaced by the averag e degree k.

Firstly, like the case in tree network, r

, r

and k should satisfy the following condi-

tion:

k ≥

1 − r

+ r

1 − r

. (7)

The approximate maximum generation G of the pyramid scheme in this case is given by

≈ ⌊1/ log

k⌋ + 1. (8)

In addition, we can also write the approximate expressions of the orga niser’s and partic-

ipants’ income which have the same for m of Eq. (

6), which we omits it here. Figure 3(a)

The analytical result and the simulative result of maximum generation G

when the av-

erage degree k changes, and we take the parameters value N = 1000, r

= 0.1, r

= 0.1.

Figure

3(b) shows the analytical result and the simulative result of maximum genera-

tion G

when the possible participants N changes, and we take the parameters value

k = 4, r

= 0.1, r

= 0.1. The simulative results in the (a) and (b) are averaged aft er

100 simulations. From Figure

3, we can ﬁnd tha t in ER random network case, the re-

lationship between maximum generation G

and mean degree k, and the relationship

between G

and potential pa rticipants N are similar to the case in tree network, where

the mean degree k represents the amount of participants that each participant can re-

cruit averagely. We can also ﬁnd that within the range of parameters we have chosen,

analytical results and simulative results are very close.

(a) (b)

Figure 3: (a) The analytical result and the simulative result of ma ximum generation G

when the average degree k changes. We take the parameters value N = 1000, r

= 0.1,

= 0 .1; (b) The analytical result and the simulative result of maximum generation

when t he possible participants N changes. We take the pa r ameters value k = 4,

= 0 .1, r

= 0 .1. The simulative results in the (a) and (b) are averaged after 100

simulations.

3.4 Small world network case

Now we consider t he pyramid scheme carries on in a SW small-world network, to some

extent this case is similar to the case in ER random network. We also randomly choose

a node as the or ganiser, other nodes represent the potential par t icipants, and r

, r

represent interest rate and reward ratio respectively. The value of the generatio n of any

participant in the pyramid scheme is the shortest path length from the node represents

the organizer. Ref. [

19] points out that the number of nodes increases exp onent ially with

the average length of the shortest path when the nodes are inﬁnite. The approximate

surface area of a sphere of radius r on the SW small-world network can be given by [

19]

A(r) = 2e

4r/ξ

, (9)

where ξ = 1/φk, and φ is r ewiring probability and k is t he degree of the corresponding

rule graph.

Change r to g, we can obtain the approximate number of participants at g-th gener-

ation. Because of the exponential form of A(g), we can deal with this case just like in

the cases of tree network and ER random network. The branching coeﬃcient α should

be replaced by e

4/ξ

, and the following conditio n should be satisﬁed:

4/ξ

≥

1 − r

+ r

1 − r

. (10)

If the nodes a re ﬁnite, the number of nodes reach the peak when the distant from

the node to the organiser is near the average length of the shortest path. If greater than

the average length of the shortest path, numbers are quickly reduce to 0, so it can be

approximately considered that the maximum generation G is close to the value of the

average length o f the shortest path. The average path length

d of the SW small-world

network is given by [19]

≈

f(φKN), (11)

where

f(u) =

(

, if u → 0.

ln u/u, if u → ∞.

(12)

The number of nodes with average shortest path length from the node representing the

organizer is the largest. So we can infer the maximum generation G

of t he pyramid

scheme is given by [20]

≈ ⌊l

⌋ + 1. (13)

In the simulation, we ﬁnd that the values of r

and r

are very important. Generally

speaking, the greater the values of r

and r

satisfy the Eq. (10), the closer the simulation

results and numerical results are. This is because when the values of r

and r

are larger,

the pyramid scheme can easily terminate when the number of generations exceeds the

value of the average shortest path length. Figure

4(a) shows t he analytical result and the

simulative result of maximum generation G

when the possible participants φ changes,

and we take the parameters value N = 1000, K = 3, r

= 0.2, r

= 0.2. Figure

4(b)

shows the analytical result and t he simulative result of ma ximum generation G

when

the possible participants N changes, and we take t he parameters value K = 3, φ = 0.1,

= 0 .2, r

= 0 .2. The simulative results in the (a) and (b) are averaged after 100

simulations. In Figure

4, we ﬁnd that within the range of parameters we selected, t he

maximum generation G

of the pyramid scheme is not very sensitive to the reconnection

probability φ and the potential participants, and the analytical results are basically in

accordance with the simulative results.

(a) (b)

Figure 4: (a) The analytical result and the simulative result of maximum generation

when the possible participants φ changes. We take the parameters value N = 1000,

K = 3, r

= 0.2, r

= 0.2; (b) The analytical result and the simulative result of maximum

generation G

when the possible participants N changes. We take the parameters value

K = 3, φ = 0.1, r

= 0.2, r

= 0.2. The simulative results in the (a) and (b) are averag ed

after 100 simulations.

3.5 Scale-free network case

If the pyra mid scheme expands in a BA scale-free network, similar to the cases in ER

random network and SW small-world network above, we also randomly choose a node

as the organiser and and other nodes represent the potentia l participants. The orga-

nizer recruits participants and the participants recruit the next generation participants

through the network connections. To ensure positive inﬂows, the following condition

must be satisﬁed:

(1 − r

)n(g + 1) ≥ r

i=1

n(g), (14)

Where n(g) represents the number of participants at g-th generation, and n(g + 1)

represents the number of participants a t g+1-th generation. The distribution o f shortest

path length approximates the norma l distribution and the position corresponding to the

highest point of normal distribution is the average shortest path length [

21]. The average

path length

s of the BA scale-free network is given by [22]

≈

ln N

ln ln N

. (15)

Figure 5: The analytical result and the simulative r esult of maximum generation G

when the possible participants N changes. We take the parameters value r

= 0.2,

= 0.2. The simulative result is averaged after 100 simulations.

Before the peak, the number of participants per g eneratio n grew faster than exponen-

tial growth. But after t hat, The number of participants per generation declined rapidly,

so the condition can not be satisﬁed any more. So we can infer the maximum generation

is close to the value of the average shortest pat h length, and is given by

≈ ⌊

⌋ + 1. (16)

Figure 5 shows the analytical result and the simulative result o f maximum generation

when the possible participants N changes. We taken the parameters value r

= 0.2,

= 0.2, and the simulative result is averaged after 100 simulations. The analytical re-

sults can basically reﬂect this characteristic. We can ﬁnd that in scale-free networks, the

maximum generation G

is not very sensitive to the potential participants in Figure

The analytical results can basically reﬂect this cha r acteristic.

4 A pyramid scheme in real world

Although real cases of pyramid scheme are easy to ﬁnd in news reports, there are few

cases that give details of the number of people involved and the pyramid generations.

Usually, when the organizer of the pyramid scheme disappears, the participants with

loss will repor t the case to the police, and the police will investigate the case. On July

23, 2018, China News Networ k Guangzhou Station reported a pyramid scheme that had

75663 account numbers and 46 generatio ns, and the pyramid scheme had amassed 76

million yuan in three months

. This is the same type of pyra mid scheme as described

in the introduction. Using the analysis in Section 3, we assume the pyramid scheme

carries on tree network, ER random network, SW small-world network and BA scale-

free network respectively, then verify which network can describe the pyramid scheme in

the real world well. We assume one account number represent a participant.

If this real pyramid scheme expands in a tree network, we can calculate the tree net-

work’ branching coeﬃcient α ≈ 1.28. This means on average, less then two participants

are recruited by each participant. But we can not know more about the connections

between the participants except branching coeﬃcient.

If this real pyramid scheme spreads in ER Random network, we can calculate the

average degree k ≈ 1.28 through Eq. (

8). So each node is connected to 1.28 nodes on

average, and the connection probability in ER random network is less then 1.28/75663 ≈

1.7 × 10

−5

, which is very small then isolated nodes and nodes with degree 1 are easy

to appear in the network. Although this case is similar to that of tree network, the

branching coeﬃcient in random network is not stable and it’s easy to end the pyramid

scheme if Eq. (??) is not sat isﬁed (the minimum value of the formula ( 1−r

)/(1−r

)

is greater than 1). So we think the pyramid scheme can hardly happen in the ER random

network.

If this real pyramid scheme carries on in a BA scale-free network, through the analysis

and simulation, we ﬁnd developing to 46 generations need far more than 75 663 partic-

ipants. Therefore, the connections between participants are impossible to f orm a BA

scale-free network.

http://www.gd.chinanews.com/2018/2018-07-24/2/397998.shtml

If this real pyramid scheme takes place in a SW small-world network and accords all

our assumptions, we could ﬁnd a simulative result to ﬁt the result o f the real pyramid

scheme. The parameters we select are N = 100000, φ = 0.02, K = 4, r

=0.1, r

= 0.1 ,

and each participant invest 23500 yuan. The simulative pyramid scheme has 7465 2

participants, and develops to 46 generations, and the fund pool of the pyra mid is about

76 million yuan. The simulation results are in good agreement with the real pyramid

scheme. Figure

6(a) and 6(b) shows the cumulative number N

cum

of participants, the

number of participants N

in each generation, and the cumulative money M

cum

changes

over g eneratio n g in the simulative pyramid scheme.

(a) (b)

Figure 6: The cumulative number N

cum

of participants, the number of participants N

in each generation, and the cumulative money M

cum

changes over generation g in the

simulatve pyramid scheme. This is one simulative result in the SW small- world case

which the parameters we select are N = 100000, φ = 0.02, r

=0.1, r

= 0.1, and each

participant invest 23500 yuan.

Figure

6 shows that, the amount of participants and the amount of accumulated

money of the pyramid scheme grow slowly in the initial stage and explosively in the

later stage. Once the growth rate slows down, the amount of the pyramid scheme’s ac-

cumulated money will reach a peak soon and the organizer will escape. The probability

of reconnection in simulation is 0.02, which can be understood according to the actual

situation and means that pa r t icipants tend to recruit new part icipants from familiar

people. In fact, according to our investigation and many news reports, such pyramid

frauds always arise in small cities, and most of the participants recruit new participants

from their familiar people. As generations go on, the network constituted by all partic-

ipants has the properties of small world: agglomeration and having some ﬂocks, which

is similar to the interpersonal network. Alt ho ugh our mo del has been simpliﬁed and

approximated, it is enlightening to explain the real case.

Through the above simula tion analysis, we can speculate that the connections be-

tween participants in the real case may constitute a SW small-world network.

5 Conclusion

In summary, we have proposed a pyramid scheme model which has the principal

characters of many pyramid schemes appeared in recent years: promising high returns

and rewarding the participants attracting the next generation of participants. Assuming

the pyramid scheme spreads in the tree network, ER random network, SW small-world

network and BA scale-free network, we give the conditions for t he continuity of the

pyramid scheme, and the analytical results of how many generations the pyramid scheme

can last if the organizer of the pyramid scheme takes all the money away when he ﬁnds

the new money is no t enough to pay interest and incent ives. We also use our model to

analyse a pyra mid scheme in the real world and the result displays the connections of

participants in the pyramid may have the feature of small wor ld.

Our work is helpful to understand the operation mechanism and characteristics of

the pyramid schemes of “ consumption rebate” type. Our model may be able to apply to

some current illegal high-interest loa ns, if these illegal projects promise a high interest

rate and reward the investors who encourage others to invest in such projects, but the

money accumulated is not actually invested in any real projects. Our wor k shows that

the pyramid schemes of “consumption rebate” type are not easy to be detected by the

supervision because of the small amount of funds and small a mo unt of participants

accumulated in the initial stage. After the rapid growth of funds and participants, it

often comes to the end of this kind of pyramid frauds, and the or ganizers oft en have

already ﬂed. So for regulators, it’s better to nip such platforms in the bud so that more

people don’t suﬀer loss. In addition, to some extent, our work provides some basis for

further study of such frauds. For example, we will further consider how the participants’

beliefs about always having enough new participants eﬀect the operation of such frauds.

6 Acknowledgement

This research was support ed by National Natural Science Foundation of China (No.71771204,

91546201) .

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