SATURATION MODEL OF NETWORK INFORMATION

DIFFUSION

Jinlou Zhao, Zhibin Liu

School of Economics and Management, Harbin Engineering University, Harbin, China

Jiannan Yu

School of Science, Harbin Engineering University, Harbin, China

Keywords: Network dimension-force, Information diffusion, Information saturation, non-homogeneous poisson

process.

Abstract: This paper first put forward information diffusion mode under the framework of network dimension-force,

then analyzed the influence of information diffusion dynamic & resistance on information saturation under

this mode，later this paper established information saturation model using non-homogeneous Poisson

Process, at last simulated information saturation model based on practical data, which proved the

effectiveness of this model.

1 INTRODUCTION

Since the theory of network dimension-force was

brought forward in 2004, some scholars have done

some primary theoretical research on network

information diffusion based on this theory. The key

of network dimension-force theory is the

information’s time-quantity effectiveness, from this

view network information diffusion can be viewed

as a process in which information’s time

effectiveness and quantity effectiveness changes

with time. In this process information’s value

increases with the increase of the relative quantity of

information which information destination obtains.

Therefore information destination is always looking

forward to obtain more sufficient information. This

sufficient degree of information that information

destination obtains can reflect quantity change of

information diffusion sufficiently, and it is more

intuitional and easy to description and modeling. In

order to make research more convenient we called

sufficient degree of information as information

saturation. The research of network information

diffusion saturation can help us find information

saturation model, which can help us find time-

quantity relation of information diffusion. Using

them, we can optimize network information

diffusion better. It is of great practical and theoretic

significance.

2 THEORETICAL ANALYSIS

2.1 Information Diffusion Mode under

the Framework of Network

Dimension-force

Information transfers from node to node in the

process of information diffusion. Obviously the

relevance of two nodes’ information is a decay

process during information transfer process. In

realistic space information transfer process is

influenced by some related factors such as distance

and time (Sinan & Don, 2007), so information

diffusion follows node-to-node mode which is two-

dimensional mode of information diffusion

obviously. While in network space information

diffusion can go without considering related

influencing factors such as distance and time under

the effect of network dimension-force, information

diffusion follows node-to-surface mode, which is

more stereotyped than information diffusion in

realistic space. This mode belongs to three-

dimensional mode of information diffusion as figure

1 shown.

In ideal state after

(0)ttΔΔ→ time the

information from source information node transfers

to thousands of information destination node

explosively. These thousands of nodes can form a

412

Zhao J., Liu Z. and Yu J..

SATURATION MODEL OF NETWORK INFORMATION DIFFUSION.

DOI: 10.5220/0003569604120417

In Proceedings of the 13th International Conference on Enterprise Information Systems (NMI-2011), pages 412-417

ISBN: 978-989-8425-53-9

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Information Diffusion Mode in Network Space.

sphere which radius is tΔ . The surface area of this

sphere is information of all information destination

nodes from source information through

time.

And after next period of

tΔ time all information

destination nodes on the first sphere will turn into

source information nodes and then information on

these nodes will transfers to thousands of new

information destination node explosively. Every

original node also forms a sphere which radius is

tΔ . Because the length of tΔ time is infinitesimal,

we can understand that the first sphere form a new

sphere which has the same center of sphere and

radius is

2 tΔ . When the length of tΔ time tends to

infinity, we can understand thousands of spheres

which has the same center of sphere and radius is

ntΔ are formed. So after tnt=Δ time a sphere

which radius is

tnt=Δ→∞ will be formed. The

process of this sphere changes with time is the

process of information diffusion under the effect of

network dimension-force. This node-to-surface

three-dimensional mode of information diffusion is

information diffusion mode under the framework of

network dimension-force.

2.2 Information Saturation Analysis

based on Dynamic & Resistance

In the theory of network dimension-force the

dynamic of information diffusion in network space

can be divided into the endogenous powers and the

exogenous powers as figure 2 shown (Deng, 2006).

The endogenous powers derive from diffusion

subject, diffusion object and their interaction in

diffusion system. The exogenous powers derive

from the effect of the network transmission media to

diffusion subject and diffusion object. These two

powers act on information diffusion through push

and pull effect on information. Obviously the

process of information quantity’s increase under the

action of information diffusion’s dynamic, which is

the process of information saturation increase, can

be viewed as the process of sphere’s surface area

increase under three-dimensional mode of

information diffusion. Information saturation

threshold is the biggest surface area of this

increasing sphere because of the existence of

diffusion resistances.

Although in network space information diffusion

can go without considering related influencing

factors such as distance and time under the effect of

network dimension-force, information diffusion will

reach stable saturation state at last. This restrictive

function comes from diffusion resistances mainly.

These resistances hinder diffusion process and

decrease diffusion efficiency. The concrete

constitutes of information diffusion resistances is as

shown in figure 2 (Deng & Xu & Zhao, 2008).

Resistances come from reaction force of endogenous

powers and nodes’ resistances. Firstly, because

nodes in network information diffusion affect as

sources and information destination each other,

sources can act on information destination through

endogenous powers. At the same time information

destination can change into sources to diffuse

information to original sources which now are

information destination. Under these circumstances

the original endogenous powers from sources will

become resistances of information diffusion (Deng

& Xu, 2008). So the endogenous powers can change

into resistances of information diffusion with the

transformation between diffusion subject and

diffusion object. Secondly, because nodes are

combination points of realistic space and network

space, information receiving and sending are

inevitably influenced by some resistances in realistic

space. Under the influence of some resistances such

as node spacing friction force, cultural difference

inertia force and the resistance of information

asymmetry in realistic space, some information

certainly will be lost in the process that nodes

process information. It will lend to information

losing. Because of these two resistances’ influence,

the information from sources can’t reach every node

on the sphere. Some nodes can’t be covered though

they are relevant with sources. We called these

nodes information vanishing points. Information

vanishing points varies with information species. A

information vanishing point can bring more

information vanishing points. So information

vanishing points can bring loss of information

SATURATION MODEL OF NETWORK INFORMATION DIFFUSION

413

quantity and this loss is random obviously (Davies,

1979). Information vanishing points reflect to sphere

model of network information diffusion in form of

sphere’s surface area decrease at

t time. At last the

final sphere will be all covered with information

vanishing points at

t →∞ time. At that time

information diffusion will reach full saturation state.

Dynamic & Resistance

of Information

Diffusion

Endogenous

Powers

Exogenous

Powers

Resistances

Information Original Driving

Force

Source Supply Propelling Force

Information Destination

Demand Traction Force

Information Potential

Propelling Force

Network Intercommunication

Force

Network Synchronization

Force

Network Clustering Force

Node Spacing Friction Force

Cultural Difference Inertia

Force

Resistance of Information

Asymmetry

Figure 2: The Sketch Map of Information Diffusion

Dynamic & Resistance.

3 MODEL DESCRIPTION

3.1 Model Description of Information

Saturation Increase

In the second theoretical analysis part, after tnt

time the information from source forms a sphere

which radius is

ntΔ . The information, which all

nodes on the sphere received in the last period of

ntΔ time, is the augmenter of information within

time. We suppose that all nodes on the sphere

receive information from source completely. In the

theory of network dimension-force nodes is equated

with information, so the augmenter of information

within

t time is equated with the quantity of nodes

on the sphere. The sphere’s surface area at

t time is

() 4 ( )

= , and time intensity parameter is

Because

()

t is integrable function in area of

[0, )

∞ , we can get following function:

()qt = ()Vt =

()

tdt

+∞

∫

4()rtdt

πτ

+∞

∫

()

(1)

In this function

()qt is total quantity of

information within

t time in ideal condition, and it

conforms to the variation of sphere’s volume which

radius is

.We can use ()qt to describe the

increase of information saturation.

3.2 Model Description of Information

Saturation Decrease

In the second theoretical analysis part, the decrease

of information saturation is equated with the

decrease of nodes which can receive information

from source availably. In the theory of network

dimension-force nodes is equated with information

(Zhao, 2006), so the decrease of available nodes

within

t time is equated with surface area which

consists of information vanishing points. In ideal

condition this process is a random process, so there

is a stable proportionality constant between surface

area

()qt

which consists of information vanishing

points and sphere’s whole surface area

()qt . It can

be described as

() ()qt qt

= . We can see that

information losing rate is a function on

t .

Meanwhile the increase of information vanishing

points can be seen as a counting process (Hudson,

1972), and the variation of information vanishing

points at

t time is mutual independent with it at

()tt

Δ time. So this counting process is also a

process with independent increments with a

information losing rate function on

. All above

proves the process of information vanishing points’

increase is a Non-homogeneous Poisson Process

which is satisfied with following functions:

(0) 0N

(2)

{(), 0}Nt t≥ have independent increments

(3)

{( ) () 2} ()pNt h Nt oh

−≥=

(4)

{( ) () 1} () ()pNt h Nt th oh

−== +

(5)

We set following condition:

() ()

mt sds

∫

(6)

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414

We can get following function:

{( ) () }

[( ) ()]

exp{ [ ( ) ( )]}

pNt s Nt n

mt s mt

+− =

+−

=−+−

(7)

It means that ()()Nt s Nt+− is of Poisson

Distribution with

()()mt s mt+− average value. Its

intensity function is following:

() /tdpdt

(8)

Because

() ()qt qt

Δ= , ()t

is variation

Probability of average information vanishing points’

appearance with following function:

() [ ()]/tdENtdt

(9)

It is of Weibull Distribution. This function can

be described with following function:

()tt

λλβ

−

(10)

In this function

> ,

is shape parameter

and

is intensity parameter. So we can describe

cumulative information decrease with following

function:

() ()

mt tdt t

∫

(11)

It conforms to the variation of sphere’s volume

which radius is

.We can use ()qt to describe the

decrease of information saturation.

3.3 Comprehensive Model Description

of Information Saturation Variation

In summation saturation function of network

information diffusion consists of following

functions:

(1) saturation increase function under the action

of information diffusion powers:

() ( )( 0, 0)

qt t t

ττ

=>≥

(12)

(2) saturation decrease function under the

action of information diffusion resistances:

() ( 0, 3, 0)mt t t

λλ β

=<≥≥

(13)

Because information diffusion will reach full

saturation state finally, the speed of saturation’s

increase will fall over time.

≥ meets this form.

So there is following information saturation function

under the action of information diffusion dynamic &

resistances:

() () ()

() ( 0, 0, 3, 0)

Qt qt mt

tt t

τλτλβ

+><≥≥

(14)

When we do diffusion experiment this function

can be described with following simple function:

() () ()

(0, 0, 3,0)

Qt qt mt

tt t

τλτ λ β

+><≥≥

(15)

4 SIMULATION AND RESULTS

4.1 Experiment Simulation

The simulation program of the proposed model is

developed in the environment of MATLAB 7.1.

In order to simulate

()Qt in realistic condition,

we adopt click-through accumulation of specific

information as simulation data. We collect monthly

click-through data of iPhone global official website

from 2007/1 to 2008/9 as table 1 shown. Because the

increase of information saturation is a process of

accumulation increase, we adopt click-through

accumulation as simulation data. The data after

processing is as table 2 shown.

Table 1: Monthly Click-through Data of iPhone Global

Official Website from 2007/1 to 2008/9 (unit: hundred

million).

Time Monthly Click-through Data

2007.1 0.0459

2007.2 0.0667

2007.3 0.3615

2007.4 0.2164

2007.5 0.3537

2007.6 0.5687

2007.7 1.4390

2007.8 1.2285

2007.9 1.5424

2007.10 1.6043

2007.11 1.1045

2007.12 1.6431

2008.1 1.4923

2008.2 0.7961

2008.3 1.0944

2008.4 0.5334

2008.5 0.4847

2008.6 0.1654

2008.7 0.2350

2008.8 0.0933

SATURATION MODEL OF NETWORK INFORMATION DIFFUSION

415

Table 2: Monthly Click-through Accumulation of iPhone

Global Official Website from 2007/1 to 2008/9 (unit:

hundred million).

Time Monthly Click-through

Accumulation

2007.1 0.0459

2007.2 0.1126

2007.3 0.4741

2007.4 0.6905

2007.5 1.0442

2007.6 1.6129

2007.7 3.0519

2007.8 4.2804

2007.9 5.8228

2007.10 7.4271

2007.11 8.5316

2007.12 10.1747

2008.1 11.6670

2008.2 12.4631

2008.3 13.5575

2008.4 14.0909

2008.5 14.5756

2008.6 14.7410

2008.7 14.9760

2008.8 15.0693

We simulate practical data through Gaussian

Fitting as figure 3 shown.

Figure 3: The Fitted Curve of Monthly Click-through

Accumulation.

In the function after simulation time intensity

parameter

0.0459

= , intensity parameter

0.02835

=− and shape parameter 3.147

= . So

information saturation function after simulation can

be described with following function:

3 3.147

( ) ( ) ( ) 0.0459 0.02835Qt qt mt t t=+ = −

(16)

4.2 Results Analysis

We found information quality pushed by

information diffusion powers outweighs it hindered

by information diffusion resistances through the

observation of figure 3 within

(0,19)t ∈ time. It

means that in this period information diffusion

powers outweigh information diffusion resistances.

When

19t

there is

(())

dQt

, so it means that

information quality pushed by information diffusion

powers equals it hindered by information diffusion

resistances, information diffusion powers equal

information diffusion resistances, and information

saturation reaches the maximum. When

(19, 26.5)t

∈

, information quality pushed by

information diffusion powers is less than it hindered

by information diffusion resistances, information

diffusion powers are fewer than information

diffusion resistances, and information saturation

begins to decrease. When

26.5t = there is

( ) 0( 0)Qt t

≠ , information quality within t time

pushed by information diffusion powers equals it

hindered by information diffusion resistances, and

information saturation dropped to 0.

Through above analysis we can find the time

point when

(())

dQt

is the turning point of

dynamic & resistances variation curve. Before the

turning point information diffusion powers occupy

leading position (James, 2005). After the turning

point information diffusion resistances occupy

leading position. Under the framework of network

dimension-force we called the time point when

(())

dQt

as saturation point of information

diffusion, and

()Qt is saturation value. After

calculation there are

19.1716t

(unit: month) and

( ) 15.0607Qt = (unit: hundred million).

The results meet market behavior of Apple

Corporation. In 2007/10 Apple Corporation declared

iPhone would be sold, and it accelerated the speed of

information saturation’s increase. After 13 months

information saturation reached the maximum, and

accordingly the sales volume of iPhone was

increasing quickly. When information saturation

began to decrease in 2008/6, Apple Corporation

developed advanced product iPhone3G in the same

month to keep the sales volume of its products.

5 CONCLUSIONS

Under the framework of network dimension-force

information diffusion mode is node-to-surface

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spherical mode. An information saturation model for

information diffusion in network space is

constructed in this study. Some dynamic &

resistance factors are involved in the model, such as

endogenous powers, exogenous powers and

diffusion resistances. From the simulation results,

we conduct three main conclusions: (a) The

tendency of information saturation is ascend in first

and descend at last in the process of information

diffusion. (b) Information saturation reaches the

maximum when information quality pushed by

information diffusion powers equals it hindered by

information diffusion resistances. (c) Information

saturation dropped to 0 when cumulative increase of

information equals its cumulative decrease. In order

to keep high information saturation we can develop

new information in advance before original time

reaches full saturation state.

ACKNOWLEDGEMENTS

This work is supported by NSFC(70971028).

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