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 modelater 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
Copyright
c
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
t
Δ
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 time a sphere
which radius is
tnt 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
=
Δ
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
t
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
2
() 4 ( )
s
tt
π
τ
= , and time intensity parameter is
τ
.
Because
()
s
t is integrable function in area of
[0, )
, we can get following function:
()qt = ()Vt =
0
()
s
tdt
+∞
=
2
0
4()rtdt
πτ
+∞
=
3
4
()
3
t
π
τ
(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
t
τ
.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
Δ
which consists of information vanishing
points and sphere’s whole surface area
()qt . It can
be described as
() ()qt qt
α
Δ
= . 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
t
. 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
λ
+
−== +
(5)
We set following condition:
0
() ()
t
mt sds
λ
=
(6)
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414
We can get following function:
{( ) () }
[( ) ()]
exp{ [ ( ) ( )]}
!
n
pNt s Nt n
mt s mt
mt s mt
n
+− =
+−
=−+
(7)
It means that ()()Nt s Nt+− is of Poisson
Distribution with
()()mt s mt+− average value. Its
intensity function is following:
() /tdpdt
λ
=
(8)
Because
() ()qt qt
α
Δ= , ()t
λ
is variation
Probability of average information vanishing points’
appearance with following function:
() [ ()]/tdENtdt
λ
=
(9)
It is of Weibull Distribution. This function can
be described with following function:
1
()tt
β
λλβ
=
(10)
In this function
,0
λ
β
> ,
β
is shape parameter
and
λ
is intensity parameter. So we can describe
cumulative information decrease with following
function:
0
() ()
t
mt tdt t
β
λ
λ
==
(11)
It conforms to the variation of sphere’s volume
which radius is
t
τ
.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:
3
4
() ( )( 0, 0)
3
qt t t
π
ττ
=>
(12)
(2) saturation decrease function under the
action of information diffusion resistances:
() ( 0, 3, 0)mt t t
β
λλ β
=<
(13)
Because information diffusion will reach full
saturation state finally, the speed of saturation’s
increase will fall over time.
3
β
meets this form.
So there is following information saturation function
under the action of information diffusion dynamic &
resistances:
3
() () ()
4
() ( 0, 0, 3, 0)
3
Qt qt mt
tt t
β
π
τλτλβ
=
+
=
+><
(14)
When we do diffusion experiment this function
can be described with following simple function:
3
() () ()
(0, 0, 3,0)
Qt qt mt
tt t
β
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
τ
= , intensity parameter
0.02835
λ
=− and shape parameter 3.147
β
= . 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
(())
0
dQt
dt
=
, 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
(())
0
dQt
dt
=
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
(())
0
dQt
dt
=
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
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
416
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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