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图与网络算法优秀论文171
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2016 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this
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2016 Mathematical Contest in Modeling (MCM) Summary Sheet
(Attach a copy of this page to each copy of your solution paper.)
How to Understand the Information
The flow of information has never been as easy or wide-ranging as it is today.
This paper proposes some models to explore the relationship between speed/flow
of information with inherent value of information.
First,to explore information flow, we build up a SIIR model which takes the me-
dia influence into consideration. This model is developed from the classical SIR
model.We introduce the concept of hot transmission nodes to highlight the impact
of media. We demonstrate our design with American media data. The results show
that our model has a significant performance in validating predicted values of the
characteristics of information network in 2050.
Second,the information filtering can be regarded as a classification problem. We
design a radial basis function network to implement the information filtering. In
order to solve sample noises and improve the computing performance,we decide
to use K-means algorithm and RLS algorithm to implement our RBF network. We
demonstrate our network in online news dataset ,with an training accuracy of
75.1%,and testing accuracy of 74.3%.we are satisfied that the accuracy rate with
information filtering as news.
Third, We have only considered the social network in the Internet and build a social
network SIR(SN-SIR) model to explore the problem-how public interest and opin-
ion can be changed through network.It simulates in the standard dataset: Facebook
social network dataset .By analyzing the factors which can implement our result-
s,we can make a plan to determinate information propagation. Then,we take infor-
mation values ,people ˛a´rs initial opinion and bias, form of the message or its source,
and topology or other reasons into account and propose a detailed scheme.
Finally, effects of external factors are considered in our models.And we analyze
the stability and sensitively of our models. Although there are some weakness in
our models,the results still demonstrate that our model can undergo disturbance in
certain extent.
数学中国
www.madio.net
本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
数学中国:www.madio.net,最专业的数学建模平台
备战美赛,美赛资料下载:http://www.madio.net/forum-108-1.html
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Team # 44173 Page 2 of 22
Contents
1 Introduction 3
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Assumptions 4
3 A SIIR Model of Information Flow 4
3.1 Social Network Information Dissemination . . . . . . . . . . . . . . . . . . . . . . 4
3.1.1 Social Network Information Dissemination Characteristics . . . . . . . . . 4
3.1.2 Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Calculating and Simplifying the Model . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2.1 The Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2.2 Rationality validation And Sensitivity analysis . . . . . . . . . . . . . . . . 7
3.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 A Model of Information Filtering 11
4.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Hybrid Learning Procedure For Networks . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.4 Sensitively Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 A SN-SIR Model 14
5.1 The Way of Information Dissemination in Network . . . . . . . . . . . . . . . . . 14
5.2 A Model of Public Opinion Dissemination in Social Network . . . . . . . . . . . . 14
5.2.1 Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 Sensitivity Analysis and Model Validation . . . . . . . . . . . . . . . . . . . . . . . 16
5.3.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3.2 Sensitivity, parameter validation analysis . . . . . . . . . . . . . . . . . . . 17
5.4 Result Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Analysis 21
6.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
数学中国
www.madio.net
本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
数学中国:www.madio.net,最专业的数学建模平台
备战美赛,美赛资料下载:http://www.madio.net/forum-108-1.html
美赛讨论加入群:2017美赛官方群287336869
Team # 44173 Page 3 of 22
1 Introduction
1.1 Background
Broadly speaking, information is one of the important ways for us to perceive the outside
world. Today it spreads quickly in tech-connected communication network. Sometimes it is
due to the information finding its way to influential or central network nodes that accelerate its
spread through social media. Our prevailing premise is that this cultural characteristic to share
information (both serious and trivial) has always been three. At present, information propa-
gation is exposed explosive growth.Beacause our social networks and medium are more and
more complex.Without effective ways to solve these problems ,it will finally cause information
propagation is more and more difficult to umderstand.Hence,our work has its unprecedented
significance in world nowadays.
1.2 Our Work
This paper propose some models to explore the relationship between speed/flow of informa-
tion with inherent value of information.
The Dilemma:
• An adaptive flow model is established with the consideration of the complex network
changes.
• Make use of the existing parameters to filter the information reasonably.
• Reasonable analysis of influencing factors in the process of information transmission.
The Approach:
• FirstˇcˇnTo explore information flow, we build up a SIIR model which takes the special fea-
tures if media influence into consideration. This model is developed from classical SIR
model.
• The Results show that our model has a significant performance in validate the value of
prediction with real value. And then ,we predict the rate of daily contact in 2050.
• Information filtering is regarded as classification problem. we design a radial basis func-
tion network to implement information filtering. In order to solve sample noise and com-
puting performance, we decide to use K-means algorithm and RLS algorithm to imple-
ment our RBF network.
• We have only considered the social network in the Internet and build a social network
SIR(SN-SIR) to explore the problem ˛a´s how public interest an opinion can be changed
through network ˛a´s.
• We can make a plan to determinate information propagation. Then ,we take information
value ,people ˛a´rs initial opinion and bias, form of the message or its source, and topology
or other reasons into account and build a detailed scheme.
数学中国
www.madio.net
本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
数学中国:www.madio.net,最专业的数学建模平台
备战美赛,美赛资料下载:http://www.madio.net/forum-108-1.html
美赛讨论加入群:2017美赛官方群287336869
Team # 44173 Page 4 of 22
2 Assumptions
The accuracy of our models rely on certain key, simplifying assumptions. These assumptions
are listed below:
• Do not consider factors such as natural birth,death,and population mobility.The total
population keeps as a constant K.
• Infected individuals once contact with susceptible individuals,Infected individuals change
to susceptible individuals at a certain probability.
• At a certain time,The number of individuals per unit time removing from the infected
individuals is proportional to the number of patients.
• The number of media is relatively fixed in the same period.
Under the above and basic assumptions, we can set out to construct our model.
3 A SIIR Model of Information Flow
Information is created by the social network users and transmits in the entire network.The
result of transmission is that the receiver of the information translates into the transmitter and
disseminate the information to others,or translates to the recovered one and does not transmit
to others.Obviously, this transmission way is similar with the way Infectious diseases spread in
the crowd. Based on the analysis of social network structure,we introduce the hot transmission
node into network,and proposes an information transmission model based on the improved
SIR model [1],which is simulated with the data-based Facebook social network.
3.1 Social Network Information Dissemination
3.1.1 Social Network Information Dissemination Characteristics
In this section,we propose a to explore a general model of information flow problems, which is
not subjected to the limitations of the times and mediums of communication.
The total number of user nodes in social networks is denoted as N.The network is assumed to
be an undirected graph and each node in the network may be in one of three states: susceptible(s),
infected(i), or recovered(r).Nodes in state i have received the message and have the ability to
transmit it.Nodes in state s have not received the message and have the possibility to receive
it.Nodes in state r have received the message and will not transmit it any more.
There are a variety of mediums of communication,such as newspapers,televisions,the Inter-
net and so on.Different mediums have different ability of transmission,but all of these medi-
ums owe stronger ability than human beings.We consider media as the hot transmission n-
odes.Their contact rate is denoted as λ
1
. Ordinary users in networks are considered as ordinary
nodes.Their contact rate is denoted as λ
2
.
The laws of Information dissemination of social networks with hot transmission nodes are as
below:[2]
数学中国
www.madio.net
本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
数学中国:www.madio.net,最专业的数学建模平台
备战美赛,美赛资料下载:http://www.madio.net/forum-108-1.html
美赛讨论加入群:2017美赛官方群287336869
Team # 44173 Page 5 of 22
• Hot transmission nodes contact with other nodes with the contact rate x1,while ordinary
nodes with the contact rate λ
2
.
• After nodes in state s received the information from nodes in state i,they change into
recovered nodes in a probability of µ,and into ordinary infected nodes in a probability of
1 − µ.
• With the propagation of information,the proportion of three types of nodes will tend to
be stable.
The schematic diagram of information dissemination of social networks with hot transmis-
sion nodes as shown in Figure 1, the solid line shows the information transmission between
two points.
Figure 1: the Information Transmission of Atrioventricular Model Structure Diagrams
Figure 2: the Improved SIR Model
3.1.2 Model Building
According to the characteristics of the transmission above,we can get the information transmis-
sion of atrioventricular model structure diagrams, as shown in Figure 1.
i
0
is the proportion of hot transmission nodes in networks,which is usually fixed.i(t) is the
proportion of ordinary infected nodes in the time of t,which consists of i
1
(t),the ordinary in-
fected nodes generated by hot transmission nodes,and i
2
(t),the ordinary nodes infected nodes
generated by ordinary infected nodes.λ
1
is the contact rate of hot transmission nodes,while λ
2
is the contact rate of ordinary infected nodes.r(t) is the proportion of nodes in state r,µ is the
rate of Immune probability.[3] We can get the model as follows:
ds(t)
dt
= −λ
1
i
0
s(t) − λ
2
s(t)i(t)
di(t)
dt
= µ[λ
1
i
0
s(t) + λ
2
s(t)i(t)]
dr(t)
dt
= (1 − µ)[λ
1
i
0
s(t) + λ
2
s(t)i(t)]
s(t) + i(t) + i
0
+ r(t) = 1
i
0
= i
10
, i(0) = i
20
, s(0) = s
0
, r(0) = r
0
(1)
数学中国
www.madio.net
本内容由645617861l同学与工作人员共同整理,恭喜同学成功完成数学中国第一期威客项目
数学中国:www.madio.net,最专业的数学建模平台
备战美赛,美赛资料下载:http://www.madio.net/forum-108-1.html
美赛讨论加入群:2017美赛官方群287336869
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