For office use only
T1

Team Control Number

43689

T2

For office use only
F1
F2

T3

Problem Chosen

F3

T4

D

F4

2016
MCM/ICM
Summary Sheet

Macroscopic and Microscopic Models of Measuring the
Efficiency of Information Networks
Summary
In this paper, we build a macroscopic model to determine the structure of the information networks in society, and a microscopic model to determine the probability of information diffusion between any two neighboring nodes.
We use the macroscopic model to simulate the process of information finding its way, considering spread medias as the nodes. Assuming that the public medias and personal medias which first send a new and same information are the source nodes, and in order to simplify the model, we simplify these nodes as one node. Then we combine series model and parallel model of information diffusion building the macroscopic model. We draw a conclusion that when information networks become extremely developed, information networks have the ability to connect the source node to anyone in society.
Moreover, based on the macroscopic model we build the microscopic model to measure the probability of information diffusion of any two neighboring nodes. We conclude that factors determining the probability of information diffusion contain two kinds: features of nodes and features of edges. We use the Fitting Method to determine the calculation model of the probability of information diffusion. We use our algorithm validate the probability of information diffusion of Paris attack event, the result is correspond to the reality. And the algorithm can be extended to many social platforms, so it has a relatively high generalization.
Last but not least, we make a sensitivity for the microscopic model discussing the impact of changing of the features of nodes and edges.
Through previous analysis, the abilities of prediction of the two model are validated.
The model and the algorithm can be extended to many information media.

Contents
1

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2
2

Introduction

Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2

2.1

Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2.2
3

Assumption

Extention Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2
2

3.1

Symbol Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3.2

Information Diffusion Model Based on Series and Parallel Connected Model

3

3.3

Conclusion Extended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.4

Analysis of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.5
4

Basic Model

Validate Reliability of Basic Model . . . . . . . . . . . . . . . . . . . . . . .

7
10

4.1

Symbol Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

4.2

Micro-model Based on the Basic Model . . . . . . . . . . . . . . . . . . . .

10

4.2.1

Features of Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

4.2.2

Features of Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

4.3

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.4

Validate Reliability of Basic Model . . . . . . . . . . . . . . . . . . . . . . .

16

4.5
5

Extension Model

Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Strengths and weaknesses

17

5.1

Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

5.2

Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Appendices

18

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1
1.1

Page 1 of 19

Introduction
Background

In 1800, people can only get local events such as weddings,storms and death. But if the events happened in the distance, they know little about those, even if the events are very important to them. Now, people can get a variety of events through media platforms, no matter how far it is, as long as he wanted to know. At the same time, the events attract almost everyone could speed in an explosive speed.
With the development of technology, how did information dissemination progress change, and whether can we predict the diffusion structures by the historical data, and what factors exactly affect the information diffusion, and how to quantitatively describe the relationship between the factors and the information diffusion. When it comes to information networks’ structure and the probability of information diffusion, we have a lot of unknowns.
This inspiration of the article comes from The Prediction of Virus Marketing Propagation Efficiency Based on the Social Network [1] and A Fine-Grained Information Diffusion Model Based on Node Attributes and Content Features [2]. Based on the Series and
Parallel Connected Model.According to the first article, we build the macroscopic model to simulate the process of information finding its way, finally we predict the communication process in the future.
But the macroscopic model ignores the diffusion probability between senders and receivers, so we build the Extension Model. In the Extension Model, we use AHP and the Stratification Method to determine the value or expression of various factors, use the
Fitting Method to determine the parameters , finally we get the Extension Model.

1.2

Our Work

To determine the structure of the information networks in society and influencing factors which affect the probability of information diffusion, and promote it to be applied in the actual, we boil down the tasks to the following five questions:
• Analyze the traditional information diffusion: when people only communicate by phone, it is a series model; when people only receive information from newspapers, it is a parallel model.
• Build a model combine series and parallel model, which reflect the information networks in real world.
• Consider how to define news according to the combine model above.
• Qualitative analysis of the diffusion probability between any two adjacent nodes in the information network, and build a diffusion probabilistic model.
• Analyze the factors which influence the diffusion probability,and determine the influence degree.

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2
2.1

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Assumption
Basic Model
• The unit of the time of influence is the same in basic model.
• The p between any two node is equal.
• Assume there is only one source node in the network,but in fact ,there may be many source nodes.But when that happens,we can merge those source nodes to one source node,as figure 1 shows:

Figure 1: Merge multiple nodes

2.2

Extention Model
• Assume there is only one source node in the network.

3
3.1

Basic Model
Symbol Description
• p: The probability of transmission and influence when an information diffused from sender to receiver.
• t: t unit time of information diffusion.
• Fs (t, m): After t,the probability that the number of affected people reaches at least
m.
• Es (t): Atter t,the expectation of the number of affected people.
• S i : The ith sub-node of S.
• f romx (t − 1, i): The influence of node X,which means after t − 1 unit time,the probability of that the number of affected people is i.

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3.2

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Information Diffusion Model Based on Series and Parallel Connected
Model

We define the nodes as user and the edges as the relationships among the users.
(1)Series Connected model of information diffusion: First,we establish a Series Connected model.According to this model, one media only receive information from one media and it only spread this information to another media.It is the most simple information diffusion model.As Figure 1 shows,effect probability among nodes is p,S reprents the source node.Then we will discuss:what’s the probability of that the number of affected people reaches m after t?

Figure 2: Series Connected Model
As table 1 shows,we can calculate the probability of that the number of affected people reaches m after t easily in probabilistic method

Table 1: Diffusion result in defferent time in Series Connected model
Time of
Diffusion m = 0 m = 1 m=2 m=3
...
m=N
Expected Numbers
1
2
3
... t 1−p
1−p
1−p
...
1−p

p p(1 − p) p(1 − p)
...
p(1 − p)

0 p2 p2 (1 − p)
...
2 (1 − p) p 0
0
p3
...
3 (1 − p) p ...
...
...
...
...

0
0
0
...
pn−1 (1 − p)

p p + p2 p + p2 + p3
...
2 + ... + pn p+p The results can expressed as:S is the source node,after t,the probability of that the number of affected people reaches m fs (t, m) is expressed as:
 m
 p (1 − p), m < t pm , m = t fs (t, m) =

0, m > t

(1)

Thus,after t the probability of that the number of affected people reaches m is as follows:
(2)

Fs (t, m) = i≥mfs (t,m)

When m<t,
Fs (t, m) = 1 − (1 − p)(1 + p + p2 + · · · + pm+1 ) m−1 = 1 − (1 − p) p(p p−1−1)
= 1 + p(pm−1 − 1)
= 1 − p + pm

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When m=t
Fs (t, m) = pm

(3)

When m>t
Fs (t, m) = 0
Atter t,the expectation of the number of affected people Es (t) is
Es (t) = p + pz + · · · + pt =

p(pt − 1) p−1 (4)

(2)Parallel Connected model of information diffusion: Suppose S is the source node,S i represents its child node.

Figure 3: Parallel Connected Model m ∀t > 1, fs (t, m) = fs (1, m) = Ck pm (1 − p)K−m

(5)

According to the formula above,we get the conclusion as follows: After t,the probability of that the number of affected people reaches at least m is: j=K K

Fs (t, m) =

j
CK pj (1 − p)k−j

fs (t, j) = j=m (6)

j=m

After t,the expectation of the number of affected people is: j=K K

j j × CK pj (1 − p)K−j = Kp

j × fs (t, j) =

Es (t) = j=0 (7)

j=0

(3)Series and Parallel Connected Model:In the actual case, we combine series and parallel connected model more often.
In the graph,node S has two sub-nodes A and B ,the dashed boxes of sub-nodes A and B represent their subtree.
Assume:
1 − p, i = 0
(8)
f romx (t − 1, i) = p × fx (t − 1, i − 1), i > 0

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Figure 4: Series and Parallel Connected Model
This formula represents the influence of node X,which means after time (t-1),the probability of the number of affected people is i. We get this formula by calculating: m f romA (t − 1, i) × f romB (t − 1, m − i)

fs (t, m) =

(9)

i=0

When the source node has random multiple sub-nodes,the formula transformes into as: i|son(s)| =1

f romsi (t − 1, nsi )

fs (t, m) =

(10)

∀ϕ

In the formula,ϕ is the arbitrary combination of ns1 , ns2 , · · · , nsson(s) which satisfies: ns1 + ns2 + · · · + nsson(s) = m i nS represents the number of affected people which is contributed from S i . f romsi (t − 1, nsi ) =

1 − p, nsi = 0 p × fsi (t − 1, nsi − 1), nsi > 0

(11)

After t,the average expectation number of affected people is: j=k j × fs (t, j) =

Es (t) =

i|son(s)| =1

j=k

j=0

j× j=0 f romsi (t − 1, nsi )

(12)

∀ϕ

In the formula,ϕ is the arbitrary combination of ns1 , ns2 , · · · , nsson(s) which satisfies: ns1 + ns2 + · · · + nsson(s) = j

3.3

Conclusion Extended
• According to the trees as follows,we define N1 is the number of affected nodes in figure 5,N2 is the number of affected nodes in figure 6.
N2 =

p(pt −1) p−1 t−1

t

−1)
N1 = pn(ES1 (t − 1) + 1) = pn( p(pp−1 + 1) = n p(p −1)
1
p−1

so
N1 = nN2

(13)

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Figure 5: Series and Parallel Connected Model

Figure 6: Simplified Model
• The processing procedure of an information diffusion problem in which informations diffuse by Figure 7 is the same as handling the same diffusion problem by
Figure 8.

Figure 7: A Series and Parallel Connected
Model

3.4

Figure 8: Symplified Model

Analysis of the Model

By analysis the functions above,we find that p and the structure of the tree influence
Es (t).Thus,we analysis the two factors:

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• When it comes to the structure of the tree,it mainly depends on the number of subnodes.Meanwhile,the number of sub-nodes depends on the main diffusion medias for the current era. For instance,at that time we only have telephone,the number of the sub-nodes is relatively small.In contrast,the number of the sub-nodes is big in the contemporary era owe to the Internet.

Figure 9: Information Diffusion Tree

• When it comes to diffusion probability p,there are two main factors.One is the information value,the other is the relationship among the nodes.When it comes to a certain tree,we assume the relationships among the nodes is definite.
By analysis,we find that generally news are diffused by large amounts of people
Es (t).Informations which are gave out by a certain media,in a certain year,in a certain region, will be diffused through the same tree,and the information that its p is high may qualifies as news.
During different periods, the informations are diffused through different information networks.The informations which is diffused through a quite developed tree regarded as news. Thus,the informations which its p is high and are diffused through a quite developed tree will be regarded news.We will probe the essence of news using our Extension Model.

3.5

Validate Reliability of Basic Model

The size of the area which we choose affects the size of the tree, but does not affect the characteristics of the tree or the rule of information diffusing. In order to simplify the simulation, we choose a university as the source node(we call it "A" University).
The total number of all the teachers and students of A is 51320,let N1 = 51320.The number of A’s followers at Wechat is 15274.All the students and teachers have 300 friends on average. A published an influential news of the campus at Wechat on December 26th,2015 .In

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one day,the number of readers reached to 28200,let N2 = 28200. The the ratio of A’s followers and the average friends of students and teachers is 50 : 1 approximately.
In order to simplify the model,we assume the sub-nodes of the first layer only have one sub-nodes each.And the news only diffuse twice.Thus,we get the Diffusion Model of the news:

Figure 10: The Diffusion Model of the news
For the convenience of simulation,according to extended conclusion ,we scale down the tree in proportion.Thus we get a new tree like this:

Figure 11: Simpified Diffusion Model of the news
Assume p = 60%,we make a simulation programme by Matlab,get Es (2) is 31 and the number of nodes N s = 60.The ratio of expectation people who receive the information and all the nodes is Es (2)/N s = 32.3/60 = 53.83%,which is almost the same as the reality
N 2/N 1 = 28200/54320 = 54.9%.
Then we will use Basic Model to predict the Information Network’s structure and capacity in 2050. We could get the following formula from the series connection formula:
N2016 =

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Figure 12: The Information Diffusion Model of 2016
From the Formula(13), we could get this:
N2016 =

N p(pt − 1) p−1 According to the speed of development of information networks,it is reasonable to forecast that the source node could affect quiet many nodes even all the other nodes in the information network in 2050.Which can berepresented as below:

Figure 13: The Information Diffusion Model of 2050
The information network is close to parallel connection,we could calculate according to
Formula 7:
N2050 = N (t − 1)p
Thus,we draw a conclusion by comparing the difference between the 2050 and the present:

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• The ratio of the number of diffusion:
(t − 1)(p − 1)
N2050
=
N2016
pt − 1
• The ratio of diffusion time:1/t

4

Extension Model

4.1

Symbol Description
Symbol
φs φi φAu φAc φr φc φe φs,r φc,r φv/r φf 1/f 2

4.2

Description
Importance of an sender’s features
Importance of an sender’s influences
Importance of an sender’s authority
Importance of an sender’s activity
Importance of an receiver’s features
Importance of the content’s features
Importance of an edge’s features
Importance of the relationship between a sender and his receiver
Importance of the relationship between a receiver and the content
Ratio of the reposts number and the views number at Weibo
Ratio of followers number and follows number of at Weibo

Micro-model Based on the Basic Model

To establish a more accurate model which considering the micro-factors based on the basic model,we implement feature extraction.
We divide features into two kinds:the features of nodes and the features of edges. We define senders represent the communicators of informations,receivers represent the accepters of informations,contents represent the information itself.
The features of nodes involves three aspects:The sender’s ability of diffusing informations,the receiver’s willing of diffusing informations,the quality and attraction of the content. The features of edges cover the degree of affinity and similarity between senders and receivers,as well as the interest measure of receivers.

4.2.1

Features of Nodes

(1)Features of senders

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Figure 14: Feature Extraction
• The influence of a sender: It measures the sender’s capabilities to affect the receivers’ behavior of passing information.It is proportional to the ratio of repost amount and view amount. φI = lg(

#reposts
+ 1)
#views

(14)

We standardize the value of φI ,φAu ,φc by Min-Max standardized method: xi − min {xj } xi =

1≤j≤n

max {xj } − min {xj }

1≤j≤n

(15)

1≤j≤n

Where xi is the eigenvalue before standardization, xi is the eigenvalue after standardization.
• The authority of a sender: It is the ratio of in-degree and out-degree.In instance,indegree represents a man’s amount of follows on facebook, out-degree represents the number of his followers on Facebook. The authority of a node is proportional to the ratio of in-degree and out-degree. φAu = lg(

#f 1
+ 1)
#f 2

(16)

Where f 1 is the number of followers, f 2 is the number of follows.
• The activity of a sender: It is proportional to the quantity of informations that a node sends average per day. φAc = N S
(17)
We decide the weight of different level of a certain feature by analytic hierarchy process(AHP). Thus we get the equation as follows: φs = 0.547φI + 0.263φAc + 0.190φAu
(2)Features of Receivers

(18)

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Table 2: The activity of a sender
Number of dissemination of information per day

Num-Score(NS)

1-6
7-12
13-21
22-28
29-35
36-42
43-49
50-56
57 and more than 57

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

Table 3: The activity of a sender
Levels

Scale

Compare the two factors, equally important
The former is slightly more important than the latter
The former is obviously more important than the latter
The former is strongly more important than the latter
The former is extremely more important than the latter
The intermediate values of the neighboring judgment
The importance scaling of the latter than the former

1
3
5
7
9
2,4,6,8
Reciprocal

Table 4: Importance of a sender’s features φs φI

φAc

φAu

Wi

Wi 0

φI φAc φAu

1

3
1

2
2
1

1.817
0.874
0.630

0.547
0.263
0.190

1
3
1
2

1
2

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• The dissemination willingness of the receiver: It measures wheather a receiver is willing to pass the informations to the next one or not.
We let w represent the dissemination willing of the receiver. The value of w is 0.1 to 0.9.The following table describes the different grades to the characteristics and the corresponding score.
Table 5: The dissemination willingness of the receiver
Level of Willingness

Score

No willingness to diffuse little willingness to diffuse
It depends
A relatively strong willingness to spread
A quite strong willingness to spread
The intermediate values of the neighboring judgment

0.1
0.3
0.5
0.7
0.9
0.2,0.4,0.6,0.8

φr = lg(

1
+ 1)
#willing

(19)

(3)Features of contents

• The attraction of an information: Different content of information has a different attraction, thereby affecting the probability of information dissemination. In general, people pay more attention to the informations about income, health, employment, medicine which is related to people’s lives.
Table 6: The attraction of an information
Level of attraction
No attraction little attraction
General attraction
A relatively strong attraction
A quite strong attraction
The intermediate values of the neighboring judgment

Score
0.1
0.3
0. 5
0.7
0.9
0.2,0.4,0.6,0.8

• Tag of an information: Tag is the keyword or the high generalization of an information online.An information with a tag is more likely to catch the attention of other people.If there is a tag in the information,we let the value of the tag is
1,otherwise the value of the tag is 0.

T ag(c) =

1, c has a T ag
0, otherwise

(20)

φc = #attraction + T ag(c)

(21)

φn = 0.105φs + 0.258φr + 0.637φc

(22)

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Table 7: The importance of features of nodes φn φr

φc

Wi

Wi 0

φs φr φc
4.2.2

φs
1
3
5

1
3

1
5
1
3

0.406
1
2.466

0.105
0.258
0.637

1
3

1

Features of Edges

• The relationship between senders and receivers: There are many factors could affect the features of the edges,such as the similar interests, the closeness, the number of mentioned times mutually and so on.
Thus,in the following table,we let the value of φs,r is 0.1 to 0.9.When φs,r is bigger,the intimate relationship between the sender and the receiver is closer.

Table 8: Relationship between senders and receivers
The relationship between senders and receivers

Score

No relationship little relationship
General relationship
A relatively strong relationship
A quite strong relationship
The intermediate values of the neighboring judgment

0.1
0.3
0.5
0.7
0.9
0.2,0.4,0.6,0.8

• The relationship between receivers and contents: There are also many factors between the receiver and the content could affect the features of the edge,like the receiver’s interest and his ability to receive informations.
Thus,in the following table,we let the value of φc,r is 0.1 to 0.9.When φc,r is bigger,the intimate relationship between the receiver and the content is closer.

Table 9: Relationship between receivers and contents
The relationship between receiver and contents

Score

No relationship little relationship
General relationship
A relatively strong relationship
A quite strong relationship
The intermediate values of the neighboring judgment

0.1
0.3
0.5
0.7
0.9
0.2,0.4,0.6,0.8

φe = 0.25φs,r + 0.75φc,r

(23)

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Table 10: The importance of the features of edges φe φs,r

φc,r

Wi

Wi 0

φs,r φc,r 1
3

1
3

0.577
1.732

0.25
0.75

1

According to Bayesian logistic function,we can caculate the information diffusion probability p: p(u, v, c) =

1
1+exp{−fa (u,v,c)}

(24) fa (u, v, c) = a0 + a1 φn + a2 φe
We choose some news from Weibo,using fitting calculative method by Matlab,get the value of a0 = −5.4352, a1 = 2.7031, a2 = 0.5630.They are constants when it comes to
Weibo.Using our algorithm,we can calculate the p of other medias.
Analyse the fitting calculative figure above,we find that when φn = 1, φe = 1, p =
0.1026.It means suppose there is a sender who has many followers,his Weibo always be reposted by other people and the news itself is very attractive etc.When he sends a
Weibo as usual,there will be 10 people in 100 approximately to repost the Weibo.
When φn = 0, φe = 0, p = 0.0043.In the same way, suppose there is a sender who has few followers,his Weibo never be reposted by other people and the information he sends is very boring etc.When he sends a Weibo as usual,there will be 0.43 people in 100 approximately to repost the Weibo.It is a little probability event.This is consistent with reality.

Figure 15: The function graph of p by fitting calculative method

4.3

Conclusions

By the Extension Model, we find that many microcosmic factors affect p , which can be devided into two kinds:the features of the nodes and the features of the edges. The features of the nodes can be devided into features of sender, features of receiver, and

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features of information content. Features of edges can be devided into features of relationship between the senders and the receivers, the relationship between receivers and the informations.
According to the Extension Model, if the details of the news is offered, we can calculate the diffusion probability between any two nodes.

4.4

Validate Reliability of Basic Model

We use the reposted amount of a Weibo named "What happened at the night of Paris violent terrorist attack" [3] which sent after that day Paris violent terrorist attacks happened in November 13, 2015, to validate our extension model.
As of November 2015,monthly active users reached 222 million at Weibo [4] .We assume that there is only one source node at Weibo, 1/3 fans could see the article from the source node. According to the Extension Model, we calculate the value of p when this information diffused three times as follows:
We calculate the number of people who might repost this artical during this progress:

Figure 16: The value of p by fitting calculative method
222 ∗ 0.1125 + 222 ∗ 0.1125 ∗ 0.0583 + 222 ∗ 0.1125 ∗ 0.0583 ∗ 0.01744 = 26.4million The result is close to 24 million which is the actual data.Thus,our model is reliable.

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4.5

Page 17 of 19

Sensitivity Analysis

Continue the example of Weibo above, when the factors below changed ∆K ,the change value of p is as follows:
Table 11: Sensitivity Analysis
Result
Reposts amount/Views amount
NS
W
Follows number/Followers number
Attraction
Tag φs,r φc,r

r
0.155*(lg( v

0.054*(lg( f1 f2 r
+ K + 1) − lg( v + 1))
0.075
+ K + 1) − lg( f1 + 1)) f2 5.563*K
1.722*K
1.722*K
0.141*K
0.422*K

From the above table we draw a conclusion that the size relationship of sensitivity is: w, Attraction, and T ag, φc,r , φs,r , N S ,except r and f 1/f 2. The bigger the coefficient of k is, the more effective to the diffusion probability is.
Due to the diffusion willing of the receiver is difficult to change, we recommend that take first consider of raising the attractiveness of news to increase the probability of spread. Based on this, senders can also increase the relationship between the senders and the receivers, such as frequent interaction etc.

5
5.1

Strengths and weaknesses
Strengths

(1)Basic Model
• Our model can explore the Information Diffusion simply and accurately , and we can estimate information transmission by calculating the expectation of the number of affected people.
(2)Extension Model
• The Diffusion probability in any stage can be predicts by the model.
• When measuring the two factors about the node and the three factors about the edge by analytic hierarchy, we can allocate reasonable weights to different factors.
• When measuring the importance of features of receivers, the relationship between sender and receiver and so on, by grading methodology, we can reduce the influence of subjective judgment quantitatively.

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5.2

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Weaknesses

(1)Basic Model
• When merge many nodes to one node ,we ignore the change of p, which may influence the result a little.

(2)Extension Model
• Ignore the time of that the sender pass the information to the receiver.

References
[1] Donghao Zhou,Wenbao Han,Yongjun Wang. The Prediction of Virus Marketing
Propagation Efficiency Based on the Social Network. 2015, 52(1): 156-166.
[2] Li Yu,Xiaoping Yang,Mingyuan Chen. A Fine-Grained Information Diffusion
Model Based on Node Attributes and Content Features. Renmin University of
China,Beijing,2012,01:87-96.
[3] http://news.sina.com.cn/w/zg/2015-11-14/doc-ifxksqiu1575754. shtml/ [4] http://www.xue163.com/news/1121/11213024.html/

Appendices
Here are simulation programmes we used in our model as follow.
Input matlab source: function f=fun1(x,tdata) f=1./(1+exp(-x(1)-x(2)*real(tdata)-x(3)*imag(tdata))) Input matlab source: tdata=[0.9506 0.8644 0.5601 0.4783 0.2834 0.3684 0.2385 0.2640 0.0438] fdata=[0.775 0.575 0.5 0.625 0.5 0.45 0.65 0.75 0.325] cdata=[0.160128 0.116519 0.116519 0.009953 0.010244 0.007443 0.006789 0.024194 0.064516] x=0:0.001:1; y=0:0.001:1;
[X,Y]=meshgrid(x,y);
Z=1./(1+exp(5.4352-2.7031*X-0.563*Y)) xlabel(’Importance of the features of node’) ylabel(’Importance of the features of edge’) zlabel(’p’) title(’p’)

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Input matlab source: tdata=[1.2 1.22 0.97 0.42 0.57 0.31 0.56 0.49 0.54 0.2] fdata=[0.775 0.575 0.5 0.575 0.625 0.5 0.45 0.65 0.75 0.325] cdata=[0.1601 0.1165 0.1088 0.0076 0.0099 0.0102 0.0074 0.0067 0.0241 0.0645] x0=[0,0,0]; a=tdata+fdata.*i; x0=[2 3 -9]; x=lsqcurvefit (’fun1’,x0,a,cdata)