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[美赛经验] 2012年二等奖论文

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2012 Mathematical Contest in Modeling (MCM) Summary Sheet
Locating the conspirators in the company
The mathematical model built in this paper can analyze the complex crime reasonably. According
to a part of known arrested suspects and their social relationship, the model can discover the
conspirator with the maximum possibility, which can accelerate the process of investigating cases
With the the suspicious information, known conspirators and innocents, a model of finding
internal crime suspects in the company is established. For a person in the company, if he has
closer contact with other members he will has more possibility to commit a crime, which is called
core degree. Making use of Matrix replacement method and Hierarchical clustering method, we
can find the group of higher core degree and sort them with descending. Next, we deem that a
member who has closer contact with a conspirator will be more possibilily a conspirator . Because
of the known conspirators in the company, to find the other suspects in the network sturcture, A
list of association degree is ordered with ascending. At last, on the basis of the computational
model, taking transmission information among members into acount, we stratified these message
topics and given the different layers of different topics of suspicious degrees. According to the
different topics among these members, we can calculate their suspicious degrees accurately and
ordered them descending. Analysis by synthesis the result of three sort, using a computer merge
the result of three sort, thereby all of the suspects is obtained accurately.
Computers is the efficient and accurate tools to handle and analyze large-scale data. Using
computers to deal with original data can reduce the error of the value of topics. We can make full
use of semantic network analysis, artificial intelligence and text analysis, and calculate the
frequency of words in original data . There are some less important but greater frequent words
among those data, such as copula and personal pronouns. Deleting this type of words and merge
them with similar semantic words, a language bank is then built with these similar words, and the
frequency of every words is computed. Sometimes words collocation in the bank are also
suspicious. therefore when these suspicious words appear and the frequency of them is high, thus
the person who transmit the type of information must be suspicious characters. It suffice to find
all of suspects accurately by this method.
The crime busting model we built can also apply to many other practical cases. For example, it
can be used on detecting difficult network-crime cases and applied to the problem of the spread of
the virus between cells in the biological network. The method can also deal with these difficulties
with high accuracy.
Locating the conspirators in the company Abstract The mathematical model built in this paper can analyze the complex crime reasonably. According to a part of known arrested suspects and their social relationship, the model can discover the conspirator with the maximum possibility, which can accelerate the process of investigating cases With the the suspicious information, known conspirators and innocents, a model of finding internal crime suspects in the company is established. For a person in the company, if he has closer contact with other members he will has more possibility to commit a crime, which is called core degree. Making use of Matrix replacement method and Hierarchical clustering method, we can find the group of higher core degree and sort them with descending. Next, we deem that a member who has closer contact with a conspirator will be more possibilily a conspirator . Because of the known conspirators in the company, to find the other suspects in the network sturcture, A list of association degree is ordered with ascending. At last, on the basis of the computational model, taking transmission information among members into acount, we stratified these message topics and given the different layers of different topics of suspicious degrees. According to the different topics among these members, we can calculate their suspicious degrees accurately and ordered them descending. Analysis by synthesis the result of three sort, using a computer merge the result of three sort, thereby all of the suspects is obtained accurately. Computers is the efficient and accurate tools to handle and analyze large-scale data. Using computers to deal with original data can reduce the error of the value of topics. We can make full use of semantic network analysis, artificial intelligence and text analysis, and calculate the frequency of words in original data . There are some less important but greater frequent words among those data, such as copula and personal pronouns. Deleting this type of words and merge them with similar semantic words, a language bank is then built with these similar words, and the frequency of every words is computed. Sometimes words collocation in the bank are also suspicious. therefore when these suspicious words appear and the frequency of them is high, thus the person who transmit the type of information must be suspicious characters. It suffice to find all of suspects accurately by this method. The crime busting model we built can also apply to many other practical cases. For example, it can be used on detecting difficult network-crime cases and applied to the problem of the spread of the virus between cells in the biological network. The method can also deal with these difficulties with high accuracy. Keywords: core degrees, suspicious degrees, association degrees, text analysis, crime busting
Team#15783 page 2 of 15
Contents
1 Introduction ............................................................................................................... 3
2 Analysis of the Problem ............................................................................................ 3
3 Crime busting ............................................................................................................ 4
3.1 Analysis ............................................................................................................ 4
3.2 Symbols ............................................................................................................ 4
3.3 Assumption ...................................................................................................... 5
3.4 Modeling .......................................................................................................... 5
3.4.1 Solving of the core degree ..................................................................... 5
3.4.2 Solving of absolutely close degree ........................................................ 6
3.4.3 Using suspected information to find criminal suspects ..................... 7
3.5 Consider the additional datas ........................................................................ 9
4 Computer processing .............................................................................................. 10
4.1 Analysis .......................................................................................................... 10
4.2 Definition ....................................................................................................... 11
4.3 Computer processing Method ...................................................................... 11
5 Model promotion ..................................................................................................... 13
5.1 Analysis of model .......................................................................................... 13
5.2 Model application.......................................................................................... 13
6 Weaknesses and Strengths of the Model ............................................................... 14
6.1 Strengths ........................................................................................................ 14
6.2 Weaknesses .................................................................................................... 14
7 Conclusion ............................................................................................................... 15
8 References ................................................................................................................ 15
Team#15783 page 3 of 15
1 Introduction
In recent years, the group frauds and economic crime problems have been very common in our daily life. Since the number of suspects is great, then it is difficult to detect the cases for the public security organs. Once the criminals escape, they will be a threat to more people and property, and even to personal security . The links between criminals posed by criminal networks is complex , and the mistake of judgment will falsely accuse innocent people and let the criminals get off . Here the problem arises: how to locate all the suspicion without a mistake? We learned many of the existing methods find that none give us a definite answer. Then we found the model of crime busting is similar to the model of social networks in the BBS in a way. What we need to do is identify the core characters and important figures in this matter. If we do the above, then we can basically determine the degree of suspicion for each person in this group combined with all kinds of the message topics, and arrange the guesswork out of suspects. It narrow the scope of criminal elements and reduce the workload of the public security organs so that they can break cases faster.
2 Analysis of the Problem
 First, we should gain a clear idea of the characteristics of internal crime. There is a wide range of the criminal subjects: the senior managers of the company can be the conspirator, and each of the staffs can be also the conspirator. So it is difficult to capture criminals. But if we use the key features of economic crime, then we can easily find the mastermind of their accomplices.
 According to the characteristics of crime and the offender's psychological, we believe that the possibility that those persons who keep close contact with criminals are the conspirators is very large. Then, by known criminals and suspicious messages, we can accurately determine criminal conspirators.
 Based on the known contacts between persons inside the company, we use the matrix displacement method and the hierarchical clustering method to identify the company's central figures who keep close contact with other employees. Then combining with these figures with the types of message topics, we can make a further judgment to identify the other suspects. In addition, using given already certain schemers and certain not schemers, we can further discuss the group which have close relationship with those persons, and so we can increase the veracity of the result. With the help of our model, we can identify the maximum possible accomplices and confidently point out the conspirators and accomplices within the company.
Team#15783 page 4 of 15
3 Crime busting
3.1 Analysis
Although criminals choose randomly the modality of crime information and the route
transmission of crime information, but their crime form still has a certain rule. Their
modus operandi in this company transmits mainly by the information transfer form, so
criminal gangs should form a network connection model. Questions have given
several already certain criminals and crime information types, then according to these
criminals' close relationship and the suspected degrees of total numbers of message
receive and message issued, we can judge the scope of the criminals.
3.2 Symbols
Table1 symbols
k A The number of the company group members
ai, k  if member i and member k have directly connected
  D C k Connection degree of each member k A
  ij g k if means of member i A and member j A pass by k A .
C k B k A ’s absolute agent degree
li, k the most short-circuit path length of member i A and member j A
  C C k the absolutely tight density
D C the average value of the absolute agent degree
B C the average value of the absolute agent degree
C C the average value of the closely degree
  D Dev C the smooth coefficient of connection degree
  B Dev C the smooth coefficient of absolute agent
  C Dev C the smooth coefficient of absolutely tight density
Team#15783 page 5 of 15
3.3 Assumption
 There is no message exchange without the message in the problem;
 Assume that all the criminals in the group is given;
 To measure the possibility of a crime, we define suspect degree of message which
is divided into four layers. Suspect degree of the information within each layer is
the same. The messages from the same layer have the same suspects degree.
3.4 Modeling
3.4.1 Solving of the core degree
In this article, we need to study the company group consisted of 83 members. There
are different degrees of interaction among people, and we can obtain the extent of this
interaction via information passing between different people. The person who has the
largest amount of messages has the greatest influence without considering message
topics, then we call this person as the central figure. His position is far above the other
opinion leaders because he has the most members of the exchange relationships. We
call the person as a loner due to the fewest numbers of information received and
information issued. This person does not influence the others, then those loners are
not criminals in our case. The central figure and important figures have a great
possibility to commit a crime.
Relationship analysis focuses on the relationship between members and interactive
behavior, and it is often used to identify the central figure of groups[4]. Let the symbol
  1 2 , , , n V  A A  A stand for the members of this group, and define as follows:
    

n
i
D C k a i k
1
, (1)
Equation (1) stands for Connection Degree of each member k A , and Connection
Degree reflects the level of activity of the group members. Let n be the number of
members of groups and n equals 83 in this problem. If ai,k 1, then member i
and member k have direct connection, that is, member i and member k have
information exchanges. If ai,k  0 , then both have no directly connection with
each other, and so have no information exchanges.
    
n
i
n
j
B ij C k g k (2)
Team#15783 page 6 of 15
The shortest circuit path number that passes member k A is called k A ’s absolute
agent degree, denoted byC k B . It describes the ability of members contacting other
members as an intermediary. If g k 1 ij , then member i A and member j A pass
by member k A .
    

n
i
C C k l i k
1
, (3)
The shortest circuit path length of member k A with all other members in the
network called absolutely tight density, denoted by C k C . The li, k means the
shortest circuit path length of member i A and member j A . What Equation (3)
describes is the close degree of a member of the small group with himself as center.
          D eCvD 
C C
D e CvB
B B
D e CvD
D D C C C C C C c o r e e e
2 / 2 / 2 /
       (4)
Equation (1) generally considers the importance of connection degree, absolute agent
degree and absolutely tight density. Using this we can measure the core degree of
character member. Given D C represent the average value of member contacts degree.
    
 
n
i
D D D Dev C C i C
1
2 means the smooth coefficient of connection degree.
    
 
n
i
B B B Dev C C i C
1
2 means the smooth coefficient of absolute agent degree.
    
 
n
i
C C C Dev C C i C
1
2 means the smooth coefficient of absolutely tight density.
Thus, we can obtain the core degree of all members by using computer program, and
then rank the core degree with decreasing order.
3.4.2 Solving of absolutely close degree
Common sense shows that an accomplice may have frequent contact with criminals，
so they will have a larger absolutely close degree. We can regard the kown
conpirators as a group. Calculate the absolutely close degree of each members to this
group.
Using the expression (3) to find the absolutely close degree with criminal suspects of
everybody in the company. Regarding all of these criminal suspects as a group,
calculate the sum of the compactness between crime groups and every individual
Team#15783 page 7 of 15
person, that is the value of number means the degree of close contact with each
individual and criminal groups. Calculate the result using computer programming.
Size down the absolutely close degree, we can find people who have closer contact
with this crime group.
3.4.3 Using suspected information to find criminal suspects
Step1: Suspected to information stratification
Except for certain suspected information, according to the characteristics of the
corporate crime, we confirm that remaining information has relationship with the
budget and economic, internal personnel dispatch and daily life. In daily life, for the
standard of stratification, we roughly divide it into the following levels:
A. 7, 11, 13 is defined certain suspected information, and set the value of suspected
degree information at 1;
B. 1, 14, 2 is defined as the larger suspected degrees information, and set the value of
suspected degree information at 0.75;
C. 5, 6, 10, 15 is defined as the smaller suspected degrees information, and set the
value of suspected degree information at 0.5;
D. 3, 4, 8, 9, 12 is defined as the smallest suspected degrees information, and set the
value of suspected degree information at 0.25;
Step2: Building matrix
According to the types of information received and information sent, we draw out the
matrix, and the row and column of the matrix mean individual numbers. 1 ij z 
means the individual staff received or sent out the information type is A, 0.75 ij z 
means the individual staff received or sent out the information type is B, 0.5 ij z 
means the individual staff received or sent out the information type is C, 0.25 ij z 
means the individual staff received or sent out the information type is D. As a result,
we get a matrix of 8383.
Team#15783 page 8 of 15
0 0 1 . 2 5 0 0 0
0 0
2.25 0
0 0
0 0
0 0 0 0 0 0
 
 
 
 
 
 
 
 
 
 
 



  



（5）
Step3: Evaluating the suspected degrees
According to the expression:
 



  83
1
83
1
i
ij
i
ij
A
a
B k （6）
Note: Bk  stands for the k th's information of suspected degrees, ij a stands for
the value of the i th row and the j th colunm in the matrix,
1 0
0 0
ij
ij
ij
if a
A
if a
 
 
 
（7）
Figure out the suspected degrees of the individual staff received or sent out the
information, and then rank the suspected degree with decreasing order.
We can size down the degree of suspicion through the above three steps. The results
as follows:
Table 2 Suspect sort
Rank No. Name
1 54 Ulf
2 81 Seeni
3 73 Carina
4 21 Alex
5 67 Yao
6 33 Kim
7 49 Harvey
8 7 Elsie
9 60 Lars
10 51 Dayi
Team#15783 page 9 of 15
continued
3.5 Consider the additional datas
According to the meaning of the question, the second question and the first question bring out the best in each other, their model are similar. Just on the basis of the original condition to put more conducive to survey the conditions, so in this question we just improve and tidy the model in the first question. The process of finding core people is similar to the first question. According to the meaning of the question, we put Chris into this crime group, and then calculate respectively the degree of correlation of the new crime group with the company internal group. After that sort them to find out more suspected of personnel. In addition to cofirming suspected information , we according to the characteristics of the corporate crime remaining information confirm that has relationship with the budget and economic , internal personnel dispatch and daily life. In daily life for the standard for stratification, roughly divided into the following levels: A.1,7,11,13 as being suspected uncertainty information, and set the value of suspected degree information at one; B. 14, 2 is definition as the larger suspected degrees information, and set the value of suspected degree information at 0.75; C.5,6,10,15 is definition as the smaller suspected degrees information, and set the value of suspected degree information at 0.5;
D. 3,4,8,9,12 is definition as the smallest suspected degrees information ,and set the value of suspected degree information at 0.25; This can create a correlation matrix as follows:
Rank
No.
Name
11
66
Melia
12
57
Sheng
13
18
Jean
14
43
Paul
15
16
Jerome
16
56
Cha
17
46
Louis
18
20
Crystal
Team#15783 page 10 of 15
0 0 1.25 0
0
2.25
0.5
0 0
0 0
 
 
 
 
 
 
 
 
 
 
 

   
   
 
（8）
Operator again with the previous methods, we can get the suspected degree of all
members . Prioritize on the results obtained above, the suspected persons are as
following table
Table 3 Suspect sort
4 Computer processing
4.1 Analysis
 The criminal process often involves a lot of people. In the detection process,
manual processing of criminal information is a very complex,and this can easily
case errors. So we need to find some other ways to reduce errors.
 With the development of computer technology , computer process data can be
Rank No. Name
1 54 Ulf
2 81 Seeni
3 21 Alex
4 73 Carina
5 67 Yao
6 33 Kim
7 49 Harvey
8 66 Melia
9 7 Elsie
10 2 Paige
11 60 Lars
12 51 Dayi
13 56 Cha
14 57 Sheng
15 43 Paul
16 16 Jerome
17 18 Jean
18 0 Chris
Team#15783 page 11 of 15
both

zan

论文, 2012, suspicious, arrested, discover