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For office use only

T1 ________________

T2 ________________

T3 ________________

T4 ________________

Team Control Number

13215

Problem Chosen

For office use only

F1 ________________

F2 ________________

F3 ________________

F4 ________________

2012 Mathematical Contest in Modeling (MCM) Summary Sheet

(Attach a copy of this page to each copy of your solution paper.)

Type a summary of your results on this page. Do not include

the name of your school, advisor, or team members on this page.

Message Network Modeling for Crime Busting

Abstract

A particularly popular and challenging problem in crime analysis is to identify

the conspirators through analysis of message networks. In this paper, using the data of

message traffic, we model to prioritize the likelihood of one’s being conspirator, and

nominate the probable conspiracy leaders.

We note a fact that any conspirator has at least one message communication with

other conspirators, and assume that sending or receiving a message has the same

effect, and then develop Model 1, 2 and 3 to make a priority list respectively and

Model 4 to nominate the conspiracy leader.

In Model 1, we take the amount of one’s suspicious messages and one’s all

messages with known conspirators into account, and define a simple composite index

to measure the likelihood of one’s being conspirator.

Then, considering probability relevance of all nodes, we develop Model 2 based

on Law of Total Probability. In this model, probability of one’s being conspirator is

the weight sum of probabilities of others directly linking to it. And we develop

Algorithm 1 to calculate probabilities of all the network nodes as direct calculation is

infeasible.

Besides, in order to better quantify one’s relationship to the known conspirators,

we develop Model 3, which brings in the concept “shortest path” of graph theory to

create an indicator evaluating the likelihood of one’s being conspirator which can be

calculated through Algorithm 2.

As a result, we compare three priority lists and conclude that the overall rankings

are similar but quite changes appear in some nodes. Additionally, when altering the

given information, we find that the priority list just changes slightly except for a few

nodes, so that we validate the models’ stability.

Afterwards, by using Freeman’s centrality method, we develop Model 4 to

nominate three most probable leaders: Paul, Elsie, Dolores (senior manager).

What’s more, we make some remarks about the models and discuss what could

be done to enhance them in the future work. In addition, we further explain

Investigation EZ through text and semantic network analysis, so to illustrate the

models’ capacity of applying to more complicated cases. Finally, we briefly state the

application of our models in other disciplines.

Team#13215Pageof 18

1

Introduction

ICM is investigating a conspiracy whose members all work for the same noted

company which majors in developing and marketing computer software for banks and

credit card companies. Conspirators commit crimes by embezzling funds from the

company and using internet fraud to steal funds from credit cards. It is a kind of

commercial fraud. Fraud is a human endeavor, involving deception, purposeful intent,

intensity of desire, risk of apprehension, violation of trust, rationalization, etc.

Psychological factors influence the behaviors of fraud perpetrators (Sridhar

Ramamoorti, 2008).

ICM provides us the following information that they have mastered

●All 83 office workers’ names;

●15 short descriptions of the topics ( Topic 7, 11, and 13 have been deemed to be

suspicious);

●400 links of the nodes that transmit messages and the topic code numbers;

●7 known conspirators: Jean, Alex, Elsie, Paul, Ulf, Yao, and Harvey;

●8 known non-conspirators: Darlene, Tran, Jia, Ellin, Gard, Chris, Paige and

Este;

●Jerome, Delores, and Gretchen are the senior managers of the company.

For crime busting, we develop models to

●Identify all conspirators as accurately as possible, make a priority list that

shows the likelihood of one’s being conspirator, so that erroneous judgments or

miss-judgments won’t happen easily;

● Nominate the conspiracy leader.

Declaration of the given data

●“Topics.xls” contains only 15 topics, but “topic 18” appears in line 215 of

“Messages.xls”. To fix this error, we decide to neglect this invalid data and delete it.

●In page 5, line 2 of “2012_ICM_Problem.pdf”, it says that “Elsie” is one of the

known conspirators. However we find two “Elsie” with node number “7” and “37”.

Throughout some basic statistics about the message traffic containing suspicious

topics, it appears that “7 Elsie” is more likely to be a known conspirator rather than

“37 Elsie”. Therefore, we assume that “Elsie” in “2012_ICM_Problem.pdf” indicates

“Elsie” with node number 7 in “names.xls”.

●As the problem paper point out, “Delores” is a senior manager. But “Delores”

can’t be found in “names.xls” while “Dolores” is found. So we consider it as

misspelling and replace “Delores” with “Dolores”.

●“Gretchen” is also one of the senior managers. But two “Gretchen” are found in

“names.xls” with different node number “4” and “32”. In consideration of this

redundancy, we determine to pick out node 32 for “Gretchen” indicated in the

problem paper artificially. In addition, our basic statistics also shows that “32

Gretchen” has more message exchanges than “4 Gretchen”, which may imply that “32

Gretchen” is more probably the senior manager than “4 Gretchen” due to managers

often contact others more than ordinary office workers.

2

Problem analysis and assumption

Commercial fraud is committed by those intelligent people who are confident

with their professional skills. Meanwhile, this kind of crime couldn’t involve only one

person, but always need cooperation of a whole group. Thus, communication with

other conspirators would be inevitable. However, they obviously know that they are

linked together and if one person discloses their secrets, none of them can get off. So

they are conscious when they communicate with their colleagues who aren’t their

companions, especially when they talk about sensitive issues. And the higher

intellectual level of perpetrators with rich society experience, the more conscious they

are (Zhigang Lin,2010). And ICM can figure out suspicious topic which stands a good

chance of being related to the conspiracy by some content analysis method. On the

one hand, although Conspirators would try to avoid involving suspicious topics in

their messages, they have to convey this kind of information sometimes due to the

business or other reason. On the other hand, trust and close relationship play an

important role in a conspiracy group, so normal messages exchanges can also reflect

the conspiracy relationship.

Based on psychology analysis above, we can state that all conspirators have at

least one message communication with other conspirators, whether suspicious or

unsuspicious message.

In addition, we make the assumption that sending and receiving messages have

same effect when we evaluate the likelihood of one’s being conspirator;

Models

Model 1

Establishment of model

According to the analysis of the problem, the likelihood of one’s being

conspirator is related to various factors, such as what topics are contained in the

worker messages, how many messages and suspicious messages are the worker

related with, who did the worker contact with, etc. To evaluate the likelihood of one’s

being conspirators, we use the following equation which combines two quantity

indexes:

, 0,1, 2,...,82

2max{ } max{ }

⎛⎞

⎜⎟

=+=

⎜⎟

⎝⎠

(1)

Where is the suspicious message number that office worker sent or received

and is message number that office worker

sent to or received by known

conspirators.

In order to get each value of and , we make data statistics and draw

Figure 1:

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