2016年美国大学生数学建模竞赛D题M奖Team42931.pdfVIP

Team Control Number For office use only For office use only 42931 T1 F1 T2 F2 T3 Problem Chosen F3 T4 F4 D 2016 Measuring the Evolution and Influence in Society s Information Networks Summary Sheet Information is spread quickly in today s tech connected communications network In order to measure the evolution and influence in society s information networks we ve determined to build four models Firstly we ve built an interpersonal information propagation model based on Cellular Automata According to the Law of Large Number we find that the average spread speed of information is of positive relevance with the strength of the information while the instantaneous propagation velocity of information is of negative relevance with the online information sharing level We ve also found a critical value of 0 6 out of 1 existing in information diffusion the information can be diffused only when the inherent value of information exceeds 0 6 otherwise it will be filtered out Secondly through fuzzy evaluation model we are able to judge the inherent value of news from different times We ve defined that only when the inherent value of the information exceeds the critical value 0 6 it will be qualified as news Thirdly we ve built a gray prediction model Based on the existing data we ve made predictions to usage rates of different media with the result of relative error for 4 5 6 5 Moreover we ve predicted the situation in 2050 based on our model by 2050 the usage rates of the several primary media will respectively reach newspapers 12 65 radio 16 65 TV 36 Internet 100 Finally we ve built the fuzzy cellular automata model to reflect people s attitudes indicated as 0 9 towards information by fuzzy inferences Then through Mamdani fuzzy reasoning algorithm we ve come to the internal probability for different cellular to influence each other and the external probability from the influence of the topology of mass media These two probabilities are independent but they together affect the flow of information Thus we can draw the following conclusions during the process of information dissemination people with similar views are likely to stay together Mass media plays a guiding role in the process of information dissemination People tend to believe what mass media report Without mass media people s opinions toward the news tend to be neutral which is to say extreme views will gradually disappear KEY WORDS Fuzzy Cellular Automata Fuzzy Inference Gray Prediction Mamdani Reasoning Algorithm Law of Large Number Contents 1Introduction1 2Task 1 Explore the Flow Filter of Information with Models1 2 1Model A Cellular Automata Information Dissemination Model 1 2 1 1Assumptions 1 2 1 2Small world Theory 1 2 1 3Cellular Automata Model 2 2 1 4 Eff ect of Inherent Value 4 2 2Model B Fuzzy Evaluation Model 6 2 2 1Establishment of Evaluation System 6 2 2 2Establishment of Evaluation Set 7 2 2 3 Overall Merit of Infl uence of News 7 2 2 4 Defuzzifi cation 7 2 3Model C Fuzzy Gray Prediction Model 8 2 4Model D Fuzzy Cellular Automata Model 9 2 4 1Assumptions 9 2 4 2 Infl uence of Internal Exposures by Infected Neighbours 9 2 4 3 The Infl uence of External Exposures by Mass Media 10 3Validity of Model11 3 1Task 2 Prediction for Situation In Contemporary 11 3 2Task 3 Prediction for Situation In Future 12 4 Infl uence of Information12 4 1Task 4 How Public Interest and Opinion Can Be Changed 12 4 1 1Only Internal Factors Was Considered 12 4 1 2 Infl uence From External Was Taken Into Consideration 13 4 2 Task 5 Other Factors That Infl uence Information Dissemination 15 4 2 1Information Value 15 4 2 2People s Initial Opinion and Bias 15 4 2 3Form of the Message or Its Source 15 4 2 4The Topology or Strength of the Information Network in A Region Country or Worldwide 15 5Sensitivity Analysis15 6Error Analysis17 7Conclusion17 References19 Team 42931page 1 of 19 1Introduction Technological and social developments have increased the speed with which informa tion can spread as well as infl uenced its content The eff ects of information dissemination includes organizational functions which consists of informing expressing explaining and guiding and social functions which consists of politic economic education and culture Information is spread quickly in today s tech connected communications network some times it is due to the inherent value of the information itself and other times it is due to the information fi nding its way to infl uential or central network nodes that accelerate its spread through social media While content has varied the prevailing premise is that this cultural characteristic to share information both serious and trivial has always been there By taking a historical perspective of fl ow of information relative to inherent value of information the Institute of Communication Media ICM seeks to understand the evolution of the methodology purpose and functionality of societys networks Our task is to analyze the relationship between speed fl ow of information vs inherent value of information 2Task 1 Explore the Flow Filter of Information with Models 2 1Model A Cellular Automata Information Dissemination Model 2 1 1Assumptions 1 In the relationship network a member receive information by the information dissemina tion process from members associated with him And the process is infl uenced by trust between members will to spread and internal value of information 2 In the original stage of dissemination only one member knows the information while the others do not know 3 The process of dissemination is unidirectional and member will not forget the information once he receives it 4 The internal value of information remain unchanged during the dissemination process 5 The member can only send information in the next step time after he received the infor mation 6 The network is sealed so there is no information from outside and to outside 7 The only way for members to receive information is the dissemination process which means he can not know the information by self propagation like learning 2 1 2Small world Theory Small world Model is a model with large clustering coeffi cients 1 and short average path length Most of the average path length of real network is much less than imagination which can be called Small world Eff ect The steps to analyze the small world model are that Firstly fi nd out the topology of abstract of social network based on complex network theory Then analyze the propagation mechanism according to certain rules Finally analyze how to aff ect this process by certain measures Team 42931page 2 of 19 2 1 3Cellular Automata Model Cellular Automata is a special methods available on computer simulation which designed according to the characteristics of complex system its space time and state of system is all discrete 2 In fact cellular automata is a discrete dynamical systems Cellular Automata is abstract of objective world which divides the continuous space into discontinuous points and use a limited number of combiations of the discrete states to characterize state of space Cellular automata can be viewed as a simple model of a spatially extended decentralized system made up of a number of individual components cells 3 The communication be tween constituent cells is limited to local interaction Each individual cell is in a specifi c state which changes over time depending on the states of its local neighbors The overall structure can be viewed as a parallel processing device Interpersonal Information Communication is information exchange between the one and another including direct face to face communication and indirect communication with by means of media According to the assumption in particular interpersonal networks when a member receive information he will spread this information towards individuals associated with him Though there are many factors that can infl uence the next member s success ful reception of information this paper mainly takes the credit between members and the will to spread into consideration Additionally while paths and speed of the information dissemination is diff erent in diff erent networks this model is based on small world networks In the network each node stands for a member Fig 1 is a simulation of interpersonal network with 20 nodes Each node is connected to the K nodes closest to it Figure 1 Interpersonal Network Cellular Automata is a set of cells with discret states that distribute in a simulated space 4 The credit credit 0 1 is the parameter to measure how close relationship two mem bers have The higher the credit is the closer the relationship will be As the network is undirected take member 1 and member 2 for example credit 1 2 credit 2 1 The will to spread indicates the intensity of one s subjective willingness to disseminate the information to other members in the network Just the same as credit the higher the value of will is the greater probability he will disseminate the information to other members In this paper we mainly consider the infl uence of these two elements to the dissemination process by studying Team 42931page 3 of 19 the process under diff erent distribution Under the structure of interpersonal network consider two situations While the credit is uniform distribution the will is normal distribution uniform dis tribution or binomial distribution While the will is uniform distribution the credit is normal distribution uniform dis tribution or binomial distribution In the simulation we use Matlab to get data meeting the required distribution at random To evaluate the dissemination we defi ne 1 Spread speed all the nodes that have received the information the step time of the dissemination 2 Average spread speed sum of spread speed of all groups number of experiment 3 Instantaneous velocity nodes that have received the information in time i nodes that have received the information in time i 1 i 1 i 20 i is integer 0 40 450 50 550 60 65 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 N 50 Interest 1 Mean Average speed Figure 2 The infl uence of means to average ve locity when credit is normal distributed Variance 0 02 00 050 10 150 2 1 35 1 4 1 45 1 5 1 55 1 6 N 50 Interest 1 Variance Average speed Figure 3 The infl uence of variance to average velocity when will is normal distributed Mean 0 5 0 40 450 50 550 60 65 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 N 50 Interest 1 Mean Average speed Figure 4 The infl uence of means to average ve locity when will is normal distributed Variance 0 02 00 050 10 150 2 1 35 1 4 1 45 1 5 1 55 1 6 1 65 1 7 N 50 Interest 1 Variance Average speed Figure 5 The infl uence of variance to average velocity when will is normal distributed Mean 0 5 The result of the simulation suggests that Team 42931page 4 of 19 In the case of a uniform distribution of trust means of will to disseminate information to the average propagation velocity is approximately in proportional relationship and the average speed of spread increases as the variance of trust increases In the case of a uniform distribution of will to disseminate means of trust to average speed of spread is approximately in proportional relationship and the average speed of spread decreases as the variance of trust increases The instantaneous velocity of information spread decreases as the information sharing level in the network increases When the information sharing level becomes saturated the instantaneous velocity is zero The interpersonal spread speed is mainly infl uenced by interpersonal network 2 1 4 Eff ect of Inherent Value We use I to represent the attractiveness of information I 0 1 It s obvious that the higher the I is the larger probability individuals can receive the information Another external factor that can infl uence the receiving of information is the eff ects of surrounding people To quantify this infl uence of surrounding people we take the number of people that have received this information into consideration The more the nearby people have received the information the larger the probability the individual will receive the information In that infl uence of diff erent people is the same so the infl uence of an individual towards other people can be represented by P P 0 1 which means the proportion of disseminators of information accounted for all the associated people Personal preference is represented by 0 1 We get the factor value P from cellular automata model Among the external factors that infl uence the receiving of information the two factors I and P play the diff erent roles C is used to represent the proportion of I towards the whole infl uence C 0 1 00 20 40 60 81 0 10 20 30 40 50 60 70 80 90 100 Interest Diffusion Rate Figure 6 The infl uence of information attractiveness to diff usion rate At the beginning we choose an individual at random and assume that he receive the information A n is the set of people that receive the information at time n Then at time n 1 the state of an individual that is not in A n is decided by state of people in time n who are associatied with him If no one around him has received the information he would not know the information Otherwise whether he receives the information or not is decided Team 42931page 5 of 19 by the whole infl uence and his own preference If the whole infl uence is larger than he will receive the information and get into set A n 1 Fig 6 shows the infl uence of information attractiveness to diff usion rate From it we can draw the conclusion that there is a threshold for information dissemination which is 0 6 Only when the attractiveness is above 0 6 can the information spread out Fig 7 to Fig 10 below shows the infl uence of information attractiveness to spread speed 00 20 40 60 81 0 10 20 30 40 50 60 70 80 90 100 Diffusion Rate 30 Interest Step Time Figure 7 The infl uence of information attrac tiveness to spread speed greater than or equal to 0 3 00 20 40 60 81 0 10 20 30 40 50 60 70 80 90 100 Diffusion Rate 50 Interest Step Time Figure 8 The infl uence of information attrac tiveness to spread speed greater than or equal to 0 5 00 20 40 60 81 0 10 20 30 40 50 60 70 80 90 100 Diffusion Rate 70 Interest Step Time Figure 9 The infl uence of information attrac tiveness to spread speed greater than or equal to 0 7 00 20 40 60 81 0 10 20 30 40 50 60 70 80 90 100 Diffusion Rate 90 Interest Step Time Figure 10 The infl uence of information at tractiveness to spread speed greater than or equal to 0 9 From fi gures that show the infl uence of information attractiveness to spread speed and time to spread we can conclude 1 The diff usivity increases as the information attractiveness increases 2 When the information attractiveness is large enough it has no eff ect on time of infor mation dissemination 3 When the information attractiveness is small enough far less than 0 2 time to dis seminate is not long However in such case information diff usion area is very small which there is nearly no diff usion of information Team 42931page 6 of 19 020406080100 0 0 5 1 1 5 2 2 5 N 50 Interest 1 Saturation Instantaneous speed Figure 11 The infl uence of saturation to instantaneous velocity 2 2Model B Fuzzy Evaluation Model 2 2 1Establishment of Evaluation System News is packaged information about current events happening somewhere else To fi gure out what qualifi es as news we should fi rstly choose fi ve properties of news Importance Signifi cance Authenticity Newness 5 and Interest We use a set U U1 U2 U3 U4 U5 to represent the fi ve elements we choose U1 to U5correspond to them respectively As for the weights of the fi ve elements we use AHP Analytic Hierarchy Process 6 to decide Considered that there are fi ve media television newspaper internet radio and mobile fi ve coeffi cient matrix is shown as below the script t n i r m represent television newspaper internet radio and mobile At 11 2433 21755 1 41 711 21 3 1 31 5211 1 31 5311 An 11 3413 31635 1 41 611 31 2 11 3221 1 31 5221 Ai 11 2343 21575 1 31 5121 1 41 71 211 2 1 31 5121 Ar 11 2434 21757 1 41 711 21 1 31 5212 1 41 711 21 Am 12464 1 21232 1 41 2121 1 61 31 211 2 1 41 2121 The weight vector can be got Ut 0 2636 0 47580 05380 09810 1087 T 1 Check the consistency CI 0 0180 CR 0 0161 Un 0 2093 0 46830 05580 13850 1282 T 2 Check the consistency CI 0 0775 CR 0 0692 Ui 0 2649 0 47870 09920 05810 0992 T 3 Team 42931page 7 of 19 Check the consistency CI 0 0067 CR 0 0060 Ur 0 2699 0 49250 06370 11010 0637 T 4 Check the consistency CI 0 0665 CR 0 0058 Um 0 4590 0 22950 12160 06840 1216 T 5 Check the consistency CI 0 0033 CR 0 0030 All the consistency of the matrix can be accepted 2 2 2Establishment of Evaluation Set A evaluation set is a collection of the overall results of the evaluation of the target V V1 V2 Vn n is integer We consider the situation n 5 which represents high relatively high medium relatively low low V1corresponds to high To determine the judgement matrix judge element i of the target in the set U Here con sider a concept called degree of membership ri which means the degree that the judgement correspond to the element j in the evaluation set V In such case the result of element ui can be represented by fuzzy set R Take the event that President Lincoln was assasinated for example In that period the main method of information diff usion is newspaper Ten judgements is taken towards the newness of the information one of which are V1 two V2 four V3 three V4 one V5 So r41 0 1 r42 0 2 r43 0 4 r44 0 3 r45 0 2 And matrix R R 10000 10000 10000 0 10 20 40 30 2 00001 6 2 2 3 Overall Merit of Infl uence of News In that the weight matrix U U1U2U3U4U5 0 1362 0 3515 0 3649 0 0901 0 0476 we can get the overall evaluation matrix Y U R 0 7472 0 02770 05540 04160 1558 7 After normalization Y 0 0 7271 0 02700 05390 04040 1516 8 The matrix Y 0 means the degree of membership to high infl uence is 0 1375 the degree of membership to relatively high infl uence is 0 3549 and so on 2 2 4 Defuzzifi cation 1 Max degree of membership Choose the max value of the