2016年美赛O奖论文

提供最新最全的美赛题目大全。数学建模美赛公布题型，ABCDEF六种，A连续 B离散 C数据见解 D运筹学／网络科学 E环境科学 F政策 根据COMAP公布的 报告 ，2016年MCM有7421支队伍参赛，其中美国参赛队389支，其他7032支队伍主要来自中国，评出Outstanding 13个（约0.18%），Finalist 22个（约0.3%），Meritorious 594个（约8%），Honorable Mention 2604个（约35%）；而2016年ICM有5025支队伍参赛，其中美国参赛队91支，评出各奖项的个数分别为14、15、935、2287，相应比例约为0.28%、0.3
Team#42220 Page 2 of 20 1 Introduction 1.1 Background Nowadays, information is spread quickly in our daily life. Due to the development of technology, no matter the big events or the trivia, people have access to information quickly and conveniently. As a result, the information garbage is everywhere which influences our life In order to manage and track the flow of information, people raise awareness of the importance of establishing a society information network Interested in information communication situation of social network, several methods have been adopted to describe it. Most models supposed that information is spread according to time sequence. For example, Independent Cascade Mode! regards that propagation probability is same among nodes all the time which has no memory. Meanwhile, Linear threshold model concentrates on cumulative probability which reflects the accepted ability of receiver. However, these models ignore such factors Attributes of nodes such as personal interests Influence of inherent value of information in propagation Propagation process is not synchronous Therefore, there is an urgent need for a complete scientific model. Based on it,we establish a doublelayer complex network 1.2 Our work To further present our solutions, we arrange our paper as follow In section 2, we give out the reliable assumptions to simplify the model e In section 3, the doublelayer network is constructed to analyze and forecast the information flow. The inner layer analyzes the information flow among a region. while the outer layer analyze the flow from region to region. Meanwhile, the sensitivity analysis is given out to validate our model o In section 4, we apply our model to solve a series of problem, such as qualification of news, prediction of information flow today and around the year 2050. Next, we establish a model to analyze the factors to influence the public interest and opinion. Later, we analyze the influence of different indexes in Spreading Information and Public opinion e At last, we discuss the strengths and weaknesses of our model in detail 2. Assumptions The data we found is authentic and reliable No invention of media in the time for prediction. That is to say, people use the same media as today, just the proportion of media changes o Ignore the firewall among countries which restrains the spread of inform Team#42220 Page 3 of 20 information in the our network is not secret o Due to the difference of development level is relatively little in a small area, we ignore the difference in media usage from a statistical point of view o To simplify the model, we only consider six media to spread information which are newspaper, telegraph, radio, television, the Internet and mobile phone The node spread the same message only once. That is to say, the indegree of a node is no more than 1 3. Task 1: Establish a Doublelaver Complex Network 3.1 Frame A large amount of factors ought to take into account to quantify the flow of information For example, due to the difference in development level among countries, the way and speed of flow must be different. Also, with the Internet technology become more sophisticated, each person plays a more and more important role in spreading information instead of original media such as newspaper and television, which the objective factors such as personal interests should be taken into consideration Based on above analysis, we can see that a simple network cannot reflect the whole ituation of the flow of information well. As a result, a complex network with multilayers whose layers have a certain interactions and independence seems to be more reasonable Therefore, we come up with a doublelayered network, global information network and point to point information network, which belongs to a complex network, to reflect the information flow more specifically The structure of our doublelayer complex network is shown in Figure 1 global network pointtopoint network Figure 1 A doublelayer Network The blue dots represent each person and small circle represents a region The directed line represents the direction of information flow. The figure vividly indicates that the information originates from a node(source), then spread quickly over a region and at last spread the whole world. The internal and external factors which influence the information flow is analyzed in detail in next two sections Team#42220 Page 4 of 20 3.2 Inner Layer: Information Flow over a region The process of information propagation can be divided into massive point to point cases Each case has three entities: sender(s), receiver(R) and content(C). Figure 2 shows the relationship SI R2 Figure 2 Relationships of three entities in the diffusion process The information spread from sender to receiver, each spread case carries a piece of information 3.2.1 Index extraction In order to describe the flow of information specifically, we choose twitter as an example. Based on the As/C Model, to improve the accuracy of our model, we take the inherent value of information, the media effect and personal subjective emotions into consideration and give five indexes to analyze the information flow Communication media Elect, Sender Characteristic, Receiver Characterislic, Content Characteristic, Relation between Sender and receiver. Suppose that node u as a sender, node v as a receiver Communication Media Characteristic(n From a historical perspective, the communication media change a lot From newspaper, and radio in the early time to television, Internet, mobile phone and many other media nowadays Information tends to spread more quickly and wide. Due to the difference in social developments, mainstream media of different regions varies a lot. Based on six media (newspaper, telegraph, radio, television, Internet and mobile phone), communication medium effect is calculated as follows fu=22(i=N, TEL,R, TV, 1, M) When n,TEL, R, TV,I is short for six media, newspaper, telegraph, radio, television the Internet and mobile phone respectively where / denotes the market share of i medium and P, denotes corresponding effect on spreading communication such as propagation speed and scope Base on the analytic Hierarchy Processs, we train the collected data and obtain the final weights as follows. The result has been approximated in order to simplify calculation Team#42220 Page 5 of 20 Meanwhile,in a specific area we assume that a, and o, is constant which represen average level. The weight of different media is listed in Table 1 Table 1 Different media and their effect on spreading communication Medium Weight Medium Weight Newspaper Television 2.5 Telegraphy 0.5 Internet Radio Mobile phone 3.5 Sender Characteristic(o) Influence (F): Influence shows the importance degree of a node. It should be InF(u=lg(rt+1) Where rt denotes the total number of forwarding times of a node Authority(A): It is defined as the ratio of indegree divides outdegree which is calculated as follows FO A(u)=lg(+1) FR Where fo denotes the number of fans while fr denotes the number of idol Activity (Ac): PO Ac(u)=lg(+1) D Where Po denotes total number of messages the user has been published. d denotes the number of the user 's active davs We take normalized form membership functions for each index so that values of all the factors can be constrained between 0 and 1. The normalization is shown as follows x;min x 3 maxim } minix 1<j<n Where x, and x respectively denotes the original value and the value after standar dization Receiver Characteristic(PR) e Willing(W): The value of willing indicates the possible trend for receivers to spr ead the information(v) is calculated as follows R W(v)=lg(+1) (6) Where denotes the number of retweet times. P denotes the number of original messages In the same way. normalize the index as v Team#42220 Page 6 of 20 Content Characteristic(o) Inherent value of content plays an important role in the information flow. For example the funnier or more important message tends to spread more quickly. To evaluate such abstract idea, we quantify it for two aspects: the value of information content and the quality DI presentation. Luckily, the article in Journal of Information Engineering University+ determines 5 indicators for the value of information content and 5 indicators for the quality of presentation Specific indexes and their weights are shown in Table 2 Table 2 Evaluation indexes and their weights General indicator Specific indicator Weight authenticity 0.318 Importance 0.174 Inherent value(0.7) Timeliness 0.247 Target 0.102 Confidentiality 0.159 Clear scheme 0.203 Careful thinking 0.063 Quality of presentation(0.3) Accurate description 0.299 Concise expression 0.232 Comprehensive factors 0.203 We give out general formulas to calculate the above two aspects below value i ∑ (8) =0.7×ae+0.3 Where a denotes the weight of above 10 indicators respectively. u denotes the value of the corresponding index, which arranges from 0 to 9. alue denotes the value of information content and re denotes the quality of presentation Relation between Sender and Receiver (OsR) o Interest similarity(Si): The message the user published indicates his or her interests which plays an important role in spreading information. TFIDF model is adopted to information retrieval and data mining. Simi(u,v)is Si(u, v) (10) where P and Q respectively file vector of uses u and v Structural similarity (Ss): It illustrates the similarity of circle of friends among users. The Jaccard Distance is applied to the calculation Team#42220 Page 7 of 20 Ss(u, v) N()N() (11) ()N(v) Where N(vi) denotes the neighboring nodes of node v, o Closeness( CL(u, v)): When the message contains information of receiver, it will be spread more easily Define Cl(u, v) as follows: when the message of u mentions v CL(u, v) (12) 0. on the contrary 3.2.2 nformation flow between two nodes Based on above analysis, we establish eigenvector of node vector(o) and side vector (o.). These two vectors are shown as follows an=[g2丁 (13) = (14) To obtain the propagation probability, fundamentally function is calculated as follows f(u,v, c)=k,m +a+ao (15) Where is a constant, a, denotes weight of nodes, characteristic, a, denotes weight of sides characteristic. The weight represents how corresponding characteristic influence in propagation probability. k, is the weight Based on Bayesian logistic function", the propagation probability is calculated plu, v,c) (16) f(u,v, c)) In the same way, propagation delay is calculated as follows (u,v,)=k2m +Bie,+B2p Where m, is constant, B, and B, respectively denotes weight of nodes and sides characteristics. k is the weight 3. 2. 3 Coefficient calculation In order to calculate the coefficient k,a, and B, (i1, 2), maximum likelihood estimate theorylsl is adopted as follows Taking time decay factor into account, the propagation probability decreases as time goes by Accumulated propagation probability from node u to node v is defined as follows (v,)(x,t)=f( (18) Where t(i=u, v) denotes the time when node i spread the information Define S((v, t(u, t,)as the probability of which node u will not spread the information to v before time t S(v,t,)(,)=1F(v,t,)(t,tn)) (19) Determine a set D=(vi, ti).(vn, t,) to represents all the node and its spreading Team#42220 Page 8 of 20 time. And the set Q=(v, ED, t, <t denotes the nodes which spread information before time t. par(v)is defined as the information source of node v, that is to say, the information spread from par(v) to v The propagation of a piece of information ck ic calculated as follows f(c2)=IF(vt,)(m,t)×S(v,)(,4) (20 Q(u)\Par(v) So the propagation probability of all pieces of information in set C=c, c,..., c") can be obtained in the following formula f(C)=∏f( (21) I≤k≤ To get the larger value of f(C), we set the object function min lg f(c) 22) Train data through computer, we can finally obtain the value of coefficient k;, a, and (i=1,2) 3.3 Outer Layer: Information Flow from Region to Region As analyzed above, the development level is unbalanced on the Earth which influences the flow of information, so the nodes we choose must be representative. to simplify the model, we choose gDP per capita as the criteria, with data from the CJDBY net 9, which is shown in Figure 3 4 10%20 Figure 3 GDP per capita Figure 4 Global network We can see that each of which has obvious difference GDP per capita is signed by a certain color. Apparently, the gdp per capita of America, Canada, Northern Europe and Australia(area with color of purple )is four times as big as the worlds average level. On the contrary, the one of area with dark blue color is relative low. According to development level and geography, we choose seven typical countries in Figure 4: America, Brazil, France, Congo Russia. China and austria In order to investigate the flow of information among nodes, we detail this phenomenon by applying the mechanism of twodimensional reactiondiffusion model(RD equation) Team#42220 Page 9 of 20 A scalar field f(e, 0) and its gradient, whose magnitude is the maximum rate of change of f per unit length of the coordinate space at the given point, could be described below in polar coordinate system V=V[f(,⊙) af (23) Therefore, this scalar field f(p, e)will transfer into a vector field V after derogating gradient. For a source flow, all the streamlines are straight lines emanating from a central point, as shown in Figure 5. Obviously, we see that the components in the radial and tangential directions are cf/ap and cf/00, respectively, where af/80=0 Figure 5 A point source flow According to the mechanism of reactiondiffusion model, the divergence of every point in this vector field V can be obtained easily div=div(vf)=dive (24) p dp dp pdp dp Where: divo denotes divergence while v() denotes gradient The second part in above equation, that is, a f/ap, has little impacts on the final value. Here, for the convenience of calculations, we overlook this part di￠f) (25) And final conclusion could be draw that divergence of gradient of a scalar field is
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