In this section we would focus on weighted edge distribution in the small-world network only. In our daily life, we had the better relationship with our family or neighbors, such that the double weighted edges might all appear in the nearest-neighbor links instead of double weighted edges randomly. By this reason, we would not discuss this issue in scale-free networks and random networks, which without nearest-neighbor links.
We took two parts A and B into the experiment. A part meant the double weighted edges were distributed randomly in the network model, and B part meant the double weighted edges were all in the nearest-neighbor links. Then as the experiment 4, we looked at the difference of double weighted edges were set at 1, 5, 10, 30, and 50 percent.
A: randomly double the weight of edges B: double the weight of nearest neighbor links
Fig. 6-14 Distribution pattern of familiar pairs in small-world networks
Results from this experiment are shown in Figure 6-14. The similarity of the curves leads us to suggest that the influence of the pattern of scattered heterogeneous
individuals is not significant.
7 Conclusion
Summarizing all results, we classified the experiments into 3 types below, network-structures, sensitive information, and insensitive information.
Network-structures information means that we have to determine what model is needed depending on various simulations. As mentioned above, we used scale-free networks for discussion of propagation of sexual disease, and used small-world networks for spreading of rumors. Besides, we could find that there was no significant difference between random networks and small-world networks with level one von Neumann Neighborhood, because of the lower clustering coefficient.
The sensitive information indicates that the parameters of epidemic simulations which might change the infectious power. Take experiment 2 for example, without nearest-neighbor links, scale-free networks and random networks are constructed by shortcuts, such that the number of shortcuts may influence the simulation so much.
Besides, in experiment 3, more heterogeneous individuals, weaker the whole society, and the influence of disease is larger. The different weighted channels are also the sensitive condition by affecting the efficiency between nodes. Say, these are all count in intensity information, and we have to configure them carefully.
Finally, insensitive information is pointing to the parameters which are side issues in epidemic experiments. As the experiment 4, the efficiency of epidemic simulation was almost equal no matter what distributions of heterogeneous individuals were configured. And the experiment 6, we configured the double weighted edge by two different ways, but the simulation results were similar. Such the insensitive information can be ignored or configured arbitrarily.
Appendix A 複雜網路
規則網路與小世界網路建構流程圖如下：
Start
For all individuals Vi in 2-D lattice Connect to 4 nearest neighbors Regular
Network
End
Set d(v) for all nodes
Determine the number of shortcuts
Pick nodes Va by its probability P(Va) Pick nodes Vb by its probability P(Vb)
Is_linked(a,b) Yes
No
Link(a,b)
Is shortcuts number enough No
Yes
End Small-world
Network
Fig. A-1 規則網路與小世界網路
隨機網路：
Start
All individuals in 2-D lattice are isolated
Determine the number of shortcuts
Pick two nodes a, b randomly
Is_linked(a,b) Yes
No
Link(a,b)
Is shortcuts number enough
No
Yes
End Random Network
Fig. A-2 隨機網路
無尺度網路：
Start
All individuals in 2-D lattice are isolated All d(V) are configured to 1
Determine the number of shortcuts
Pick nodes Va by its probability P(Va) Pick nodes Vb by its probability P(Vb)
Is_linked(a,b) Yes
No Link(a,b) Increase d(Va) by 1 Increase d(Vb) by 1
Is shortcuts number enough No
Yes
End Scale-Free
Network
Fig. A-3 無尺度網路
而觀察各網路模型的節點分支度可發現，本論文所使用的三種小世界網路模 型經過設定各節點的權重參數後得到的節點分支度分佈的確如我們所要，另外隨 機網路當然呈現完全隨機的樣子，而無尺度網路也遵照最特殊的冪次定律來呈現 他的分支度分佈。
Appendix B 傳播問題建模
傳播問題主要是指傳播者透過傳播途徑使接受者受到影響而改變。以流 行病傳染為例，一般病人為傳播者透過直接接觸或是空氣傳染等途徑，使得正常 人受到感染而成為病人。再以信仰傳播為例，傳教士為傳播者透過口頭傳教或是 書報雜誌等傳播途徑使得一些人受到感招而信教。因此可以說傳播問題主要是需 由傳播者、傳播途徑以及接受者三個部份所組成。
在本論文中將使用各種社會網路模型來當作傳播問題模擬平台，由於不同性 質的傳播問題所適用的網路模型都有所差異，因此無論是無尺度網路，小世界網 路甚至是簡化的隨機網路都將在本研究中討論，而其中的隨機網路與規則網路則 主要是做個比較，可以看看結構上的差異會造成多少不同的結果。另一方面本論 文將使用 SIR 模型來模擬個體狀態改變的動態，使得互動過程呈現出傳播者透過 網路模型上的連結來當作傳播途徑以達到影響並改變接受者的效果。
檢視個體的狀態改變可以發現，當一個 Susceptible 個體經過連結與 Infectious 個體互動的過程，將有一個機率會把本身的狀態從 S 轉變成 I，稱之為 ，而隨 著時間的經過，Infectious 個體將會以 的機率被設定成移除或痊癒，若是個體形 成痊癒狀態則因為產生抗體而比 Susceptible 個體來的不易被感染。
以 SIR 模型來討論疾病傳染是較常被使用以及易被接受的，而文化的傳播或 謠言擴散是否也能用相同的模型來做模擬呢？想像在選舉前的拉票動作，某個候 選人在初始的狀況下擁有一些擁護者支持這個候選人的政見，假設這些擁護者為 Infectious 狀態，即是被候選人的政治魅力給感染了；而這些擁護者將會在親朋 好友之間替這個候選人的政見做宣傳，因此也許有些機率讓原本沒有支持特定候 選人的朋友 Susceptible 轉而支持相同的候選人；但是即使已經變成擁護者，仍
有特定的機率去相信一些所支持候選人的負面消息而放棄支持立場 Remove。
融合 SIR 模型與複雜網路就成了本論文探討傳播問題所使用的平台了。
檢視傳播問題架構流程圖(圖 B-2)可以發現個體間的互動需要透過人際關係網路 的連結而非個體獨立且隨機的改變狀態，以此模型來探討傳播問題更能貼近真實 世界，因此近來討論傳播問題的相關研究都使用類似的模型，本論文也使用此流 程探討各種區域資訊的差異所造成的影響。
實驗流程圖：
Start
Generate Network Model
Apply Local Information
Time Step Start
Next Individual
Interacts with all neighbors
Update all individuals states Last Individual?
No
Yes
Limit of Time Step?
No
Yes End
Fig. B-2 實驗流程圖
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