Fig. 1. A simple network example: (a) The network diagram; (b) the corresponding propagation tree of (a).一个简单网络例子 (a)网络图; (b)图(a)对应的传播树
Fig. 2. (a) Approximate propagation expectation and the degree of propagation over time for different simulation times; (b) variance of the approximate propagation expectation and the degree of propagation of different simulation times.(a)不同仿真次数下的近似传播期望值和传播度随时间的变化; (b)不同仿真次数的近似传播期望值与传播度的方差变化
Fig. 3. Changes of the 4th degree of propagation (normali-zed) of network node in Fig. 1 with the propagation pro-bability.
图1中的网络各节点4阶传播度(归一化)随传播概率变化
Fig. 4. Flow chart of node influence maximization algorithm based on propagation degree.基于传播度的节点影响力极大化算法流程图
Fig. 5. (a)−(c): Respectively reflects the relationship between the propagation capacity and degree, second-order propagation, and third-order propagation of several nodes randomly selected by the Deezer network, where
β = 0.06; (d)−(f): Respectively reflects the global propagation probability of randomly selected nodes by the Deezer network (see reference[
12]) Correspondence with degrees, second-order propagation, and third-order propagation, where
β = 0.12 (here, it is not difficult to find through the simulation experiments that the percolation threshold of the network is between 0.06 and 0.12)
(a)−(c) 分别反映了Deezer网络随机挑选若干节点的传播能力与度, 二阶传播度, 三阶传播度的关系, 其中
β = 0.06; (d)−(f) 分别反映了Deezer网络随机挑选若干节点全局传播概率(参见文献[
12])与度, 二阶传播度, 三阶传播度的对应情况, 其中
β = 0.12(这里通过仿真实验不难发现该网络的渗流阈值介于0.06和0.12之间)
Fig. 6. Kendall’s tau coefficient for different propagation probabilities, second-order, third-order propagation and simulation propagation ability.不同传播概率下度, 二阶、三阶传播度与仿真传播能力的kendall’s tau系数
Fig. 7. (a) The relationship between the second-order propagation degree and the global propagation probability p in the case of the Email-Enron network with a propagation probability of 0.02; (b) the kendall's tau coefficient of each indicator with different propagation rates and propagation capabilities.
(a)在Email-Enron网络中, 传播概率为0.02的情况下二阶传播度和全局传播概率p的关系; (b)各指标在不同概率下与传播能力的kendall’s tau系数
Fig. 8. (a) The relationship between the second-order propagation degree and the global propagation probability p in the case of a Facebook network with a propagation probability of 0.09; (b) the kendall's tau coefficient of each indicator with different propagation rates and propagation capabilities.
(a)在Facebook网络中, 传播概率为0.09的情况下二阶传播度和全局传播概率p的关系; (b)各指标在不同概率下与传播能力的kendall’s tau系数
Fig. 9. In the Deezer network, (a) the variation of the propagation range with the number of selected seed nodes under different algorithms; (b) the variation of the global propagation probability with the number of seed nodes under different algorithms. The candidate nodes are 9792 nodes with a degree of 6, 7 and 8.在Deezer网络中(a)不同算法下传播范围随所选种子节点数量的变化和(b)不同算法下全局传播概率随种子节点数量的变化, 其中候选节点为度为6, 7, 8的9792个节点
Fig. 10. (a), (b), and (c), (d), respectively, compare the propagation performance of Email-Enron, the selected seed node of the Facebook network under different algorithms.(a), (b)和(c), (d)分别比较了Email-Enron, Facebook网络在不同算法下所选种子节点的传播性能
辅助算法二 | 联合概率算法 | 输入 | 时间t, 节点u以及对于u的序号集W | 输出 | 联合概率
${{{f}}_{{t}}}\left( {{{{u}}_{{W}}}} \right)$![]() ![]() | 1. | if W只含一个元素 then | 2. | 找到该节点的双亲节点
${{{v}}_{\left( {{i}} \right)}}$![]() ![]() | 3. | ${{{f}}_{{t}}}\left( {{{{u}}_{{W}}}} \right) = {{{f}}_{{{t}} - {{1}}}}\left( {{{{v}}_{\left( {{i}} \right)}}} \right) \times {{\beta }} \times \left( {{{1}} - {{{p}}_{{{t}} - {{1}}}}\left( {{u}} \right)} \right)$![]() ![]() | 4. | else | 5. | 找到传播树种
${{{P}}_{{t}}}$![]() 代序号集为W的u节点, 记其双亲节点为
${{{v}}_{{1}}}, {{{v}}_{{2}}}, \cdots , {{{v}}_{{l}}}$![]() , 出现次数为
$ m_1, $![]() $ m_2, \cdots , m_l$![]() , 这些路径对于双亲节点的序号集分别为
${{{X}}_{{1}}}, {{{X}}_{{2}}}, \cdots , {{{X}}_{{l}}}$![]() , 对于节点u的序号集分别为
${{{Y}}_{{1}}}, {{{Y}}_{{2}}}, \cdots {{{Y}}_{{l}}}$![]() .
| 6. | for${{j}} = 0 \to {{l}}$![]() do | 7. | ${{{f}}_{{t}}}\left( {{{{u}}_{{{{Y}}_{{j}}}}}} \right) = {{F}}\left( {{{t}} - {{1}}, {{{v}}_{{j}}}, {{{X}}_{{j}}}} \right) \times {{\beta }}$![]() ![]() | 8. | end for | 9. | ${{{f}}_{{t}}}\left( {{{{u}}_{{W}}}} \right) = \left[ {{{1}} - \mathop \prod \nolimits_{{{j}} = {{1}}}^{{l}} \left( {{{1}} - {{{f}}_{{t}}}\left( {{{{u}}_{{{{Y}}_{{j}}}}}} \right)} \right)} \right] \times \left( {{{1}} - {{{p}}_{{{t}} - {{1}}}}\left( {{u}} \right)} \right)$![]() ![]() | 10. | end if | 11. | return${{{f}}_{{t}}}\left( {{{{u}}_{{W}}}} \right)$![]() ![]() |
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Table 1. Joint probability algorithm
联合概率算法
递推算法 | 求种子节点s阶传播度 | 输入 | seed,${{\beta }}$![]() , s | 输出 | ${{{d}}_{{s}}}$![]() ![]() | 1. | ${\rm Tree }_{s} = {\rm FTree }(\rm seed)$![]() %获取节点
${{seed}}$![]() 的s阶传播树
| 2. | 赋初值
${ { {p} }_{ {0} } }\left( { { {\rm seed} } } \right) = { { {f} }_{ {0} } }\left( { { {\rm seed} } } \right) = { {1} }$![]() , 其他情况
${{{p}}_{{t}}}\left( {{u}} \right) = {{{f}}_{{t}}}\left( {{u}} \right) = {{0}}$![]() ![]() | 3. | for${{t}} = {{1}} \to {{s}}$![]() do | 4. | for${{{u}}_{{y}}} \in {{{P}}_{{t}}}$![]() do %遍历传播树中
${{{P}}_{{t}}}$![]() 代的个体
| 5. | 获得
${{{u}}_{{y}}}$![]() 的双亲节点
${{{v}}_{{x}}}$![]() %表示节点u所在传播树层中第x个u节点
| 6. | ${{{f}}_{{t}}}\left( {{{{u}}_{{y}}}} \right) = {{{f}}_{{{t}} - {{1}}}}\left( {{{{v}}_{{x}}}} \right) \times {{\beta }} \times \left( {{{1}} - {{{p}}_{{{t}} - {{1}}}}\left( {{u}} \right)} \right)$![]() ![]() | 7. | end for | 8. | for${{u}} \in {{{P}}_{{t}}}$![]() do | 9. | 获得u的序号集W %指
${{{P}}_{{t}}}$![]() 代节点u出现次序构成的集合
| 10. | 计算联合概率
${{{f}}_{{t}}}\left( {{u}} \right) = {{F}}\left( {{{t}}, {{u}}, {{W}}} \right)$![]() ![]() | 11. | end for | 12. | for$u \in {\rm Tree}_{s}$![]() do | 13. | ${{{p}}_{{t}}}\left( {{u}} \right) = {{{p}}_{{{t}} - 1}}\left( {{u}} \right) + {{{f}}_{{t}}}\left( {{u}} \right)$![]() ![]() | 14. | end for | 15. | end for | 16. | ${ { {d} }_{ {s} } }\left( { { \rm{seed} } } \right) = \mathop \sum \nolimits_{ {u} } { { {p} }_{ {s} } }\left( { {u} } \right)$![]() ![]() | 17. | return${{{d}}_{{s}}}$![]() ![]() |
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Table 2. Pseudocode of the algorithm flow for s order progpagation of the seed node.
种子节点s阶传播度的算法流程伪代码