# Influence Maximization in Social Network Considering Memory Effect and Social Reinforcement Effect

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Influence Maximization under DCM4VM

#### 3.1. Traditional Influence Maximization Problem

#### 3.2. Dependent Cascade Model for Viral Marketing

_{1}($v,t$) and m

_{2}($v,t$) are the numbers of neighbors of node v at state interested and accepted until time t respectively, which reflect the memory effect. r1 > 0 and r2 > 0 are parameters which reflect the social reinforcement effect for interested and accepted states respectively. ${P}_{v}^{max}$ is the upper bound of the probability indicating maximal purchasing probability, which is used to control the maximal effect of social reinforcement effect and memory effect when increasing m

_{1}($v,t$) and m

_{2}($v,t$), Pv(t) will gradually approach ${P}_{v}^{max}$ and the speed is determined by the parameters r1 > 0 and r2 > 0.

#### 3.3. Influence Maximization under DCM4VM

## 4. Algorithms

#### 4.1. General Greedy Algorithm

Algorithm 1: Propagation Simulation${\sigma}_{tar}\left(G,S\right)$ | |

Input: network graph$G$, initial seeds$S$Output: accepted seeds number${\sigma}_{tar}$ | |

1 | ${\sigma}_{tar}=0$, $t=0$; |

2 | for each node$\upsilon \in S$do |

3 | $\mathrm{state}\left(u,t\right)=accepted$; |

4 | ${P}_{u}=1$; |

5 | end |

6 | while$S$not empty do |

7 | $t++$; |

8 | $nexS\leftarrow \varphi ,accS\leftarrow \varphi $; |

9 | //assume all newly interested or accepted nodes spread immediately and simultaneously, ${N}_{out}\left(u\right)$ is the out neighbors of $u$; |

10 | for each node$\upsilon \in S$do |

11 | for each node$\upsilon \in {N}_{out}\left(\upsilon \right)$do |

12 | if$\mathrm{state}\left(\upsilon ,t\right)\in \left\{\mathrm{interested},\text{}\mathrm{accepted},\text{}\mathrm{exhausted}\right\}$then |

13 | continue; |

14 | end |

15 | if$\mathrm{state}\left(\upsilon ,t-1\right)\in \left\{\mathrm{unknown},\text{}\mathrm{known}\right\}$then |

16 | $\mathrm{P}\sim \mathrm{Uniform}(0,1)$; |

17 | If${P}_{u,\upsilon}<P$then |

18 | $\mathrm{state}\left(\upsilon ,t\right)=\mathrm{interested}$; |

19 | $newS=newS\cup \left\{\upsilon \right\},accS=accS\cup \left\{\upsilon \right\}$; |

20 | end |

21 | else |

22 | $state\left(\upsilon ,t\right)=known$; |

23 | end |

24 | end |

25 | else |

26 | $\mathrm{state}\left(\upsilon ,t\right)=\mathrm{state}\left(\upsilon ,t-1\right)$; |

27 | end |

28 | If$\mathrm{state}\left(\upsilon ,t-1\right)\text{}=\text{}\mathrm{interested}$then |

29 | $accS=accS\cup \left\{\upsilon \right\}$; |

30 | end |

31 | end |

32 | end |

33 | //whether the newly exposed interested nodes at time t will turn to accepted; |

34 | for each$\upsilon \in accS$do |

35 | recalculate${P}_{\upsilon}\left(t\right)$according to (2); |

36 | $\mathrm{P}\sim \mathrm{Uniform}(0,1)$; |

37 | If${P}_{\upsilon}\left(t\right)<P$then |

38 | $\mathrm{state}\left(\upsilon ,t\right)\text{}=\text{}\mathrm{accepted}$; |

39 | ${\sigma}_{tar}++$; |

40 | end |

41 | end |

42 | $S=newS$; |

43 | end |

44 | return${\sigma}_{tar}$; |

Algorithm 2:$GG\left(G,k\right)$ | |

Input: network graph$G$, initial seeds size$k$Output: accepted seeds$S$ | |

1 | Initialize$S\leftarrow \varphi $and$R=10000$; |

2 | for$t=1$to$k$do |

3 | for each node v$\upsilon \in V\backslash S$do |

4 | ${\sigma}_{\upsilon}=0$; |

5 | for$i=1$to$R$do |

6 | ${\sigma}_{\upsilon}+=\left|{\sigma}_{tar}\left(G,S\cup \left\{\upsilon \right\}\right)\right|$; |

7 | |

8 | ${\sigma}_{\upsilon}={\sigma}_{\upsilon}/R$; |

9 | end |

10 | $S=S\cup \left\{\mathrm{arg}{\mathrm{max}}_{\upsilon \in V\backslash S}{\sigma}_{\upsilon}\right\}$; |

11 | end |

12 | return$S$; |

#### 4.2. Degree Discount Algorithm

Algorithm 3:$DD\left(G,k\right)$ | |

Input: network graph $G$, initial seeds size $k$Output: accepted seeds $S$ | |

1 | Initialize $S\leftarrow \varphi $; |

2 | for eachnode$\upsilon \in V$do |

3 | compute its degree ${d}_{\upsilon}$; |

4 | $d{d}_{\upsilon}={d}_{\upsilon}$; |

5 | Initialize ${t}_{\upsilon}$ to 0; |

6 | end |

7 | for$t=1$to$k$do |

8 | ${\upsilon}^{+}=\mathrm{arg}{\mathrm{max}}_{\upsilon \in V\backslash S}d{d}_{\upsilon}$; |

9 | $S=S\cup \left\{{\upsilon}^{+}\right\}$ |

10 | //recalculate ${p}_{V}$; |

11 | update ${P}_{V}$ according to (4); |

12 | //discount degree; |

13 | for eachnode$\upsilon \in {N}_{out}\left(u\right)\backslash S$do |

14 | ${t}_{\upsilon}={t}_{\upsilon}+1$; |

15 | $d{d}_{\upsilon}={d}_{\upsilon}-2{t}_{\upsilon}-\left({d}_{\upsilon}-{t}_{\upsilon}\right){t}_{\upsilon}{P}_{u,\upsilon}{P}_{\upsilon}$; |

16 | end |

17 | end |

18 | return$S$; |

#### 4.3. Exchangeable Greedy Algorithm

Algorithm 4:$EG\left(G,k\right)$ | |

Input: network graph $G$, initial seeds size $k$Output: accepted seeds $S$ | |

1 | Initialize $S\leftarrow \varphi $ $R=10000,c=0$; |

2 | Initialize $S$ with $k$ nodes using heuristic algorithms; |

3 | while${\sigma}_{tar}\left(S\right)$not converge and$c<m$do |

4 | pick one node $u$ out of $S$ with heuristic strategies or simply by order; |

5 | $S=S\backslash \left\{u\right\}$; |

6 | for each node $\upsilon \in V\backslash S$ do |

7 | ${\sigma}_{\upsilon}=0$; |

8 | for $i=1$ to $R$ do |

9 | ${\sigma}_{\upsilon}+=\left|{\sigma}_{tar}\left(G,S\cup \left\{\upsilon \right\}\right)\right|$; |

10 | end |

11 | ${\sigma}_{\upsilon}={\sigma}_{\upsilon}/R$; |

12 | end |

13 | $S=S\cup \left\{\mathrm{arg}{\mathrm{max}}_{\upsilon \in V\backslash S}{\sigma}_{\upsilon}\right\}$; |

14 | $c++$; |

15 | end |

16 | return$S$; |

Algorithm 5:$EG-SV\left(G,k,m,n\right)$ | |

Input: network graph $G$, initial seeds size $k$, number of iterations $m$, number of nodes sampled each time $n$Output: accepted seeds $S$ | |

1 | Initialize $S\leftarrow \varphi $ $R=10000,c=0$; |

2 | Initialize $S$ with $k$ nodes using heuristic algorithms; |

3 | Initialize $RC$ according to (5); |

4 | while${\sigma}_{tar}\left(S\right)$not converge and$c<m$do |

5 | pick one node $u$ out of $S$ with heuristic strategies or simply by order; |

6 | $S=S\backslash \left\{u\right\}$; |

7 | Sampling stage: |

8 | sample $\left|RS\right|=n$ out of $RC$ randomly; |

9 | Verification stage: |

10 | for each node $\upsilon \in RS$ do |

11 | ${\sigma}_{\upsilon}=0$; |

12 | For$i=1$to$R$do |

13 | ${\sigma}_{\upsilon}+=\left|{\sigma}_{tar}\left(G,S\cup \left\{\upsilon \right\}\right)\right|$; |

14 | end |

15 | ${\sigma}_{\upsilon}={\sigma}_{\upsilon}/R$; |

16 | update $RC$ according to (6); |

17 | end |

18 | ${\upsilon}^{+}=\mathrm{arg}{\mathrm{max}}_{\upsilon \in V\backslash S}\left\{{\sigma}_{\upsilon}\right\}$; |

19 | $S=S\cup \left\{{\upsilon}^{+}\right\}$; |

20 | remove ${\upsilon}^{+}$ from $RC$; |

21 | $c++$; |

22 | end |

23 | return$S$; |

## 5. Experiment

#### 5.1. Analysis of DCM4VM

#### 5.2. Analysis of Algorithms

## 6. Comparison of Propagation Performance

#### 6.1. Experimental Data

#### 6.2. Experimental Parameters

#### 6.3. Evaluation Index

#### 6.4. Results and Analysis of Experiment

## 7. Conclusions

- In the degree discount algorithm, a method is adopted to compute and update the expectations of accepted nodes (which is influenced nodes in the original algorithm) in each round of selection considering only one-step neighbor nodes. This calculation method has a certain randomness. In the next step, empirical value and experimental value can be considered to replace the existing method and reduce the complexity of calculation.
- This paper adopts a combination of greedy and heuristic algorithms to improve the problem of impact maximization, and divides the whole process into two parts. In future research, these two parts could be iterated to further improve the effect. This method also has a hidden danger of falling into local optimization. The next step is to consider the most effective method to prevent local entanglement, such as some bionic algorithms.
- The propagation problem studied in this paper does not take into account the time delay. In fact, time delay has a very important impact on this study, which will also be the focus of future research.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Interested and Accepted Nodes: (

**a**) Interested Nodes P = 0.05 in CA-GrQc (

**a**,

**b**); (

**b**) Accepted Nodes with P = 0.05 in CA-GrQc (

**a**,

**b**); (

**c**) Interested Nodes with P = 0.1 in NetHEPT (

**c**,

**d**); (

**d**) Accepted Nodes with P = 0.1 in NetHEPT (

**c**,

**d**).

Algorithms | Time Complexity |
---|---|

GG | O(kNT) |

EG-SV | O(kmnT) |

DD | O(kE) |

Network Name | Node Number | Side Number | Average Node Degree | Maximum Node Degree | Clustering Coefficient | Correlation Coefficient of Degree |
---|---|---|---|---|---|---|

Epinions | 22,437 | 212,970 | 9.49 | 2031 | 0.09717 | 0.052 |

145,942 | 203,152 | 1.392 | 7079 | 0.00014 | −0.1114 | |

Sina microblog | 146,091 | 205,408 | 1.406 | 2000 | 0.00024 | −0.2446 |

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**MDPI and ACS Style**

Wang, F.; Zhu, Z.; Liu, P.; Wang, P.
Influence Maximization in Social Network Considering Memory Effect and Social Reinforcement Effect. *Future Internet* **2019**, *11*, 95.
https://doi.org/10.3390/fi11040095

**AMA Style**

Wang F, Zhu Z, Liu P, Wang P.
Influence Maximization in Social Network Considering Memory Effect and Social Reinforcement Effect. *Future Internet*. 2019; 11(4):95.
https://doi.org/10.3390/fi11040095

**Chicago/Turabian Style**

Wang, Fei, Zhenfang Zhu, Peiyu Liu, and Peipei Wang.
2019. "Influence Maximization in Social Network Considering Memory Effect and Social Reinforcement Effect" *Future Internet* 11, no. 4: 95.
https://doi.org/10.3390/fi11040095