SCIENCE CHINA Information Sciences, Volume 62 , Issue 5 : 052103(2019) https://doi.org/10.1007/s11432-018-9609-7

Cumulative activation in social networks

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  • ReceivedApr 3, 2018
  • AcceptedSep 11, 2018
  • PublishedApr 3, 2019



This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61433014, 61502449, 61602440), National Basic Research Program of China (973) (Grant No. 2016YFB1000201).


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  • Figure 1

    (Color online) Illustration of multiple cascades. (a) $T=0$; (b) $T=1$; (c) $T=2$; (d) $T=3$.

  • Figure 2

    Figures for understanding the model. (a) CA & IC; (b) nonsubmoduarity; (c) $\eta<n$.

  • Table 1   Datasets
    Name # Node #Edge Type AOD
    Flixster 29 K 174 K Directed 6.0
    NetPHY 37 K 348 K Undirected 18.8
    DBLP 655 K 2 M Undirected 6.1

    Algorithm 1 Estimate $f(S)$ by Monte Carlo


    Output:$\hat{f}(S)$: the estimation of $f(S)$;


    $\hat{P}_u(S)=0$; $t_u=0$ for all $u~\in~V$;

    for $i=1$ to $R$

    Simulate IC diffusion from seed set $S$;

    if $u$ is activated then


    end if

    end for

    for $u\in~U$


    if $\hat{P}_u(S)\geq\tau_u$ then




    end if

    end for

    return $\hat{f}(S)$.


    Algorithm 2 Greedy algorithm for SM-CA with $\eta=n$

    Require:$G=(V,~E),~\{p_{uv}\}_{(u,~v)\in~E},\{\tau_u\}_{u\in~V},~U$, $\varepsilon$;

    Output:Seed set $S$

    $S=\emptyset$, $\hat{f}(S)=0$;

    while $\hat{f}(S)<\sum_{u\in~V}\tau_u-\varepsilon$ do

    Choose $v=\arg\max_{u\in~V\setminus~S}~[\hat{f}(S\cup~\{u\})-\hat{f}(S)]$;


    end while

    return $S$.

  • Table 2   Running time ($\tau=0.3$, $k=500$) (s)
    sf TIM$^+$ ADG-IM-CA BTG-IM-CA
    Flixster39 87 138
    NetPHY54 112 142
    DBLP 509 8865 8685

    Algorithm 3 Framework of greedy algorithm for IM-CA problem


    Output:Seed set $S$;

    Set $S=\emptyset$;

    Generate $\theta$ RR sets for each node $u\in~V$: $\{\mathcal~R_u\}_{u\in~V}$;

    Set ${\rm~req}(u)=\tau_u\theta~$ for each node $u\in~V$;

    for $j=1$ to $k$

    $x=$ SS($G,~\{p_{uv}\}_{(u,v)\in~E},~\{{\rm~req}(u)\}_{u\in~V},~~\{\mathcal~R_u\}_{u\in~V}$);


    Remove all RR Sets containing $x$;

    for each $u$ in $V$

    ${\rm~rem}(u)$: the number of RR Sets removed from $\mathcal~R_u$;


    end for

    end for

    return $S$.