Dynamic top-k influence maximization in social networks

Abstract

The problem of top-k influence maximization is to find the set of k users in a social network that can maximize the spread of influence under certain influence propagation model. This paper studies the influence maximization problem together with network dynamics. For example, given a real-life social network that evolves over time, we want to find k most influential users on everyday basis. This dynamic influence maximization problem has wide applications in practice. However, to our best knowledge, there is little prior work that studies this problem. Applying existing influence maximization algorithms at every time step provides a straightforward solution to the dynamic top-k influence maximization problem. Such a solution is, however, inefficient as it completely ignores the smoothness of network change. By analyzing two real social networks, Brightkite and Gowalla, we observe that the top-k influential set, as well as its influence value, does not change dramatically over time. Hence, it is possible to find the new top-k influential set by updating the previous one. We propose an efficient incremental update framework that takes advantage of such smoothness of network change. The proposed method achieves the same approximation ratio of 1 − e^− 1 as its state-of-the-art static counterparts. Our experiments show that the proposed method outperforms the straightforward solution by a wide margin.

Appendix: : Proof of Lemma 3

Consider the set G_s of all possible random subgraphs of G. The size of G_s is n = 2^|E| since each edge is considered independently of the others. By definition, the influence of $U\subseteq V$ is

$$ \begin{array}{@{}rcl@{}} I(U|G)=\sum\limits_{i=1}^{n}p_{i}\cdot r\left( U\left|G_{i}\right.\right), \end{array} $$

where G_i = (V,E_i) ∈G_s and p_i is its occurring probability,

$$ \begin{array}{@{}rcl@{}} p_{i}=\underset{e_{u,v}\in E_{i}}{\prod}p_{u,v}\cdot\underset{e_{u,v}\not\in E_{i}}{\prod}\left( 1-p_{u,v}\right). \end{array} $$

Similarly, there is a set $\mathbf {G}_{s}^{+}$ containing all the $2^{\left |E^{+}\right |}$ possible random graphs of G⁺. Each G_i ∈G_s is also a possible random subgraph of G⁺, though its occurring probability may change since p_u,v’s may change.

$\mathbf {G}_{s}^{+}$ can be constructed by augmenting G_i’s. Consider the set of newly added edges in G⁺. (There are in total $c=\left |E^{+}\setminus E\right |$ such edges.) In each G_i, we enumerate all possible m = 2^c occurrence patterns of these newly added edges. As a consequence, each G_i ∈G_s corresponds to a clutter of m subgraphs in $\mathbf {G}_{s}^{+}$, denoted as $\left \{G^{+}_{i1},G^{+}_{i2},\cdots ,G^{+}_{im}\right \}$. The probability distribution within the clutter is the same for every G_i; let q_j be the occurring probability of $G^{+}_{ij}$ given G_i. Now we get

$$ \begin{array}{@{}rcl@{}} I(U|G^{+})=\sum\limits_{i=1}^{n}p^{\prime}_{i}\left( \sum\limits_{j=1}^{m} q_{j}\cdot r\left( U\left|G^{+}_{ij}\right.\right)\right), \end{array} $$

and thus

$$ \begin{array}{@{}rcl@{}} {\Delta} I(U)=\sum\limits_{i=1}^{n}p^{\prime}_{i}\left( \sum\limits_{j=1}^{m} q_{j}\cdot r\left( U\left|G^{+}_{ij}\right.\right)\right)-\sum\limits_{i=1}^{n}p_{i}\cdot r\left( U\left|G_{i}\right.\right). \end{array} $$

Based on the above equation, we can revise ΔI(S ∪ T) ≤ I(S) + I(T) into the following inequality.

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{n} p^{\prime}_{i}\left( \sum\limits_{j=1}^{m} q_{j}\cdot\beta(S,T,G^{+}_{ij})\right)\geq \sum\limits_{i=1}^{n} p_{i}\cdot\beta(S,T,G_{i}), \end{array} $$

(2)

where $\beta (S,T,\cdot )=r\left (S\left |\cdot \right .\right )+r\left (T\left |\cdot \right .\right )-r\left (S\cup T\left |\cdot \right .\right )$ gives the number of vertices that are commonly reachable from S and T in a given graph. One intuitive and yet essential property of β is that if $G\subseteq G^{\prime }$ then $\beta (S,T,G)\leq \beta (S,T,G^{\prime })$.

In Eq. 2 we notice that $p^{\prime }_{i}$ could be smaller than p_i. However, whenever this happens, there must exist some ℓ such that (i) $p_{\ell }>p^{\prime }_{i}$ and (ii) $G_{i}\subseteq G_{\ell }$ thus β(S,T,G_i) ≤ β(S,T,G_ℓ). In general, graph update ΔE⁺ tends to shift the probability distribution towards subgraphs that have more edges. Therefore, we have

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{n} p^{\prime}_{i}\left( \sum\limits_{j=1}^{m} q_{j}\cdot\beta(S,T,G^{+}_{ij})\right)&\geq& \sum\limits_{i=1}^{n} p_{i}\left( \sum\limits_{j=1}^{m} q_{j}\cdot\beta(S,T,G^{+}_{ij})\right)\\ &\geq&\sum\limits_{i=1}^{n} p_{i}\cdot\beta(S,T,G_{i}). \end{array} $$

The last inequality holds because all $G^{+}_{ij}$’s are supergraphs of G_i, therefore $\beta (S,T,G^{+}_{ij})\geq \beta (S,T,G_{i})$ and so is their non-negative linear combination.