Dynamic influence maximization

1−o ( 1 ) -approximation unless the amortized running time is n1−o ( 1 ) . On the technical side , we exploit novel adaptive sampling approaches that reduce DIM to the dynamic MAX-k coverage problem , and design an efficient ( 1−1/e− ) -approximation algorithm for it . Our lower bound leverages the recent developed distributed PCP framework . 1 Introduction . Influence maximization ( IM ) is the algorithmic task of given a social network and a stochastic diffusion model , finding a seed set of at most k nodes with the largest expected influence spread over the network [ 35 ] . Influence maximization and its variants have been extensively studied in the literature over past decades [ 35 , 13 , 29 , 45 , 49 , 38 , 4 , 22 , 8 ] , and it has applications in viral market , rumor control , advertising , etc . Social influence can be highly dynamic and the propagation tendencies between users can alter dramatically over time . For example , in a Twitter network , new users join in and existing users drop out in real time , pop stars arise instantly for breaking news and trending topics ; in a DBLP network , scientific co-authorship is built up and expands gradually over time . The classic IM algorithms make crucial assumptions on a stationary social network and they fail to capture the elastic nature of social networks . As a consequence , their seeding set could become outdated rapidly in a constantly involving social network . To mitigate the issue , one designs a dynamic influence maximization algorithm , which maintains a feasible seed set with high influence impact over time , and at the same time , saturates low average computation cost per update of the social network . In this paper , we initiate a systematic study on the dynamic influence maximization ( DIM ) problem . Two types of evolution models are of interests to us . In an incremental model1 , the social networks keep growing : new users join in and new social relationship is built up or strengthened over time . The motivating example is the DBLP network , where co-authorship gets expanded over time . Another justification is the preferential attachment involving model of social network [ 12 , 41 ] . In a fully 1These terms are standard notions in the dynamic algorithm literature , e.g. , see [ 30 , 1 ] 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . dynamic model , the social network involves over time , users can join in and leave out , social links emerge and disappear , influence impacts get strengthened or weakened . The motivating examples are social media networks like Twitter or Facebook , and the advertising market . There are some pioneer works on the topic of dynamic influence maximization and various heuristic algorithms have been proposed in the literature [ 52 , 44 , 24 ] . These algorithms have different kinds of approximation guarantee on the seed set , but the crucial drawback is that the amortized running time per update is no better than Ω ( n ) , here n is the total number of nodes in a social network . This is extremely unsatisfactory from a theoretical view , as it indicates these algorithm do no better than re-run the state-of-art IM algorithm upon every update . Hence , the central question over the field is that , can we achieve the optimal approximation guarantee with no significant overhead on amortized running time ? Our contribution We address the above questions and provide clear resolutions . In the incremental model , we prove it is possible to maintain a seed set with ( 1 − 1/e − ) -approximation ratio in amortized running time of k · poly ( log n , −1 ) , which matches the state-of-art offline algorithm up to poly-logarithmic factors ( Theorem 3.1 ) . While in the fully dynamic algorithm , assuming the Strong Exponential Time Hypothesis ( SETH ) , we prove no algorithm can achieve 2− ( logn ) 1−o ( 1 ) approximation unless the amortized running time is n1−o ( 1 ) ( Theorem 4.3 and Theorem 4.4 ) . This computational barrier draws a sharp separation between these two dynamic models and delivers the following surprising message : There is no hope to achieve any meaningful approximation guarantee for DIM , even one only aims for a slightly reduced running time than the naive approach of re-instantiating an IM algorithm upon every update . On the technical side , our DIM algorithm in the incremental model uses novel adaptive sampling strategies and reduces the DIM problem to the dynamic MAX-k coverage problem . We provide an efficient dynamic algorithm to the later problem . Both the adaptive sampling strategy and the combinatorial algorithm could be of independent interests to the influence maximization community and the submodular maximization community . For the computaional hardness result , we exploit a novel application of the recent developed distribution PCP framework for fine grained complexity . Related work Influence maximization as a discrete optimization task is first proposed in the seminal work of Kempe et al . [ 35 ] , who propose the Independent Cascade ( IC ) model and the Linear Threshold ( LT ) model , prove the submodularity property and study the performance of greedy approximation algorithm . Upon then , influence maximization and its variant has been extensively studied in the literature , including scalable algorithms [ 21 , 13 , 51 , 50 ] , adaptive algorithms [ 29 , 28 , 45 , 19 , 49 , 6 ] , learning algorithms [ 22 , 8 , 39 , 32 ] , etc . For detailed coverage over the area , we refer interested readers to the survey of [ 18 , 40 ] . Efficient and scalable influence maximization has been the central focus of the research . Influence maximization is known to be NP-hard to approximate within ( 1 − 1/e ) under the Independent Cascade model [ 35 ] , and APX-hard under the Linear Threshold model [ 48 ] , while at the same , the simple greedy algorithm achieves ( 1− 1/e ) approximation under both diffusion models , but with running time Ω ( mnk ) . The breakthrough comes from Borgs et al . [ 13 ] , who propose the Reverse influence sampling ( RIS ) approach and their algorithm has running time O ( ( m + n ) k −3 log2 n ) . The running time has been improved to O ( ( m+ n ) k −2 log n ) and made practical in [ 51 , 50 ] . There are some initial efforts on the problem of dynamic influence maximization [ 52 , 44 , 24 , 3 , 42 ] . All of them provide heuristics and none of them has rigorous theoretical guarantee on both the approximation ratio and the amortized running time . Chen et al . [ 24 ] propose Upper Bound Interchange ( UBI ) method with 1/2-approximation ratio . Ohsaka et al . [ 44 ] design a RIS based algorithm that achieves ( 1− 1/e ) -approximation . Inspired by recent advance on streaming submodular maximization , Wang et al . [ 52 ] give a practical efficient algorithm that maintains a constant approximated solution . However , none of the above mentioned algorithm has rigorous guarantees on the amortized running time , and they can be as worse as Ω ( n ) . The failure on their theoretical guarantees are not accidental , as our hardness result indicates that Ω ( n1− ) amortized running time is essential to achieve any meaningful approximation . Meanwhile , our algorithm for incremental model substantially deviates from these approaches and provides rigorous guarantee on amortized running time . Influence maximization is closely related to submodular maximization [ 15 , 5 , 33 , 11 , 10 , 7 , 14 , 9 , 25 , 27 , 26 , 16 , 25 , 34 , 43 , 37 , 23 ] , a central area of discrete optimization . Ours is especially related to the dynamic submodular maximization problem , which has been recently studied in [ 43 , 37 , 23 ] . The work of [ 43 ] and [ 37 ] gives ( 1/2 − ) -approximation to the fully dynamic submodular under cardinality constraints , with amortized query complexity O ( k2 log2 n · −3 ) and O ( log8 n · −6 ) . The approximation ratio of 1/2 is known to be tight , as Chen and Peng [ 23 ] prove that any dynamic algorithm achieves ( 1 2 + ) approximation must have amortized query complexity of at least nΩ̃ ( ) /k3 . The dynamic submodular maximization is studied under the query model and measured in query complexity , while we consider time complexity in dynamic influence maximization . These two are generally incomparable . 2 Preliminary . We consider the well-studied Independent Cascade ( IC ) model and the Linear Threshold ( LT ) model . Independent Cascade model In the IC model , the social network is described by a directed influence graph G = ( V , E , p ) , where V ( |V | = n ) is the set of nodes and E ⊆ V × V ( |E| = m ) describes the set of directed edges . There is a probability pu , v associated with each directed edge ( u , v ) ∈ E. In the information diffusion process , each activated node u has one chance to activate its out-going neighbor v , with independent success probability pu , v . The live-edge graph L = ( V , L ( E ) ) is a random subgraph of G , where each edge ( u , v ) ∈ E is included in L ( E ) with an independent probability pu , v . The diffusion process can also be seen as follow . At time τ = 0 , a live-edge graph L is sampled and nodes in seed set S ⊆ V are activated . At every discrete time τ = 1 , 2 , . . . , if a node u was activated at time τ − 1 , then its out-going neighbors in L are activated at time τ . The propagation continues until no more activated nodes appears at a time step . Linear Threshold model In the LT model , the social network is a directed graph G = ( V , E , w ) , with V denotes the set of nodes and E denotes the set of edge . There is a weight wu , v > 0 associate with each edge ( u , v ) ∈ E and the weight satisfies ∑ u∈Nin ( v ) wu , v ≤ 1 for every node v , where Nin ( v ) contains all incoming neighbors of node v. In the information diffusion process , a threshold tv is sampled uniformly and independently from [ 0 , 1 ] for each node v , and a node v becomes active if its active in-coming neighbors have their weights exceed the threshold tv . In the LT model , the live-edge graph L = ( V , L ( E ) ) can be obtained as follow . Each node v samples an incoming neighbor from Nin ( v ) , where the node u is sampled with probability wu , v and the edge ( u , v ) is included in L ( E ) . With probability 1− ∑ u∈Nin ( v ) wu , v , no edge is added . The diffusion process can also be described by the live-edge graph . That is , a live-edge graph L is sampled at the beginning and the influence is spread along the graph . Influence maximization Given a seed set S , the influence spread of S , denoted as σ ( S ) , is the expected number of nodes activated from seed set S , i.e. , σ ( S ) = EL∼G [ |Γ ( S , L ) | ] , and Γ ( S , L ) is the set of nodes reachable from S in graph L. In the influence maximization problem , our goal is to find a seed set S of size at most k that maximizes the expected influence , i.e. , finding S ? ∈ arg maxS⊆V , |S|≤k σ ( S ) . The reverse reachable set [ 13 ] has been the key concept for all near-linear time IM algorithms [ 13 , 51 , 50 ] . Definition 2.1 ( Reverse Reachable Set ) . Under the IC model and the LT model , a reverse reachable ( RR ) set with a root node v , denoted Rv , is the random set of nodes that node v reaches in one reverse propagation . Concretely , Rv can be derived by randomly sampling a live-edge graph L = ( V , L ( E ) ) and include all nodes in V that can reach v under the live-edge graph L. Dynamics of network Social networks are subject to changes and involve over time . We consider the following dynamic model of a social network . At each time step t , one of the following four types of changes could happen . ( 1 ) Introduction of a new user . This corresponds to insert a new node to influence graph ; ( 2 ) Establishment of a new relationship . It is equivalent to insert a new directed edge to the graph2 ; ( 3 ) Diminishing of an old relationship . It is equivalent to remove an existing edge of the graph ; ( 4 ) Leave of an old user . This means to remove an existing node in the influence graph . In the incremental model , we only allow the first two types of changes . That is , we assume the social network gets enlarged over time and we only consider the introduction of new users and new relations . In the fully dynamic model , we allow all four types of changes . The theoretical results developed in this paper also adapts to other forms of change on the social network , including ( a ) strengthen of influence , which means the increase of the propagation probability pe for some edge under IC model or the increase of the weight we under LT model ; ( b ) weaken the influence . which means the decrease of propagation probability pe under IC model or the decrease of the weight we under LT model . Dynamic influence maximization Let Gt = ( Vt , St ) be the social network at time t. Define the expected influence to be σt ( S ) = EL∼Gt [ |Γ ( S , L ) | ] . The goal of the dynamic influence maximization problem is to maintain a seed set St of size at most k that maximizes the expected influence at every time step t. That is to say , we aim to find S ? t ∈ arg maxSt⊆Vt , |St|≤k σ ( St ) for all t. The ( expected ) amortized running time of an algorithm is defined as the ( expected ) average running time per update . Dynamic MAX-k coverage The ( dynamic ) influence maximization problem is closely related to the ( dynamic ) MAX-k coverage problem . In a MAX-k coverage problem , there is a collection of n sets A defined on the ground element [ m ] . The goal is to find k sets A1 , · · · , Ak such that their coverage is maximized , i.e . finding arg maxA1 , ... , Ak∈A | ∪i∈ [ k ] Ai| . The problem can also be formulated in terms of a bipartite graph G = ( VL , VR , E ) , named coverage graph , where VL ( |VL| = n ) corresponds to set in A and VR ( |VR| = m ) corresponds to element in [ m ] , for any i ∈ [ n ] , j ∈ [ m ] , there is an edge between VL , i and VR , j if and only if j ∈ Ai , here VL , i is the i-th node in VL and VR , j is the j-th node in VR . In the dynamic MAX-k coverage problem , nodes and edges arrive or leave one after another . Let V ⊆ VL . At any time step t , we use ft ( S ) to denote the number of neighbors of nodes in S. Our goal is to maintain a set of node S∗t ⊆ VL that maximizes the coverage for every time step t. That is , finding S∗t = arg maxSt⊆VL , |St|≤k ft ( St ) . Submodular functions Let V be a finite ground set and f : 2V → R be a set function . Given two sets X , Y ⊆ V , the marginal gain of Y with respect to X is defined as fX ( Y ) : = f ( X ∪ Y ) − f ( X ) The function f is monotone , if for any element e ∈ V and any set X ⊆ V , it holds that fX ( e ) ≥ 0 . The function f is submodular , if for any two sets X , Y satisfy X ⊆ Y ⊆ V and any element e ∈ V \Y , one has fX ( e ) ≥ fY ( e ) . The influence spread function σ is proved to be monotone and submodular under both the IC model and the LT model in the seminar work of [ 35 ] .

Summary: This paper studies influence maximization (IM) in a dynamically evolving social network, where at each time step, one node/edge may emerge or disappear. The authors consider two evolution models: the incremental model (only node/edge insertion is allowed; e.g., the DBLP network) and the full model (both insertion and deletion are allowed; e.g., the Twitter network).   Under the incremental model, a $(1-1/e-\epsilon)$-approximation algorithm is proposed with $k\cdot \textrm{poly}(\log n, \epsilon^{-1})$ amortized running time. This is an algorithmic breakthrough because previous dynamic IM algorithms [44, 36, 20] all need $\Omega(n)$ amortized running time in the worst cases.  Under the full model, it is proved that no algorithm can achieve a meaningful approximation ratio within $O(n^{1-o(1)})$ amortized running time under the Strong Exponential Time Hypothesis.  Both theoretical results are proved under the classical IC and LT models.