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 As a consequence , their seeding set could become outdated rapidly in a constantly involving social network .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: The paper presents new algorithms and hardness results for dynamic influence maximization, where dynamics come from one of two models: (1) the incremental model, which consider only the addition of new nodes and edges in a graph, and (2) the fully dynamic model, where nodes and edges can also be removed. Previous work has considered only heuristics that satisfy certain approximation guarantees but do not come with good amortized cost for updating the see set in the dynamic case. This paper provides a (1-1/e-eps) approximation for the optimal solution with k*poly(log n, 1/eps) approximation for the incremental model. For the fully dynamic model, the paper presents a hardness of approximation result if the Strong Exponential Time Hypothesis is true.  The proof for the approximation algorithm for the incremental model is based on two major pieces. First, the paper provides a reduction from dynamic influence maximization to dynamic MAX-k coverage. Then, a new algorithm for dynamic Max-k coverage is given. This follows a previous strategy for the static version of the problem, though a significant amount of theoretical work is required to extend both parts to the dynamic case.  The work is primarily theoretical and no implementations or experimental results are given.