Evolving Influence Maximization

Evolving Influence Maximization

Xudong Wu

, Luoyi Fu

, Jingfan Meng

, Xinbing Wang

and Songwu Lu

1Shanghai Jiao Tong University

2University of California, Los Angeles (UCLA)

{xudongwu, yiluofu, JeffMeng, xwang8}@sjtu.edu.cn¹, [email protected]²

ABSTRACT

Influence Maximization (IM) aims to maximize the number of peo- ple that become aware of a product by finding the ‘best’ set of ‘seed’

users to initiate the product advertisement. Unlike prior arts on static social networks containing fixed number of users, we un- dertake the first study of IM in more realistic evolving networks with temporally growing topology. The task of evolving IM (EIM), however, is far more challenging over static cases in the sense that seed selection should consider its impact on future users and the probabilities that users influence one another also evolve over time.

We address the challenges through EIM, a newly proposed bandit-based framework that alternates between seed nodes se- lection and knowledge (i.e., nodes’ growing speed and evolving influences) learning during network evolution. Remarkably, EIM in- volves three novel components to handle the uncertainties brought by evolution: (1) A fully adaptive particle learning of nodes’ grow- ing speed for accurately estimating future influenced size, with real growing behaviors delineated by a set of weighted particles. (2) A bandit-based refining method with growing arms to cope with the evolving influences via growing edges from previous influence diffusion feedbacks. (3)Evo-IMM, a priority based seed selection algorithm in hope of maximizing the influence spread to highly attractive users during evolution. Theoretically, EIM returns a re- gret bound that provably maintains its sublinearity to the growing network size. Empirically, the effectiveness of EIM are also vali- dated, with three notable million-scale evolving network datasets possessing complete social relationships and nodes’ joining time.

The results confirm the superiority of EIM in terms of an up to 50%

larger influenced size over four static baselines.

1 INTRODUCTION

With the development of massive social networks (e.g., Facebook, Wechat and Twitter, etc), Influence Maximization (IM) has become a key technology of viral marketing in modern business [1–3]. Given a social networkG and an integer K, the goal of IM is to select K seed users inG in hope that their adoptions of a promoted product or idea can maximize the expected number of final adopted users through word-of-mouth effect [4, 5]. Initially put forwarded by Kempe et al.

[6], the problem of IM has been intensively studied by a plethora of subsequent works, proposing improvements or modifications from multiple aspects, including influence size estimation [3, 7–9], adaptive seeding [2, 5], boosting seeding [1], and many others.

The fundamental task in IM, as we noted above, lies in estimating the expected influenced size of each alternative seed set based on each user’s activation probabilities, referring to the the probability that a user successfully influences his social neighbors after having been influenced himself. And theinfluences among users are quan- tified by those activation probabilities. While existing literature works well in finding the most influential seed users, they are all

constrained to the assumption that the number of nodes in the net- work, along with their edges in between, are fixed during influence diffusion. Consequently, it violates real practices as many realistic social networks are usually growing over time. Take Wechat [10], the most popular social media app in China as an example. The number of Wechat accounts (nodes) grew from zero to 300 million during its early two years, with 410 thousand new users per day on average, and are continuing to fastly approach almost 1 billion ones [11]. And Facebook also exhibits a fast growth with roughly 340K new users per day [12]. Similar phenomena also hold in a wide range of other real social applications including Twitter, Academic networks, and etc. Meanwhile, a viral marketing action such as the web advertisements via messages or emails propagation may consume up weeks to months [13]. Thus, given an evolving net- workGtat timet and time span T for a viral marketing action, Gt

have greatly evolved toG_{t +T}during influence diffused from seed users to the expected maximal size. Consequently, the expected influenced size estimated by existing IM techniques overG_tcannot reflect the influence of seed set overG_{t +T}, which severely impacts the quality of selected seed users.

The above issue motivates the study of evolving influence maximization (EIM), whose problem formulation should in- corporate the evolutionary nature ofG during propagation.

Interpreted technically, given an instance of evolving social network G_t at timet and an integer K, the goal of EIM is to select K seed users to maximize the influence diffused to both existing users and those will join during timet to t +T . Different from the well investi- gated existing IM problems, the task ofEIM turns out to be highly non-trivial due to the following three challenges in reality: (1) The growing speed of a specific network exhibits uncertainties due to multiple external factors (e.g., the number of potential users, user interests and peer competitions). Such uncertain growing speed hinders accurately predicting how the network evolve during time t to t + T . (2) There is no prior knowledge about the influences via newly emerged edges, and they may also evolve over time with the changes of social relations among users (e.g., from friends to strangers or on the contrast). Although some recent efforts [4, 5, 14–

16] have been dedicated to online IM where influences among users are uncertain, the underlying network topology is still assumed to be completely known, thus inapplicable to the situations with both growing nodes and edges. (3) In evolving networks, newly added users are more inclined to establish relationship with those of higher popularity. Thus users inG_thave different attractiveness to new users, as opposed to existing IM studies which treat each user equally. Unfortunately, as far as we know, no directions have been directed toward IM in temporally growing networks. Consequently, it remains open how to effectively resolveEIM that can jointly deal with the unknown influences, uncertain growing speed and heterogeneous attractiveness.

This motivates us to present a first look into EIM prob- lem. By proving its NP-hardness, we attempt to solve the above three challenges inEIM by EIM, a new and novel bandit-based Evolving Influence Maximization framework with multiple periods of IM campaigns¹. Each period amounts to an IM campaign which chooses seeds that improve the knowledge and/or that lead to a large spread to both existing users and those that will join till the end of this period, and incurs aregret in such influence spread due to the lack of network knowledge. Different from prior IM studies, here the network knowledge includes the nodes’ growing speed and evolving influences via continuously emerging new edges in network evolution. Thus EIM seeks to minimize the accumulated regret incurred by choosing suboptimal seeds over multiple periods.

While we defer the details of EIM design in later sections (Sec- tions 4, 5, 6), here we briefly unfold its three novel components in addressing the aforementioned three challenges inEIM:

(1) It is unrealistic to assume the complete network topology is known in advance, thus a fully adaptive particle learning method is proposed to capture the uncertain network growing speed, with real growing function of nodes explicitly represented by a set of weighted particles. By modeling network evolution via the popular Preferential Attachment (PA) rule (i.e., new users prefer connecting to higher degree nodes), we are able to predict potential added users during influence diffusion with weighted particles (Section 4).

(2) Considering the evolving influences among users, we model the influences via continuously emerging edges as the growing arms in the bandits, thus ensuring the applicability of EIM to the evolving network with growing nodes and edges (Section 5). By modeling the activating probabilities as the dynamic rewards distribution of the arms, the reward of each edge as the edge-level feedback can then be taken to adaptively refine the estimating values of the evolving influences.

(3) Aiming to maximize the influence diffused to both existing and future users, we introduce a novel priority based seed selection algorithmEvo-IMM that incorporates the heterogeneity of users’

attractiveness formed by PA rule (Section 6). InEvo-IMM, users with higher attractiveness to future ones are sampled with higher priority in seeds selection.Evo-IMM turns out to provably enjoy comparable performance such as approximation ratio and time complexity with the static counterparts.

We validate the performance of EIM from both theoretical and empirical perspectives. Theoretically, although the growing size and successive emerging orders of the arms duo to network evolution further challenges the knowledge learning compared to classical bandits, the regret bound of EIM still provably maintains to be sublinear to the number of trials under the growing network size (Section 7). Empirically, the effectiveness of EIM is validated on both synthetic and real world evolving networks, with up to 200 years of time span and million scale data size respectively (Sec- tion 8). Notably, the real evolving networks are extracted from the true academic networks with complete co-authorship, citation and joining time of all authors and papers, which is severely lacking in existing IM works. Experimental results demonstrate the superior- ity of EIM. For example, EIM achieves a 50% lager influenced size

1We shall elaborate in Section 3 the reason for the choosing bandit-based framework and the incorporation of multiple periods.

than four static baselines in an evolving Co-author network with 1.7 million nodes.

2 RELATED WORKS

2.1 Static Influence Maximization Problem

Kempe. et al. [6] are the first to formulate influence maximization problem over a given network as a combinatorial optimization prob lem. Particularly, in their seminal work [6], they treat the network as a graphG = (V , E), where there is an influence cascade process triggered by a small number of influenced users that are calledseed users. The influence diffusion process is then characterized by the later widely adopted Influence Cascade (IC) model [1]-[6], whose definition is given as follows:

Definition 2.1. (Influence Cascade (IC) model.) In the IC model, the influences among users are characterized by the activation probabilities. Specifically, once userui is influenced, he has a single chance to activate his social neighborujsuccessfully with activation probabilityp_ijvia edge between usersu_i andu_j). And whether or notui can influenceuj successfully is independent of the history of information diffusion.

For a given seed setS, let I (S,G) be the expected number of users that are finally influenced by the seed users inS estimated under the IC model. The objective of IM is to find a set ofK seed users (i.e.,S^opt) who can maximizeI (S,G) among all the sets of users with sizeK. That is,

S^opt = arg maxS ⊆V , |S |=KI (S,G). (1) Based on the above formulation, Kempe. et al. prove the NP- hardness of the IM problem, and design the greedy algorithm that provably returns a(1 − 1/e)-approximate solution for seed selec- tion. Since then, a large number of subsequent works have emerged to improve the efficiency and quality of IM designing. For some representative examples, [3], [8] and [9] focus on achieving reason- able complexity in seed selection over million or even billion-scale networks. Besides, different costs for seeding different users are considered in [2] and [17] for the cost-aware IM problems, with the corresponding near optimal budget allocation methods proposed.

The objectives of the above works are all set to select the seed set with the maximumI (S,G) estimated over the static network G.

As a result, over evolving networks where new users continuously join in and influences evolve over time, it is difficult for classical IM techniques to return high quality seeds sinceI (S,G) estimated by them fails to include the future users and their influences.

2.2 Dynamic Influence Maximization Problem

As a step ahead of classical IM problems, some recent attempts are made in evolving networks. For example, considering the network with dynamically changing edges, [14] takes multiple specific ex- amples to show the effect of network evolution on IM design, and highlights the importance of seeding time. Similarly, [18] focuses on the effect of dynamic user availability, and also experimentally shows the effect of seeding time. [19] aims to maximize the influ- ence diffused to multiple given network snapshots. However, they assume that future network is known in advance, which violates the real practices.

Meanwhile, there emerge a class of online IM techniques that periodically seed one or more users in dynamic networks, in a similar manner to our settings that will be described later. To un- fold, [20] proposes to successively select seed users with influence

diffusion over dynamic networks, while just considers changing edges among fixed users. Besides, [15] and [5] adopt the bandit- based learning framework to refine unknown influences from the feedbacks of previous influence diffusion, and periodically select a set of seed users under the refined influences. Regardless of their progress, those online IMs still considers the uncertain influences over static network topology, where the estimatedI (S,G) also fails to include the influence diffused to the future users. Thus it is still difficult for the seeds to be repeatedly selected at different time to meet requirement of high quality.

As far as we know, the only work that shares the closest correla- tion with us belongs to Li et. al. [12], who simulate the network growth based on the Forest Fire Model and then run the existing static IM algorithms over the simulated network. However, under the unknown growing speed, it is difficult for the simulation to capture the real network evolution. Furthermore, influences among users are still preset as known constants. The limitations of the state-of-art IM techniques motivates us to study evolving influence maximization, which will be formally defined in next section.

3 EVOLVING INFLUENCE MAXIMIZATION 3.1 Problem Formulation

Evolving IM problem (EIM). We assume that time is divided into different time stamps. And an evolving network at time stampt is modeled as a graphGt = (Vt, Et), where VtandEtrespectively denote users and their relationships inGt. Given an IM campaign that takesT time (which is called as survival time later), the net- work may evolve fromGttoG_{T +t}during influence diffusion with newly added nodes and edges. Thus, different from the classical IM problem defined in Eqn. (1), we redefine the evolving IM problem overG_t = (Vt, E_t) as follows.

Definition 3.1. (EIM problem.) Given an evolving network at timestampt, i.e., G_t = (Vt, E_t) and the survival time T of an IM campaign, the objective ofEIM is to find a set of usersS^optwith sizeK to maximize the influence spread to both users in Vtand those that will join duringt to t + T . That is, we aim to solve

S^opt = arg max_{S ⊆Vt, |S |=K}I (S,G_{t +T}). (2) Note that in Definition 3.1, the seeds are selected from the current networkGt instead of the future instancesG_t⁰(t < t⁰ ≤T ). The reason behind is that the existing networkG_tis known, while it is difficlt to know which users will be in the future network instances and how they will be connected to each other. Since assuming the future instancesGt⁰(t < t⁰≤T ) known at time t is unrealistic, it is more reasonable to select the seed set from the currentG_t, with the objective being maximizing the influence diffused overG_{t +T}. Similar to the classical IM problem, theEIM defined above is also NP-hard. Lemma 3.2 states the hardness ofEIM problem and the submodularity of its objective functionI (S,G_{t +T}).

Lemma 3.2. TheEIM problem is NP-hard. The computation of I (S,G_{t +T}) is #P-hard. And the objective function I (S,G_{t +T}) is mono- tone and submodular².

Proof. The NP-hardness and #P-hardness can be respectively proved by the reductions of NP-completedSet Cover problem and

2A set functionI (·) is monotone if I (A) ≤ I (B) for all A ⊆ B, and I (·) is submodular ifI (A ∪ x ) − I (A) ≥ I (B ∪ x ) − I (B) for all A ⊆ B.

#P-completedS-D connectivities counting problem. And the sub- modularity ofI (S,G_{t +T}) can be proved by modeling the additional influence brought by a new seed as the marginal gain from adding an element to the setS. We leave the detailed analysis in the Ap-

pendix D. 

Challenges of solving EIM. The NP-hardness of EIM implies the necessity to seek for approximate algorithms for seed selection.

However, as noted in Section 1, solvingEIM is far more challeng- ing due to the evolving nature of the network included. Under Definition 3.1, the three challenges can be reproduced as: (1) The unknown growing speed makes it difficult to predict how many new users inV_{t +T}will connect to existing users inVt; (2) The in- fluences among users evolve over time, which, together with the unknown growing speed, renders it impossible to accurately esti- mateI (S,G_{t +T}). (3) The heterogeneous attractiveness infers that users inVtcannot be equally treated in seed selection.

3.2 Overview of EIM

Regarding the three challenges above, we propose a new framework that can better incorporate the evolving nature in solvingEIM. We note that what is built upon the three challenges, as also indicated in Section 1, is that the survival time of an IM campaign only varies from weeks to months in reality, leading to users joining the network several months later unable to be influenced by this early IM campaign. Consequently, triggering an IM campaign only once under the uncertain network knowledge will severely restrict the long term profits obtained from viral marketing.

3.2.1 Basic idea of solving EIM. We thus try to maximize the influence diffusion size over such evolving network by solvingEIM inmultiple periods, with one period corresponding to the survival timeT of an IM campaign. Given the initial network is G_tandT , the objective ofEIM in the first period is to select a setS of seeds from V_tto maximizeI (S,G_{t +T}) defined in Definition 3.1. And the objec- tive in the second period is to select a setS from V_{T +t}to maximize I (S,G_{t +2T}). Similar manner holds in subsequent periods. Thus successively IM campaigns in multiple periods give chance to maxi- mize the number of influenced users in a long term. Meanwhile, the periodical seed selection also enables us to cope with the three chal- lenges. To elaborate, the sizes of users join during pervious periods are the natural samples to learn the growing speed at a given period.

And the evolving influences among users can be learnt from the activating results during previous influence diffusion. Thus to sys- tematically resolve the above three challenges, each period consists of the following three steps: (1) Learning network growing speed from the feedbacks of observed newly added users. (2) Learning evolving influences from previous influence diffusion feedbacks. (3) Selecting seed set for triggering an IM campaign under the refined network knowledge in above two steps.

While we unfold the details of the three steps in Sections 4, 5 and 6 respectively, we remark that the idea of periodical seed selection in EIM cannot be trivially extended from that in recent online IM studies. As pointed out in Section 2.2, it is because the dynamic influences are restricted among fixed number of users in online IM, while seeds inEIM are selected from continuously joining users and the objective is to maximize the influence diffused to both the existing and future users. With this regard, existing online IM can

Triggered edges New users

Learning Network Evolution

Learning Evolving Influence

Learning Decision

Reward

Feedback Feedback

Evolution prediction

Influenced size estimation Step (2):

Step (1): Step (3):

Evolving Seed Selection

Figure 1: Overview of EIM in r -th trial

be reduced as a special case ofEIM by simply letting users in the network remain static over time.

3.2.2 Adaption to Combinatorial Multi Arm Bandits (CMAB).

Note that the above three steps in each period naturally forms a learning-decision process, where we first learn the growing speed and evolving influences from previous period and then decide which users to seed. To this end, we design a novel framework EIM to coordinate the above three steps in multiple periods, as illustrated in Figure 1. EIM allows to convert the EIM problem into a Combi- natorial Multi Arm Bandits (CMAB) one reviewed below:

In general CMAB, there arem arms with unknown reward dis- tributions and, in each trial, it makes a decision that chooses a set of arms with maximum expected rewards to trigger. Then the reward obtained from each arm is taken as the feedback to update its reward distribution, and in next trial, the decision is made under the updated rewards distribution. Given the total number of trials R, the objective of CMAB is to design an arm selection strategy to maximize the long term rewards obtained from the trials.

Regarding this, in EIM we model an IM campaign as one trial and totallyR trials will be performed. The decision in the r-th (0< r < R) trial is to select the seeds from the evolving network G^r = (V^r, E^r) at time³T^r . The triggered arms correspond to the activated edges in the influence diffusion starting from the selected seed nodes under IC model duringT^r toT^{r +1}. By modeling the activation probabilities as the reward distributions, we consider the edge-level feedback in EIM where we can observe wether the activation via an edge is successful or not.

Table 1 lists the mapping of the various components of CMAB to EIM framework. Different from the general CMAB, here the number of arms in EIM grows with the continuously emerging new edges during network evolution.

Table 1: Mappings between CMAB and EIM

CMAB # Symbol # EIM

r -th trial r r -th IM period

Arm e Influence via edgee

Reward of arme ze Activating result of edgee Reward ofr -th trial I (S^r, G^{r +1}) Influenced size duringT^rtoT^{r +1}

Bandits feedback ∆n(T^r) Observed new users duringT^rtoT^{r +1}

ze Reward of edgee

Example. We further give an example to facilitate the under- standing of EIM. Consider the budget for a viral marketing is seeding 60 users and the survival time for an IM campaign is one month. EIM divides it into multiple trials by seeding 5 users one

3Throughout the rest of the paper, we have the following relations: The initial network isG (t ) at time t , where we set T¹= t. Let T^rrepresent the time when ther -th trial occurs. And the evolved network atT^ris denoted asG^r= (V^r, E^r), with V^rand E^rbeing the corresponding evolved node and edge sets atT^r.

month. Here, two consecutive trials are one month apart. Sup- pose that the initial network starts at 1st, May, and the objective of the first trial is to select 5 seeds from current users to maxi- mize the influence among those joining before1st, May and during 1st, May to 31st, May. Then the second trial is on 1st June with the corresponding objective being maximizing influences among users joining until 30th, June, etc. In ther-th trial, EIM first learns the network knowledge from influence diffusion feedbacks during previous(r − 1) months, and then selects 5 users to maximize the influence during ther-th month based on the refined growing speed and evolving influences. The reward of EIM in this example is the influenced size during the 12 months.

Remark. In the present work, we focus on the case where the network exhibits fast growth while the promoted information remains effective in a far longer period. However, we do not need to rely on any correlation between the speed of newly added users and that of influence propagation. As long as the network is evolving, EIM can adaptively capture its growing speed, and then selects seed users under the learnt growing speed in each trial. Even the network is static, EIM is also applicable by setting Gt +T = Gt.

4 LEARNING NETWORK EVOLUTION

In this section, we dive into the first step, i.e, learning the future netowrk evolution during influence diffusion in the proposed EIM framework. To unfold, we need to address the following two ques- tions: (1) How the newly added users connect with existing users;

(2) How many new users will join in during influence diffusion.

4.1 Preferential Attachment (PA) Rule

For the first question, we adopt the well-known Barab ´asi-Albert (BA) model [21, 22] to characterize the evolution of social networks.

BA model is capable of well capturing the typical features such as power-law degree distribution and shrinking diameter that exit in most real social networks. The evolution under BA model is interpreted as follows: a new node joins the network at each time slot∆t, and establishes m new edges with the existing nodes (m is a constant) [11, 22]. LetVt denote the set of users at timet, and d_n^t denote the current degree of nodevn∈Vt. For a newly added user at timet, it establishes a new edge with a chosen existing uservs in each time slot∆t according to the rule of Preferential Attachment (PA), meaning that the probability ofv_sbeing chosen is proportional to its current degree at∆t. Then the remaining (m − 1) edges are respectively established in next(m − 1) time slots in the same manner.

Remark. Althoughm is set as a constant in the BA model [21, 22], it can still capture the evolution of most networks since each newly added node expectedly establishes a same number of new edges [11]. The BA model will also be empirically justified in Section 8.1 under various real datasets, all of which exhibit the phenomenon of “Richer gets richer”.

Under the PA rule, the expected degree of nodevnat time slot t + ∆t is equal to

E(d_n^{t +∆t}) = d^tn· * ,

1+ 1

Pvj∈Vtd^t_j+ 1+ -

. (3)

Given the number of users in evolution at timet is n(t ), the time spanT contains m[n(t + T ) − n(t)] evolving slots since there are [n(t + T ) − n(t)] newly added users and each user brings m new

edges. Based on the PA rule, Lemma 4.1 gives the expected degree of a given node in evolution.

Lemma 4.1. Given the degree of nodevnat timet is d_n^t, we have E(d^{T +t}_n ) = d_n^t ·

m[n(t +T )−n(t )]

Y

s=1

* ,

1+ 1

Pvj∈Vtd^t_j + (2s − 1)+ - .

Proof. We first consider a special case whenm = 1. According to Eqn. (3), at each evolving time slot, we have the following deduction whenm = 1,

E(d_n^{t +∆t}) = d^t_n· *. ,

1+ 1

Pvj∈Vtd^l_j+ 1 + / -

. (4)

Since a new edge establishes in time slott + ∆t, the total degrees of nodes after time slott + ∆t becomes Pvj∈Vtd^l_j+ 2. Then the expected degree of nodevnat time slott + 2∆t is

E(d_n^{t +2∆t}) = E(d^{t +∆t}_n ) · *. ,

1+ 1

Pvj∈Vtd^l_j+ 3 + / -

(5)

= d^t_n·

2

Y

s=1

* ,

1+ 1

Pvj∈Vtd^t_j + (2s − 1)+ -

. (6)

Here,∆t denotes an evolving slot. Under the growing speed n(t ), there aren(T + t) − n(t) new nodes joining the network during t tot +T . Thus there are n(T + t) − n(t) evolving time slots during t tot + T , and we have

E(d^{T +t}_n ) = d^t_n·

n(t +T )−n(t )

Y

s=1

* ,

1+ 1

Pvj∈Vtd^t_j + (2s − 1)+ -

. (7)

Then we consider the general cases whenm ≥ 2. Under the PA rule, them new edges brought by a same new node are respectively established inm evolving time slots. Thus there arem[n(T +t)−n(t)]

evolving time slots duringt to t +T in the general cases. Then Eqn.

(7) inductively becomes E(d^{T +t}_n ) = d_n^t ·

m[n(t +T )−n(t )]

Y

s=1

* ,

1+ 1

Pvj∈Vtd^t_j + (2s − 1)+ - .

Thus we end the proof for Lemma 4.1. 

Under PA rule, Lemma 4.1 returns the expected degrees of the existing users determined by the given growing speedn(t ), while the real degrees of users can be extracted from the bandits feedback in each trial. Next, we will show how to refinen(t ) for a specific network from the feedbacks.

4.2 Learning Nodes’ Growing Speed

Now we proceed to answer the second question posted at the be- ginning of this section. In reality, as noted in Section 1, the network growing speedn(t ) is affected by multiple factors. To elaborate, at timet, the n(t ) existing users prefer to attract new users to join the network, while the total populationN of who can join is limited [11]. As a result, the growing speed is constrained by the term [N − n(t )]. On the other hand, users exhibit decaying interests

β

t^θ in attracting users to join [11], in a similar manner to the sus- ceptible infected (SI) model in epidemiology [23]. The exponentθ reflects the growing dynamics such as power law, linear, sub-linear,

etc. Jointly considering the above factors, we adopt the Nettide- node model [11] to characterize the nodes’ growing speed, which is expressed as

dn(t ) dt =

β

t^θn(t )[N − n(t )]. (8) The Nettide-node model has been previously empirically justified over real social network data (e.g., Facebook, Wechat, Google-plus and arXiv, etc) [11] in terms of its effectiveness in capturing nodes’

growing speed, with an error of less than 3%. However, under the assumption of unknown future network topology, the parameters (i.e.,β, θ and N ) of a specific evolving network are unknown in advance. To deal with this dilemma, we propose a fully adaptive particle learning method to adaptively capture the nodes growing speed. The definition of particles is given below.

Definition 4.2. (Particle.) Each particleρi represents a growing speed function with given prior parameters (β_i,θ_i andN_i), i.e.,

dni(t )

dt =_t^β_θiⁱ ·ni(t )[Ni −ni(t )]. And they will be resampled based on their weightswi in each trial.

Given the definition, the particle learning is initialized by a set of particles with randomly sampled prior parameters (β, θ and N ) from their possible ranges, which will also be empirically presented in Section 8.5. And the number of sampled particles is set asM. Since the real growing functionn(t ) for a specific network is unknown in advance, we utilize the possible growing functionsni(t )(1 ≤ i ≤ M) to simulate the real network evolution in EIM. Over the R trials in EIM, the evolving process under each particle are proceeded in parallel, which is described as follows.

d=m d=m

Influenced nodes Influenced nodes

2 1

( ) ( )

i i

n T -n T

V1

3 2

( ) ( )

i i

n T -n T

2 1

( ),

i e e

E d vÎV E di( e³),veÎVi²

V1 ²

Vi

Figure 2: A sketch of evolving process under particleρ_i in the first three trials.

Evolving process under particleρi. The process starts from the given initial nodes set in the first trialV¹, which is same for the all particles. A sketch that contains the first three trials is shown in Figure 2. LetV_i^rdenote the evolving nodes set under particleρ_i until timeT^rwith|V_i^r|= ni(T^r). In the first trial, given the initial degree of a nodeve ∈V¹beingd_e¹, according to Lemma 4.1, its expected degree until timeT²can be estimated as

E_i(d_e²) = de¹·

m[ni(T²)−ni(T¹)]

Y s=1

* . ,

1+ 1

Pvj∈V¹d¹_j+ (2s − 1)+ / - .

Besides, from timeT¹toT², there aren_i(T²) − n_i(T¹) newly added nodes in expectation under particleρ_i, and the degrees of such nodes inV_i²\V¹ are uniformly expected asm, as shown in the middle part of Figure 2. Here,|V_i²\V¹|= ni(T²) − n_i(T¹). Then in the 2-nd trial, given influenced nodes setO (T²) during T¹toT², the degrees of nodes inO (T²) are updated by their real observed degrees, while the others inV_i²\O (T²) still reserve their estimated degrees. Thus forv_e∈V_i², its expected degree untilT³equals

Ei(d³_e) = Hd_e²·

m[n_i(T³)−n_i(T²)]

Y s=1

* . . ,

1+ 1

Pv_j∈V_i²dH_j²+ (2s − 1) + / / - .

And Hd_e²,v_e∈V_i²(the updated degrees of nodes inV_i²) is defined as

dH_e²=















d_e², v_e∈O (T²) E_i(d_e²), v_e∈V¹\O (T²) m, v_e ∈V_i²\V¹SO (T²) .

Based on the analysis above, we can inductively obtain the expected degrees until timeT^r in the(r − 1)-th trial. Specially,

Ei(d_e^r) = Hd_e^{r −1}·

m[n_i(T^r)−n_i(T^{r −1})]

Y s=1

* . . ,

1+ 1

Pv_j∈V_i^{r −1}dH^{r −1}_j + (2s − 1) + / / - .

(9) Accordingly, Hd_e^{r −1}(v_e ∈V_i^{r −1}) is defined as

dH_e^{r −1}=















d^{r −1}_e , ve∈O (T^{r −1})

Ei(d_e^{r −1}), ve∈V_i^{r −2}\O (T^{r −1}) m, ve ∈V_i^{r −1}\V_i^{r −2}SO (T^{r −1}) .

Upon the evolving process under each particle, we design the learn- ing process based on the general resample-propagate process, which is considered as an optimal and fully adaptive framework in parti- cle learning [24]. The objective of resampling phase is to resample the particles whose growing functions near the ground truth as more new particles, and simultaneously kill those with large de- viations from the ground truth. Following the resampling phase, the propagation phase is conducted to predict the real network evolution based on the simulation under each sampled particles.

Thus in correspondence to theEIM problem, the resampling and propagation phase respectively refer to the growing speed learning and evolution prediction.

The details of each phase are unfolded as follows:

Growing speed learning: This phase is to determine the weight of each particle with observed ground truth from bandits feedbacks, and then resample the particles set based on the their weights.

Due to the partial monitoring, the influenced users and only the influenced users in ther-th trial (i.e., O (T^r)) can be observed in influence diffusion, since social medias (e.g., Twitter and Weibo) can track the activities of their users such as one user retweeting a tweet forwarded by another user [5][15].

Given the last observed time for nodeve ∈ O (T^r) ∩

∪^{r −1}_i=1 O (Tⁱ)

isT^(e,0), with the corresponding degree beingd^(e,0), its real incremental degreed^r_e−d^(e,0)serves as the ground truth in the resampling phase. Then under the particleρi, its expected degree at timeT^r is equal to

E_i(d_e^r) = de^(e,0)·

m[n_i(T^r)−n_i(T^(e,0))]

Y s=1

* . . . ,

1+ 1

P

v_j∈V_i^(e,0)dH_j^(e,0)+ (2s − 1) + / / / - .

(10) Here,V_i^(e,0)denotes the set of nodes under particleρiat timeT^(e,0). ThusEi(d_e^r) − d_e^(e,0)denotes the expected incremental degree of nodee under particle ρi. By summing incremental degrees of nodes inO (T^r) ∩

∪^{r −1}_i=1O (Tⁱ)

, the prior value of particleρi is equal to

∆n_i(T^r) = X O (T^r)∩

∪^{r −1}_i=1O (Tⁱ)

Ei(d_e^r) − d_e^(e,0) . (11)

And the ground truth∆n(T^r) in the r-th trial is determined as

∆n(T^r) = X

O (T^r)∩

∪^{r −1}_i=1O (Tⁱ)

d_e^r−d^(e,0) . (12)

Given the ground truth and the prior value of each particle, the weight of particleρiis inversely proportional to the square error between∆n(T^r) and ∆n_i(T^r), i.e.,

wi(T^r) ∝ 1/(∆n(T^r) − ∆ni(T^r))², (13) Then a resampling process is conducted to resample particles setP^r from those inP^{r −1}with the number proportional to their weights, and the total number always satisfies|P^r|= M (0 ≤ r ≤ R).

Evolution prediction: Following the resampling phase, we com- pute the expected incremental degrees (i.e., Ei(∆d_e^{r +1})) of nodes in V^r until timeT^{r +1}under each resampled particle, i.e.,

E_i(d_e^{r +1}) = Hd_e^r ·

m[n_i(T^{r +1})−n_i(T^r)]

Y s=1

* . ,

1+ 1

Pvj∈V_i^rdH^r_j + (2s − 1) + / -

. (14)

And the expected incremental degree ofvefromT^(e,0)toT^{r +1}can be computed as

Ei(∆d_e^{r +1}) = Ei(d^{r +1}_e ) − d^(e,0). (15) Then we set the incremental degree of nodeveas the average of ex- pectation under each particle. That is, E(∆d^{r +1}e ) = P_i=1^M _M¹Ei(∆d^{r +1}_e )).

And the prediction of network evolution is represented by the in- cremental degree of each node inV^r.

//Particle learning in ther -th trial

Input: Influenced nodes setO (T^r), particles set: P^{r −1};

Output: Particles set: P^r, incremental degree: E(∆de^{r +1})(ve ∈V^r);

//Resample phase

Count the number of newly added nodes:∆n(T^r);

for eachρi ∈ P^{r −1}do for eachve∈O (T^r) ∩

∪^{r −1}_i=1O (Tⁱ) do Compute expected degree:Ei(d^r_e) (Eqn. (10));

end

Compute prior value:∆n_i(T^r) (Eqn. (11));

Compute its weight:w_i(T^r) (Eqn. (13));

end

Resample particles with weights:P^{r −1}→ P^r; //Propagate phase

for eachve∈V^rdo

Compute incremental degree: E(∆d^{r +1}e ) (Eqn. (15));

return E(∆d^{r +1}e )(ve∈V^r) and P^r. end

Algorithm 1: Learning network evolution (Evo-NE).

The pseudo code of the above particle learning process, which is mainly composed of resampling-propagation phases, is further summarized in Algorithm 1 calledEvo-NE. Algorithm 1 takes the influenced nodes duringT^{r −1}toT^r and particles setP^{r −1}as the input, and the prior value of each particle is computed as Eqn.

(11). Then the particles inP^{r −1}are resampled as new particlesP^r based on their weights determined by Eqn. (13). Following the resampling phase, we compute the expected incremental degrees of nodes inV^r with the resampled phase inP^rwhich represent the predicted network evolution duringT^rtoT^{r +1}. And the complexity of Algorithm 1 is shown as below.

Complexity. In resampling phase, Evo-NE needs to traverse all the nodes under each particle inP^{r −1}to compute the prior value and weight of each particle. Then in the propagation phase, the expected incremental degree of each node inV^r under each resampled particle is computed to predict the network evolution. In ther-th trial, the number of particles is M and the number of nodes under each particle is scaled asO (V^r), thus the network evolution learning algorithmEvo-NE inr-th trial costs O (M |V^r|) time.

This section illustrates the first step in EIM for learning net- work evolution, and in next section, we will show the solutions for learning evolving influences which is the second step in EIM.

5 LEARNING EVOLVING INFLUENCES

In this section, we move to the second step of EIM framework illus- trated in Figure 1. That is, we need to learn the unknown influences I (S,G_{t +T}) among users to facilitate the accurate influenced size estimation over the predicted networkG_{t +T}. Our methodology of influence learning is presented as below.

Evolving influences modeling. In ther-th trial, the objective is to maximize the influenced size over the target networkG^{r +1}. For any node pairu_iandu_jinG^{r +1}, we usew_eto denote evlweight of edge fromuitouj. Built upon the widely used IC model depicted by Definition 2.1,u_ican successfully activatesu_jwith a probability equal towe. However, the traditional IC model cannot be directly applied to determine the weights inEIM problem since: (1) The weights of newly established edges remain unknown in advance; (2) The weights may exhibit random dynamics with network evolution.

The reason behind is that real-world factors such as users’ interests of propagated contextual information and the closeness of friend- ship may be dynamic in evolution [5]. For example, new edges are established when users make new friends, and the weights of edges may strengthen over time until they become stable close friends. In contrast, a pair of partners may gradually become strangers after their cooperation has ended. Thus the weights of edges in evolution may randomly become larger or lower over time with decaying fluctuations. To jointly consider these features, we characterize each weight as a Gaussian random walk presented below, where its fluctuation during each period can be represented by a Gaussian random variable.

Evolving weights of edges. Letwe,rdenote the value ofwein ther-th trial. For a new edge e that establishes during (T^{r −1}, T^r], under the Gaussian random walk mode, we set the initial value of the weightw_efollows a Gaussian distribution withw_{e,r −1} ∼ N(w_{e,r −1}⁰ , Σe,r −1) = N (w⁰, Σ⁰) and re,0 = r − 1. Here, w_{e,r −1}⁰ andΣe,r −1respectively denote mean and variance ofwe,r −1’s dis- tribution. Then the variation ofweis defined with a Markov process as below

we,r = we,r −1+ ve,r, ve,r ∼ N(0, ∆Σe,r), (16) whereve,r denotes the Gaussian random noise to characterize the variation ofw_ein ther-th trial and ∆Σ_e,r =_{(r −r}^Σ0

e,0)^k(k > 0).

For the above evolving influences, recall that in Section 3.2, under the bandit-based framework of EIM, we treat them as the arms and leverage the edge-level feedbacks to update their estimating values. In detail, letwedenotewij. For a userui being influenced in ther-th trial, he will try to influence his neighbor u_jsuccessfully with probabilitywe,r, thus edgee is triggered. Since a successful influence can bring a more influenced user, we model the reward obtained from edgee as a binary reward ze,r with success denoted by 1 and failure denoted by 0, which is leveraged as the feedback to refine the distribution ofw_e,r. Since the weightw_echanges over time with a Gaussian random walk, in ther-th trial, it follows a Gaussian distribution after accumulating pervious random walks, which is denoted bywe,r ∼ N(w_e,r⁰ , Σe,r). Based on the Gaussian statistical properties, we adopt the Kalman Filter as the refining

method for the distributions of evolving influences, as described below.

Kalman Filter based refining method. Let a binary variable ze,r denote the reward obtained from the triggered edgee in the r-th trial, referring Kalman Filter theory [25], the mean w⁰_e,r and varianceΣe,r of the weight of edgee in the r-th trial is refined with w⁰_e,r = w⁰_{e,r −1}+ Ge,r·(z_e,r−w_{e,r −1}⁰ ); (17) Σe,r = Σe,r −1+ ∆Σe,r −Ge,rQe,r. (18) Here,Ge,r(z_e,r −w⁰_{e,r −1}) and G_e,rQe,r are the correction from Kalman filte. AndGe,ris the Kalman Gain in refinement to quantify the correction from the new observationze,r, which is determined as follows.

//Edge weight refining in ther -th trial Input: Observed edges from timeT^{r −1}toT^r;

Output: Refined distribution of each edge’ s weightw_e,r⁰ ; Process:

Setw_e,r∼ N(w0, Σ0) for each first observed edge;

for each observed edgee in E^r do ComputeQe,r = Σe,r+ 1;

ComputeG_e,r = (Σe,r −1+ ∆Σe,r) · Q⁻¹_e,r; Updatew⁰_e,r = w⁰e,r −1+ Ge,r·(z_e,r−w⁰_{e,r −1});

UpdateΣe,r= Σe,r −1+ ∆Σe,r−Ge,rQe,r; end

for each edge unobserved edgee in E^rdo Updatew⁰_e,r = w⁰e,r −1;

UpdateΣ_e,r= Σe,r −1+ ∆Σe,r; end

returnw⁰_e,r,Σ_e,rfor each edge.

Algorithm 2: Evolving influence learning(Evo-IL).

Lemma 5.1. The Kalman Gain in the refinement ofw_ein ther-th trial is determined by

Ge,r = (Σe,r −1+ ∆Σe,r) · Q_e,r⁻¹,

where Q_e,r = Σe,r+ 1 denotes variance of the activating result via e.

Proof. In the refinement ofweinr-th trial, the Kalman Gain G_{e,r +1}is determined by minimizing the mean square estimation error ofwe,r, i.e.,

Ge,r = arg min

M ∈R E

w⁰_e,r−w_e,r²

where w⁰_e,r = w⁰_{e,r −1}+ M ·z_e,r −w⁰_{e,r −1} .

(19)

Here, R represents the set of real numbers. By minimizing the objec- tive function in Eqn. (19), the Kalman Gain in refinement is formu- lated asGe,r = (Σ⁰_{e,r −1}+ Σr) · Q⁻¹_e,r, whereQe,r = Σ⁰_e,r+ 1 denotes the variance of the activating result via edgee, i.e., Qe,r = Σ_{e,r −1}⁰ + σ², whereσ²denotes the square observing error of Bernoulli dis- tributionB(we,r) with 0 ≤ σ²≤ 1. And we setσ²as its maximum value 1 (e.g.,w_e,r = 0 and ze,r = 1). Then the distributions of observed edges’ weights are refined withGe,randQe,ras Eqn. (17)

and Eqn. (18). 

On the other hand, in case that useru_i is not influenced, edgee is not triggered. Then the distributions for the non-triggered edges in ther-th trial evolve as:

w_e,r⁰ = w⁰_{e,r −1}, Σ_e,r = Σe,r −1+ ∆Σe,r. (20)