Adaptive Submodular Influence Maximization with Myopic ... - arXiv

arXiv:1704.06905v3 [cs.SI] 18 May 2017

Adaptive Submodular Influence Maximization with Myopic Feedback

Guillaume Salha∗ LIX, École Polytechnique [email protected]

Nikolaos Tziortziotis∗ LIX, École Polytechnique [email protected]

Michalis Vazirgiannis LIX, École Polytechnique [email protected]

Abstract This paper examines the problem of adaptive influence maximization in social networks. As adaptive decision making is a time-critical task, we consider a realistic feedback model, called as myopic. In this direction, we propose the myopic adaptive greedy policy that is guaranteed to provide an (1 − 1/e)-approximation of the optimal policy under both the standard independent cascade diffusion model as well as a variant of it. This strategy maximizes an alternative IM utility function that has been proved to be adaptive monotonic and adaptive submodular. Furthermore, we show that the adaptive submodularity property does not hold if the nodes are allowed to be deactivated randomly over time. Our empirical analysis on realworld social networks reveals the benefits of the myopic adaptive greedy strategy to the influence maximization problem, validating our theoretical results.

1 Introduction Graphs are useful models for specifying relationships within a collection of objects. Numerous real-life situations could be represented as nodes linked by edges, including social, biological or computer networks. Discovering the most influential nodes in such networks has been the objective of considerable research in ML and AI community. One of the most practical applications is that of product placement or viral marketing. Consider a directed social network in which nodes correspond to potential customers. If a customer owns a product then can recommend it to his friends, according to a given diffusion model that simulates the word-of-mouth effect. Given a fixed budget, our objective is to select a set of customers to give a product for free, in order to maximize the spread of influence through the network, i.e., to maximize the number of peoples that will finally buy this product. Influence maximization (IM) in social networks was first studied by Domingos and Richardson [4]. Kempe et al. [11] reformulated influence maximization as a discrete optimization problem by introducing two diffusion models: Independent Cascade (IC) and Linear Threshold (LT) model. They demonstrated that finding an optimal set of at most k seed nodes, with k to represent our budget, that maximizes influence in the network is NP-hard under both models. Nevertheless, they proved that the utility function to maximize, which is the expected number of influenced nodes, is monotone and submodular. These properties in conjunction with the results of Nemhauser et al. [17] imply that the greedy algorithm is guaranteed to provide an (1 − 1/e)-approximation of the optimal set. Feige [5] highlighted that this result is tight if P 6= N P , and considered as near-optimal [16, 21]. The ∗

Equal Contribution

aforementioned seminal works have inspired a large part of other research works, either to provide alternative frameworks [22, 15, 1, 9, 19], or to speed up the greedy algorithm via heuristics providing theoretical results [3, 8, 2] or scalability guarantees [14, 10, 12]. Most of the works on influence maximization are restricted to the non-adaptive setting, where all seed nodes must be selected in advance. The main drawback of this assumption is that the particular choice of seed nodes is completely driven by the diffusion model and the edge probability assignment. Apparently, it may leads to a severe overestimation of the actual spread resulting from the chosen seed nodes [8]. Under this prism, we focus on the adaptive setting of the IM problem. Instead of selecting a number of seed nodes in advance, we select one (or more) node at a time, then we observe how its activation propagates through the network and, based on the observations made so far, we adaptively select the next seed node(s). Actually, it constitutes a sequential decision making problem where we need to design a policy that specifies which is the most appropriate node(s) to be selected at a given time. It can be verified, even on small graphs, that the adaptive setting leads to higher spreads compared to the non-adaptive one, since we gradually gain more knowledge about the ground truth influence graph (see Appendix A for a toy example). Adaptive submodularity [6] constitutes a natural generalization of submodularity to adaptive policies. Similar to Kempe et al. [11], Golovin and Krause [6] showed that, when the objective function under consideration is adaptive monotone and adaptive submodular, a simple adaptive greedy policy performs near-optimally. Adaptive submodularity has been verified to be useful on several practical applications such as active learning, sensor placement, etc.. However, in the adaptive IM task, the adaptive submodularity property of the utility function holds only in the case of the unrealistic Full feedback model. Tang [18] has introduced the partial-feedback model that captures the trade-off between delay and performance. An (α, β)−greedy policy has also been proposed that guarantees a constant approximation ratio under this model. Nevertheless, the question of whether adaptive submodularity property can be proved for more realistic feedback models, has not been answered yet. In this paper, we introduce an alternative utility function that instead of the total number of the active nodes at the end of the diffusion, considers the cumulative number of active nodes through the time. In order to present our theoretical analysis in a strict way, we resort to a layered graph representation, similar to the one presented by Kempe et al. [11], with each one of the graph’s layers to illustrate the diffusion in network at a particular time stamp. Additionally, we propose a new diffusion model, which constitutes a variant of the standard IC diffusion model. In the modified IC model, an active node has more than one opportunities to activate each one of its neighbors. The main contribution of this work is that we achieve to prove rigorously that the considered objective function is adaptive monotonic and adaptive submodular under both the standard and modified IC model. Therefore, the proposed myopic adaptive greedy policy is theoretically guaranteed to reach an (1 − 1/e)-approximation ratio in terms of the expected utility of the optimal adaptive policy. We also prove that the assumption of the IC model under which the active nodes cannot be deactivated through the time constitutes a necessary condition to verify the adaptive submodularity property of the proposed utility function in both diffusion models. Finally, the superiority of the myopic adaptive greedy strategy over other adaptive heuristic strategies and a non-adaptive greedy strategy to the IM problem has been demonstrated on three real-life social networks.

2 Preliminaries A social network is typically modeled as a directed graph G = (V, E) with each node v ∈ V to represent a person, and the edges E ⊂ V × V to reflect the relationships among them. To simulate the diffusion process in a social network we consider the IC model. It is a discrete-time model where only the seed nodes are initially active. Afterwards, each time where a node v first becomes active, it has a single chance to activate/influence each of its inactive neighbors u, succeeding with known influence probability pvu . The diffusion process continues until no further activations are possible. We consider that each edge e ∈ E is associated with a particular state o ∈ O, with O to be a set of possible states (whether an edge is live or dead). We denote by φ : E → O a particular realization of the influence graph, indicating the status of edges in a particular world’s state. It is also assumed that the realization Φ is a random variable with known probability distribution, p(φ) , P[Φ = φ]. 2

In the adaptive setting, after selecting a seed node v ∈ V, we get a partial observation of the ground truth influence graph φ [6]. More specifically, after each step, our knowledge so far will be represented as a partial realization ψ ⊆ E × O, which is a function from a subset of E to their states. We use the notation dom(ψ), called as domain of ψ, to refer to the set of nodes that are observed to be active through ψ. More clearly, we say that a partial realization observes an edge e, if some node u ∈ dom(ψ) has revealed its status. A partial realization ψ is said to be consistent with φ, denoted by φ ∼ ψ, if the state of all edges observed by ψ are the same in φ. Also, we say that ψ is a subrealization of ψ ′ , ψ ⊆ ψ ′ , if both of them are consistent with some φ, and dom(ψ) ⊆ dom(ψ ′ ). Adaptive influence maximization constitutes a sequential decision making problem where we have to design a policy π, determining sequentially which node(s) must be selected as seed(s) at each time step, given ψ. We call as E(π, Φ) ⊆ V the seed nodes that have been selected following policy π under realization φ. The standard IM utility function is defined as f (S, φ) , |σ(S, φ))|, with σ(S, φ) to be the set of the influenced nodes at the end of the process under realization φ, and given the seed set S. Actually, our objective is the discovering of an optimal policy π ∗ that maximizes the expected utility, favg (π) , EΦ [f (E(π, Φ), Φ)]. This can be written more concretely as: π ∗ ∈ arg max favg (π)

s.t. |E(π, φ)| ≤ k, ∀φ.

π

In general, this is an NP-hard optimization problem [6]. In the non-adaptive case, we can easily derive near-optimal policies if the utility function is monotonic and submodular [17, 11]. To provide generalizations of monotonicity and submodularity in such an adaptive setting, Golovin and Krause [6] adopt the expected marginal gain notion. Definition 1. The conditional expected marginal benefit of v ∈ V, conditioned on h i partial realization ψ, is given as: ∆f (v|ψ) , EΦ f (dom(ψ) ∪ {v}, Φ) − f (dom(ψ), Φ)|Φ ∼ ψ . This leads us to the following definitions of adaptive monotonicity and adaptive submodularity, defined w.r.t. to the distribution p(φ) over realizations. Definition 2. Function f is adaptive monotone iff ∆f (v|ψ) ≥ 0 for all v ∈ V and ψ such that P(Φ ∼ ψ) > 0. Definition 3. Function f is adaptive submodular iff ∆f (v|ψ) ≥ ∆(v|ψ ′ ), for all v ∈ V \ dom(ψ ′ ) and ψ ⊆ ψ ′ . Let π g be the adaptive greedy policy that given the partial realization ψ selects the node v ∈ V \ dom(ψ) with the highest expected marginal gain, ∆f (v|ψ). Golovin and Krause [6] proved that, if the utility function f is adaptive monotone and adaptive submodular w.r.t. p(φ), then π g is an (1 − 1/e)-approximation of π ∗ , favg (π g ) ≥ (1 − 1/e)favg (π ∗ ). This constitutes a direct extension of the non-adaptive bound, which was proved to be near-optimal [17]. In the adaptive IM problem, the following two concrete feedbacks can be considered: • Full-adoption feedback: activating a seed node, we observe the entire propagation (cascade) in graph, and then we select the next seed node; • Myopic feedback: activating a seed node at time t, we only observe the status (active or not) of the neighbors of the seed nodes at time t + 1. Therefore, in myopic feedback model, selecting a node at time t has an impact at time t + 2, t + 3, and so on. Nevertheless, it has been shown [6] that the standard utility function f holds its adaptive submodular property only under the full-adoption feedback model (counterexamples are reported in [6, 20]). Thus, there is no guarantee that we can discover a policy able to approximate the expected utility of the best policy within a reasonable factor in the case of the myopic feedback model.

3 Myopic Feedback through Layered Graphs The limitations of the full-adoption feedback (i.e., in most applications the propagation in the network is not instantaneous) motivates us to focus on the myopic feedback model that fits better on real world. However, the result of Golovin and Krause [6] is not applicable under the IC model with myopic feedback, as the adaptive submodular property of f is violated. To deal with this situation, we introduce an alternative utility that considers the cumulative number of active nodes over time 3

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Figure 1: Influence propagation representation over (a) Original graph G and (b) Layered graph G L . The shaded nodes illustrate the active nodes in the graph. pvu represents the propagation probability between v and u. In both cases, the nodes can only switch from being inactive to being active.

instead of the total number of active nodes at the end of the diffusion process. More precisely, given a finite horizon T , the proposed utility function is defined as: f˜(S, φ) ,

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where σt (S, φ) represents the set of active nodes at time t if the seed set S has been selected under realization φ. According to f˜, if a node is active for three time steps, it will yield a reward equal to 3 instead of 1 as in the case of standard IM utility function f . The proposed utility function is consistent with many real life situations. Consider, for instance, the case of platforms with a monthly subscription, like Netflix or Amazon. Those services charge each active user every month on the date he signed up. Thus, the companies’ profit increase as the users be active for longer periods. Thus, the value of an active node is additive over time. Modified IC model Let us now to introduce a slight modification of the standard IC model, which is still consistent with most real-world applications. In contrast to the standard IC model where an active node has a single change to influence its neighbors, in the modified IC model each active node has multiple opportunities to influence its inactive neighbors. In Section 4, we prove that the proposed utility function, f˜, is adaptive submodular under both the modified and standard IC model with myopic feedback. Layered graph representation To represent the evolution of the network over time, we resort to a layered graph representation, denoted as G L . A graph’s layer corresponds to the representation of the original graph at a particular time step, with Lt to denote the set of nodes on layer t. Consider for example the original graph illustrated at Fig. 1(a) and its evolution over three successive time steps. We retrieve the same amount of information as in the case of the layered graph, Fig. 1(b). Indeed, node v is active at time t if and only if vt is active in the layered graph. Then, it influences its neighbor u at time t + 1 with probability pvu . Thus, there is a possibly live edge from vt to ut+1 . For the sake of simplicity, in the rest of the paper we use the next indexing fG or f˜G in order to explicitly declare that function f or f˜ is estimated on graph G. It can be easily verified that the two networks, the original and the layered one, are closely linked. The following lemma highlights the fact that computing f˜G is equivalent to computing f on the layered graph, i.e. fG L . Lemma 1. For seed set S (with time indices) and realization φ, it holds that f˜G (S, φ) = fG L (S, φ). Proof. It suffices to remark that the number of active nodes on layer Lt is equal to the number of active nodes on G at time t. Summing up the active nodes of each layer Li is the same by applying f on G L , which is equivalent to summing up the number of active nodes on G at each time-step. In our model, the time dependency is even stronger compared to previous models. Partial realizations ψ should now indicate the status of observed nodes and edges as well as the corresponding timesteps, as nodes can be activated over multiple timesteps and edges can be crossed multiple times. 4

Actually, we need to know up to which time step, the ψ contains observations. This leads to the next definition. Definition 4. Let Ψ be the set of all possible partial realizations. Time function T : Ψ → {1, . . . , T } returns, for a particular ψ, the largest time index from observed nodes and edges, and 1 if ψ = ∅. In a nutshell, choosing v as a seed node having observed ψ with T (ψ) = t ≤ T , is the same as choosing vt as a seed node in the layered graph, since the process is now at time t. In this point, let us provide a last definition. Definition 5. The marginal gain of choosing v as a seed node, having observed ψ with T (ψ) = t, and for the ground truth realization φ of the network, is defined as: δφ (v|ψ) , f˜G (dom(ψ) ∪ {vt }, φ) − f˜G (dom(ψ), φ). The aforementioned definition is useful for the analysis of the next two lemmas. Firstly, Lemma 2 is a markovian result on layers. It shows that, to evaluate δφ (v|ψ), we only need information from the current layer, LT (ψ) . Information from previous layers, L1 , . . . , LT (ψ)−1 , have no impact on the marginal gain of adding v to seed nodes at time T (ψ). Secondly, Lemma 3 is a submodularity result on δφ (·|ψ), that will be central in proofs of Section 4. Lemma 2. The marginal gain of choosing v as a seed node on G L , under partial realization ψ with T (ψ) = t, is given by: δφ (v|ψ) = fG L ([Lt ∩ dom(ψ)] ∪ {vt }, φ) − fG L (Lt ∩ dom(ψ), φ). Proof. Based on Def. 5 and Lem. 1, it holds that: δφ (v|ψ) = fG L (dom(ψ) ∪ {vt }, φ) − fG L (dom(ψ), φ). Given a set S of seed nodes on the G L , the utility function f is given by PT fG L (S, φ) = t′ =1 |σ(S, φ) ∩ Lt′ |. Then we get that: δφ (v|ψ) =

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= fG L ([Lt ∩ dom(ψ)] ∪ {vt }, φ) − fG L (Lt ∩ dom(ψ), φ). The second equality holds due to the fact that the network G L is feedforward, which means that the node vt can only influence nodes on the subsequent layers: Lt+1 , . . . , LT . Lemma 3. For partial realizations ψ ⊆ ψ ′ with T (ψ) = T (ψ ′ ) = t and any v ∈ V , we get δφ (v|ψ) ≥ δφ (v|ψ ′ ). Proof. Let R(vt , φ) denotes the set of nodes that can be reached from node vt via a path consisting of live edges, under realization φ. For any A ⊆ Lt (layer t of G L ), we have fG L (A, φ) = | ∪v∈A R(v, φ)|. Let us now consider the quantity fG L (A ∪ {vt }, φ) − fG L (A, φ) to be equal to the number of elements of R(vt , φ) that are not already contained in ∪v∈A R(v, φ). Clearly, this quantity is larger or equal to the number of elements of R(vt , φ) that are not contained in the bigger set ∪v∈B R(v, φ), for any A ⊆ B ⊆ Lt . Therefore, it holds that: fG L (A ∪ {vt }, φ) − fG L (A, φ) ≥ fG L (B ∪ {vt }, φ) − fG L (B, φ). Setting A = Lt ∩ dom(ψ), B = Lt ∩ dom(ψ ′ ) and using Lem. 2, we get: δφ (v|ψ) ≥ δφ (v|ψ ′ ).

4 Theoretical Guarantees for the Myopic Adaptive Greedy Strategy The myopic adaptive greedy policy starts with an empty set S = ∅, and repeatedly chooses as seed the node that gives the maximum expected marginal gain under partial realization ψ. For simplicity reasons, we assume that only one seed node is selected at each time step. Furthermore, in the case of the standard IC model, only the seed set nodes S that have been activated for a first time, are allowed to influence their neighbors. A sketch of our policy is presented in Alg. 1. 5

Algorithm 1 Myopic adaptive greedy policy Input: G, T 1: ψ ← ∅, S ← ∅ 2: for t = 1 to T do 3: Compute ∆f˜(v|ψ), ∀v ∈ V \ S 4: Select v ∗ ∈ arg max ∆f˜(v|ψ) v∈V\S

5: S ← S ∪ {v ∗ } 6: Update ψ observing (one-step) myopic feedback 7: S ← S ∪ dom(ψ) 8: end for 9: return S (final set of influenced nodes)

We are now ready to formally state our main results that constitute approximation guarantees for the proposed strategy. Actually, we would like to prove that the proposed utility function f˜G is adaptive monotonic and adaptive submodular w.r.t. p(φ) under both the standard and modified IC model with myopic feedback. These properties in conjunction with the result of [6] will allow us to complete our proofs (the proofs of Theorems 1 and 2 could be found at Appendix B and C, respectively). Theorem 1. The adaptive greedy policy π g obtains at least (1 − 1/e) of the value of the best policy for the adaptive influence maximization problem under the modified IC model with myopic feedback and f˜ as utility function. In other words, if f˜avg (π g ) , EΦ [f˜G (E(π, Φ), Φ)], we get that: f˜avg (π g ) ≥ (1 − 1/e)f˜avg (π ∗ ). 4.1 Standard IC diffusion model analysis Our analysis so far has been focused on the modified IC model where an active node has multiple opportunities to influence its neighbors. In this section, we prove that the same guarantees are hold in the case of the standard IC diffusion model with myopic feedback [6]. The main difficulty now is that the influence weights among two nodes in the layered graph representation G L are not the same along the time anymore. More specifically, it is no longer true that the influence probabilities between vt and ut+1 with t ∈ {1, . . . , T − 1} in the G L are equal to pvu ∈ [0, 1]. Let us define the function 1ψ (vt ) = 1{vi ∈ / dom(ψ), ∀i ∈ {1, . . . , t − 1}} that indicates whether the node v has been activated before time t (0) or not (1). Therefore, the influence probabilities on the edges between vt to ut+1 (∀t ∈ {1, . . . , T − 1}) will now be equal to pvu 1ψ (vt ). Actually, it is like adding randomness into the edges’ probabilities that is strongly related to the decisions taken by the following policy. As a consequence, it becomes difficult to get results for all realizations φ of the ground truth graphs as in Lemmas 2 and 3. To overcome this handicap, we assume w.l.o.g. that the status of the edges from node v to u in the G L are revealed right after the first and last attempt of the v to activate u. Actually, all edges have the same status (live or dead) that depends on whether this attempt was successful or not. It can be easily verified that this assumption has no impact on the diffusion process. For instance, let us consider a particular state of the world under which we observe that the first and single chance of node v to influence u is successful. Then, assuming that the status of all edges between vt and ut+1 with t ∈ {1, . . . , T − 1} in G L are all equal to 1 (live) will not modify the final spread. Indeed, the edges prior to the one actually crossed to influence u has no longer impact on the spread as the network G L is feedforward. On the other hand, the status of the edges that belong on subsequent layers has no impact at the diffusion as nodes v and u are both active, and we assumed that nodes cannot be deactivated through the time. The same reasoning can be also applied in the case where the first and single opportunity of node v to activate u is failed. Now, we can easily derive that Lemmas 1-3 are still valid for our new framework. First of all, the results of Lemma 1 remain unmodified as it is not related to the diffusion model. Furthermore, Lemmas 2 and 3 have been validated for all ground truth realizations φ of the layered graph. Therefore, they are still valid for the subset of these realizations that satisfy the previous constraint (i.e., all edges in the layered graph, associated to the same pair of nodes, have the same status). In this way, we conclude to the next Theorem that constitutes an extension of Theorem 1 to the standard IC model. 6

Theorem 2. The adaptive greedy policy π g obtains at least (1 − 1/e) of the value of the best policy for the adaptive influence maximization problem, in the standard IC model with myopic feedback and f˜ as utility function. Therefore, it holds that: f˜avg (π g ) ≥ (1 − 1/e)f˜avg (π ∗ ). The aforementioned results are a direct extension of the bound of the full-adoption feedback model, which has been proven to be near-optimal [6]. This is the first time that such inequalities are demonstrated on the adaptive setting under myopic feedback model. Using the generalization of the result of Golovin and Krause [6], we also retrieve the (1−e−ℓ/αk ) bound for any α-approximate (ℓ-truncated) greedy policies. Moreover, it can be easily verified that the bounds (Theorems 1 and 2) are still valid even if we select more than one seed nodes at each time step.

5 Non-Progressive Adaptive Submodular IM In this section, we examine the scenario where the main hypothesis of the IC model (active nodes can not be deactivated randomly) does not hold anymore. Actually, the application itself can determine if this hypothesis is realistic or not. In the case of our layered graph representation, we can easily relax this assumption, by replacing the “1” with a random probability over the edges between the same nodes. Our model along with the main notations are still well defined under this relaxation. However, it appears that it destroys the reasoning of the proofs of our results, as the utility function f˜G is no longer adaptive submodular. Additionally, we show that the adaptive submodularity property is also violated even in the case of the full-adoption feedback by using the standard IM utility function fG . Lemma 4. Forcing active nodes to remain active throughout the process constitutes a necessary condition to verify the adaptive submodularity property of: i) f˜G in the modified IC model with myopic feedback; ii) f˜G in the standard IC model with myopic feedback; iii) fG in the standard IC model with full-adoption feedback. Therefore, the theoretical results presented in our paper and those of [6] are not directly applicable in the case where the active nodes can be deactivated. However, the hypothesis of active nodes deactivation may be consistent with many applications, including some versions of the product placement problem (e.g. customers could reject the product). In this direction, we are still able to prove a weaker inequality at each time step. We still consider the previous framework, but now active nodes are allowed to be deactivated randomly. At each step t, we choose kt seed nodes from layer Lt of G L , in order to maximize the expected spread in the future having observed which nodes are currently active, i.e. active nodes on Lt . Then, we get the next result (proved in Appendix E). Theorem 3. Let t ∈ {1, . . . , T }, let St ⊆ Lt the (observed) set of active nodes at time t, and consider the following problem: A∗ ∈ arg maxA⊆Lt \St ,|A|≤k EΦ [fLt ∪···∪LT (A ∪ St , Φ)]. Then, a greedily constructed set Ag ⊆ Lt \ St is guaranteed to achieve a (1 − 1/e)-approximation of the optimal set: EΦ [fLt ∪···∪LT (Ag ∪ St , Φ)] ≥ (1 − 1/e)EΦ [fLt ∪···∪LT (A∗ ∪ St , Φ)]. This result is weaker than those of Theorems 1 and 2, since it is simply a “step-by-step” inequality on each seeding, but not anymore on the entire policy. However, it is free from the assumption that active nodes should remain active.

6 Empirical Analysis We conducted experiments on three social networks obtained from the Stanford’s SNAP database [13]. The first one is a small directed ego network from Twitter (|V| = 228, |E| = 9.938). We also study two medium-size (undirected) real networks, a social network from Facebook (|V| = 4.039, |E| = 88.234) and a collaboration network from Arxiv General Relativity and Quantum Cosmology section (|V| = 5.242, |E| = 28.980) (more details could be found at Appendix F). Throughout our analysis, we considered both the standard and the modified IC diffusion model with myopic feedback. Our primary objective is the adaptive selection of k seed nodes, one at each time. The time horizon is defined as T = k+1, i.e. the diffusion process stops one step right after selecting the last seed. Similar to [11, 7], we set an identical influence probability at each edge, p = 0.1. As it is not possible to actually compute the optimal set of influential nodes, we compare the performance of the adaptive greedy strategy w.r.t. three alternative heuristics to identify influential seed 7

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nodes. These heuristics adaptively choose: (i) the node with highest betweenness centrality; (ii) the node with highest degree; and (iii) a random node among inactive nodes. Comparisons have been also made with the non-adaptive standard greedy strategy [11]. The specific policy chooses the k seed nodes in advance and activate each one of them sequentially (one at each time step). In both the proposed policy and the vanilla non-adaptive greedy one, the expected marginal gains were approximated via MC sampling (1.000 simulations) (see Appendix G for a detailed description). Figure 2 illustrates the empirical means of the expected utility f˜ as well as the ±1 standard deviation intervals over 100 runs in both the modified and standard IC models. As expected, the adaptive greedy strategy outperforms the other strategies in all cases. Actually, in the case of the modified IC model (Fig. 2(a)) the adaptivity is more profitable as we gradually gain more knowledge about the ground truth influence graph. This happens due to the fact that each node in the modified IC model has more than one chances to influence its neighbors. Therefore, the ground truth influence graph is revealed in a much faster rate than in the case of the standard IC model. Actually, the performance of the non-adaptive greedy strategy is worse even when it is compared with that of the simple adaptive degree or centrality strategies under the modified myopic feedback. Moreover, the improvement on the performance that the adaptive greedy strategy achieves compared to that of the non-adaptive greedy depends on the network’s properties. This becomes apparent in the case where we consider the standard IC model in a dense network, like facebook (Fig. 2(b)). In that case, the performance of the non-adaptive strategy is quite close to that of the adaptive one. Summing up, it should be noticed that our analysis validates our initial claim that the performance of the proposed myopic adaptive greedy policy will be at least as good as that of the non-adaptive greedy policy. 8

7 Conclusions We proposed the myopic adaptive greedy strategy for the adaptive influence maximization task. It is the first time where a policy like this one offers provable approximation guarantees under both the standard IC model and a variant of it with myopic feedback. This is achieved by proposing an alternative utility function that returns the cumulative number of active nodes over time, which corresponds to the number of active nodes on a layered graph representation. Nevertheless, several interesting directions for future work remain. So far, we assumed that the influence graph was fully known, which may be a strong assumption in practice. We intend to relax this assumption, studying alternative problems where influence probabilities must be adaptively learned in order to maximize influence. Last but not least, we plan to examine an even more realistic version of the modified IC model. It that case, the influence probabilities between an active node and its inactive neighbors will be decreased by a predefined factor right after each failure of the first node to influence the rest ones.

References [1] Çigdem Aslay, Nicola Barbieri, Francesco Bonchi, and Ricardo A. Baeza-Yates. Online topicaware influence maximization queries. In EDBT, 2014. [2] Christian Borgs, Michael Brautbar, Jennifer Chayes, and Brendan Lucier. Maximizing social influence in nearly optimal time. In SODA, 2014. [3] Wei Chen, Yajun Wang, and Siyu Yang. Efficient influence maximization in social networks. In KDD, 2009. [4] Pedro Domingos and Matt Richardson. Mining the network value of customers. In KDD, 2001. [5] Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, JACM, 45(4):634–652, 1998. [6] Daniel Golovin and Andreas Krause. Adaptive submodularity: Theory and applications in active learning and stochastic optimization. JAIR, 42(1):427–486, 2011. [7] Alkis Gotovos, Amin Karbasi, and Andreas Krause. Non-monotone adaptive submodular maximization. In IJCAI, 2015. [8] Amit Goyal, Francesco Bonchi, and Laks V. S. Lakshmanan. A data-based approach to social influence maximization. Proc. VLDB Endow., 5(1):73–84, 2011. [9] Xinran He and David Kempe. Robust influence maximization. In KDD, 2016. [10] Kyomin Jung, Wooram Heo, and Wei Chen. Irie: Scalable and robust influence maximization in social networks. In ICDM, 2012. [11] David Kempe, Jon M. Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In KDD, 2003. [12] Jinha Kim, Seung-Keol Kim, and Hwanjo Yu. Scalable and parallelizable processing of influence maximization for large-scale social networks? In ICDM, pages 266–277, 2013. [13] Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, June 2014. [14] Jure Leskovec, Andreas Krause, Carlos Guestrin, Christos Faloutsos, Jeanne VanBriesen, and Natalie Glance. Cost-effective outbreak detection in networks. In KDD, 2007. [15] Wei Lu, Francesco Bonchi, Amit Goyal, and Laks V.S. Lakshmanan. The bang for the buck: Fair competitive viral marketing from the host perspective. In KDD, 2013. [16] G. L. Nemhauser and L. A. Wolsey. Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res., 3(3):177–188, 1978. [17] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functions—i. Mathematical Programming, 14(1):265–294, 1978. 9

[18] Shaojie Tang. No time to observe: Adaptive influence maximization with partial feedback. CoRR, 2016. [19] Shaojie Tang and Jing Yuan. Going viral: Optimizing discount allocation in social networks for influence maximization. CoRR, abs/1606.07916, 2016. [20] Sharan Vaswani and Laks V. S. Lakshmanan. Adaptive influence maximization in social networks: Why commit when you can adapt? CoRR, abs/1604.08171, 2016. [21] Jan Vondrak. Submodularity and curvature : The optimal algorithm. RIMS Kokyuroku Bessatsu, B23, 2010. [22] Yu Wang, Gao Cong, Guojie Song, and Kunqing Xie. Community-based greedy algorithm for mining top-k influential nodes in mobile social networks. In KDD, 2010.

A. Adaptive setting leads to higher spreads Claim 1. The adaptive setting leads to higher spreads compared to the non-adaptive one, since we gradually gain more knowledge about the ground truth influence graph. To defend our claim, we give a simple example. Consider the network shown in Figure. 3(a) with influence probabilities pvu = 0.9 and pvw = 0.1. Let k = 2 (seed nodes - our budget). The nonadaptive greedy algorithm will select as seed nodes the v (t = 1) and w (t = 2). Nevertheless, based on the true world (see Fig. 3(b)), we observe that nodes v and w are active at time t = 2. Hence, we will infer that the edges (v, w) and (v, u) are live and dead, respectively. Therefore, the non-adaptive strategy will lead to a reward equal to 2, as only nodes v and w will be activated finally, but not u. Roughly speaking, we are going to make an offer at an already influenced user. On the other hand, the adaptive myopic strategy will first choose the node v and then will observe the status of the outgoing edges of node v. In other words, he will observe that v managed to influence w but not u. Hence, he will choose node u as the second seed node, since it is the only one which is not activated at this point. This returns a reward equal to 3, which is higher than that returned by the non-adaptive policy, since all nodes are finally activated. v

0.9

u

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0

u

1 w

w

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(b)

Figure 3: (a) Graph network (b) True world at time t = 2

B. Proof of Theorem 1 Theorem 1. The adaptive greedy policy π g obtains at least (1 − 1/e) of the value of the best policy for the adaptive influence maximization problem under the modified IC model with myopic feedback and f˜ as utility function. In other words, if f˜avg (π g ) , EΦ [f˜G (E(π, Φ), Φ)], we get that: f˜avg (π g ) ≥ (1 − 1/e)f˜avg (π ∗ ). Proof. Actually, we want to prove that the utility function f˜G is adaptive monotonic and adaptive submodular w.r.t. p(φ). This will allow us to apply the result of Golovin and Krause [6] in order to complete the proof. Adaptive monotonicity can be easily verified, since f˜G (·, φ) is itself monotonic for each φ. 10

At this point, let us choose two subrealizations ψ, ψ ′ with ψ ⊆ ψ ′ and v ∈ / dom(ψ ′ ). To prove that the proposed utility function f˜G is adaptive submodularity in a rigorous way, i.e. to prove that ∆f˜(v|ψ) ≥ ∆f˜(v|ψ ′ ), we define a coupled distribution µ over pairs of realizations φ ∼ ψ and φ′ ∼ ψ ′ similar to Golovin and Krause [6]. Let X , {Xvu : (v, u) ∈ E} denote the Bernoulli r.v. of the model, and let us explicit the realizations’ dependency over these r.v.: we indeed have φ(E) = φ(E(X)) that we denote by φ(X) for conciseness. Using this formalism, we define µ implicitly in terms of a joint distribution µ ˆ on X × X′ where φ = φ(X) and φ′ = φ′ (X′ ) (the realizations generated from these sets of random edge statuses), so µ(φ(X), φ′ (X′ )) = µ ˆ(X, X′ ). ′ Due to the fact that ψ ⊆ ψ , we derive that all edges observed by ψ have the same state - 0 or 1 - in φ and φ′ , since φ ∼ ψ and φ′ ∼ ψ ′ . All (X, X′ ) ∈ support(ˆ µ) have these properties. Similarly to Golovin and Krause [6], we also define µ ˆ s.t. the status of all edges not observed by ψ ′ and ψ ′ are the same in X and X′ , i.e. Xvu = Xvu for all edges (v, u), or µ ˆ(X, X′ ) = 0. Here, we have randomness for all edges unobserved by ψ. Selecting them independently with E[Xvu ] = pvu according to p(φ), we end up with Y 1−Xvu vu , pX µ ˆ(X, X′ ) = vu (1 − pvu ) (v,u) unobserved by ψ

for (X, X′ ) satisfying previous constraints, and µ ˆ(X, X′ ) = 0 otherwise (see [6] for details). Then, let us remark that δφ (v|ψ) ≥ δφ′ (v|ψ ′ ) for all (φ, φ′ ) ∈ support(µ). There are three different situations: i) The first one corresponds to the case where T (ψ) = t < T (ψ ′ ) and ψ ⊆ ψ ′ : Let us consider w.l.o.g. that T (ψ ′ ) = t + 1. Node vt is activated in G L , after observing ψ. Let ψ+ denotes the partial realization combining ψ and observing one more step of the process - from layer t to layer t + 1 - without adding any seed node, w.r.t. φ. Also, let A denotes the set of activated nodes of layer t + 1, observed by ψ+ and that would not have been activated if vt was not selected as seed node, except vt+1 . Therefore, we get that: δφ (v|ψ) =(1) 1 + δφ (v ∪ A|ψ+ ) ≥ δφ (v ∪ A|ψ+ ) =(2) δφ′ (v ∪ A|ψ+ ) ≥(3) δφ′ (v|ψ+ ) ≥(4) δφ′ (v|ψ ′ ). (1) This equality comes from the fact that G L is feedforward, therefore activating v brings a reward of 1 at time t, plus the reward from the future; (2) It is verified since (φ, φ′ ) ∈ support(µ) and quantities depend only on layers from t + 1 to T (Lemma 2), i.e. on edges unobserved by ψ+ and ψ ′ ; (3) It comes due to the monotonicity of the set function δφ′ (·|ψ+ ); (4) It is a direct application of Lemma 3 since ψ+ ⊆ ψ ′ is always true under the given realizations. This result is still valid for T (ψ ′ ) = t + x, ∀x > 1. Indeed, one can easily verify that we still get that δφ (v|ψ) ≥ δφ (v ∪ A|ψ+ ) with A to be defined as before and ψ+ to denote the partial realization combining ψ and observing x more step of the process without adding any seed node, w.r.t. φ. Then, the remaining steps of the proof are identical with those of the case where x = 1. ii) If T (ψ) = T (ψ ′ ) = t, a direct application of Lemmas 2 and 3 leads us to δφ (v|ψ) ≥ δφ′ (v|ψ ′ ). iii) The third case where T (ψ) > T (ψ ′ ) is impossible, since it would violate the ψ ⊆ ψ ′ hypothesis. Having proved that δφ (v|ψ) ≥ δφ′ (v|ψ ′ ) for all (φ, φ′ ) ∈ support(µ), we use the trick of [6] to conclude our proof. Since by definition δφ′ (v|ψ ′ ) = f˜G (dom(ψ ′ )∪{vT (ψ′ ) }, φ′ )− f˜G (dom(ψ ′ ), φ′ ), we get that: X X µ(φ, φ′ )δφ (v|ψ) = ∆f˜(v|φ) µ(φ, φ′ )δφ′ (v|ψ ′ ) ≤ ∆f˜(v|φ′ ) = (φ,φ′ )

(φ,φ′ )

11

proving that f˜G is adaptive submodular.

C. Proof of Theorem 2 Theorem 2. The adaptive greedy policy π g obtains at least (1 − 1/e) of the value of the best policy for the adaptive influence maximization problem, in the standard IC model with myopic feedback and f˜ as utility function. Therefore, it holds that: f˜avg (π g ) ≥ (1 − 1/e)f˜avg (π ∗ ). Proof. Adaptive monotoniticy follows from the fact that f˜G (·, φ) is monotone for each φ. To prove adaptive submodularity, we once again use the methodology of [6]. The key point of the proof of Theorem 1 was to show that δφ (v|ψ) ≥ δφ′ (v|ψ ′ ) for all (φ, φ′ ) ∈ support(µ). This result is still verified here, since it relies on Lemmas 2 and 3 (that are still valid), on the monotonicity of δφ′ (·|ψ+ ) and on the fact that the layered graph is a feedforward network. This result is true for all (φ, φ′ ) ∈ support(µ), therefore it is also true for the subset of all (φ, φ′ ) ∈ support(µ) with regards to the constraint that the edges associated to the same pair of nodes must have the same status. By summing up all these (φ, φ′ ), we finally obtain that: X X µ(φ, φ′ )δφ (v|ψ) = ∆f˜(v|φ) µ(φ, φ′ )δφ′ (v|ψ ′ ) ≤ ∆f˜(v|φ′ ) = (φ,φ′ )

(φ,φ′ )

that proves that the adaptive submodularity property holds.

D. Proof of Lemma 4 Lemma 4. Forcing active nodes to remain active throughout the process constitutes a necessary condition to verify the adaptive submodularity property of: i) f˜G in the modified IC model with myopic feedback; ii) f˜G in the standard IC model with myopic feedback; iii) fG in the standard IC model with full-adoption feedback. Proof. i) In the case of the modified myopic feedback model, consider the layered graph of Fig. 4(a) that consists of six random edges. There are 26 = 64 ground truth graphs, each of them being obtained with probability 1/64 since edges are independent Bernoulli r.v., B(1/2). We want to add v to the set of seed nodes. Now, consider ψ where we only know that u is activated at t = 1 (T (ψ) = 1), and ψ ′ where we also observed that (u1 , u2 ) and (u1 , v2 ) are dead edges (T (ψ ′ ) = 2). Clearly, ψ ⊆ ψ ′ . A simple decomposition of all possible ground truth graphs leads to ∆(v|ψ) = 10 48 6 + 1 × 64 + 0 × 64 ] = 86 1 + [2 × 64 64 : a reward of 1 for activating v1 and possibly a marginal gain of adding v2 and v3 (0 in 48 ground truth realizations, 1 in 10 of them, 2 in 6 of them). We also get that ∆(v|ψ ′ ) = 1 + 12 = 1.5: v2 is active (seed) while v3 is active with probability 1/2. Therefore, ∆(v|ψ ′ ) > ∆(v|ψ) that violates the adaptive submodularity property. ii) We consider again the same scenario as previously, with ψ ⊆ ψ ′ . Similarly, a simple decomposition of all possible ground truth graphs gives ∆(v|ψ) = 1 + 14 × [1 + 12 ] = 11 8 . On the other hand, we get that ∆(v|ψ ′ ) = 1 + 12 = 1.5 as v2 is a seed node and v3 remains active with probability 1/2. Once again, we conclude that the adaptive submodularity property does not hold since ∆(v|ψ ′ ) > ∆(v|ψ). iii) Let us consider the graph of Fig. 4(b), where active nodes have a probability of 1/2 to be deactivated at each time step. Recall that our utility function is now the number of activated nodes at the end of the process (standard IC utility function), and let T = 2. Suppose also that we want to choose v as seed node under the next two scenarios. At the first one we are at time step t = 1, so ψ = ∅. The second scenario assumes that we are at time step t = 2 having chosen node u at t = 1, so ψ ′ only contains the information that u is activated. Therefore, we get that ∆(v|ψ) < 3, since v, w and z are active at t = 1, but they have a non-null probability to deactivate at t = 2. On the other hand, ∆(v|ψ ′ ) = 3 as the process ends right after nodes v, w and z are activated via choosing v as seed node. Since ψ ⊆ ψ ′ and ∆(v|ψ ′ ) = 3 > ∆(v|ψ), adaptive submodularity is once again violated.

12

u1

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Figure 4: Counterexamples to verify that the spread is not adaptive submodular when the active nodes are allowed to be deactivated.

E. Proof of Theorem 3 Theorem 3. The adaptive greedy policy π g obtains at least (1 − 1/e) of the value of the best policy for the adaptive influence maximization problem, in the standard IC model with myopic feedback and f˜ as utility function. Therefore, it holds that: f˜avg (π g ) ≥ (1 − 1/e)f˜avg (π ∗ ). Proof. Adaptive monotoniticy stems from the fact that f˜G (·, φ) is monotone for each φ. To prove adaptive submodularity, we once again use the methodology of [6]. The key point of the proof of Theorem 1 was to show that δφ (v|ψ) ≥ δφ′ (v|ψ ′ ) for all (φ, φ′ ) ∈ support(µ). This result is still verified here, since it relies on Lemmas 2 and 3 (that are still valid), on the monotonicity of δφ′ (·|ψ+ ) and on the fact that the layered graph is a feedforward network. This result is true for all (φ, φ′ ) ∈ support(µ), therefore it is also true for the subset of all (φ, φ′ ) ∈ support(µ) with regards to the constraint that the edges associated to the same pair of nodes must have the same status. By summing up all these (φ, φ′ ), we finally obtain that: X X µ(φ, φ′ )δφ (v|ψ) = ∆(v|φ), µ(φ, φ′ )δφ′ (v|ψ ′ ) ≤ ∆(v|φ′ ) = (φ,φ′ )

(φ,φ′ )

which proves that the adaptive submodularity property holds.

F. Networks Description Experiments have been conducted on three networks obtained from the Stanford’s SNAP database [13]. The first one is a small graph thar corresponds to an ego network from Twitter. Actually, the dataset is a subset - a “circle” - from the list of social circles from Twitter, crawled from public sources. The graph consists of 228 nodes and 9.938 edges. The second one is a social network extracted from Facebook. Data were anonymously collected from survey participants using the Facebook app. The graph is undirected, and has 4.039 nodes and 88.234 edges. The third graph is that of the Arxiv General Relativity and Quantum Cosmology collaboration network. In this graph, we have an undirected edge from i to j, if author i co-authored a ArXiv paper with author j (between 1993 and 2003). This graph has 5.242 nodes and 28.980 edges. Table 1 summarizes a number of usefuel about the aforementioned networks. Mean degree is the mean number of edges exiting nodes. A.P.L. stands for Average Path Length, which is the average number of nodes in the shortest path between two nodes of the graph. Moreover, the diameter of a graph is the length of the longest shortest path between two nodes.

G. Adaptive Greedy Myopic Policy and Alternative Heuristics The performance of the proposed myopic adaptive greedy strategy has been compared with that of the next four alternative adaptive heuristics. 13

Network Twitter ArXiv GR-QC Facebook

Nodes 228 5.242 4.039

Edges 9.938 28.980 88.234

Mean degree 43.6 11.1 43.7

Max degree 125 162 1.045

A.P.L. 2.1 6.1 3.7

Diameter 6 17 8

Type Directed Undirected Undirected

Table 1: Statistics of the real-world networks used through the experimental analysis.

• Degree: The node with thehighest degree (i.e., the node with the highest number of outgoing edges) has been chosen as a seed node at each time; • Centrality: The node(s) with the highest centrality measure among the inactive nodes has been selected as a seed node at each time step. In our analysis, we adopted the betweenness centrality measure, which is equal to the number of shortest paths from all nodes to all others that pass through a node; • Random: Selects randomly an inactive node as a seed node at each time step; • Non-adaptive: The k seed nodes have been selected in advance by using the standard greedy algorithm [11]. Then, we activate each of them sequentially, at each time step, starting from the one with the maximum expected marginal gain. Finally, it should be stressed that in the case of the modified myopic feedback model, we chose to implement the improved accelerated version of the adaptive greedy strategy [6], for computational reasons. The algorithm is based on so-called lazy evaluations, i.e. on a clever use of the adaptive submodularity inequality to significantly reduce running times in practice by diminishing the number of nodes on which Monte Carlo simulations should be performed. The pseudocode and the justification of this acccelerated adaptive greedy algorithm is reported in [6].

14