Original: L EARNING G RAPH R EPRESENTATIONS FOR I NFLUENCE M AXIMIZATION
Revised: L EARNING G RAPH R EPRESENTATIONS FOR I NFLUENCE M AXIMIZATION

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: As the ﬁeld of machine learning for combinatorial optimization advances, traditional problems are resurfaced and readdressed through this new perspective.
Revised: As the ﬁeld of machine learning for combinatorial optimization advances, traditional problems are resurfaced and readdressed through this new perspective.

Original: The overwhelming majority of the literature focuses on small graph problems, while several real-world problems are devoted to large graphs.
Revised: The overwhelming majority of the literature focuses on small graph problems, while several real-world problems are devoted to large graphs.

Original: Here, we focus on two such problems: inﬂuence estimation, a #P-hard counting problem, and inﬂuence maximization, an NP-hard problem.
Revised: Here, we focus on two such problems: inﬂuence estimation, a #P-hard counting problem, and inﬂuence maximization, an NP-hard problem.

Original: We develop G LIE , a Graph Neural Network (GNN) that inherently parameterizes an upper bound of inﬂuence estimation and train it on small simulated graphs.
Revised: We develop G LIE , a Graph Neural Network (GNN) that inherently parameterizes an upper bound of inﬂuence estimation and train it on small simulated graphs.

Original: Experiments show that G LIE provides accurate inﬂuence estimation for real graphs up to 10 times larger than the train set.
Revised: Experiments show that G LIE provides accurate inﬂuence estimation for real graphs up to 10 times larger than the train set.

Original: More importantly, it can be used for inﬂuence maximization on considerably larger graphs, as the predictions ranking is not effected by the drop of accuracy.
Revised: More importantly, it can be used for inﬂuence maximization on considerably larger graphs, as the predictions ranking is not effected by the drop of accuracy.

Original: We develop a version of Cost Effective Lazy Forward optimization with G LIE instead of simulated inﬂuence estimation, surpassing the benchmark for inﬂuence maximization, although with a computational overhead.
Revised: We develop a version of Cost Effective Lazy Forward optimization with G LIE instead of simulated inﬂuence estimation, surpassing the benchmark for inﬂuence maximization, although with a computational overhead.

Original: To balance the time complexity and quality of inﬂuence, we propose two different approaches.
Revised: To balance the time complexity and quality of inﬂuence, we propose two different approaches.

Original: The ﬁrst is a Q-network that learns to choose seeds sequentially using G LIE ’s predictions.
Revised: The ﬁrst is a Q-network that learns to choose seeds sequentially using G LIE ’s predictions.

Original: The second deﬁnes a provably submodular function based on G LIE ’s representations to rank nodes fast while building the seed set.
Revised: The second deﬁnes a provably submodular function based on G LIE ’s representations to rank nodes fast while building the seed set.

Original: The latter provides the best combination of time efﬁciency and inﬂuence spread, outperforming SOTA benchmarks.
Revised: The latter provides the best combination of time efﬁciency and inﬂuence spread, outperforming SOTA benchmarks.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Several real-world problems can be cast as a combinatorial optimization problem over a graph.
Revised: Several real-world problems can be cast as a combinatorial optimization problem over a graph.

Original: From distributing packages Mathew et al. (2015) to improving the general health Wilder et al. and vehicles’ management Touati-Moungla & Jost (2012), optimization on graphs lies in the core of many real-world applications that are vital to our way of living.
Revised: From distributing packages (Mathew et al., 2015) to improving the general health (Wilder et al., 2018) and vehicles’ management (Touati-Moungla & Jost, 2012), optimization on graphs lies in the core of many real-world applications that are vital to our way of living.

Original: Unfortunately, the majority of these problems are NP-hard, and hence we can only approximate their solution in a satisfactory time limit that matches the real world requirements.
Revised: Unfortunately, the majority of these problems are NP-hard, and hence we can only approximate their solution in a satisfactory time limit that matches the real world requirements.

Original: Recent machine learning methods have emerged as a promising solution to develop heuristic methods that provide fast and accurate approximations Bengio et al.
Revised: Recent machine learning methods have emerged as a promising solution to develop heuristic methods that provide fast and accurate approximations (Bengio et al., 2020).

Original: The general idea is to train a supervised or unsupervised learning model to infer the solution given an unseen graph and the problem constraints.
Revised: The general idea is to train a supervised or unsupervised learning model to infer the solution given an unseen graph and the problem constraints.

Original: The models tend to consist of Graph Neural Networks (GNNs) to encode the graph and the nodes, Q-learning Watkins & Dayan (1992); Sutton & Barto (2018) to produce sequential predictions, or a combination of both.
Revised: The models tend to consist of Graph Neural Networks (GNNs) to encode the graph and the nodes, Q-learning (Watkins & Dayan, 1992; Sutton & Barto, 2018) to produce sequential predictions, or a combination of both.

Original: The practical motivation behind learning to solve combinatorial optimization problems, is that inference time is faster than running an exact combinatorial solver Joshi et al.
Revised: The practical motivation behind learning to solve combinatorial optimization problems, is that inference time is faster than running an exact combinatorial solver (Joshi et al., 2019).

Original: That said, specialized combinatorial algorithms like C ONCORDE for the Traveling Salesman Problem (TSP) or G UROBI in general, cannot be surpassed yet Kool et al.
Revised: That said, specialized combinatorial algorithms like C ONCORDE for the Traveling Salesman Problem (TSP) or G UROBI in general, cannot be surpassed yet (Kool et al., 2018).

Original: Though many such methods have been proposed for a plethora of problems, inﬂuence maximization (IM) has not been addressed yet extensively.
Revised: Though many such methods have been proposed for a plethora of problems, inﬂuence maximization (IM) has not been addressed yet extensively.

Original: IM addresses the problem of ﬁnding the set of nodes in a network that would maximize the number of nodes reached by starting a diffusion from them Kempe et al.
Revised: IM addresses the problem of ﬁnding the set of nodes in a network that would maximize the number of nodes reached by starting a diffusion from them (Kempe et al., 2003).

Original: The problem is proved to be NP-hard, from a reduction to the set-cover problem.
Revised: The problem is proved to be NP-hard, from a reduction to the set-cover problem.

Original: Moreover, the inﬂuence estimation (IE) problem that is embedded in IM, i.e., estimating the number of nodes inﬂuenced by a given seed set, is #P-hard as it is analogous to counting s-t connectedness and would require 2 | E | possible combinations to compute exactly, where | E | is the number of network edges Wang et al.
Revised: Moreover, the inﬂuence estimation (IE) problem that is embedded in IM, i.e., estimating the number of nodes inﬂuenced by a given seed set, is #P-hard as it is analogous to counting s-t connectedness and would require 2 | E | possible combinations to compute exactly, where | E | is the number of network edges (Wang et al., 2012).

Original: Typically, inﬂuence estimation is approximated using repetitive MonteCarlo (MC) simulations of the independent cascade (IC) diffusion model.
Revised: Typically, inﬂuence estimation is approximated using repetitive Monte-Carlo (MC) simulations of the independent cascade (IC) diffusion model.

Original: In general, the seed set is built greedily, taking advantage of the submodularity of the inﬂuence function which guarantees an at least (1 − 1 e ) approximation to the optimal.
Revised: In general, the seed set is built greedily, taking advantage of the submodularity of the inﬂuence function which guarantees an at least (1 − 1 e ) approximation to the optimal.

Original: Although the latter lacks of efﬁciency as one still has to estimate inﬂuence for every candidate seed in every step of building the seed sets.
Revised: Although the latter lacks of efﬁciency as one still has to estimate inﬂuence for every candidate seed in every step of building the seed sets.

Original: Hence, several scalable algorithms Borgs et al.
Revised: Hence, several scalable algorithms Borgs et al.

Original: ; Tang et al.
Revised: ; Tang et al.

Original: and heuristics Chen et al.
Revised: and heuristics Chen et al.

Original:  Jung et al.
Revised: ; Jung et al.

Original: (2012) were developed capitalizing on sketches or the structure of the graph to produce more efﬁcient solutions.
Revised: (2012) were developed capitalizing on sketches or the structure of the graph to produce more efﬁcient solutions.

Original: IM can be applied on a plethora of real-world tasks, such as epidemic containment Wilder et al. (2017), diminishing fake news Budak et al. (2011), and running viral marketing campaigns Domingos & Richardson (2001).
Revised: IM can be applied on a plethora of real-world tasks, such as epidemic containment (Wilder et al., 2017), diminishing fake news (Budak et al., 2011), and running viral marketing campaigns (Domingos & Richardson, 2001).

Original: We address IM through the perspective of deep learning for combinatorial optimization.
Revised: We address IM using neural networks, to capitalize on the aforementioned advantages as well as their ability to easily incorporate contextual information such as user proﬁles and topics (Tian et al., 2020), a task that remains unsolvable for non-specialized IM algorithms and heuristics.

Original: More speciﬁcally, we propose G LIE , a GNN that provides efﬁcient IE for a given seed set and a graph with inﬂuence probabilities.
Revised: We propose G LIE , a GNN that provides efﬁcient IE for a given seed set and a graph with inﬂuence probabilities.

Original: It can be used as a standalone inﬂuence predictor with competitive results for graphs up to 10 times larger than the train set.
Revised: It can be used as a standalone inﬂuence predictor with competitive results for graphs up to 10 times larger than the train set.

Original: Moreover, we leverage G LIE for IM, combining it with C ELF Leskovec et al. (2007), that typically does not scale beyond networks with thousands of edges.
Revised: Moreover, we leverage G LIE for IM, combining it with C ELF (Leskovec et al., 2007), that typically does not scale beyond networks with thousands of edges.

Original: The proposed method runs in networks with millions of edges in seconds, and exhibits better inﬂuence spread than a state-of-the-art algorithm and previous GNN-RL methods for IM.
Revised: The proposed method runs in networks with millions of edges in seconds, and exhibits better inﬂuence spread than a state-of-the-art algorithm and previous GNN-RL methods for IM.

Original: In addition, we develop G RIM  a Q-learning architecture that utilizes G LIE ’s representations and predictions to obtain seeds sequentially, while minimizing the number of inﬂuence estimations throughout steps.
Revised: In addition, we develop G RIM , a Q-learning architecture that utilizes G LIE ’s representations and predictions to obtain seeds sequentially, while minimizing the number of inﬂuence estimations throughout steps.

Original: Finally, we propose  a heuristic based on G LIE ’s representations that represents the number of neighbors predicted to be uninﬂuenced and acts as an approximation to the marginal gain, P UN .
Revised: Finally, we propose P UN , a method that uses G LIE ’s representations to compute the number of neighbors predicted to be uninﬂuenced and uses it as an approximation to the marginal gain.

Original: We prove that it is submodular and  hence can be optimized greedily.
Revised: We prove P UN ’s inﬂuence spread is submodular and monotone, and hence can be optimized greedily with a guarantee, in contrast to prior learning-based methods.

Original: The experiments indicate  P UN provides the best balance between inﬂuence quality and computational efﬁciency.
Revised: The experiments indicate that P UN provides the best balance between inﬂuence quality and computational efﬁciency.

Original: The paper is organized as follows.
Revised: The paper is organized as follows.

Original: Section 2 presents an overview of relevant approaches and clariﬁes the advantage of the our models.
Revised: Section 2 presents an overview of relevant approaches and clariﬁes the advantage of the proposed models.

Original: Section 3 describes thefour proposed methods, starting with IE and advancing progressively towards faster methods for IM.
Revised: Section 3 describes the proposed methods, starting with IE and advancing progressively towards faster methods for IM.

Original: Section 4 exhibits and interprets the experimental results for IE and IM.
Revised: Section 4 exhibits and interprets the experimental results for IE and IM.

Original: Finally Section 5 summarizes the contribution and presents future steps.
Revised: Finally, Section 5 summarizes the contribution and presents future steps.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: The ﬁrst approach to solving combinatorial optimization (CO) using neural networks was based on attention-based NNs for discrete structures, P OINTER N ETS Vinyals et al. (2015), followed by an architecture that combines P OINTER N ETS with an actor-critic training to ﬁnd the best route for TSP Bello et al.
Revised: The ﬁrst approach to solving combinatorial optimization (CO) using neural networks was based on attention-based NNs for discrete structures, P OINTER N ETS (Vinyals et al., 2015), followed by an architecture that combines P OINTER N ETS with an actor-critic training to ﬁnd the best route for TSP (Bello et al., 2016).

Original: The ﬁrst architecture that utilized graph-based learning was S2 N -D QN Dai et al. (2017), which used S TRUCT 2V EC to encode the states of the nodes and the graph, and uses them to train a Q-learning modelSutton & Barto (2018) that chooses the right node to add in a solution given the current state.
Revised: The ﬁrst architecture that utilized graph-based learning was S2 N -D QN (Dai et al., 2017), which used S TRUCT 2V EC to encode the states of the nodes and the graph, and uses them to train a Q-learning model that chooses the right node to add in a solution given the current state.

Original: Based on S2 V -D QN , a D QN Mnih et al. (2015) for the network dismantling problem was recently proposed Fan et al.
Revised: Based on S2 V -D QN , a D QN (Mnih et al., 2015) for the network dismantling problem was recently proposed (Fan et al., 2020).

Original: The model, named F INDER , uses a deep Q-learning architecture along with three G RAPH S AGE layers.
Revised: The model, named F INDER , uses a deep Q-learning architecture where the representations are derived by three G RAPH S AGE layers.

Original: The reward is based on size of the giant connected component size, i.e., every new node (seed) chosen, aims to dismantle the network as much as possible.
Revised: The reward is based on size of the giant connected component size, i.e., every new node (seed) chosen, aims to dismantle the network as much as possible.

Original: The main advantage of F INDER is that it is trained on small synthetic data while it can extrapolate to relatively large graphs.
Revised: Some of the main advantages of F INDER is that it is trained on small synthetic data, which are easy to make, and can extrapolate to relatively large graphs.

Original: On the other hand, one of its core disadvantages is that it can not work with directed graphs and weighted edges. Moreover it addresses a proxy problem Morone & Makse (2015) rather than traditional IM.
Revised: On the other hand, one of the core disadvantages is that it can not work with directed graphs and weighted edges and that it addresses a proxy problem (Morone & Makse, 2015) rather than traditional inﬂuence maximization.

Original: Another recent supervised deep learning approach on IM, G COMB Manchanda et al. (2020), leverages a probabilistic greedy method to produce scores on graphs of the training set and uses them to teach a GNN to predict such scores.
Revised: A recent supervised deep learning approach on inﬂuence maximization, G COMB (Manchanda et al., 2020), utilizes a probabilistic greedy method to produce scores on graphs of the training set and uses them to learn a GNN that predicts such scores.

Original: A Q-network receives the score along with the maximum score from all nodes as graph state, and an approximate calculation of the seed’s neighborhood correlation with the seed set, to predict the Q-value of a node.
Revised: Once these scores are computed, a Q-network receives the score along with the maximum score between all nodes as graph state, and an approximate calculation of the seed’s neighborhood correlation with the seed set, to predict the Q-value (Sutton &

Original: Though very scalable and comparable to state-of-the-art in inﬂuence, G COMB has to be trained on a large random subset of the test graph.
Revised: Barto, 2018) of a node. The authors propose as well a noise estimation method to reduce the number of candidate seeds based on the degree. This approach, though very scalable and comparable to state-of-the-art in accuracy, has to be trained on a large random subset of the graph ( 30% of it) and tested on the rest.

Original: This means the model  has to be retrained for every new test graph, which imposes a serious overhead considering the time required for producing the scores using MC ICs and training. To be speciﬁc, it needs to be trained on 30 % of the graph and tested on the rest, a preprocessing that in practice takes hours, as noted in the supplementary material G Manchanda et al.
Revised: This makes the model graph-speciﬁc, i.e., it has to be retrained to perform well on a new large graph. This imposes a serious overhead, considering the time required for training but mostly the time to label these samples using the probabilistic greedy method with traditional inﬂuence estimation.

Original: Hence
Revised: In this paper, we propose an approach that combines the advantages of the aforementioned methods, in that it is only trained on small simulated data once and generalizes to larger graphs, and it addresses the problem of inﬂuence maximization in weighted directed networks. Furthermore, the approach can be broken down to a GNN (G LIE ) and a Q-learning part. The former can act alone as inﬂuence predictor and be competitive with relevant methods, such as D MP (Lokhov & Saad, 2019) for graphs up to one scale larger than the train set, as we will present in Section 4.1. G LIE is used to propose two inﬂuence maximization methods: C ELF with G LIE as inﬂuence estimator, and a Q-network that learns how to choose seeds using the G LIE ’s estimations and hidden representations.

Original:  G COMB is optimized for scalability but not for time efﬁciency, and is not comparable to the current works which run in seconds.
Revised: Both proposed inﬂuence maximization methods are compared in Section 4.2 with a state-of-the-art IM algorithm I MM (Tang et al., 2015), with F INDER and indirectly through a common dataset and benchmark with G COMB .

Original: We note here that the majority of the relevant literature on DL for CO address very small graphs Vinyals et al. (2015); Kool et al.
Revised: We note here that the majority of the relevant literature on DL for CO address very small graphs (Vinyals et al., 2015; Dai et al., 2017; Kool et al., 2018;

Original: ; Prates et al. (2019) which renders themas well not applicable to our task.
Revised: Prates et al., 2019) which makes them not applicable to our task.

Original: More scalable, unsupervised methods Karalias & Loukas (2020) are tailored to speciﬁc problems and is non-trivial to adjust them to our problem. The current work combines the advantages of the aforementioned methods, in that it is only trained on small simulated data once and generalizes to larger graphs, while it addresses the problem of IM in weighted directed networks. The main idea is separating IE and IM, and train a supervised model for IE. Afterwards, we have the freedom to use the trained model to develop various approaches, ranging from a new version of a classical algorithm, a Q-learning approach that resembles the recent literature, and a fast submodular function to rank nodes based on the model’s representations.
Revised: More scalable, unsupervised methods (Karalias & Loukas, 2020) are tailored to speciﬁc problems and is non-trivial to adjust them to our problem, with the exception of (Li et al., 2018) which was compared with G COMB and found signiﬁcantly worse than the SOTA algorithm we compare with (Tang et al., 2015).

Original:  M ETHODOLOGY 3. G RAPH L EARNING FOR I NFLUENCE E STIMATION (G LIE )
Revised:  M ETHODOLOGY 3. G RAPH L EARNING FOR I NFLUENCE E STIMATION (G LIE )

Original: In this section, we introduce G LIE ( G NN L earning I nﬂuence E stimation), which aims to learn how to estimate the inﬂuence of a seed set S over a graph G =
Revised: In this section, we introduce G LIE ( G NN L earning I nﬂuence E stimation), which aims to learn how to estimate the inﬂuence of a seed set S over a graph G =

Original: ( V, E ) .
Revised: ( V, E ) .

Original: Let A ∈ R n × n be the adjacency matrix and X ∈ R n × d be the features of nodes, representing which nodes belong to the seed set byand 0 otherwise:
Revised: Let A ∈ R n × n be the adjacency matrix and X ∈ R n × d be the features of nodes, representing which nodes belong to the seed set byand 0 otherwise:

Original: [Equation]
Revised: [Equation]

Original: For the derivation of the corollary, we set d = 1 .
Revised: For the derivation of the corollary, we set d = 1 .

Original: More dimensions will become meaningful when we parameterize the problem.
Revised: More dimensions will become meaningful when we parameterize the problem.

Original: If we normalize A by each row, we form a row-stochastic transition matrix, as:
Revised: If we normalize A by each row, we form a row-stochastic transition matrix, as:

Original: [Equation]
Revised: [Equation]

Original: where deg ( u ) is the in-degree of node u and N ( u ) is the set of neighbors of u .
Revised: where deg ( u ) is the in-degree of node u and N ( u ) is the set of neighbors of u .

Original: Based on the weighted cascade Kempe et al. (2003), each row u stores the probability of node u being inﬂuenced by each of the other nodes that are connected to it by a directed link v → u .
Revised: Based on the weighted cascade (Kempe et al., 2003), each row u stores the probability of node u being inﬂuenced by each of the other nodes that are connected to it by a directed link v → u .

Original: Note that, in case of directed inﬂuence graphs, A should correspond to the transpose of the adjacency matrix.
Revised: Note that, in case of directed inﬂuence graphs, A should correspond to the transpose of the adjacency matrix.

Original: The inﬂuence probability p ( u | S ) resembles the probability of a node u getting inﬂuenced if its neighbors belong in the seed set, i.e., during the ﬁrst step of the diffusion.
Revised: The inﬂuence probability p ( u | S ) resembles the probability of a node u getting inﬂuenced if its neighbors belong in the seed set, i.e., during the ﬁrst step of the diffusion.

Original: We can use message passing to compute a well-known upper bound ˆ p ( u | S ) of p ( u | S ) for node u :
Revised: We can use message passing to compute a well-known upper bound ˆ p ( u | S ) of p ( u | S ) for node u :

Original: [Equation]
Revised: [Equation]

Original: where the second equality stems from the deﬁnition of the weighted cascade and the inequality from the proof in Zhou et al.
Revised: where the second equality stems from the deﬁnition of the weighted cascade and the inequality from the proof in Zhou et al.

Original:  App. A.
Revised: , App. A.

Original: As the diffusion covers more than one-hop, the derivation requires repeating the multiplication to approximate the total inﬂuence spread.
Revised: As the diffusion covers more than one-hop, the derivation requires repeating the multiplication to approximate the total inﬂuence spread.

Original: To be speciﬁc, computing the inﬂuence probability of nodes that are not adjacent to the seed set requires estimating recursively the probability of their neighbors being inﬂuenced by the seeds.
Revised: To be speciﬁc, computing the inﬂuence probability of nodes that are not adjacent to the seed set requires estimating recursively the probability of their neighbors being inﬂuenced by the seeds.

Original: If we let
Revised: If we let

Original: H 1 =
Revised: H 1 =

Original: A · X , and we assume the new seed set S t to be the nodes inﬂuenced in the step t − 1 , their probabilities are stored in H t , much like a diffusion in discrete time.
Revised: A · X , and we assume the new seed set S t to be the nodes inﬂuenced in the step t − 1 , their probabilities are stored in H t , much like a diffusion in discrete time.

Original: We can then recompute the new inﬂuence probabilities with H t +1 =
Revised: We can then recompute the new inﬂuence probabilities with H t +1 =

Original: A · H t ,
Revised: A · H t ,

Original: Corollary 3.0.1.
Revised: Corollary 3.0.1.

Original: The repeated product H t +1 =
Revised: The repeated product H t +1 =

Original: A · H t computes an upper bound to the real inﬂuence probabilities of each infected node at step t + 1 .
Revised: A · H t computes an upper bound to the real inﬂuence probabilities of each infected node at step t + 1 .

Original: The proof can be found in Appendix A.2.
Revised: The proof can be found in Appendix A.2.

Original: In reality, due to the existence of cycles, two problems arise with this computation.
Revised: In reality, due to the existence of cycles, two problems arise with this computation.

Original: Firstly, if the process is repeated the inﬂuence of the original seeds may increase again, which comes in contrast with the independent cascade model.
Revised: Firstly, if the process is repeated the inﬂuence of the original seeds may increase again, which comes in contrast with the independent cascade model.

Original: This can be controlled by minimizing the repetitions, e.g., four repetitions cause the original seeds to be able to reinfect other nodes in a network with triangles.
Revised: This can be controlled by minimizing the repetitions, e.g., four repetitions cause the original seeds to be able to reinfect other nodes in a network with triangles.

Original: To this end, we leverage up to three neural network layers.
Revised: To this end, we leverage up to three neural network layers.

Original: Another problem due to cycles pertains to the probability of neighbors inﬂuencing each other.
Revised: Another problem due to cycles pertains to the probability of neighbors inﬂuencing each other.

Original: In this case, the product of the complementary probabilities in Eq.
Revised: In this case, the product of the complementary probabilities in Eq.

Original: (3) does not factorize for the non-independent neighbors.
Revised: (3) does not factorize for the non-independent neighbors.

Original: This effect was analyzed extensively in Lokhov & Saad (2019) App. B and proved that the inﬂuence probability computed by p ( u | S ) is itself an upper bound on the real inﬂuence probability for graphs with cycles.
Revised: This effect was analyzed extensively in Lokhov & Saad (2019), App. B, and proved that the inﬂuence probability computed by p ( u | S ) is itself an upper bound on the real inﬂuence probability for graphs with cycles.

Original: Intuitively, the product that represents non-independent probabilities is larger than the product of independent ones.
Revised: Intuitively, the product that represents non-independent probabilities is larger than the product of independent ones.

Original: This renders the real inﬂuence probability, which is complementary to the product, smaller than what we compute.
Revised: This renders the real inﬂuence probability, which is complementary to the product, smaller than what we compute.

Original: We can thus contend that the estimation ˆ p ( u | S ) provides an upper bound on the real inﬂuence probability—and we can use it to compute an upper bound to the real inﬂuence spread of a given seed set i.e., the total number of nodes inﬂuenced by the diffusion.
Revised: We can thus contend that the estimation ˆ p ( u | S ) provides an upper bound on the real inﬂuence probability—and we can use it to compute an upper bound to the real inﬂuence spread of a given seed set i.e., the total number of nodes inﬂuenced by the diffusion.

Original: Since message passing can compute inherently an approximation of inﬂuence estimation, we can parameterize it to learn a function that tightens this approximation based on supervision.
Revised: Since message passing can compute inherently an approximation of inﬂuence estimation, we can parameterize it to learn a function that tightens this approximation based on supervision.

Original: In our neural network architecture, each layer consists of a GNN with a batchnorm and dropout omitted here, and starting from H 

Revised: In our neural network architecture, each layer consists of a GNN with a batchnorm and dropout omitted here, and starting from H

Original: = X ∈ R n × d we have:
Revised: = X ∈ R n × d we have:

Original: [Equation]
Revised: [Equation]

Original: The readout function that summarizes the graph representation based on all nodes’ representations is a summation over all the ﬁnal representations with skip connections:
Revised: The readout function that summarizes the graph representation based on all nodes’ representations is a summation over all the ﬁnal representations with skip connections:

Original: [Equation]
Revised: [Equation]

Original: This representation captures the probability of all nodes being active throughout each layer.
Revised: This representation captures the probability of all nodes being active throughout each layer.

Original: The output that represents the predicted inﬂuence spread is derived by:
Revised: The output that represents the predicted inﬂuence spread is derived by:

Original: [Equation]
Revised: [Equation]

Original: 3. C OST E FFECTIVE L AZY F ORWARD WITH G LIE (C ELF -G LIE )
Revised: 3. C OST E FFECTIVE L AZY F ORWARD WITH G LIE (C ELF -G LIE )

Original: The original Cost Effective Lazy Forward Leskovec et al. (2007) is an acceleration to the original greedy algorithm that is based on the constraint that a seed’s spread will never get bigger in subsequent steps.
Revised: The original Cost Effective Lazy Forward (Leskovec et al., 2007) is an acceleration to the original greedy algorithm that is based on the constraint that a seed’s spread will never get bigger in subsequent steps.

Original: The inﬂuence spread is computed for every node in the ﬁrst iteration and kept in a sorted list.
Revised: The inﬂuence spread is computed for every node in the ﬁrst iteration and kept in a sorted list.

Original: In each step, the marginal gain is computed for the node with the best inﬂuence spread in the previous round.
Revised: In each step, the marginal gain is computed for the node with the best inﬂuence spread in the previous round.

Original: If it is better than the previously second node, it is chosen as the next seed because it is necessarily bigger than the rest.
Revised: If it is better than the previously second node, it is chosen as the next seed because it is necessarily bigger than the rest.

Original: This property stems from submodularity, i.e., the marginal gain can never increase with the size of the seed set.
Revised: This property stems from submodularity, i.e., the marginal gain can never increase with the size of the seed set.

Original: If the ﬁrst node’s current gain is smaller than the second previous gain, the list is resorted and the process is repeated until the best node is found.
Revised: If the ﬁrst node’s current gain is smaller than the second previous gain, the list is resorted and the process is repeated until the best node is found.

Original: The worst-case complexity is similar to greedy but in practice it can be hundreds of times faster, while retaining greedy’s original guarantee of (1 − 1 /e ) .
Revised: The worst-case complexity is similar to greedy but in practice it can be hundreds of times faster, while retaining greedy’s original guarantee.

Original: In our case, we propose a straight forward adaptation where we substitute the original C ELF IE based on MC IC with the output of G LIE .
Revised: In our case, we propose a straight forward adaptation where we substitute the original C ELF IE based on MC IC with the output of G LIE .

Original: We redirect the reader in the Appendix B, where we show that the estimations of G LIE are monotonous and submodular in practice, and hence ˆ σ is suitable for optimizing with C ELF .
Revised: We redirect the reader in the Appendix B, where we show that the estimations of G LIE are monotonous and submodular in practice, and hence ˆ σ is suitable for optimizing with C ELF .

Revised: Since we do not prove the submodularity of ˆ σ , we can not contend that the theoretical guarantee is retained.

Original: C ELF -G LIE has two main computational bottlenecks.
Revised: C ELF -G LIE has two main computational bottlenecks.

Original: First, although it alleviates the need to test every node in every step, in practice it still requires IE for more than one nodes in each step.
Revised: First, although it alleviates the need to test every node in every step, in practice it still requires IE for more than one nodes in each step.

Original: Second, it requires computing the initial IE for every node in the ﬁrst step.
Revised: Second, it requires computing the initial IE for every node in the ﬁrst step.

Original: We will try to alleviate both with the two subsequent methods.
Revised: We will try to alleviate both with the two subsequent methods.

Original: 3. G RAPH R EINFORCEMENT L EARNING FOR I NFLUENCE M AZIMIZATION (G RIM )
Revised: 3. G RAPH R EINFORCEMENT L EARNING FOR I NFLUENCE M AZIMIZATION (G RIM )

Original: We develop a method that computes only one IE in every step, along with the initial IE for all nodes.
Revised: We develop a method that computes only one IE in every step, along with the initial IE for all nodes.

Original: We ﬁrst deﬁne the inﬂuence set representation L S ∈ { 0 , 1 } n , which can be computed by adding theactivations of each layer H t , summing along the axis of the hidden layer size, and thresholding to get a binary vector:
Revised: We ﬁrst deﬁne the inﬂuence set representation L S ∈ { 0 , 1 } n , which can be computed by adding the

Original: [Equation]
Revised: [Figure]

Original: where T is the number of layers, and H it ∈

Original: R n × 1 is a column from H t . This vector contains a label for each node whose sign indicates if it is predicted to be inﬂuenced. We utilize this to compute the difference between the inﬂuence of the current seed set and the initial inﬂuence of each other node. We aim to build a method that learns how to pick seeds sequentially.
Revised: [Table]

Original: The model needs to receive information from G LIE regarding the state (graph and seed set), and decide on the next action

Original: [Figure]
Revised: [Figure]

Original: [Table][Figure] [Table]
Revised: [Table]

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Original: /43z+AGEXnAA=</latexit>
Revised: /43z+AGEXnAA=</latexit>

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Original: <latexit sha1_base64="CCEqGwEzfGQVsDkBjHs6QbEGVkg=">AAACDnicbVDLTgIxFL2DL8QX6tJNIyFxRWYMiS6JblxiIo8IhHRKBxo6nUl7x0gI/+DCrX6GO+PWX/Ar/AULTAyCN2lycs69OafHj6Uw6LpfTmZtfWNzK7ud29nd2z/IHx7VTZRoxmsskpFu
Revised: <latexit sha1_base64="CCEqGwEzfGQVsDkBjHs6QbEGVkg=">AAACDnicbVDLTgIxFL2DL8QX6tJNIyFxRWYMiS6JblxiIo8IhHRKBxo6nUl7x0gI/+DCrX6GO+PWX/Ar/AULTAyCN2lycs69OafHj6Uw6LpfTmZtfWNzK7ud29nd2z/IHx7VTZRoxmsskpFu

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Original: +bWxDXnLfayC+nnJK5fKt+
Revised: +bWxDXnLfayC+nnJK5fKt+

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Original: /43z+AGEXnAA=</latexit> i = 1 ,...,n <latexit sha1_base64="BBxCLSQbs0Bgi09BthHFKIh8KSI=">AAACEHicbVDLSsNAFJ3UV62vqks3g6XgopRECroRim5cVrAPaEKZTCbt0MlMmJmIJfQnXLjVz3Anbv0Dv8JfcNIGqa0XLhzOuZdzOH7MqNK2/WUV1tY3NreK26Wd3b39g/LhUUeJRGLSxoIJ2fORIoxy0tZUM9KLJUGRz0jXH99keveBSEUFv9eTmHgRGnIaUoy0oVx65dRcFgitanxQrth1ezZwFTg5qIB8WoPytxsInESEa8yQUn3HjrWXIqkpZmRachNFYoTHaEj6BnIUEeWls8xTWDVMAEMhzXINZ+ziR4oipSaRby4jpEdqWcvI/7R+osNLL6U8TjTheG4UJgxqAbMCYEAlwZpNDEBYUpMV4hGSCGtTU6m6aBM/ZtnUr41pyFnuYxV0zutOo964a1Sa13lXRXACTsEZcMAFaIJb0AJtgEEMnsELeLWerDfr3fqYnxas
Revised: /43z+AGEXnAA=</latexit> i = 1 ,...,n <latexit sha1_base64="BBxCLSQbs0Bgi09BthHFKIh8KSI=">AAACEHicbVDLSsNAFJ3UV62vqks3g6XgopRECroRim5cVrAPaEKZTCbt0MlMmJmIJfQnXLjVz3Anbv0Dv8JfcNIGqa0XLhzOuZdzOH7MqNK2/WUV1tY3NreK26Wd3b39g/LhUUeJRGLSxoIJ2fORIoxy0tZUM9KLJUGRz0jXH99keveBSEUFv9eTmHgRGnIaUoy0oVx65dRcFgitanxQrth1ezZwFTg5qIB8WoPytxsInESEa8yQUn3HjrWXIqkpZmRachNFYoTHaEj6BnIUEeWls8xTWDVMAEMhzXINZ+ziR4oipSaRby4jpEdqWcvI/7R+osNLL6U8TjTheG4UJgxqAbMCYEAlwZpNDEBYUpMV4hGSCGtTU6m6aBM/ZtnUr41pyFnuYxV0zutOo964a1Sa13lXRXACTsEZcMAFaIJb0AJtgEEMnsELeLWerDfr3fqYnxas

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Original: Glie
Revised: Glie

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Original: /PIqaVyUvUq5cl8pVa+zOPLkhJySc+KRS1Ild6RG6oSTR/JMXsmbM3ZenHfnY9Gac7KZY/IHzucPTLSScw==</latexit
Revised: /PIqaVyUvUq5cl8pVa+zOPLkhJySc+KRS1Ild6RG6oSTR/JMXsmbM3ZenHfnY9Gac7KZY/IHzucPTLSScw==</latexit

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Original: Glie <
Revised: Glie <

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Original: Sum
Revised: Sum

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Original: ⊙ n ⇥
Revised: ⊙ n ⇥

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Original: <latexit sha1_base64="Hq2FkOTYn0FfM19J2f5XONWKY7g=">AAACFXicbVA9SwNBEN3zM8avqKXNYhAsJNxJQEvRQsuIRoXccextJnFxd+/YnQuGI/4IG/+KjYUitoKd/8ZNTOHXg4HHezPMzEsyKSz6/oc3MTk1PTNbmivPLywuLVdWVs9tmhsOTZ7K1FwmzIIUGpooUMJlZoCpRMJFcn049C96YKxI9Rn2M4gU62rREZyhk+LKdohwgwUzXcVuBnHRC4WmR5SGFlAJnVt6OrilKj5t9SIaV6p+zR+B/iXBmFTJGI248h62U54r0Mgls7YV+BlGbhsKLmFQDnMLGePXrAstRzVTYKNi9NWAbjqlTTupcaWRjtTvEwVT1vZV4joVwyv72xuK/3mtHDt7USF0liNo/rWok0uKKR1GRNvCAEfZd4RxI9ytlF8xwzi6IMsuhOD3y3/J+U4tqNfqJ/Xq/sE4jhJZJxtkiwRkl+yTY9IgTcLJHXkgT+TZu/cevRfv9at1whvPrJEf8N4+AXuOnvw=</latexit> 
Revised: <latexit sha1_base64="Hq2FkOTYn0FfM19J2f5XONWKY7g=">AAACFXicbVA9SwNBEN3zM8avqKXNYhAsJNxJQEvRQsuIRoXccextJnFxd+/YnQuGI/4IG/+KjYUitoKd/8ZNTOHXg4HHezPMzEsyKSz6/oc3MTk1PTNbmivPLywuLVdWVs9tmhsOTZ7K1FwmzIIUGpooUMJlZoCpRMJFcn049C96YKxI9Rn2M4gU62rREZyhk+LKdohwgwUzXcVuBnHRC4WmR5SGFlAJnVt6OrilKj5t9SIaV6p+zR+B/iXBmFTJGI248h62U54r0Mgls7YV+BlGbhsKLmFQDnMLGePXrAstRzVTYKNi9NWAbjqlTTupcaWRjtTvEwVT1vZV4joVwyv72xuK/3mtHDt7USF0liNo/rWok0uKKR1GRNvCAEfZd4RxI9ytlF8xwzi6IMsuhOD3y3/J+U4tqNfqJ/Xq/sE4jhJZJxtkiwRkl+yTY9IgTcLJHXkgT+TZu/cevRfv9at1whvPrJEf8N4+AXuOnvw=</latexit> activations of each layer H t , summing along the axis of the hidden layer size, and thresholding to get a binary vector: [Equation] where T is the number of layers, and H it ∈ R n × 1 is a column from H t .

Original:  (seed).
Revised: This vector contains a label for each node whose sign indicates if it is predicted to be inﬂuenced. We utilize this to compute the difference between the inﬂuence of the current seed set and the initial inﬂuence of each other node. We aim to build a method that learns how to pick seeds sequentially. The model needs to receive information from G LIE regarding the state (graph and seed set), and decide on the next action (seed).

Original: Note that, G LIE can not provide a direct estimate of a new candidate’s s marginal gain without rerunning G LIE ( S ∪ s, G ) , which is what we try to avoid.
Revised: Note that, G LIE can not provide a direct estimate of a new candidate’s s marginal gain without rerunning G LIE ( S ∪ s, G ) , which is what we try to avoid.

Original: To this end, we utilize a double
Revised: To this end, we utilize a double

Original: Q-network Van Hasselt et al. and the model is depicted in Fig.
Revised: Q-network (Van Hasselt et al., 2016) and the model is depicted in Fig.

Original: 1 (middle and right part).
Revised: 1 (middle and right part).

Original: During the ﬁrst step, G LIE provides an IE for all candidate seeds, and the node with the highest is added to the seed set, similar to G LIE -C ELF .
Revised: During the ﬁrst step, G LIE provides an IE for all candidate seeds, and the node with the highest is added to the seed set, similar to G LIE -C ELF .

Original: We also keep a list of each node’s initial inﬂuence set
Revised: We also keep a list of each node’s initial inﬂuence set

Original: L s .
Revised: L s .

Original: Subsequently, the Q-network produces a Q-value for each node s using as input the estimated inﬂuence of the current seed set ˆ σ ( S ) , the initial inﬂuence of the node ˆ σ ( s ) , and the interaction between them.
Revised: Subsequently, the Q-network produces a Q-value for each node s using as input the estimated inﬂuence of the current seed set ˆ σ ( S ) , the initial inﬂuence of the node ˆ σ ( s ) , and the interaction between them.

Original: The interaction is deﬁned as the difference between their corresponding inﬂuence sets
Revised: The interaction is deﬁned as the difference between their corresponding inﬂuence sets

Original: O ( S, s ) =
Revised: O ( S, s ) =

Original: [Equation]
Revised: (cid:80) ni

Original: ni =0 1 { L is − L iS ≥ 0 } , as predicted by G LIE .
Revised: =0 1 { L is − L iS ≥ 0 } , as predicted by G LIE .

Original: The latter aims to measure how different is the candidate node from the seed set, in order to quantify the potential gain of adding it.
Revised: The latter aims to measure how different is the candidate node from the seed set, in order to quantify the potential gain of adding it.

Original: The
Revised: The

Original: Q-network is called G raph R einforcement for I nﬂuence M aximization (G RIM ), and its architecture is composed by two layers:
Revised: Q-network is called G raph R einforcement for I nﬂuence M aximization (G RIM ), and its architecture is composed by two layers:

Original: [Equation]
Revised: [Equation]

Original: where W q ∈ R hd × 1 , and hd is the hidden layer size.
Revised: where W q ∈ R hd × 1 , and hd is the hidden layer size.

Original: We utilize a greedy policy to choose the next seed, similar to Dai et al.
Revised: We utilize a greedy policy to choose the next seed, similar to Dai et al.

Original: (2017): π ( u | S )
Revised: (2017): π ( u | S )

Original: = arg max u ∈ S Q ( u, S, G ) .
Revised: = arg max u ∈ S Q ( u, S, G ) .

Original: Given the chosen action u , the reward is computed based on the marginal gain, i.e., the estimated inﬂuence of the new seed set minus the inﬂuence of the seed set before the action, as computed by G LIE in:
Revised: Given the chosen action u , the reward is computed based on the marginal gain, i.e., the estimated inﬂuence of the new seed set minus the inﬂuence of the seed set before the action, as computed by G LIE in:

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: During training, we use epsilon-greedy to simulate an IM “game” and balance between exploration and exploitation.
Revised: During training, we use epsilon-greedy to simulate an IM “game” and balance between exploration and exploitation.

Original: We store as a train tuple the current state-action embedding { ˆ σ ( S ) , ˆ σ ( s (cid:48) ) ,
Revised: We store as a train tuple the current state-action embedding { ˆ σ ( S ) , ˆ σ ( s (cid:48) ) ,

Original: O ( S, s (cid:48) ) } , the new state embedding along with all next possible actions { ˆ σ ( S ∪ s ) , ˆ σ ( s ) ,
Revised: O ( S, s (cid:48) ) } , the new state embedding along with all next possible actions { ˆ σ ( S ∪ s ) , ˆ σ ( s ) ,

Original: O ( S, s ) } , s ∈ V and the reward r .
Revised: O ( S, s ) } , s ∈ V and the reward r .

Original: Throughout the IM, we randomly sample from the memory and train the parameters of
Revised: Throughout the IM, we randomly sample from the memory and train the parameters of

Original: G RIM .
Revised: G RIM .

Original: G LIE is frozen, because apart from providing graph and node embeddings, it is only used for the computation of the reward—thus, it is not updated.
Revised: G LIE is frozen, because apart from providing graph and node embeddings, it is only used for the computation of the reward—thus, it is not updated.

Original: The strategy to pretrain a graph encoding layer on a supervised task and use it as part of the Q-network has proven beneﬁcial in similar works Mirhoseini et al.
Revised: The strategy to pretrain a graph encoding layer on a supervised task and use it as part of the Q-network has proven beneﬁcial in similar works (Mirhoseini et al., 2021).

Original: In our case though we found out that without the overlap O and with simpler features such as the degree, the model performs rather poorly, which means we can not alleviate the burden of computing IE for every node in the ﬁrst step.
Revised: In our case though we found out that without the overlap O and with simpler features such as the degree, the model performs rather poorly, which means we can not alleviate the burden of computing IE for every node in the ﬁrst step.

Original: 3. P OTENTIALLY U NINFLUENCED N EIGHBORS (P UN )
Revised: 3. P OTENTIALLY U NINFLUENCED N EIGHBORS (P UN )

Original: Computing the inﬂuence spread of every (or even some selected) nodes in the ﬁrst step is very computationally demanding, as we will also empirically examine in the experiments section.
Revised: Computing the inﬂuence spread of every (or even some selected) nodes in the ﬁrst step is very computationally demanding, as we will also empirically examine in the experiments section.

Original: We thus seek a method that can surpass this hinder and provide adequate performance.
Revised: We thus seek a method that can surpass this hinder and provide adequate performance.

Original: We ﬁrst need to redeﬁne a simpler inﬂuence set representation based on G LIE ’s representations.
Revised: We ﬁrst need to redeﬁne a simpler inﬂuence set representation based on G LIE ’s representations.

Original: Let ˆ L S ∈ { 0 , 1 } n be the binary vector with 1 s in nodes predicted to be uninﬂuenced and 0 s elsewhere: 
Revised: Let ˆ L S ∈ { 0 , 1 } n be the binary vector with 1 s in nodes predicted to be uninﬂuenced and 0 s elsewhere: [Equation]

Original: [Equation]
Revised: [Equation]

Original: ˆ L S is obviously simpler than L S deﬁned in Eq.
Revised: ˆ L S is obviously simpler than L S deﬁned in Eq.

Original: (7), and provides a more rough estimate, but we need this deﬁnition to prove that our function is submodular and optimize it greedily.
Revised: (7), and provides a more rough estimate, but we need this deﬁnition to prove that our function is submodular and optimize it greedily.

Original: Using these, we can use message passing to predict the amount of a node’s neighborhood that remains uninﬂuenced, i.e., the P otentially U ninﬂuenced N eighbors (P UN ), weighted by the respective probability of inﬂuence: 
Revised: Using these, we can use message passing to predict the amount of a node’s neighborhood that remains uninﬂuenced, i.e., the P otentially U ninﬂuenced N eighbors (P UN ), weighted by the respective probability of inﬂuence: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Maximizing this heuristic can be considered an approximation to a seed’s marginal gain on its immediate neighbors.
Revised: Maximizing this heuristic can be considered an approximation to a seed’s marginal gain on its immediate neighbors.

Original: In accordance to the deﬁnition of the marginal gain, the inﬂuence spread is the summation of the marginal gains of every incremental seed set i , up to S :
Revised: In accordance to the deﬁnition of the marginal gain, the inﬂuence spread is the summation of the marginal gains of every incremental seed set i , up to S :

Original: [Equation]
Revised: [Equation]

Original:  In order to establish that σ m can be optimized greedily with a theoretical guarantee of (1 − 1 e ) OPT, we need to prove its monotonocity and submodularity.
Revised: [Equation] In order to establish that σ m can be optimized greedily with a theoretical guarantee of (1 − 1 e ) OPT, we need to prove its monotonocity and submodularity.

Original: The monotonicty is straight forward as it is a cumulative quantity of a non negative measure m .
Revised: The monotonicty is straight forward as it is a cumulative quantity of a non negative measure m .

Original: Theorem 1.
Revised: Theorem 1.

Original: The inﬂuence spread σ m is submodular.
Revised: The inﬂuence spread σ m is submodular.

Original: The proof can be found in the Appendix A.2.
Revised: The proof can be found in the Appendix A.2.

Original: P UN can be seen in the left part of Fig.
Revised: P UN can be seen in the left part of Fig.

Original: We start by setting the ﬁrst seed as the node with the highest degree, which can be considered a safe assumption as in practice it is always part of seed sets.
Revised: We start by setting the ﬁrst seed as the node with the highest degree, which can be considered a safe assumption as in practice it is always part of seed sets.

Original: We use G LIE ( S, G ) to retrieve ˆ L S , which we use to ﬁnd the next node based on arg max v ∈ G \ S m S [ v ] and the new ˆ L S ∪{ v } .
Revised: We use G LIE ( S, G ) to retrieve ˆ L S , which we use to ﬁnd the next node based on arg max v ∈ G \ S m S [ v ] and the new ˆ L S ∪{ v } .

Original: One disadvantage of P UN is that σ m is an underestimation of the predicted inﬂuence, as can be seen in Fig.
Revised: One disadvantage of P UN is that σ m is an underestimation of the predicted inﬂuence, as can be seen in Fig.

Original: Contrasted with the upper bound, D MP , σ m is not as accurate as ˆ σ , but allows us to compute efﬁciently a submodular proxy for the marginal gain.
Revised: Contrasted with the upper bound, D MP , σ m is not as accurate as ˆ σ , but allows us to compute efﬁciently a submodular proxy for the marginal gain.

Original: This underestimation means that a part of the network considered uninﬂuenced in ˆ L S is measured as potential gain for their neighbors, hence the ranking based on m S can be effected negatively.
Revised: This underestimation means that a part of the network considered uninﬂuenced in ˆ L S is measured as potential gain for their neighbors, hence the ranking based on m S can be effected negatively.

Original: As we observed in Fig.
Revised: As we observed in Fig.

Original: 2, the divergence of σ m increases with the size of the seed set.
Revised: 2, the divergence of σ m increases with the size of the seed set.

Original: [Figure] For this purpose, we will use adaptive full-feedback selection (AFF), where after selecting a new seed node, we remove it from the network along with nodes predicted to be inﬂuenced.
Revised: For this purpose, we will use adaptive full-feedback selection (AFF), where after selecting a new seed node, we remove it from the network along with nodes predicted to be inﬂuenced.

Original: It has been proved in the seminal work of Golovin & Krause (2011) that an AFF greedy algorithm for a submodular and monotonic function is guaranteed to have a competitive performance with the optimal policy.
Revised: It has been proved in the seminal work of Golovin & Krause (2011) that an AFF greedy algorithm for a submodular and monotonic function is guaranteed to have a competitive performance with the optimal policy.

Original: In our case, we will use an AFF update every k seeds, as it adds a small computational overhead if we do it in every step.
Revised: In our case, we will use an AFF update every k seeds, as it adds a small computational overhead if we do it in every step.

Original: The beneﬁt to P UN is twofold.
Revised: The beneﬁt to P UN is twofold.

Original: Firstly, as we remove the inﬂuenced node and truncate  the seed set, G LIE produces a more valid estimate because it performs better when the graph and seed set are smaller.
Revised: Firstly, as we remove the inﬂuenced node and truncate [Figure] the seed set, G LIE produces a more valid estimate because it performs better when the graph and seed set are smaller.

Original: Secondly, as the neighborhood size decreases, the effect of missed inﬂuenced nodes is diminished in m S .
Revised: Secondly, as the neighborhood size decreases, the effect of missed inﬂuenced nodes is diminished in m S .

Original: These can be observed in the “Adaptive” plots of Fig.
Revised: These can be observed in the “Adaptive” plots of Fig.

Original: 2 (b), (d) and in other related ﬁgures in Appendix B, where we employ an AFF every 10 seeds and we contrast the aforementioned gap between the upper bound D MP and σ m ( S ) .[Table]
Revised: 2 (b), (d) and in other related ﬁgures in Appendix B, where we employ an AFF every 10 seeds and we contrast the aforementioned gap between the upper bound D MP and σ m ( S ) .

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: All the experiments are performed in a PC with an NVIDIA GPU TITAN V (12GB RAM), 256GB
Revised: All the experiments are performed in a PC with an NVIDIA GPU TITAN V (12GB RAM), 256GB

Original: RAM and an Intel(R) Xeon(R) W-2145 CPU @ 3.70GHz.
Revised: RAM and an Intel(R) Xeon(R) W-2145 CPU @ 3.70GHz.

Original: The python implementation can be found in the supplementary material.
Revised: The python implementation can be found in the supplementary material.

Original: 4. I NFLUENCE E STIMATION
Revised: 4. I NFLUENCE E STIMATION

Original: For training in the inﬂuence estimation task, we create a set of labeled samples, each consisting of the seed set S and the corresponding inﬂuence spread σ ( S ) .
Revised: For training in the inﬂuence estimation task, we create a set of labeled samples, each consisting of the seed set S and the corresponding inﬂuence spread σ ( S ) .

Original: We create 100 Barabasi-Albert Barabási & Albert (1999) and Holme-Kim Holme & Kim (2002) undirected graphs ranging from 100 tonodes and 30 from 300 to 500 nodes.
Revised: We create 100 Barabasi-Albert Barabási & Albert (1999) and Holme-Kim Holme & Kim (2002) undirected graphs ranging from 100 tonodes and 30 from 300 to 500 nodes.

Original: 60% are used for training, 20% for validation and 20% for testing.
Revised: 60% are used for training, 20% for validation and 20% for testing.

Original: We have used these network models because the degree distribution resembles the one of real world networks.
Revised: We have used these network models because the degree distribution resembles the one of real world networks.

Original: The inﬂuence probabilities are assigned based on the weighted cascade model, i.e., a node u has equal probability 1 / deg ( u ) to be inﬂuenced by each of her N ( u ) nodes.
Revised: The inﬂuence probabilities are assigned based on the weighted cascade model, i.e., a node u has equal probability 1 / deg ( u ) to be inﬂuenced by each of her N ( u ) nodes.

Original: This model requires a directed graph, hence we turn all undirected graphs to directed ones by appending reverse edges.
Revised: This model requires a directed graph, hence we turn all undirected graphs to directed ones by appending reverse edges.

Original: Though estimating inﬂuence probabilities is a problem on its own Panagopoulos et al. (2020), in the absence of extra data, the weighted cascade is considered more realistic than pure random assignments Kempe et al.
Revised: Though estimating inﬂuence probabilities is a problem on its own (Panagopoulos et al., 2020), in the absence of extra data, the weighted cascade is considered more realistic than pure random assignments (Kempe et al., 2003).

Original: To label the samples, we run the C ELF algorithm using 1 , 000 Monte Carlo (MC) ICs for inﬂuence estimation, for up to 5 seeds.
Revised: To label the samples, we run the C ELF algorithm using 1 , 000 Monte Carlo (MC) ICs for inﬂuence estimation, for up to 5 seeds.

Original: The optimum seed set for size 1 to 5 is stored, along with 30 random negative samples for each seed set size.
Revised: The optimum seed set for size 1 to 5 is stored, along with 30 random negative samples for each seed set size.

Original: Each sample is accompanied with its ground truth inﬂuence spread computed with MC ICs.
Revised: Each sample is accompanied with its ground truth inﬂuence spread computed with MC ICs.

Original: This amounts to a total of 20 , 150 training samples.
Revised: This amounts to a total of 20 , 150 training samples.

Original: The rest of the training parameters are described in 5.
Revised: The rest of the training parameters are described in 5.

Revised: [Table]

Original: We evaluate the models in three different types of graphs.
Revised: We evaluate the models in three different types of graphs.

Original: The ﬁrst is the test set of the dataset mentioned above.
Revised: The ﬁrst is the test set of the dataset mentioned above.

Original: The second is a set of 10 power-law large graphs ( 1 , 000 – 2 , 000 nodes) to evaluate the capability of the model to generalize in networks that are larger by one factor.
Revised: The second is a set of 10 power-law large graphs ( 1 , 000 – 2 , 000 nodes) to evaluate the capability of the model to generalize in networks that are larger by one factor.

Original: The third is a set of three real-world graphs, namely the Crime (CR), HI-II-14 (HI), and GR collaborations (GR).
Revised: The third is a set of three real-world graphs, namely the Crime (CR), HI-II14 (HI), and GR collaborations (GR).

Original: More information about the datasets is given in Table 1.
Revised: More information about the datasets is given in Table 1.

Original: The real graphs are evaluated for varying seed set sizes, from 2 to 10, to test our model’s capacity to extrapolate to larger seed set sizes.
Revised: The real graphs are evaluated for varying seed set sizes, from 2 to 10, to test our model’s capacity to extrapolate to larger seed set sizes.

Original: Due to the size of the latter two graphs (HI and GR), we take for each seed set size the top nodes based on the degree as the optimum seed set along with a 30 random seed sets for the large simulated graphs and 3 for the real graphs, to validate the accuracy of the model in non-signiﬁcant sets of nodes.
Revised: Due to the size of the latter two graphs (HI and GR), we take for each seed set size the top nodes based on the degree as the optimum seed set along with arandom seed sets for the large simulated graphs and 3 for the real graphs, to validate the accuracy of the model in non-signiﬁcant sets of nodes.

Original: We have compared the accuracy of inﬂuence estimation with D MP Lokhov & Saad (2019).
Revised: We have compared the accuracy of inﬂuence estimation with D MP (Lokhov & Saad, 2019).

Original: We could not utilize the inﬂuence estimation of U BLF Zhou et al. (2015) because its central condition is violated by the weighted cascade model and the computed inﬂuence is exaggerated to the point it surpasses the nodes of the network.
Revised: We could not utilize the inﬂuence estimation of U BLF (Zhou et al., 2015) because its central condition is violated by the weighted cascade model and the computed inﬂuence is exaggerated to the point it surpasses the nodes of the network.

Original: The average error throughout all datasets and the average inﬂuence can be seen in Table 2, along with the average time.
Revised: The average error throughout all datasets and the average inﬂuence can be seen in Table 2, along with the average time.

Original: We evaluate the retrieved seed set using the independent cascade, and the results are shown in Table 3.
Revised: We evaluate the retrieved seed set using the independent cascade, and the results are shown in Table 3.

Original: We should underline here that this task would require more then 3 hours for the Crime dataset and days for GR using the traditional approach with 1 , 000 MC IC.
Revised: We should underline here that this task would require more then 3 hours for the Crime dataset and days for GR using the traditional approach with 1 , 000 MC IC.

Original: As we can see in Table 3, G LIE -C ELF allows for a signiﬁcant acceleration in computational time, while the retrieved seeds are more effective. 
Revised: As we can see in Table 3, G LIE -C ELF allows for a signiﬁcant acceleration in computational time, while the retrieved seeds are more effective. Moreover, in C ELF , the majority of [Table]

Original: Moreover, in C ELF , the majority of time is consumed in the initial computation of the inﬂuence spread, i.e., the overhead to compute 100 instead of the 20 seeds shown in Table 3, amounts to 0 .
Revised: [Table] time is consumed in the initial computation of the inﬂuence spread, i [Table] e., the overhead to compute 100 instead of the 20 seeds shown in Table 3, amounts to 0 .

Original: 22 and0 .
Revised: 22 and

Original: 19 seconds for the three datasets respectively.
Revised: 19 seconds for the three datasets respectively.

Original: [Table]

Original: [Table]

Original: 4. I NFLUENCE M AXIMIZATION
Revised: 4. I NFLUENCE M AXIMIZATION

Original: G RIM is trained on a dataset that consists of 50 random BA graphs of 500 – 2 , 000 nodes.
Revised: G RIM is trained on a dataset that consists of 50 random BA graphs of 500 – 2 , 000 nodes.

Original: It is trained by choosing 100 seeds sequentially, to maximize the reward (delay = 2 steps) for each network.
Revised: It is trained by choosing 100 seeds sequentially, to maximize the reward (delay = 2 steps) for each network.

Original: Since the immediate reward corresponds to the marginal gain, the sum of these rewards at the end of the “game” corresponds to the total inﬂuence of the seed set.
Revised: Since the immediate reward corresponds to the marginal gain, the sum of these rewards at the end of the “game” corresponds to the total inﬂuence of the seed set.

Original: An episode corresponds to completing the game for all 50 graphs; we play 500 episodes, taking roughly 40 seconds each.
Revised: An episode corresponds to completing the game for all 50 graphs; we play 500 episodes, taking roughly 40 seconds each.

Original: The exploration is set to 0 .
Revised: The exploration is set to 0 .

Original: 3 and declines with a factor of 0 . 99 .
Revised: 3 and declines with a factor of 0 . 99 .

Original: The model is optimized using ADAM, as in G LIE .
Revised: The model is optimized using ADAM, as in G LIE .

Original: We store the model that has the best average inﬂuence over all train graphs in a training episode.
Revised: We store the model that has the best average inﬂuence over all train graphs in a training episode.

Original: In order to diminish the computational time of the ﬁrst step in G LIE -C ELF and G RIM , we focus on candidate seeds that surpass a certain degree threshold based on the distribution, a common practice in the literature Chen et al. ; Manchanda et al.
Revised: In order to diminish the computational time of the ﬁrst step in G LIE -C ELF and G RIM , we focus on candidate seeds that surpass a certain degree threshold based on the distribution, a common practice in the literature (Chen et al., 2009; Manchanda et al., 2020) .

Original: For comparison, we use a state-of-the-art IM method, I MM Tang et al. (2015) which capitalizes on reverse reachable sets Borgs et al. (2014) to estimate inﬂuence.
Revised: For comparison, we use a state-of-the-art IM method, I MM (Tang et al., 2015) which capitalizes on reverse reachable sets (Borgs et al., 2014) to estimate inﬂuence.

Original: Speciﬁcally, it produces a series of such inﬂuence sketches and uses them to approximate the inﬂuence spread without any simulation when building the seed set.
Revised: Speciﬁcally, it produces a series of such inﬂuence sketches and uses them to approximate the inﬂuence spread without any simulation when building the seed set.

Original: This results in remarkable acceleration while retaining a theoretical guarantee with high probability.
Revised: This results in remarkable acceleration while retaining a theoretical guarantee with high probability.

Original: Note that, I MM is considered state-of-the-art and has similar inﬂuence spreads with Tang et al.
Revised: Note that, I MM is considered state-of-the-art and has similar inﬂuence spreads with Tang et al.

Original: (2014), while surpassing various heuristics Jung et al.
Revised: (2014), while surpassing various heuristics (Jung et al., 2012).

Original: We set e = 0 .as proposed by the authors.
Revised: We set e = 0 . 5 as proposed by the authors.

Original: We also compare with F INDER , which is analyzed in Section 2. 
Revised: We also compare with F INDER , which is analyzed in Section 2. [Table] [Table] HI

Original: [Table]
Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: EN 14 ,14 ,14 ,

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Original:  The results for the inﬂuence spread of20 and 100  seeds as computed by simulations of MC ICs can be seen in Table 4, while the time resultsfor 100 seeds are shown in Table 5.
Revised: [Table] [Table] [Table] The results for the inﬂuence spread of 100 and 200 seeds as computed by simulations of MC ICs can be seen in Table (4) and (5), while the time results are shown in Table 6.

Revised: The top result is in bold and the second best is underlined.

Revised: The inﬂuence spread for smaller seed sets and the time for heuristics are attached in Appendix C, along with comparisons of P UN without the use of GPU.

Original: One can see  that G LIE -C ELF exhibits overall superior inﬂuence quality compared to the rest of the methods, but is quite slower.
Revised: One can see [Table] [Table] that G LIE -C ELF exhibits overall superior inﬂuence quality compared to the rest of the methods, but is quite slower.

Original: G RIM is slightly faster than G LIE -C ELF but is the second slowest method.
Revised: G RIM is slightly faster than G LIE -C ELF but is the second slowest method.

Original: This quantiﬁes the substantial overhead caused by computing the inﬂuence spread of all candidate seeds in[Table] the ﬁrst step.
Revised: This quantiﬁes the substantial overhead caused by computing the inﬂuence spread of all candidate seeds in the ﬁrst step.

Original: Their time difference amounts to how many more IEs G LIE -C ELF performs in every step compared to G RIM , which performs only one.
Revised: Their time difference amounts to how many more IEs G LIE -C ELF performs in every step compared to G RIM , which performs only one.

Original: This is more obvious with P UN , which requires only one IE in every step and no initial computation.
Revised: This is more obvious with P UN , which requires only one IE in every step and no initial computation.

Original: It is from 3 to 60 times faster than I MM while its computational overhead moving from smaller to larger graphs is less than linear to the number of nodes.
Revised: It is from 3 to 60 times faster than I MM while its computational overhead moving from smaller to larger graphs is less than linear to the number of nodes.

Original: In terms of inﬂuence quality, P UN is ﬁrst or second in half of the datasets in both seed set sizes.
Revised: In terms of inﬂuence quality, P UN is ﬁrst or second in the majority of the datasets in both seed set sizes.

Original: We can thus contend that it provides the best accuracy-efﬁciency tradeoff from the examined methods.
Revised: We can thus contend that it provides the best accuracy-efﬁciency tradeoff from the examined methods.

Original: I MM is the third slowest method, but it is very accurate, specially for smaller seed set sizes.
Revised: I MM is the third slowest method, but it is very accurate, specially for smaller seed set sizes.

Original: F INDER exhibits the least accurate performance, which is understandable given that it solves a relevant problem and not exactly IM for IC.
Revised: F INDER exhibits the least accurate performance, which is understandable given that it solves a relevant problem and not exactly IM for IC.

Original: The computational time presented is the time required to solve the node percolation, in which case it may retrieve a bigger seed set than 100 nodes.
Revised: The computational time presented is the time required to solve the node percolation, in which case it may retrieve a bigger seed set than 100 nodes.

Original: Thus, we can hypothesize it is quite faster for a limited seed set, but the quality of the retrieved seeds is the least accurate among all methods.
Revised: Thus, we can hypothesize it is quite faster for a limited seed set, but the quality of the retrieved seeds is the least accurate among all methods.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We have proposed G LIE , a GNN-based solution for the problem of inﬂuence estimation.
Revised: We have proposed G LIE , a GNN-based solution for the problem of inﬂuence estimation.

Original: We showcase its accuracy in the respective task and utilized it to addressmore effectively the problem of inﬂuence maximization.
Revised: We showcase its accuracy in that task and utilized it to address the problem of inﬂuence maximization.

Original: We developed three methods based on the representations and the predictions of G LIE .
Revised: We developed three methods based on the representations and the predictions of G LIE .

Original: G LIE -C ELF , an adaptation of a classical algorithm that surpasses SOTA but with signiﬁcant computational overhead.
Revised: G LIE -C ELF , an adaptation of a classical algorithm that surpasses SOTA but with signiﬁcant computational overhead.

Original: G RIM , a Q-learning model that learns to retrieve seeds sequentially using G LIE ’s predictions and representations.
Revised: G RIM , a Qlearning model that learns to retrieve seeds sequentially using G LIE ’s predictions and representations.

Original: And P UN , a submodular function that acts as proxy for the marginal gain and can be optimized adaptively, striking a balance between efﬁciency and accuracy.
Revised: And P UN , a submodular function that acts as proxy for the marginal gain and can be optimized adaptively, striking a balance between efﬁciency and accuracy.

Original: Inﬂuence maximization in the real world could be beneﬁted by supplementary information regarding the properties of the network.

Original: A typical IM algorithm needs a signiﬁcant contribution in order to take into account the topic of the information shared or the user’s characteristics Chen et al. (2016) i.e., conditional diffusion.
Revised: A typical IM algorithm needs a signiﬁcant contribution in order to take into account the topic of the information shared or the user’s characteristics (Chen et al., 2016) i.e., conditional diffusion.

Original: An important practical advantage of a neural network approach is the easy incorporation of such complementary data by adding the corresponding embeddings in the input, as has been done in similar settings Tian et al.
Revised: An important practical advantage of a neural network approach is the easy incorporation of such complementary data by adding the corresponding embeddings in the input, as has been done in similar settings (Tian et al., 2020).

Original: We thus deem an experiment with contextual information a natural next step, given a proper dataset.
Revised: We thus deem an experiment with contextual information a natural next step, given a proper dataset.

Original: Our approach can also be utilized to address the minimum vertex cover in large graphs, as it is a problem related to inﬂuence maximization and there exists models that work well in both Manchanda et al.
Revised: Our approach can also be utilized to address the minimum vertex cover in large graphs, as it is a problem related to inﬂuence maximization and there exists models that work well in both (Manchanda et al., 2020).

Original: Finally, we also plan to examine the potential of training online the reinforcement learning, i.e., receiving real feedback from each step of the diffusion that could update both, the Q-N ET and G LIE .
Revised: Finally, we also plan to examine the potential of training online the reinforcement learning, i.e., receiving real feedback from each step of the diffusion that could update both, the Q-N ET and G LIE .

Original: This would allow the model to adjust its decisions based on the partial feedback received during the diffusion.
Revised: This would allow the model to adjust its decisions based on the partial feedback received during the diffusion.

Original: Ethics and Reproducibility Our inﬂuence maximization methods can potentially be used for large scale manipulation in social media.
Revised: Ethics and Reproducibility Our inﬂuence maximization methods can potentially be used for large scale manipulation in social media.

Original: For example, targeting the right susceptible users with speciﬁc political advertisements might maximize the effect of the campaign on the general intent to vote.
Revised: For example, targeting the right susceptible users with speciﬁc political advertisements might maximize the effect of the campaign on the general intent to vote.

Original: On the bright side, there has been extensive research on battling such effects Tu et al. (2020), by producing campaigns that are equally effective and hence balancing each other Bharathi et al.
Revised: On the bright side, there has been extensive research on battling such effects (Tu et al., 2020), by producing campaigns that are equally effective and hence balancing each other (Bharathi et al., 2007).

Original: More importantly, G LIE can be used as part of the mitigation strategy Tu et al. (2020), as a black box that substitutes the method’s RR-set based inﬂuence estimation, to actually increase the balance of political exposure in the given social network for the running campaigns.
Revised: More importantly, G LIE can be used as part of the mitigation strategy (Tu et al., 2020), as a black box that substitutes the method’s RR-set based inﬂuence estimation, to actually increase the balance of political exposure in the given social network for the running campaigns.

Original: Moreover, our methods can inherently be utilized in the context of limiting fake news spreading similar to other inﬂuence maximization algorithms Budak et al.
Revised: Moreover, our methods can inherently be utilized in the context of limiting fake news spreading similar to other inﬂuence maximization algorithms (Budak et al., 2011).

Original: Regarding the reproducibility of our experiments, we have attached all codes in the supplementary ﬁles along with detailed instructions on how to reproduce the results.
Revised: Regarding the reproducibility of our experiments, we have attached all codes in the supplementary ﬁles along with detailed instructions on how to reproduce the results.

Original: We also point to the codes we utilized to run the benchmarks.
Revised: We also point to the codes we utilized to run the benchmarks.

Original: All data can be downloaded through an anonymous repository.
Revised: All data can be downloaded through an anonymous repository.

Original: S UPPLEMENTARY MATERIAL : L EARNING G RAPH R EPRESENTATIONS FOR I NFLUENCE M AXIMIZATION
Revised: S UPPLEMENTARY MATERIAL : L EARNING G RAPH R EPRESENTATIONS FOR I NFLUENCE M AXIMIZATION

Original: Section 5 gathers the proofs of theorems and corollaries.
Revised: Section 5 gathers the proofs of theorems and corollaries.

Original: Section 5 includes plots in support of the arguements in the main text.
Revised: Section 5 includes plots in support of the arguments in the main text.

Original: A PPENDIX A
Revised: A PPENDIX A

Original: We recall the ﬁrst corollary from Sec. 3 and detail the proof below.
Revised: We recall the ﬁrst corollary from Sec. 3 and detail the proof below.

Original: Corollary 3.0.2.
Revised: Corollary 3.0.2.

Original: The repeated product H t +1 =
Revised: The repeated product H t +1 =

Original: A · H t computes an upper bound to the real inﬂuence probabilities of each infected node at step t + 1 .
Revised: A · H t computes an upper bound to the real inﬂuence probabilities of each infected node at step t + 1 .

Original: Proof.
Revised: Proof.

Original: We have:
Revised: We have:

Original: [Equation]
Revised: [Equation]

Original: • (15) stems from Eq.
Revised: • (15) stems from Eq.

Original: 3 in the manuscript:
Revised: 3 in the manuscript:

Original: [Equation]
Revised: [Equation]

Original: • (16) can be proved by induction similar to Zhou et al.
Revised: • (16) can be proved by induction similar to Zhou et al.

Original: For every p v ≤ 1 , the base case (cid:80) v ∈X p
Revised: For every p v ≤ 1 , the base case (cid:80) v ∈X p

Original: [Equation]
Revised: [Equation]

Original: p
Revised: p

Original: [Equation]
Revised: [Equation]

Original: −
Revised: −

Original: [Equation]
Revised: [Equation]

Original: v ∈X (1 − p v p vu ) is obvious for |X| = 1 .
Revised: v ∈X (1 − p v p vu ) is obvious for |X| = 1 .

Original: For |X| > 1 , we have:
Revised: For |X| > 1 , we have:

Original: [Equation]
Revised: [Equation]

Original: • (17), we have p ( u | v ) =
Revised: • (17), we have p ( u | v ) =

Original: p v p vu per deﬁnition of the independent cascade, and consequently p ( u | S ) = 1 − (cid:81) v ∈N ( u ) ∩ S (1 − p v p vu ) , where p v = 1 for v ∈ S 1 , which are the initial seed set that are activated deterministically.
Revised: p v p vu per deﬁnition of the independent cascade, and consequently p ( u | S ) = 1 − (cid:81) v ∈N ( u ) ∩ S (1 − p v p vu ) , where p v = 1 for v ∈ S 1 , which are the initial seed set that are activated deterministically.

Original: We can thus contend that Eq.
Revised: We can thus contend that Eq.

Original: equation 18 stands, and the computed probabilities are an upper bound of the real inﬂuence probabilities.
Revised: equation 18 stands, and the computed probabilities are an upper bound of the real inﬂuence probabilities.

Original: Hence the inﬂuence spread, which is computed as ˆ σ ( S )
Revised: Hence the inﬂuence spread, which is computed as ˆ σ ( S )

Original: = (cid:80) ( u,v ) ∈
Revised: = (cid:80) ( u,v ) ∈

Original: E p uv is also an upper bound to the real σ ( S ) .
Revised: E p uv is also an upper bound to the real σ ( S ) .

Original: Now we address the theorem from Sec. 3.4 and detail the proof below.
Revised: Now we address the theorem from Sec. 3.4 and detail the proof below.

Original: Theorem 1.
Revised: Theorem 1.

Original: The inﬂuence spread σ m is submodular.
Revised: The inﬂuence spread σ m is submodular.

Original: For the purposes of the proof, X i ∈ { 0 , 1 } n × d is the input and H i ∈ R n × hd is the output of the ﬁrst neural layer for the input seed set S i , and P ∈ { 1 } hd × 1 .
Revised: For the purposes of the proof, X i ∈ { 0 , 1 } n × d is the input and H i ∈ R n × hd is the output of the ﬁrst neural layer for the input seed set S i , and P ∈ { 1 } hd × 1 .

Original: Moreover we deﬁne the support function supp ( v ) =
Revised: Moreover we deﬁne the support function supp ( v ) =

Original: { i ∈ [1 , d ] , v i (cid:54) = 0 } James et al.
Revised: { i ∈ [1 , d ] , v i (cid:54) = 0 } James et al.

Original: as the set of indices of non zero elements in vector v ∈ R d .
Revised: as the set of indices of non zero elements in vector v ∈ R d .

Original: Finally let R represent ReLU and b tr , st tr the mean and standard deviation computed by the batchnorm.
Revised: Finally let R represent ReLU and b tr , st tr the mean and standard deviation computed by the batchnorm.

Original: Each step is justiﬁed further below.
Revised: Each step is justiﬁed further below.

Original: Proof.
Revised: Proof

Original: [Equation]

Original: i

Original: [Equation]
Revised: [Equation]

Original: (21) stems by the deﬁnition of X in Eq.
Revised: (21) stems by the deﬁnition of X in Eq.

Original: (22) X j is a convex hull that contains X i Boyd et al.
Revised: (22) X j is a convex hull that contains X i (Boyd et al., 2004).

Original: We multiply both sides by a real matrix W ∈ R d × hd which can equally dilate both convex hulls in terms of direction and norm.
Revised: We multiply both sides by a real matrix W ∈ R d × hd which can equally dilate both convex hulls in terms of direction and norm.

Original: This equal transformation cannot change the sign of the difference between the elements of X i and X j and hence cannot interfere with the support of X j over X i .
Revised: This equal transformation cannot change the sign of the difference between the elements of X i and X j and hence cannot interfere with the support of X j over X i .

Original: The statement becomes more obvious for X ∈ { 0 ,  n ×  and W ∈ R 1 ×
Revised: The statement becomes more obvious for X ∈ { 0 , 1 } n × 1 and W ∈ R 1 ×

Original: Note that both can result in zero matrices so
Revised: Note that both can result in zero matrices so

Original: we
Revised: we

Original: use
Revised: use

Original: subset or equal.
Revised: subset or equal.

Original: A is a non-negative matrix.
Revised: A is a non-negative matrix.

Original: (24) ReLU is a non negative monotonically increasing function.
Revised: (24) ReLU is a non negative monotonically increasing function.

Original: (25) Subtract by the same number and divide by the same positive number.
Revised: (25) Subtract by the same number and divide by the same positive number.

Original: (26) Deﬁnition in Eq.
Revised: (26) Deﬁnition in Eq.

Original: (27) P is positive.
Revised: (27) P is positive.

Original: By deﬁnition of the support.
Revised: By deﬁnition of the support.

Original: (29) Complementarity between elements that are ≤ 0 and elements > 0 .
Revised: (29) Complementarity between elements that are ≤ 0 and elements > 0 .

Original: (30) Deﬁnition in Eq.
Revised: (30) Deﬁnition in Eq.

Original: (10) and multiply with non-negative row u from matrix A (cid:48) .
Revised: (10) and multiply with non-negative row u from matrix A (cid:48) .

Original: (31) Deﬁnition in Eq.
Revised: (31) Deﬁnition in Eq.

Original: (32) Adding and subtracting σ m ( S i ) and σ m ( S j ) .
Revised: (32) Adding and subtracting σ m ( S i ) and σ m ( S j ) .

Original: (33) By deﬁnition of σ m in Eq.
Revised: (33) By deﬁnition of σ m in Eq.

Original: (12) we arrive at submodularity in Eq.
Revised: (12) we arrive at submodularity in Eq.

Original: A PPENDIX B B. G LIE TRAINING
Revised: A PPENDIX B B. G LIE TRAINING

Original: We observed that 1 , 000  has similar results to 10 , 000 , due to the small size of the networks.
Revised: We observed that 1 , 000 simulations for labeling has similar results to 10 , 000 , due to the small size of the networks.

Original: The training takes 500 up to 1 , 200 seconds, depending on when the validation error converges.
Revised: The training takes 500 up to 1 , 200 seconds, depending on when the validation error converges.

Original: We have used a small scale grid-search to ﬁnd the optimum batch size 64 , dropout 0 . 4 , number of layers 2 , hidden layer size 64 , and feature dimension 50 .
Revised: We have used a small scale grid-search to ﬁnd the optimum batch size 64 , dropout 0 . 4 , number of layers 2 , hidden layer size 64 , and feature dimension 50 .

Original: More importantly, we observed that it is beneﬁcial to decrease the hidden layer size (by a factor of 2 ) as the depth increases, i.e., go from 32 to 16 .
Revised: More importantly, we observed that it is beneﬁcial to decrease the hidden layer size (by a factor of 2 ) as the depth increases, i.e., go from 32 to 16 .

Original: This means that the 1 -hop node representations are more useful compared to the 2 -hop ones and so on—validating the aforementioned conclusion that the approximation to the inﬂuence estimation in Eq.
Revised: This means that the 1 -hop node representations are more useful compared to the 2 -hop ones and so on—validating the aforementioned conclusion that the approximation to the inﬂuence estimation in Eq.

Original: (3), diverges more as the message passing depth progresses.
Revised: (3), diverges more as the message passing depth progresses.

Original: B. A DAPTIVE F ULL F EEDBACK
Revised: B. A DAPTIVE F ULL F EEDBACK

Original: We see here that the pattern that is prevalent in the two datasets described in section 3.4 is retained in other graphs.
Revised: We see here that the pattern that is prevalent in the two datasets described in section 3.4 is retained in other graphs.

Original: [Figure]
Revised: [Figure]

Original: The results of P UN without AFF can be seen Tab. The computational time is similar to the original P UN .
Revised: [Figure]

Original: B. G LIE S UBMODULARITY AND MONOTONICITY
Revised: . G LIE S UBMODULARITY AND MONOTONICITY

Original: In this section we design an experiment to empirically prove that G LIE ’s output is submodular and monotonous.
Revised: In this section we design an experiment to empirically prove that G LIE ’s output is submodular and monotonous.

Original: For each of the real datasets, we use the seed set retrieved by G LIE -C ELF and a random seed set to quantify the differences between subsequent estimations.
Revised: For each of the real datasets, we use the seed set retrieved by G LIE -C ELF and a random seed set to quantify the differences between subsequent estimations.

Original: To be speciﬁc, we have a sequence S that represents the seed set and a sequence R that represents the random nodes, with
Revised: To be speciﬁc, we have a sequence S that represents the seed set and a sequence R that represents the random nodes, with

Original: [Table]

Original: S j being the seed set up to j th element and s j being the j th element, and similarly r j for R .
Revised: S j being the seed set up to j th element and s j being the j th element, and similarly r j for R .

Original: We compute the marginal gain to check for monotonicity:
Revised: We compute the marginal gain to check for monotonicity:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: 36) and for submodularity, we have , with i =
Revised: 36) and for submodularity, we have , with i =

Original: j − 1 :
Revised: j − 1 :

Original: [Equation]
Revised: [Equation]

Original: In Figures 4 to 7, we plot m and s for some of our datasets.
Revised: In Figures 4 to [Figure] we plot m and s for some of our datasets.

Original: Regarding s , since we require a constant node, we randomly sample one of the seeds s j and a random node r j and visualize the sequences of both s with regard to adding them in every step.
Revised: Regarding s [Figure] since we require a constant node, we randomly sample one of the seeds s j and a random node r j and visualize the sequences of both s with regard to adding them in every step.

Original: The values of these functions correspond to nodes, and range from tens to thousands, depending on the datasets.
Revised: The values of these functions correspond to nodes, and range from tens to thousands, depending on the datasets.

Original: For monotonicity and submodularity, we verify that m
Revised: For monotonicity and submodularity, we verify that m

Original: and s are always more than zero.
Revised: and s are always more than zero.

Original: Moreover, we see that they decrease with the size of the seed set, as well as the observation that adding a random seed provides worst marginal gains ( in monotonicity plots) than adding the chosen seed.
Revised: Moreover, we see that they decrease with the size of the seed set, as well as the observation that adding a random seed provides worst marginal gains ( in monotonicity plots) than adding the chosen seed [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Revised: (b)

Original: [Figure]
Revised: [Figure]

Original: d e s Submodularity HI s ss s sr (b)
Revised: We compare I MM and P UN on the same graphs with uniform inﬂuence probabilities p = 0 . 01 in

Original: [Figure]
Revised: Figure 8.

Revised: We clearly observe that P UN outperforms I MM .

Original: [Figure]
Revised: [Figure]

Revised: A PPENDIX C C. R ESULTS

Revised: SMALLER SEED SETS

Revised: [Table]

Revised: [Table]

Revised: C. C OMPUTATIONAL TIME OF HEURISTICS

Revised: [Table]

Revised: C. P UN ’ S COMPUTATIONAL TIME WITHOUT GPU

Revised: We performed experiments to compare P UN without the use of GPU in Table (10)

Revised: [Table]

Revised: where it is visible that GPU provides a substantial acceleration

Revised: [Table]

Revised: but P UN remains the faster option even without it.

Revised: [Table]

Revised: C. R ELATIVE ERROR OF G LIE FOR LARGER SEED SETS .

Revised: To quantify the potential of G LIE for larger seed sets, we sample 9 random seed sets and 1 with the highest degree nodes and compute the error of D MP and G LIE , with the ground truth inﬂuence divided by the average inﬂuence in Table (11).

Revised: We see that the error does not increase signiﬁcantly as the seed set increases, and that G LIE outperforms D MP in GR while the reverse happens in CR and

Revised: HT.

Revised: [Table]