Given a social network$G=(V, E)$, the unconstrained profit maximization problem aims to identify a subset$S \subseteq V$that maximizes the net profit, defined as the expected influence spread$\Gamma (S)$of set$S$minus the associated cost$c(S)$, i.e.,$\Gamma (S) - c(S)$. However, this problem presupposes an unlimited budget, which is often impractical in real scenarios. Motivated by this, we investigate the budgeted profit maximization (BPM) problem by adding a budget constraint. Unfortunately, addressing the BPM problem with a theoretical approximation guarantee remains relatively under-explored in the literature. In response, assuming$\Gamma (S)$is known for any$S \subseteq V$, we propose an algorithm that guarantees returning a set$S^{o}$such that$\Gamma (S^{o}) - c(S^{o}) \geq (1 - \frac{1}{e}) \frac{\Gamma (S^*)}{2} - \frac{c(S^*)}{2}$, where$S^*$denotes an optimal solution for BPM. Then, we develop a practical solution, which uses the reverse reachable set (RR-set) technique for influence estimation, without assuming knowledge of$\Gamma (S)$, while still maintaining a strong approximation guarantee. Additionally, similar to existing RR-set-based solutions for influence cascade-related problems, our RR-set-based solution relies on generating a large number of random RR-sets to accurately estimate$\Gamma (S)$. However, the existing RR-set generation method suffers from high memory stall rates due to its irregular memory access patterns, leaving room for further efficiency improvement. Therefore, we propose a new RR-set generation method that utilizes batch execution and cache prefetching. When a memory access is required, instead of stalling while waiting for data, the CPU first issues an asynchronous prefetch request to load the target data into the cache, and then switches to processing the generation of other RR-sets within the same batch, effectively hiding memory access latency. This method can be seamlessly integrated into existing RR-set-based solutions to improve their efficiency. Finally, we conduct extensive experiments on real, large-scale datasets to demonstrate the effectiveness and efficiency of our proposed solutions.

Graph clustering is a fundamental data mining task that clusters vertices into different groups. The structural graph clustering algorithm ($SCAN$) is a widely used graph clustering algorithm that derives not only clustering results, but also special roles of vertices like hubs and outliers. In this paper, we consider structural graph clustering on dynamic graphs under Jaccard similarity. The state-of-the-art index-based solution focuses on static graphs and incurs prohibitive update costs to maintain indices. Lately, an efficient approximate dynamic structural graph clustering algorithm DynStrClu under Jaccard similarity is proposed. However, their solution needs to fix input parameters while parameter settings of SCAN usually need to be fine-tuned to achieve good clustering results. Motivated by these limitations, we present a study on devising effective index structures for SCAN algorithm on dynamic graphs. Similar to the state-of-the-art dynamic scheme, our main idea to reduce the time complexity is still by bringing approximation to clustering results. However, our solution does not need to fix the input parameters. To achieve this, our solution includes two key components. The first is to maintain a bottom-$k$sketch for each vertex so that the similarities of affected vertices can be easily updated. The second key is a bucketing strategy that allows us to update clustering results and roles of vertices efficiently. Our theoretical analysis shows that our proposed algorithm achieves$O(\log \log {\frac{M+m}{p_{f}}}\cdot \log {\frac{M+m}{p_{f}}})$expected update cost and guarantees to return approximate clustering results with probability$1-p_{f}$after up to$M$updates. Extensive experiments show that our solution is up to two orders of magnitude faster than the state-of-the-art index-based solution while still achieving high-quality clustering results.

Graphs are a ubiquitous data structure in diverse domains such as machine learning, social networks, and data mining. As real-world graphs continue to grow beyond the memory capacity of single machines, out-of-core graph processing systems have emerged as a viable solution. Yet, existing systems that rely on strictly synchronous, iteration-by-iteration execution incur significant overheads. In particular, their scheduling mechanisms lead to I/O inefficiencies, stemming from read and work amplification, and induce costly synchronization stalls hindering sustained disk utilization. To overcome these limitations, we present ACGraph, a novel asynchronous graph processing system optimized for SSD-based environments with constrained memory resources. ACGraph employs a dynamic, block-centric priority scheduler that adjusts in real time based on workload, along with an online asynchronous worklist that minimizes redundant disk accesses by efficiently reusing active blocks in memory. Moreover, ACGraph unifies asynchronous I/O with computation in a pipelined execution model that maintains sustained I/O activation, and leverages a highly optimized hybrid storage format to expedite access to low-degree vertices. We implement popular graph algorithms, such as Breadth-First Search (BFS), Weakly Connected Components (WCC), personalized PageRank (PPR), PageRank (PR), and k -core on ACGraph and demonstrate that ACGraph substantially outperforms state-of-the-art out-of-core graph processing systems in both runtime and I/O efficiency.

Given a social network G , a cost associated with each user, and an influence threshold η, the minimum cost seed selection problem (MCSS) aims to find a set of seeds that minimizes the total cost to reach η users. Existing works are mainly devoted to providing an expected coverage guarantee on reaching η, classified as MCSS-ECG, where their solutions either rely on an impractical influence oracle or cannot attain the expected influence threshold. More importantly, due to the expected coverage guarantee, the actual influence in a campaign may drift from the threshold evidently. Thus, the advertisers would like to request for a probability guarantee of reaching η. This motivates us to further solve the MCSS problem with a probabilistic coverage guarantee, termed MCSS-PCG. In this paper, we first propose our algorithm CLEAR to solve MCSS-ECG, which reaches the expected influence threshold without any influence oracle or influence shortfall but a practical approximation ratio. However, the ratio involves an unknown term (i.e., the optimal cost). Thus, we further devise the STAR method to derive a lower bound of the optimal cost and then obtain the first explicit approximation ratio for MCSS-ECG. In MCSS-PCG, it is necessary to estimate the probability that the current seeds reach η, to decide when to stop seed selection. To achieve this, we design a new technique named MRR, which provides efficient probability estimation with a theoretical guarantee. With MRR in hand, we propose our algorithm SCORE for MCSS-PCG, whose performance guarantee is derived by measuring the gap between MCSS-ECG and MCSS-PCG, and applying the theoretical results in MCSS-ECG. Finally, extensive experiments demonstrate that our algorithms achieve up to two orders of magnitude speed-up compared to alternatives while meeting the requirement of MCSS-PCG with the smallest cost.

Given a source node$s$and a target node$t$, the hitting probability tells us how likely an$\alpha$-terminating random walk (which stops with probability$\alpha$at each step) starting from$s$can hit$t$before it stops. This concept originates from the hitting time, a classic concept in random walks. In this paper, we focus on the group hitting probability (GHP) where the target is a set of nodes, measuring the node-to-group structural proximity. For this group version of the hitting probability, we present efficient algorithms for two types of GHP queries: the pairwise query which returns the GHP value of a target set$T$with respect to (w.r.t.) a source node$s$, and the top-$k$query which returns the top-$k$target sets with the largest GHP value w.r.t. a source node$s$. We first develop an efficient algorithm named SAMBA for the pairwise query, which is built on a group local push algorithm tailored for GHP, with rigorous analysis for correctness. Next, we show how to speed up SAMBA by combining the group local push algorithm with the Monte Carlo approach, where GHP brings new challenges as it might need to consider every hop of the random walk. We tackle this issue with a new formulation of the GHP and show how to provide approximation guarantees with a detailed theoretical analysis. With SAMBA as the backbone, we develop an iterative algorithm for top-$k$queries, which adaptively refines the bounds for the candidate target sets, and terminates as soon as it meets the stopping condition, thus saving unnecessary computational costs. We further present an optimization technique to accelerate the top-$k$query, improving its practical performance. Extensive experiments show that our solutions are orders of magnitude faster than their competitors.

Given a graph$\mathcal {G}$with a set of attributes, theattributed network embedding (ANE)aims to learn low-dimensional representations of nodes that preserve both graph topology and node attribute proximity. ANE is shown to be more effective than plain network embedding methods (using only graph topology) on many graph mining tasks. However, existing ANE solutions still provide inferior performance on tasks like node classification and link prediction, as will be shown in our experiments. The key issue is that when combining graph topology and attribute information, most existing solutions take attributes with equal importance, while in real scenarios, different attribute exerts distinct influence over the network due to the heterogeneous nature among attributes. Motivated by this, we presentLATAM, a learning-based framework for ANE via trainable proximity matrix factorization. To capture the node-attribute relationships, we first construct the attribute-augmented graph by adding attribute nodes (resp. edges) to the original graph. Then, we define the attribute-augmented random walk and proximity on the attribute-augmented graph, where the weights of different attributes can be learned automatically by our designed loss functions so that more indicative attributes tend to have higher weights, imposing a higher impact on the node connectivity. To achieve this, we incorporate a differentiable SVD to back-propagate gradients of attribute weights in an end-to-end process. To scale our LATAM to large graphs, we further propose sampling techniques to learn attribute weights and an efficient attribute-augmented push algorithm to compute the proximity matrix. Extensive experiments on 8 public attributed networks against 11 existing methods show the effectiveness of our LATAM.

Given a graph G, a cost associated with each node, and a budget B, the budgeted influence maximization (BIM) aims to find the optimal set S of seed nodes that maximizes the influence among all possible sets such that the total cost of nodes in S is no larger than B. Existing solutions mainly follow the non-adaptive idea, i.e., determining all the seeds before observing any actual diffusion. Due to the absence of actual diffusion information, they may result in unsatisfactory influence spread. Motivated by the limitation of existing solutions, in this paper, we make the first attempt to solve the BIM problem under the adaptive setting, where seed nodes are iteratively selected after observing the diffusion result of the previous seeds. We design the first practical algorithm which achieves an expected approximation guarantee by probabilistically adopting a cost-aware greedy idea or a single influential node. Further, we develop an optimized version to improve its practical performance in terms of influence spread. Besides, the scalability issues of the adaptive IM-related problems still remain open. It is because they usually involve multiple rounds (e.g., equal to the number of seeds) and in each round, they have to construct sufficient new reverse-reachable set (RR-set) samples such that the claimed approximation guarantee can actually hold. However, this incurs prohibitive computation, imposing limitations on real applications. To solve this dilemma, we propose an incremental update approach. Specifically, it maintains extra construction information when building RR-sets, and then it can quickly correct a problematic RR-set from the very step where it is first affected. As a result, we recycle the RR-sets at a small computational cost, while still providing correctness guarantee. Finally, extensive experiments on large-scale real graphs demonstrate the superiority of our algorithms over baselines in terms of both influence spread and running time.

\em Personalized PageRank (PPR) stands as a fundamental proximity measure in graph mining. Given an input graph G with the probability of decay α, a source node s and a target node t, the PPR score π(s,t) of target t with respect to source s is the probability that an α-decay random walk starting from s stops at t. A \em single-source PPR (SSPPR) query takes an input graph G with decay probability α and a source s, and then returns the PPR π(s,v) for each node v ∈ V. Since computing an exact SSPPR query answer is prohibitive, most existing solutions turn to approximate queries with guarantees. The state-of-the-art solutions for approximate SSPPR queries are index-based and mainly focus on static graphs, while real-world graphs are usually dynamically changing. However, existing index-update schemes can not achieve a sub-linear update time. Motivated by this, we present an efficient indexing scheme for single-source PPR queries on evolving graphs. Our proposed solution is based on a classic framework that combines the forward-push technique with a random walk index for approximate PPR queries. Thus, our indexing scheme is similar to existing solutions in the sense that we store pre-sampled random walks for efficient query processing. One of our main contributions is an incremental updating scheme to maintain indexed random walks in expected O(1) time after each graph update. To achieve O(1) update cost, we need to maintain auxiliary data structures for both vertices and edges. To reduce the space consumption, we further revisit the sampling methods and propose a new sampling scheme to remove the auxiliary data structure for vertices while still supporting O(1) index update cost on evolving graphs. Extensive experiments show that our update scheme achieves orders of magnitude speed-up on update performance over existing index-based dynamic schemes without sacrificing the query efficiency.

Given a social network G with n nodes and m edges, a positive integer k , and a cascade model C , the influence maximization (IM) problem asks for k nodes in G such that the expected number of nodes influenced by the k nodes under cascade model C is maximized. The state-of-the-art approximate solutions run in O(k(n+m) log n/ ε 2 ) expected time while returning a (1 - 1/ e - ε) approximate solution with at least 1 - 1/ n probability. A key phase of these IM algorithms is the random reverse reachable (RR) set generation, and this phase significantly affects the efficiency and scalability of the state-of-the-art IM algorithms. In this article, we present a study on this key phase and propose an efficient random RR set generation algorithm under IC model. With the new algorithm, we show that the expected running time of existing IM algorithms under IC model can be improved to O(k ċ n log n ċ 2 ), when for any node v , the total weight of its incoming edges is no larger than a constant. For the general IC model where the weights are skewed, we present a sampling algorithm SKIP. To the best of our knowledge, it is the first index-free algorithm that achieves the optimal time complexity of the sorted subset sampling problem. Moreover, existing approximate IM algorithms suffer from scalability issues in high influence networks where the size of random RR sets is usually quite large. We tackle this challenging issue by reducing the average size of random RR sets without sacrificing the approximation guarantee. The proposed solution is orders of magnitude faster than states of the art as shown in our experiment. Besides, we investigate the issues of forward propagation and derive its time complexity with our proposed subset sampling techniques. We also present a heuristic condition to indicate when the forward propagation approach should be utilized to estimate the expected influence of a given seed set.

Given a social network G with n nodes and m edges, a positive integer k, and a cascade model C, the influence maximization (IM) problem asks for k nodes in G such that the expected number of nodes influenced by the k nodes under cascade model C is maximized. The state-of-the-art approximate solutions run in O(k(n+m)log(n)/ε2) expected time while returning a (1-1/e -ε) approximate solution with at least 1-1/n probability. A key phase of these IM algorithms is the random reverse reachable (RR) set generation, and this phase significantly affects the efficiency and scalability of the state-of-the-art IM algorithms. In this paper, we present a study on this key phase and propose an efficient random RR set generation algorithm under IC model. With the new algorithm, we show that the expected running time of existing IM algorithms under IC model can be improved to O(k· n log(n)/ε2), when for any node v, the total weight of its incoming edges is no larger than a constant. Moreover, existing approximate IM algorithms suffer from scalability issues in high influence networks where the size of random RR sets is usually quite large. We tackle this challenging issue by reducing the average size of random RR sets without sacrificing the approximation guarantee. The proposed solution is orders of magnitude faster than states of the art as shown in our experiment.

Given a social network G , a cost associated with each node, and a budget B , the budgeted influence maximization (BIM) problem aims to find a set S of nodes, denoted as the seed set, that maximizes the expected number of influenced users under the constraint that the total cost of the users in S is no larger than B. The current state-of-the-art practical solution for BIM problem provides a (1-1/ e /2 --- ε)-approximate (≈ 0.316 --- ε) result and is still inefficient on large networks. We first show that we can improve the approximation guarantee to 1 --- 1/ e β --- ε where 1 --- 1/ e β = (1 --- β) (1 --- 1/ e ), achieving a better approximation guarantee (≈ 0.355 --- ε). Next, we apply the reverse sampling based technique, a popular technique for classic influence maximization, to our studied BIM problem. However, it is non-trivial to design efficient solutions for large scale networks even the reverse sampling based technique is applied. On one hand, it is unclear how to derive tight bounds for the nodes selected by the greedy algorithm under the budgeted scenario, where each time it selects the seed node with the highest benefit-cost ratio. With tighter bounds, the algorithm can terminate as soon as the approximation ratio is satisfied, thus saving the running cost. On the other hand, the number of nodes selected under BIM problem may be quite large since it may greedily select many nodes with large benefit-cost ratio but with low costs. The time complexity of existing influence maximization algorithms heavily depends on the size of the seed set. To tackle such challenging issues, we first present new bound estimation techniques for the BIM problem. Next, we present new node selection strategies to alleviate the dependency to the size of the seed set. Extensive experiments show that our proposed solution is far more efficient than alternatives.