A Multi-Objective Influence Maximization Method Based on Cost Constraints in Hypergraphs

Abstract

This invention discloses a cost-constrained multi-objective influence maximization method in hypergraphs, belonging to the field of social network analysis technology. The method includes the following steps: S1, selecting an independent cascaded IC model as the base model; S2, employing a multi-objective optimization function to evaluate the two objectives; S3, a three-mode initialization strategy: generating diverse initial individuals in different regions of the Pareto front (PF) through an HCI-based initialization module, a unit collective influence-based initialization module, and a random initialization module; S4, operator selection: a fast non-dominated sorting algorithm, a two-point crossover operator, a mutation operator, and an adaptive adjustment mechanism for crossover and mutation rates. Through these methods, this invention solves the core technical problems of the inability to simultaneously consider influence diffusion and seed cost in hypergraph scenarios, as well as the low efficiency of evaluation and search, providing an efficient and scalable evolutionary multi-objective solution.

Description

Technical Field

[0001] This invention relates to the field of social network analysis technology, and specifically to a method for maximizing multi-objective influence in a hypergraph based on cost constraints. Background Technology

[0002] Complex networks are a fundamental tool for modeling complex systems and have received increasing research and attention in recent years. Social networks, as a typical type of complex network, have garnered significant attention due to their wide-ranging applications in information dissemination and product marketing, particularly in identifying the set of nodes that maximizes influence. The Influence Maximization (IM) problem is typically defined as selecting a set of seed nodes in a network to maximize the spread of information. Traditional IM research often uses binary graphs to represent networks, capturing only point-to-point relationships. However, in collaborative networks, transportation systems, and social groups, information often propagates in higher-order ways, such as one-to-many or many-to-many relationships. Hypergraphs allow a single hyperedge to connect any number of nodes simultaneously, providing more accurate modeling of higher-order interactions and thus becoming a powerful tool for studying cascading propagation and identifying key nodes. In practical applications such as product recommendations and news distribution, it is crucial to maximize the diffusion effect while controlling the "seed selection cost" (e.g., the operational investment corresponding to the number of nodes participating in a group). Existing hypergraph IM research often focuses on a single objective (diffusion) without systematically considering cost factors; achieving a balance between these two objectives becomes a key challenge in resource-constrained scenarios. To simulate cascading processes in high-order systems, research typically employs three types of models: Susceptible-Infected (SI), Linear Threshold (LT), and Independent Cascade (IC). Among these, the IC model, which considers both bidirectional propagation probabilities between nodes and hyperedges, has been proven to more accurately characterize the fault / information diffusion mechanism in hypergraphs, and thus serves as the modeling foundation for this invention. Complex dual-objective conflicts make the problem difficult to solve analytically. Evolutionary Multi-Objective Optimization (EMO) algorithms, with their advantages of requiring no differentiability assumptions and possessing strong search capabilities, have been successfully applied in the IM domain of graph models (such as NSGA-II), providing a feasible path to solving dual-objective problems in hypergraph scenarios.

[0003] In existing research, the technical solutions most similar to this invention can be roughly divided into the following categories: The first category is a fairness-perceived influence maximization method proposed by Hao et al. based on an evolutionary multi-objective algorithm. By introducing a new definition of "fairness," prior knowledge is integrated into a specific region of the Pareto front (PF), thereby improving the quality of the solution in that region. Fabían et al. reconstructed the IM problem into an extreme adversarial problem of "maximizing the seed set and minimizing the diffusion capacity," and solved it using a swarm intelligence metaheuristic method. Perrault et al. introduced a budget framework with total advertising cost constraints for online IM, and designed a new algorithm based on the IC model and edge-level semi-bandit feedback. Although it has theoretical analysis and experimental proof, many assumptions are difficult to verify in practical applications. The second category is single-objective heuristic algorithms based on hypergraphs, including High Hyperdegree Adaptive Algorithm (HHDA), Eigenvector Centrality, Hypergraph Collective Influence Algorithm (HCI), and k-kernel decomposition. Xie et al.'s High Hyperscale Adaptive (HHDA) method, Kovalenko et al.'s extension of graph eigenvector centrality to hypergraphs, and Lee et al.'s analysis of hypergraph seepage characteristics through (k,q)-kernel decomposition and self-consistent equations, etc.

[0004] Methods for maximizing the objective influence of ordinary graphs are difficult to extend directly to hypergraph scenarios because they only support binary edge relationships. They also suffer from drawbacks such as uneven distribution of solutions across the graph floor and slow convergence. While single-objective heuristic algorithms for hypergraphs utilize the high-order structure of the hypergraph to identify key nodes, they only focus on diffusion capacity, neglecting seed selection costs, and exhibit poor adaptability to weak nodes or highly heterogeneous structures. Although these methods have made some progress in hypergraph IM research, they generally suffer from shortcomings such as neglecting weak nodes, poor adaptability, and coarse influence ranking due to strongly coupled structures and complex cascading dynamics. Furthermore, these studies focus only on a single objective—influence diffusion—without considering seed node selection costs, rendering them ineffective in multi-objective scenarios. Therefore, this paper introduces two conflicting objectives—"diffusion amount" and "selection cost"—into the hypergraph model and conducts research based on the NSGA-II framework. Summary of the Invention

[0005] To address the aforementioned shortcomings of existing technologies, this invention provides a method for maximizing multi-objective influence in hypergraphs based on cost constraints.

[0006] To achieve the above objectives, the present invention is implemented through the following technical solutions:

[0007] A multi-objective influence maximization method based on cost constraints in a hypergraph includes the following:

[0008] I. Select the independent cascaded IC model as the basic model;

[0009] II. Evaluating the dual objectives using a multi-objective optimization function;

[0010] III. Three-Model Initialization Strategy: Diverse initial individuals are generated in different regions of the Pareto front (PF) through an HCI-based initialization module, a unit collective influence-based initialization module, and a random initialization module, respectively.

[0011] IV. Operator Selection:

[0012] 4.1. Use the fast non-dominated sorting algorithm to divide the current population into several non-dominated levels according to the dominance relationship, and then select individuals based on the non-dominated level and crowding distance to construct the next generation population;

[0013] 4.2. Use a two-point crossover operator to generate offspring;

[0014] 4.3 When a mutation occurs, random mutation, HCI mutation, or UCI mutation is performed randomly;

[0015] 4.4 Introduce an adaptive adjustment mechanism for crossover and mutation rates to adjust the crossover and mutation rates based on current cutting-edge diversity.

[0016] 5. Output the Pareto front solution set.

[0017] Furthermore, we choose the independent cascaded IC model as the basic model. Let the hypergraph H(V, E) have N nodes and M hyperedges; where V represents the set of nodes and E represents the set of hyperedges; the association between nodes and hyperedges can be described by the association matrix H: when node i is associated with a hyperedge... When there is a relationship, ;otherwise, The number of hyperedges associated with node i is denoted as ki, and is called the superdegree of that node; The number of associated nodes is denoted as That is, the cardinality of the superedge.

[0018] Furthermore, the independent cascaded IC model is used to characterize the cascading failure process in high-order systems: when node i fails, it will probabilistically cause a chain reaction (CRD) to occur. The associated hyperedges Ineffective; conversely, when the edge is superimposed. When it fails, it will then proceed with a probability. This causes its associated node i to fail.

[0019] Furthermore, the multi-objective optimization function is defined as follows:

[0020]

[0021] Where σ0(S) represents the number of seed nodes, σ1(S) represents the expected number of activations of one-hop neighbor nodes other than the seed set S, and σ2(S) represents the expected number of activations of two-hop neighbor nodes.

[0022]

[0023] Where |S| represents the number of seed nodes, This indicates that node j is a one-hop neighbor of seed set S, N S （1） Represents the set of one-hop neighbor nodes of seed set S; Indicates the first A super edge, Indicates the superedge Includes nodes j and N S （2） This represents the set of two-hop neighbor nodes of the seed set S. This indicates that node k is a two-hop neighbor, but neither a one-hop neighbor nor a seed node. Indicates the superedge e μ Includes two-hop neighbor node k, Indicates the direction from any seed node to the superedge The probability of propagation, Indicates the superedge The number of seed nodes that intersect with the seed set S; Indicates that in the superedge There is When there are 1 seed node, the probability that the superedge will successfully propagate to its one-hop neighbor node j is given; q μk Indicates the superedge The number of active nodes that intersect with the set of one-hop neighbors, p eμ Indicates the superedge The probability of being activated, p eμ q μk Indicates that in the superedge There is When there are already activated nodes, the probability that the superedge will successfully propagate to its two-hop neighbor node k.

[0024]

[0025] in, This indicates that a one-hop neighbor node j also belongs to the superedge e. μ p j s represents the probability that node j is activated. jeμ This indicates that node j is activated via hyperedge e. μ The probability of successful propagation to a second-hop neighbor;

[0026]

[0027] in, Indicates the superedge A one-hop neighbor superedge belonging to seed set S and containing node j. Indicates the superedge The probability of being activated. Indicates activation of the superedge The probability of successful propagation to a one-hop neighbor node j;

[0028]

[0029] in, Indicates the superedge from seed node i The probability of propagating and activating the superedge. This indicates that node i belongs to the seed set S.

[0030] Furthermore, the HCI-based initialization module is used to calculate the first-level collective influence (HCI) value of all nodes in the hypergraph:

[0031]

[0032] in, - k represents the collective influence of node i in first-order propagation (ICM). i This represents the degree of node i (the number of super edges connected to i). Indicates the superedge cardinality (and) (Number of connected nodes), ∂ i Let r represent the set of hyperedges associated with node i. i Indicates the hyperedge from node i The probability of propagating and activating the superedge;

[0033] Then, the HCI value of each node is multiplied by a random number between [0, 1], sorted in descending order, and the first n nodes are taken to form the initial individual.

[0034] Furthermore, the initialization module based on the collective influence of units is used to define the collective influence of unit cost as follows:

[0035]

[0036] Then, the UCI value of each node is multiplied by a random number in the range [0, 1] and sorted in descending order. The first n nodes are selected to generate individuals.

[0037] The random initialization module is used to directly select n different nodes from the entire hypergraph to form an individual.

[0038] Furthermore, a two-point crossover operator is used to generate offspring. This operator randomly selects two crossover points on two parent individuals and exchanges these two gene segments to generate two offspring of higher quality. If duplicate nodes appear in the offspring after crossover, they are randomly replaced with nodes that have not yet been used by that individual.

[0039] Furthermore, random mutation: each gene in each individual mutates with a probability of 1 / len, and can be replaced by any node that has not appeared before; if a duplicate node appears after mutation, it is randomly replaced by a node that has not yet been included in the individual.

[0040] HCI mutation: Calculate the first-level HCI value of the node corresponding to each gene in the individual and multiply the value by a random number in the range [0, 1]; after all genes have obtained HCI values, select the gene position with the smallest HCI as the mutation point; randomly select n nodes from the remaining nodes, calculate their first-level HCI values, and replace the gene at the mutation point with the node with the largest HCI.

[0041] UCI mutation: Calculate the UCI value of each gene in the individual and multiply the value by a random number in the range [0, 1]; select the gene position with the smallest UCI as the mutation point; randomly select n nodes from the remaining nodes, calculate their UCI values, and replace the gene at the mutation point with the node with the largest UCI.

[0042] Furthermore, given the Pareto front (PF), its diversity is defined as:

[0043]

[0044] Where, x i Let x represent the seed set corresponding to the i-th solution in the Pareto front (PF). j This represents the seed set corresponding to the j-th solution;

[0045] Initially, let the initial crossover rate C init =0.8, initial mutation rate M init =0.2; When population diversity declines, dynamically increase the mutation rate M. p To enhance exploration while reducing the crossover rate C p Set the minimum crossover rate C. min =0.6, maximum variation rate M max =0.4, and the crossover rate and mutation rate for each generation are calculated using the following formulas:

[0046] .

[0047] Furthermore, it also includes S6, result evaluation: Monte Carlo simulation performs a precise influence evaluation on the Pareto front solution set obtained by the above method: For each candidate seed set S, 10,000 independent IC cascade propagation simulations are performed, and the average number of activated nodes in each simulation is counted as the expected influence σ(S) of the solution, which is used as the final influence score to sort the solution set within the Pareto front and select the optimal solution.

[0048] Compared with the prior art, the beneficial effects of this invention are as follows: 1. Dual-objective modeling: This invention is the first to simultaneously incorporate two conflicting objectives, "influence diffusion" and "seed cost," into a hypergraph IM, forming a multi-objective optimization task, which fundamentally makes up for the shortcomings of single-objective research.

[0049] 2. Use of low-complexity evaluation function: This invention proposes an approximate calculation of the influence of two-hop local structures, which significantly reduces the dependence on Monte Carlo simulation and improves evaluation efficiency.

[0050] 3. This invention adopts the NSGA-II improved framework (EMOIM-HG) for hypergraphs: First, three modes (HCI / UCI / random) initialization, respectively covering high-impact-high-cost, unit-impact-priority and low-cost regions, to ensure the initial diversity of the population; Second, a crossover-mutation operator based on diversity adaptive adjustment, combined with dedicated UCI / HCI mutation, avoids getting trapped in local optima and accelerates convergence.

[0051] 4. This invention is easy to deploy and can be easily distributed or edge-computed. Evaluation does not require full-graph multi-round simulation and can be quickly executed on social platforms or local IoT devices, reducing communication and computing power overhead.

[0052] 5. Experiments have verified that, on both synthetic and real hypergraph datasets, the Pareto solution set generated by the method of this invention is more uniform and has greater diffusion at the same cost, which is significantly better than existing single-objective centrality algorithms and general multi-objective frameworks. Attached Figure Description

[0053] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.

[0054] Figure 1 This is a schematic diagram of a multi-objective influence maximization method based on cost constraints in a hypergraph according to the present invention.

[0055] Figure 2 Results of cost-influence experiments on six simulated hypergraphs.

[0056] Figure 3 Results of the cost-influence experiment on four real hypergraphs.

[0057] Figure 4 For each generation of the three algorithms, IGD (non-generational distance) is given.

[0058] Figure 5 This is a schematic diagram of the three-mode initialization strategy.

[0059] Figure 6 This is a schematic diagram of the mutation operator.

[0060] Figure 7 This is a schematic diagram of the two-point intersection operator. Detailed Implementation

[0061] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0062] Example 1: In some embodiments, please refer to the accompanying drawings. Figure 1 A cost-constrained multi-objective influence maximization method for hypergraphs, named EMOIM-HG, includes the following:

[0063] I. In hypergraphs, the influence maximization (IM) problem aims to maximize the spread of influence by selecting a fixed number of n seed nodes given a propagation model. This problem is often considered a single-objective optimization problem and has been proven to be NP-hard. Minimizing the selection cost of seed nodes is also crucial in many practical applications. For example, in news dissemination, maximizing influence while reducing costs is a typical multi-objective optimization problem, where the goal is to minimize seed selection costs while maximizing propagation influence. This single-objective optimization problem can be formalized as:

[0064]

[0065] Where S is the set of seed nodes, σ(S) represents the expected propagation scale of the influence under the selected set of seed nodes S, and c i Let C(S) represent the selection cost of node i, and let C(S) represent the total selection cost of the selected seed set S. In this invention, the selection cost of a seed node can be quantified using its superdegree (the number of superedges to which the node belongs)—the greater the superdegree, the more groups the node participates in, and the higher its selection cost.

[0066] II. Model Selection: Since there is strong correlation and coupling between nodes and hyperedges during information propagation in a hypergraph, this invention introduces an Independent Cascade (IC) model to describe the information propagation rules in the hypergraph. This helps EMOIM-HG accurately measure the collective influence of multiple nodes.

[0067] Let H(V, E) be a hypergraph with N nodes and M hyperedges. Here, V represents the set of nodes, and E represents the set of hyperedges. The association between a node and a hyperedge can be described by the association matrix H: when node i is associated with a hyperedge... When there is a relationship, ;otherwise, The number of hyperedges associated with node i is denoted as ki, and is called the degree of that node; the number of hyperedges associated with node i is called the degree of that node. The number of associated nodes is denoted as , that is, the cardinality of the hyperedge. In a hypergraph, nodes can be viewed as "agents," and hyperedges are equivalent to functional modules composed of multiple agents. The IC model is used to characterize the cascading failure process in high-order systems: when node (agent) i fails, it will fail with probability This leads to the associated hyperedges (functional modules). Ineffective; conversely, when the edge is superimposed. When it fails, it will then proceed with a probability. This causes its associated node i to fail.

[0068] III. Designing a Low-Complexity Influence Evaluation Function to Evaluate Two Objectives: A low-complexity, high-precision multi-objective optimization function can improve algorithm efficiency and guide the population towards a better solution. In independent cascade (IC) models, a large number of Monte Carlo simulations are usually required to estimate the expected influence propagation, taking into account the randomness of the propagation process, resulting in high computational complexity. To address this issue, this invention proposes a novel multi-objective optimization function that utilizes local structural information to evaluate the collective influence of the seed set. Given the significance of this function in practical applications, its value is not normalized in this paper. This multi-objective optimization function consists of two objectives: the first objective uses the local structural information of two-hop neighborhoods in the hypergraph to approximate the influence of the seed set; the second objective evaluates the seed selection cost based on the node's hyperdegree ki.

[0069] The multi-objective optimization function is defined as follows:

[0070]

[0071] Where σ0(S) represents the number of seed nodes, σ1(S) represents the expected number of activations of one-hop neighbor nodes other than the seed set S, and σ2(S) represents the expected number of activations of two-hop neighbor nodes.

[0072]

[0073] Where |S| represents the number of seed nodes, This indicates that node j is a one-hop neighbor of the seed set S (excluding the seed node itself), N S （1） This represents the set of one-hop neighbor nodes of the seed set S (excluding the seed set itself). Indicates the first A super edge, Indicates the superedge Includes nodes j and N S （2） This represents the set of two-hop neighbor nodes of the seed set S (excluding one-hop neighbors and the seed set). This indicates that node k is a two-hop neighbor, but neither a one-hop neighbor nor a seed node. Indicates the superedge e μ Includes two-hop neighbor node k, Indicates the direction from any seed node to the superedge The probability of propagation, Indicates the superedge The number of seed nodes that intersect with the seed set S (used when affecting one-hop neighbor j); Indicates that in the superedge There is When there are q seed nodes, the probability that the superedge will successfully propagate (activate) to its one-hop neighbor node j. μk Indicates the superedge The number of active nodes that intersect with the set of one-hop neighbors (used when affecting the set of two-hop neighbors k), p eμ Indicates the superedge The probability of being activated, p eμ q μk Indicates that in the superedge There is When there are 1 activated node, the probability that the superedge will successfully propagate to its two-hop neighbor node k.

[0074]

[0075] in, This indicates that a one-hop neighbor node j also belongs to the superedge e. μ p j s represents the probability that node j is activated. jeμ This indicates that node j is activated via hyperedge e. μ The probability of successful propagation to a second-hop neighbor;

[0076]

[0077] in, Indicates the superedge A one-hop neighbor superedge belonging to seed set S and containing node j. Indicates the superedge The probability of being activated. Indicates activation of the superedge The probability of successful propagation to a one-hop neighbor node j;

[0078]

[0079] in, Indicates the superedge from seed node i The probability of propagating and activating the superedge. This indicates that node i belongs to the seed set S.

[0080] This invention employs a low-complexity impact evaluation function and introduces a two-hop local structure approximation. Under the IC model, σ(S) is calculated using only the closed-form probability of the 0 / 1 / 2-hop neighborhood of the seed node (without global Monte Carlo), reducing the complexity to O(|S|d). 2 ).

[0081] IV. Three-mode initialization strategy ( Figure 5 In EMOIM-HG, a candidate solution represents a seed set, denoted by a vector of length n. Each gene corresponds to an index of a node in the hypergraph, and nodes within the same vector are guaranteed to be unique. A high-quality initial population typically accelerates convergence and improves algorithm performance. To ensure uniformly distributed solutions on the Pareto front (PF), EMOIM-HG generates diverse initial individuals in different regions of the PF using three modules:

[0082] 4.1. An HCI-based initialization module is used to calculate the first-order collective influence (HCI) value of all nodes in the hypergraph:

[0083]

[0084] in, - k represents the collective influence of node i in first-order propagation (ICM). i This represents the degree of node i (the number of super edges connected to i). Indicates the superedge cardinality (and) (Number of connected nodes), ∂ i Let r represent the set of hyperedges associated with node i. i Indicates the hyperedge from node i The probability of propagating and activating the superedge;

[0085] Then, the HCI value of each node is multiplied by a random number between [0, 1], sorted in descending order, and the first n nodes (n is the number of seeds to be selected) are taken to form the initial individuals.

[0086] This module aims to generate solutions that have high impact diffusion but also relatively high seed selection costs.

[0087] 4.2 Initialization module based on Unit Collective Influence (UCI), used to define the collective influence of unit cost as:

[0088]

[0089] Then, the UCI value of each node is multiplied by a random number in the range [0, 1] and sorted in descending order. The first n nodes are selected to generate individuals.

[0090] The solution generated by this module maintains a high degree of influence diffusion while having a moderate seed selection cost, thus achieving a balance between influence and cost.

[0091] 4.3 Random initialization module, used to directly select n different nodes from the entire hypergraph to form an individual.

[0092] This module can generate solutions with low impact diffusion and low seed cost, thus enhancing population diversity.

[0093] This invention employs a three-mode initialization strategy: the HCI module prioritizes the generation of high-impact, high-cost individuals; the UCI module generates individuals with high unit return (diffusion / cost); and the random module covers low-cost regions. This improves convergence speed and solution set uniformity.

[0094] V. Operator Selection:

[0095] 5.1. The Fast Non-dominated Sorting algorithm is used to divide the current population into several non-dominated levels (fronts) according to dominance relationships. Then, individuals are selected based on the non-dominated level and crowding distance to construct the next generation population, stopping after a given number of iterations. Non-dominated sorting and crowding distance calculations are performed after evaluating the bi-objective.

[0096] The fast non-dominated rank ranking method is as follows: For each individual, calculate the number of other individuals dominating that individual and the set of other individuals dominated by that individual. Individuals not dominated by any individual are assigned to the first non-dominated rank. Remove all individuals of the first rank from the population, and repeat the above process for the remaining individuals to generate the second rank, the third rank, and so on, until all individuals are assigned a rank.

[0097] Crowding distance is used to measure the density of an individual's solution set in the target space. If an individual has a large Euclidean distance between its adjacent solutions (sorted according to their target values), it indicates that the solution region is relatively sparse; conversely, it indicates that the solutions are relatively concentrated. During the selection phase, individuals are first filled in order of non-dominance level from low to high. If including all individuals at a certain level would exceed the population size, then individuals within that level are sorted in descending order of crowding distance, and those with larger distances are selected for the next generation, thus ensuring the diversity and uniformity of solution distribution.

[0098] 5.2, such as Figure 7 As shown, a two-point crossover operator is used to generate offspring. This operator randomly selects two crossover points on two parent individuals and swaps these two gene segments to generate two offspring of higher quality. If duplicate nodes appear in the offspring after crossover, they are randomly replaced with nodes that have not yet been used by that individual.

[0099] 5.3, such as Figure 6 As shown, a mutation operator is selected when mutation occurs. To prevent the algorithm from getting trapped in local optima, this invention designs a novel mutation operator consisting of three modules: random mutation, HCI mutation, and UCI mutation. During mutation, one of these modules is randomly selected for execution.

[0100] Random mutation: Each gene in each individual mutates with a probability of 1 / len (individual), and can be replaced by any node that has not appeared before; if a duplicate node appears after mutation, it is randomly replaced by a node that has not yet been included in the individual.

[0101] HCI mutation: Calculate the first-level HCI value of the node corresponding to each gene in the individual and multiply the value by a random number in the range [0, 1]. After all genes have obtained HCI values, select the gene position with the smallest HCI as the mutation point to increase the probability of generating low-influence nodes. Randomly select n nodes from the remaining nodes, calculate their first-level HCI values, and replace the gene at the mutation point with the node with the largest HCI.

[0102] UCI mutation: Calculate the UCI value of each gene in the individual and multiply the value by a random number in the range [0, 1]; select the gene position with the smallest UCI as the mutation point to increase the probability of generating low unit influence nodes; randomly select n nodes from the remaining nodes, calculate their UCI values, and replace the gene at the mutation point with the node with the largest UCI.

[0103] The aforementioned mutation operators can guide the population to evolve in a direction that achieves a better balance between influence diffusion and seed selection costs while ensuring search diversity.

[0104] 5.4 Adaptive Adjustment Mechanism for Crossover and Mutation Rates: To further prevent getting trapped in local optima, EMOIM-HG introduces an adaptive probability adjustment strategy based on the genotypic diversity of individuals at the Pareto front (PF). Given a Pareto front PF, its diversity is defined as:

[0105]

[0106] Where, x i Let x represent the seed set corresponding to the i-th solution in the Pareto front (PF). j This represents the seed set corresponding to the j-th solution;

[0107] Initially, let the initial crossover rate C init =0.8, initial mutation rate M init =0.2. When population diversity declines, the mutation rate M is dynamically increased. p To enhance exploration while reducing the crossover rate C p (and satisfy M) p +C p =1). To ensure convergence, a minimum crossover rate C is set. min =0.6, maximum variation rate M max =0.4, and the crossover rate and mutation rate for each generation are calculated using the following formulas:

[0108]

[0109] This invention employs an adaptive evolution operator, dynamically selecting from three operators—random, UCI, and HCI—based on individual scores to increase the probability of escaping local optima. The crossover rate C is adjusted using the current frontier diversity D. p Variation rate M p To avoid premature development or shocks.

[0110] VI. Results Evaluation

[0111] In the EMOIM-HG framework, Monte Carlo simulation is used to accurately evaluate the influence of the Pareto front solution set obtained by the above method: for each candidate seed set S, 10,000 independent IC cascade propagation simulations are performed, and the average number of activated nodes in each simulation is counted as the expected influence σ(S) of the solution. This is used as the final influence score to sort the solution set within the Pareto front and select the optimal solution.

[0112] This invention performs Monte Carlo simulations to calculate the range of influence and find the Pareto front, taking the Pareto front as the final result.

[0113] Figure 2As can be seen, on 6 composite and 4 real hypergraphs, EMOIM-HG outputs over 600 non-dominated solutions, covering high-, medium-, and low-cost regions, while traditional centrality or single-objective algorithms only provide solutions for a single point. This demonstrates that the present invention simultaneously considers high influence diffusion and low seed cost, providing decision-makers with a complete Pareto compromise solution set.

[0114] Figure 3 As can be seen, the present invention has a diffusion-cost advantage of over 25% in both Restaurant-reviews (commercial) and Geometry-questions (scientific research) datasets, and exhibits high generalization and scalability across different scenarios.

[0115] Figure 4 It can be seen that the present invention accelerates convergence and has a more uniform front distribution. The IGD (inverse generational distance) index decreases by an average of 40-60% in ER, SF, and UF networks, and approaches the final front within 10 generations, while random initialization of NSGA-II requires more than 30 generations.

[0116] This invention has the following advantages:

[0117] Dual-objective modeling: This invention is the first to simultaneously incorporate two conflicting objectives, "influence diffusion" and "seed cost," into a hypergraph IM, forming a multi-objective optimization task and fundamentally making up for the shortcomings of single-objective research.

[0118] Using a low-complexity evaluation function: This invention proposes an approximate calculation of the impact of two-hop local structures, which significantly reduces the dependence on Monte Carlo simulation and improves evaluation efficiency.

[0119] This invention adopts the NSGA-II improved framework (EMOIM-HG) for hypergraphs: First, three modes (HCI / UCI / random) initialization, respectively covering high-impact-high-cost, unit-impact-priority, and low-cost regions, to ensure the initial diversity of the population; Second, a crossover-mutation operator based on diversity adaptive adjustment, combined with dedicated UCI / HCI mutation, avoids getting trapped in local optima and accelerates convergence.

[0120] This invention is deployment-friendly, easy to distribute or edge compute, evaluation does not require full graph multi-round simulation, and can be quickly executed on social platforms or local IoT devices, reducing communication and computing power overhead.

[0121] Through experiments, this invention demonstrates that, on both synthetic and real hypergraph datasets, the Pareto solution set generated by the method of this invention is more uniform and has greater diffusion at the same cost, significantly outperforming existing single-objective centrality algorithms and general multi-objective frameworks.

[0122] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions will not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for maximizing multi-objective influence in a hypergraph based on cost constraints, characterized in that, Includes the following: S1. Select the independent cascaded IC model as the basic model; S2. Use a multi-objective optimization function to evaluate the dual objectives; S3. Three-mode initialization strategy: Diverse initial individuals are generated in different regions of the Pareto front (PF) through an HCI-based initialization module, a unit collective influence-based initialization module, and a random initialization module. S4. Operator Selection: S4.

1. Use the fast non-dominated sorting algorithm to divide the current population into several non-dominated levels according to the dominance relationship, and then select individuals based on the non-dominated level and crowding distance to construct the next generation population; S4.

2. Use the two-point crossover operator to generate offspring; S4.3 When a mutation occurs, perform random mutation, HCI mutation, or UCI mutation randomly. S4.4 Introduce an adaptive adjustment mechanism for crossover and mutation rates to adjust the crossover and mutation rates based on current cutting-edge diversity. S5, Output the Pareto front solution set; Choosing the independent cascaded IC model as the basic model, let the hypergraph H(V,E) have N nodes and M hyperedges; where V represents the set of nodes and E represents the set of hyperedges; the association between nodes and hyperedges can be described by the association matrix H: when node i is associated with hyperedge e γ When there is a correlation, H i e γ =1; otherwise, H i e γ =0; the number of hyperedges associated with node i is denoted as k. i This is called the degree of the node; and the degree of the hyperedge e. γ The number of associated nodes is denoted as m. γ That is, the cardinality of the superedge; Independent cascaded IC models are used to characterize cascading failure processes in high-order systems: when node i fails, it will fail with probability r. i e γ The associated hyperedge e γ Failure; conversely, when the superedge e γ When it fails, it will then proceed with probability qe. γ i causes its associated node i to fail; The multi-objective optimization function is defined as follows: ； Where σ0(S) represents the number of seed nodes, σ1(S) represents the expected number of activations of one-hop neighbor nodes other than the seed set S, and σ2(S) represents the expected number of activations of two-hop neighbor nodes. ； Where |S| represents the number of seed nodes, This indicates that node j is a one-hop neighbor of seed set S, N S （1） Represents the set of one-hop neighbor nodes of the seed set SSS; e γ Let e ​​represent the γth hyperedge. γ ∋j represents the superedge e γ Includes nodes j and N S （2） This represents the set of two-hop neighbor nodes of the seed set S. This indicates that node k is a two-hop neighbor, but neither a one-hop neighbor nor a seed node. Indicates the superedge e μ Includes two-hop neighbor nodes k, r eγ This indicates that any seed node extends to the superedge e. γ The propagation probability, q γj Indicates the superedge e γ The number of seed nodes that intersect with the seed set S; r eγ q γj Indicates that at the hyperedge e γ There is q γj When there are 1 seed node, the probability that the superedge will successfully propagate to its one-hop neighbor node j is given; q μk Indicates the superedge e γ The number of active nodes that intersect with the set of one-hop neighbors, p eμ Indicates the superedge e γ The probability of being activated, p eμ q μk Indicates that at the hyperedge e γ There is q γk When there are already activated nodes, the probability that the superedge will successfully propagate to its two-hop neighbor node k. ； in, This indicates that a one-hop neighbor node j also belongs to the superedge e. μ p j s represents the probability that node j is activated. jeμ This indicates that node j is activated via hyperedge e. μ The probability of successful propagation to a second-hop neighbor; ； in, Indicates the superedge e γ A one-hop neighbor superedge belonging to the seed set S and containing node j, p eγ Indicates the superedge e γ The probability of being activated, s eγj Indicates activation of the hyperedge e γ The probability of successful propagation to a one-hop neighbor node j; ； Where, r ieγ This indicates that the seed node i is directed to the superedge e. γ The probability of propagating and activating the superedge. This indicates that node i belongs to the seed set S.

2. The method for maximizing multi-objective influence in a hypergraph based on cost constraints according to claim 1, characterized in that, The HCI-based initialization module is used to calculate the first-level collective influence (HCI) value of all nodes in the hypergraph: ； in, - k represents the collective influence of node i under the first-level propagation ICM. i Let m represent the degree of node i. γ Indicates the superedge e γ The cardinality, ∂ i Let r represent the set of hyperedges associated with node i. i This indicates that node i is directed to the hyperedge e. γ The probability of propagating and activating the superedge; Then, the HCI value of each node is multiplied by a random number between [0, 1], sorted in descending order, and the first n nodes are taken to form the initial individual.

3. The method for maximizing multi-objective influence in a hypergraph based on cost constraints according to claim 2, characterized in that, The initialization module based on the collective influence of units is used to define the collective influence of unit cost as follows: ； Then, the UCI value of each node is multiplied by a random number in the range [0, 1] and sorted in descending order. The first n nodes are selected to generate individuals. The random initialization module is used to directly select n different nodes from the entire hypergraph to form an individual.

4. The method for maximizing multi-objective influence in a hypergraph based on cost constraints according to claim 3, characterized in that, A two-point crossover operator is used to generate offspring. This operator randomly selects two crossover points on two parent individuals and exchanges these two gene segments to generate two offspring of better quality. If duplicate nodes appear in the offspring after crossover, they are randomly replaced with nodes that have not yet been used by that individual.

5. The method for maximizing multi-objective influence in a hypergraph based on cost constraints according to claim 4, characterized in that, Random mutation: Each gene in each individual mutates with a probability of 1 / len, and can be replaced by any node that has not appeared before; if a duplicate node appears after mutation, it is randomly replaced by a node that has not yet been included in the individual. HCI mutation: Calculate the first-level HCI value of the node corresponding to each gene in the individual and multiply the value by a random number in the range [0, 1]; after all genes have obtained HCI values, select the gene position with the smallest HCI as the mutation point; randomly select n nodes from the remaining nodes, calculate their first-level HCI values, and replace the gene at the mutation point with the node with the largest HCI. UCI mutation: Calculate the UCI value of each gene in the individual and multiply the value by a random number in the range [0, 1]; select the gene position with the smallest UCI as the mutation point; randomly select n nodes from the remaining nodes, calculate their UCI values, and replace the gene at the mutation point with the node with the largest UCI.

6. The method for maximizing multi-objective influence in a hypergraph based on cost constraints according to claim 5, characterized in that, Given a Pareto front (PF), its diversity is defined as: ； Where, x i Let x represent the seed set corresponding to the i-th solution in the Pareto front (PF). j This represents the seed set corresponding to the j-th solution; Initially, let the initial crossover rate C init =0.8, initial mutation rate M init =0.2; When population diversity declines, dynamically increase the mutation rate M. p To enhance exploration while reducing the crossover rate C p Set the minimum crossover rate C. min =0.6, maximum variation rate M max =0.4, and the crossover rate and mutation rate for each generation are calculated using the following formulas: 。 7. The method for maximizing multi-objective influence in a hypergraph based on cost constraints according to claim 6, characterized in that, It also includes S6, result evaluation: Monte Carlo simulation performs an accurate influence evaluation on the Pareto front solution set obtained by the above method: For each candidate seed set S, 10,000 independent IC cascade propagation simulations are performed, and the average number of activated nodes in each simulation is counted as the expected influence σ(S) of the solution, which is used as the final influence score to sort the solution set within the Pareto front and select the optimal solution.

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