Graph-structured combinatorial problems in complex networks are prevalent in many domains, and are computationally demanding due to their complexity and non-linear nature. Traditional evolutionary algorithms (EAs), while robust, often face obstacles due to content-shallow encoding limitations and lack of structural awareness, necessitating hand-crafted modifications for effective application. In this work, we introduce an original framework, visual evolutionary optimization (VEO), leveraging multimodal large language models (MLLMs) as the backbone evolutionary optimizer in this context. Specifically, we propose a context-aware encoding scheme, representing the solution of the network as an image. In this manner, we can utilize MLLMs’ image processing capabilities to intuitively comprehend network configurations, thus enabling machines to solve these problems in a human-like way. We develop MLLM-based operators tailored for various evolutionary optimization stages, including initialization, crossover, and mutation. Furthermore, we propose that graph sparsification can effectively enhance the applicability and scalability of VEO on large-scale networks, owing to the scale-free nature of real-world networks. We demonstrate the effectiveness of our method using a well-known task in complex networks, influence maximization, and validate it on eight different realworld networks of various structures. The results confirm VEO’s reliability and enhanced effectiveness compared to traditional evolutionary optimization.

Real-world complex systems usually share structural characteristics such as the small-world property, power-law distributions, and community structure. Multidomain evolutionary optimization (MDEO) leverages these commonalities to search for optimal solutions in multiple domains simultaneously. However, it still faces challenges in achieving effective cooperation when the networks involved are of highly imbalanced sizes. To address this issue, we propose the harmonized MDEO (HMDEO), in which two graph coarsening strategies are developed to jointly coarsen the large network to a fine level, acting as the bridge between the large network and the small network for knowledge exchange. Following that, we propose a bidirectional optimization framework incorporating two cases:large-to-smallandsmall-to-large, allowing solutions optimized in one domain to be seamlessly transferred to another domain of varied scales. To enhance applicability, we also tailor the measurement of network similarity and the network alignment strategy targeted to imbalanced-size scenarios. The effectiveness of HMDEO is demonstrated through experiments on several pairs of networks of differing scales, where HMDEO outperforms other optimization approaches in addressing adversarial edge perturbation against community detection in complex systems.

Evolutionary algorithms (EAs) have proven effective in exploring the vast solution spaces typical of graph-structured combinatorial problems. However, traditional encoding schemes, such as binary or numerical representations, often fail to straightforwardly capture the intricate structural properties of networks. Through employing the image-based encoding to preserve topological context, this study utilizes multimodal large language models (MLLMs) as evolutionary operators to facilitate structure-aware optimization over graph data. To address the visual clutter inherent in large-scale network visualizations, we leverage graph sparsification techniques to simplify structures while maintaining essential structural features. To further improve robustness and mitigate bias from different sparsification views, we propose a cooperative evolutionary optimization framework that facilitates cross-domain knowledge transfer and unifies multiple sparsified variants of diverse structures. Additionally, recognizing the sensitivity of MLLMs to network layout, we introduce an ensemble strategy that aggregates outputs from various layout configurations through consensus voting. Finally, experiments on real-world networks through various tasks demonstrate that our approach improves both the quality and reliability of solutions in MLLM-driven evolutionary optimization.

Knowledge transfer-based evolutionary optimization has garnered significant attention, such as in multitask evolutionary optimization (MTEO), which aims to solve complex problems by simultaneously optimizing multiple tasks. While this emerging paradigm has been primarily focusing on task similarity, there remains a hugely untapped potential in harnessing the shared characteristics between different domains. For example, real-world complex systems usually share the same characteristics, such as the power-law rule, small-world property and community structure, thus making it possible to transfer solutions optimized in one system to another to facilitate the optimization. Drawing inspiration from this observation of shared characteristics within complex systems, we present a novel framework, multidomain evolutionary optimization (MDEO). First, we propose a community-level measurement of graph similarity to manage the knowledge transfer among domains. Furthermore, we develop a graph-learning-based network alignment model that serves as the conduit for effectively transferring solutions between different domains. Moreover, we devise a self-adaptive mechanism to determine the number of transferred solutions from different domains, and introduce a knowledge-guided mutation mechanism that adaptively redefines mutation candidates to facilitate the utilization of knowledge from other domains. To evaluate its performance, we use a challenging combinatorial problem known as adversarial link perturbation as the primary illustrative optimization task. Experiments on multiple real-world networks of different domains demonstrate the superiority of the proposed framework in efficacy compared to classical evolutionary optimization.

Identifying the diffusion origin within networks is critically important for controlling the spread of information, diseases, or other contagions. In this work, we study how to recognize the diffusion source based on limited observational knowledge. To date, the applicability of existing methods is often challenged when dealing with large-scale networks. To improve the scalability, we here develop a community-based source localization (CSL) model by splitting the network into several clusters from the community perspective. In addition, we study an issue that has received less attention but is rather important, i.e., observer deployment. Specifically, we categorize three types of observers and propose a two-stage placement strategy to enhance the accuracy of localization. Our findings suggest that the optimized deployed observers tend to situate at key positions and have higher betweenness compared to non-observer nodes. Based on the information obtained from deployed observers, we further develop five different strategies to locate the community where the source belongs, and our study shows that the diffusion often occurs within a small number of communities so that the search scope can be effectively refined. We validate our method through simulations on various networks, and the experimental results demonstrate the excellent efficacy and efficiency of CSL in early source localization.

Identifying the transmission source based on network topology is of great significance for timely blocking migration from epidemic areas. To date, several sensor-based works have effectively focused on knowledge related to a single aspect, such as correlating the deterministic delay and observational delay or calculating the infection density. In this work, we study this problem from a new direction and develop a multi-attribute source estimator (MASE) and its extension Fuzzy-MASE. The proposed framework allows us to transform source localization into a multi-attribute decision-making (MADM) problem, ensuring better flexibility and extensibility. In addition, we propose a novel method, i.e., source dimension which can be taken as a supplementary attribute to MASE or used as an independent estimator called source dimension estimator (SDE). Crucially, we also study a previously underappreciated yet vital issue—how to effectively manage the influence of noisy observation sources. To broaden the generality and applicability of our research, we examine arbitrary graph structures rather than any specific configurations. As part of our validation, we have simulated diffusion on networks of different scales. The experimental results demonstrate the strengths of our proposed method over other existing methods in localization accuracy.

As the number of social network users grows exponentially with increasingly complex profiles, community detection algorithms play a critical role in user portrait analysis. The associated privacy concerns, however, have not sufficiently received the attention that it deserves. In this work, we investigate methods for obfuscating the original community structure by modifying a small number of connections imperceptibly so as to protect the privacy of users. The existing evolutionary models have some successes in this type of NP-hard problem but can only be applied to small-scale datasets, rendering them inadequate for real-world applications. To alleviate this problem, we propose an original and novel CoeCo, a cooperative evolutionary community obfuscation model. In CoeCo, we leverage the divide-and-conquer strategy and put forward a co-evolutionary optimization algorithm suitable for community structure, in which two different fitness functions promote each other to find the optimal edge set. In addition, the motif hypergraph and permanence are used to improve population initialization. The experimental results indicate that our proposed method can achieve excellent efficacy in obfuscating community structure and also greatly reduces running time.

The rapid development of community detection algorithms, while serving users in social networks, also brings about certain privacy problems. In this work, we study community deception, which aims to counter malicious community detection attacks by imperceptibly modifying a small part of the connections. However, it is computationally challenging to find an optimal edge set since it is an NP-hard problem. To address this issue, we propose a self-adaptive evolutionary deception (SAEP) framework. In SAEP, a novel fitness function that is able to capture local and global community change is being proposed. SAEP also provides a well-designed initialization mechanism to reduce the size of the solution space. In addition, we assign an indicator to each gene to reflect its strength within the chromosome that it belongs to, thereby a set of self-adaptive operations can be defined to enhance the algorithm’s stability and efficacy. Furthermore, we define a new “edge distance” to conserve the limited modification resource on the graph. In the experiment, the proposed method is tested against different community detection methods using various real-world datasets, and the experimental results demonstrate that SAEP improves significantly over state-of-the-art approaches in terms of effectiveness.

The identification of influential nodes in complex networks is an open issue. To address it, many centrality measures have been proposed, among which the most representative iteration algorithm is the PageRank algorithm. However, it ignores the correlation between nodes and assumes that the jumping probability from a node to its adjacent nodes is the same. To make up it, we proposed a method to improve the PageRank based on the structural similarity of nodes calculated by Kullback–Leibler divergence. The Susceptible-infected (SI) model was used in six real networks, and the results of comparison experiments demonstrate the effectiveness of the proposed method.

Because of the advantages of graphs in visualizing the relationship between individuals, complex networks have been widely used and greatly developed. In real-world applications of Dempster–Shafer evidence theory, there are usually thousands of sensors collecting information. It is easy to be overwhelmed by the mass of information and ignore the connections between them. The rise of the semisupervised learning method graph convolutional network makes it possible to address this issue. In this article, inspired by complex network, the basic probability assignment function, the base function of evidence theory, is modeled in a novel form of the network graph. Some typical issues of evidence theory, such as conflicting evidence, multiclass evidence clustering, and computational complexity for large-scale fusion are systematically addressed in the framework of the proposed network model. What's more, a new combination rule is presented from the point view of the graph. The empirical results of experiments on real data set demonstrate the potential and feasibility of complex networks in traditional evidence theory.