Structure-Aware Cooperative Ensemble Evolutionary Optimization on Combinatorial Problems with Multimodal Large Language Models
Authors: Jie Zhao
Kang Hao Cheong
Venue: NeurIPS 2025 poster ()
Keywords: Evolutionary Optimization
Multimodal Large Language Models
Complex Networks
Combinatorial Optimization
Primary area: machine_learning_for_sciences

Abstract: 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.

Structure-Aware Cooperative Ensemble Evolutionary Optimization on Combinatorial Problems with Multimodal Large Language Models
Jie Zhao', Kang Hao Cheong 1.2+
School of Physical and Mathematical Sciences, Nanyang Technological University Cooleee of Computing and Data Science, Nanyang Technological University
Abstract
Evolutionary algorithms (EAs) have proven effective in exploring the vast solu- tion spaces typical of graph-structured combinatorial problems. However, tradi- tional encoding schemes, such as binary or numerical representations, often fail 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) evolutionary operators facilitate Atructure-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, introduce ensemble strategy that aggregates outputs from various layout figurations 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.
1 Introduction
Graph-structured combinatorial problems in real-world complex systems occupy central role in various domains .T] These problems require selecting an optimal set of nodes within a network and are inherently NP-hard due to the combinatorial explosion of possible node combinations 3).
Evolutionary algorithms (EAs) have been extensively employed to address such challenges because of their ability to effectively deal with non-linearity and explore discrete search spaces 44.
The encoding scheme plays a fundamental role addressing discrete optimization problems, as it dictates how solutions are structured and operated within the EA framework [5] 61. Different combinatorial tasks have unique requirements, for example, permutation and path encodings are commonly used for scheduling [7] and routing problems (8), while binary and
encodings are often applied to node or edge subset selection tasks (]. Despite their widespread use, conventional encoding schemes represent data in an abstract way, overlooking their contextual significance within the network. Therefore, evolutionary operators like crossover and mutation are applied blindly, without recognizing the underlying structural relationships of solutions.
"Corresponding author: kanghao. Sanghao.cheeng@@tt. eew.eetttuuuuuu.....
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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An image-basee based encoding framework, combined with multimodal large language models (MLLMs) as evolutionary operators, offers a practical approach for complex network optimization [TO. Visual representations can preserve structural and contextual nuances that are often lost in string-based encodings. By virtue of their capacity to interpret both textual and visual information [[1, MILMs facilitate the integration of context awareness into the optimization process, rendering them especially effective for tackling combinatorial problems in networks.
However, visualizing large-scale networks often leads to cluttered plots, hindering structural interpre- tation. Graph sparsification can help reduce this complexity while preserving key topological features [3. Yet, any single sparsified view may introduce bias by overemphasizing specific structures. Therefore, we distribute the optimization process across multiple sparsified versions, allowing diverse structural insights to emerge and reducing reliance on any single simplification.
While cooperative evolutionary optimization has shown success in handling complex tasks [14 13), traditional approaches generally rely on the assumption of a shared domain, which incompat- ible with our multi-sparsification setting. To make the cooperative framework adaptable to the multi-domain case, we implement a master-worker architecture, where a central coordinator directs subprocessors operating on distinct sparsified networks. Moreover, we establish a cross-domain mapping mechanism to enable effective knowledge transfer across these subprocessors.
Recent studies indicate that MLLMs are sensitive variations in network layouts [16], which can influence optimization outcomes. To address this limitation, we propose an ensemble model that integrates multiple layout styles. By employing consensus voting mechanism to fuse their outputs, we are able to enhance robustness and mitigate layout-induced bias. To demonstrate the effectiveness of the proposed method, an important combinatorial problem, influence maximization, utilized as the main case study 7 Our key contributions are summarized as follows:
To counteract the bias of relying on any single simplified network, we distribute the op- timization process across multiple sparsified versions. This approach preserves diverse structural insights and improves optimization performance.
In the cooperative framework, we adopt the master master-worker model to coordinate optimization across heterogencous sparsified networks. By establishing the cross-domain mapping mech- anism, knowledge is effectively transferred among subprocessors operating on networks of different scales and characteristics.
Recognizing the sensitivity of MLLMS to network visualization layouts, we develop an ensemble model that incorporates multiple layouts. A consensus voting mechanism is then proposed aggregate these representations mitigate layout-induced variance and improve the optimization robustness.
Illustrative task: The focus of this paper not on advancing the state-of-the-art in the influence maximization problem, but rather on using this challenging problem as representative task to demonstrate how MLLMs can function as structure-aware evolutionary optimization operators. Moreover, several additional toy tasks are presented in Section demonstrate the generalizability and potential structure-aware optimization. The implementation code is available online at StructureAAarry EO
2 Related Work
2.1
Evolutionary Optimization on Combinatorial Problems
EAs have proven effective across various combinatorial tasks, including routing g scheduling
[19]. network robustness enhancement (20] and network reconstruction [21]. Their adaptability
also extends to tasks like community deception 222, sensor selection (23 and dynamic community
detection [24] In EAs, the encoding strategies are problem-specific, e.g., perautationn/path encodings for scheduling and routing 25), and binary/label--asee encoding for influence maximization (26).
However, such encoding schemes often lack structural awareness, leading to naive modifications.

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2.2 LLM-Assisted Evolutionary Optimization
Recent work has integrated LLMs into evolutionary frameworks 27 28 particularly as search operators [29 30] and automation tools [3]. Specifically, LLMs could be used to implement mutation and crossover 32 [3], and support multi-objective optimization 34 351. However, LLMs struggle eapture higher-order structural information, and representing networks natural language will incur substantial computational costs, making them less suitable for structure- aware optimization. An alternative is offered by MILMs with image-based encodings, which address such problems intuitively in human-like manner [10].
Cooperative and Ensemble MLLM-based Evolutionary Optimization
In this section, we provide detailed explanation of the proposed framework, including graph sparsification and cooperative//nsemble MLLM-based evolutionary optimization.
3.1 Problem Formulation
Given network G EV, the goal of the influence maximization problem is to select seed set S V of size that maximizes the expected influence spread:
max SC SCV.S o(S).
where (S) denotes the expected number of nodes activated under a diffusion model. Estimating (SS) typically requires costly Monte Carlo simulations, posing scalability issues. Therefore, adopt the expected diffusion value (EDV) (36] as a surrogate:
EDV(S) k
Σ
(1-1)p))
BEN(S)S
where p is the influence probability, 6(b) counts the edges from to b, and M(S) the one-hop neighborhood of S, including S itself.
3.2 Graph Sparsification
Visualizing large-scale labeled networks result in excessive clutter, affecting MLLMs' recognition on graph structure. To address this, we apply graph sparsification to obtain a miniature version the original networks that preserves essential structures while reducing noise. Let the original network be = (V, E); a sparsified version is produced via a sparsification operator S(;:),, yielding:
G = SGG
where 0; is the sparsification strategy. Different sparsification methods preserve different structural properties, with some emphasizing global connectivity and others focusing local clustering. By applying multiple strategies, we can generate a set of simplified networks capturing diverse aspects of the original:
G, = (G,,
where G² C G.
In this work, employ two heuristic graph sparsification techniques to reduce network size while preserving essential structural properties from various perspectives.
3.2.1 Degree-based Sparsification
In this method, we simplify the graph by retaining nodes with high degree. Given graph G = (V, E), we define N as the number of nodes retain, and the subset V,, of retained nodes is:
V. = {or E V d(o)) 2 d(us). E V.) with VV = N
where d(v) represents the degree of the node

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3.2.2 Community-based Sparsification
Given partition of G into communities C= [C1, C2,...,,Cmn Cm the number of selected nodes each community is determined by:
C V
CC|
=
N
Within each community, the nodes chosen based their betweenness centrality values, denoted as b(v) for v E V. The selected nodes V, are given by:
m
V. Ute e C, b(o,) 2 b(0r), Vey C, \ V33 with IV, [C]]-
In the community- based sparsification method, the initial community distribution of network is obtained by the FastGreedy algorithm [37]
3.2.3 Subgraph Refinement
After selecting the nodes, we construct the subgraph G. (Va, E.), where
E = f [(u, v) E E u V.).
If the number of edges in E, exceeds the predefined edge number No, prune edges randomly until only Nº remain. Finally, we remove the isolated nodes and retain only the largest connected component to ensure structural coherence.
3.3 Knowledge Transter-based Cooperative Evolutionary Optimization
In our cooperative optimization, each processor optimizes over different sparsified networks G (V; E with the knowledge transfer from each other (see Figure for the illustration), optimizing candidate solutions V. During the optimization, each processor evolves population P., where
Ys E P.. 8 EE V
The goal of the i-th processor to find the individual sf with the best performance:
arg max (s) =EP:
where [s; G) is the fitness function evaluate solution regarding the original network G. Every candidate solution optimized in G E G, will be mapped back to the domain of network G via function for fitness evaluation to ensure that the solutions relevant and effective with respect to the full network. Following that, the elite solution optimized in one sparsified domain will be transferred into another sparsified domain via the original domain. Specifically, we define two mapping functions:
Projection to Original: A function 6; Vi V maps nodes from a sparsified network G; to their corresponding nodes in G. A candidate solution C V is projected to the domain of the original network (s) = (v()) u }} where (x(8) E V.
Master
(Original network)
Injection (Reverse Projection): A function Φ7
Vi maps candidate solutions C V from the original network into the domain of given sparsified network G i.e., of (s) = { oF (v) E s where o (s) V1.
(Sparsified network)
network)
(Sparsified
Remark 1. Let G and G. be two simplified rep- resentations of original network G, obtained via distinct graph sparsification methods. Although Φz(v) is valid node in G for any Vi. there is guar- antee that (x(v) E V is contained within the vertex set V3 of G¹, when attempting map via the injec- tion function 67- This potential infeasibility arises because different sparsification methods may retain distinct subsets of nodes, even though both sparsified networks originate from G.
Figure 1: Diagram of the cooperative evolu- tionary optimization. The master unit stores the elite solutions and is responsible for knowledge transfer across subprocessors.

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The master process coordinates the search by collecting high-quality solutions from the subprocessors. Each subprocessor periodically sends its best candidate (projected via () to the master, which aggregates them into global elite pool &. Specifically, if the i-th subprocessor provides its best candidate sj from its sparsified network, the corresponding candidate in the original domain is:
5 = (s?):
where the mapped candidate S; will be added into the clite pool
3.3.1 Update of Master
In our cooperative evolutionary optimization framework, each candidate stored in the global clite pool & is represented as (S, i.e.,
m {( f(S,)),
1, (
where St, the best solution the i-th subprocessor the j-th generation. f(S) is its corresponding fitness (e.g., estimated influence spread), and is the generation number at which the candidate was added (serving as timestamp). Then, we can determine the current generation Jeurrent by taking the maximum generation value in E:
Jeurrnnt = max 9j (() EEJ.
Then, threshold parameter Ag is defined to specify the maximum allowed age (in terms of genera- tions) for candidates to remain in the pool. The elite pool & is pruned follows:
& { (S.,5, ((S..), 9n) E Jeurret - 9j S Ag }
which ensures that only those candidates whose age is at most Ag are retained.
Note that & is sorted in descending order based solely on fitness. Specifically, for any two candidates (S f(S..)..)) 9j) and (S :.v.''f( ((Sy.)) gy) in E, if (S.g.f(S.,)) 9j) appears before ((. in the sorted order, then f(St.)) 2 f(Su.r). Finally, to prevent unbounded growth, the pool & is limited size Ng. ensuring that the global elite pool focuses on recent and high-quality solutions.
3.3.2 Update of Subprocessor
When the i-th subprocessor requests elite candidate, the master may send candidate Sy E E, expressed in the original domain. Because the sparsified networks differ in size, S, may not directly map onto the domain of G² E Gg. To address this issue, we define projection mechanism for cross-domain injection. Given solution S and sparsified network G, let
o (S) = {Ø, (v) E V | vu E S, SV..
where o, maps the node E V into the domain of sparsified network G The projected candidate then obtained by
if (TT((S) k; oF (S) if (T()) k:
(T
Proj(S, G.)
where & the predefined size of solutions. Fill(S, G.)) (strategically or randomly) selects additional nodes from V to ensure that the final candidate contains exactly i nodes. This mechanism guarantees that elite solutions from the master can be feasibly injected into any subprocessor's local search space, despite differences among the sparsified networks.
shared global elite pool & is maintained by all processors. Every T generations (the elite injection interval), each processor sends its best candidate (after projection to the original network) into E. Conversely, candidate from & is injected into the local population of subprocessor after projection into the sparsified network whenever that candidate performs better than the weakest solution currently in the local population.
3.4 Ensemble -LLM- Evolutionary Optimization
Here, we introduce an ensemble framework leveraging MILLMs for evolutionary optimization on graph-structured data by incorporating multiple visual layouts.
5

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3.4.1 Visualization
Since each graph layout imposes distinct structural biases [6], the performance of MLLMs is inherently sensitive layout choice, and reliance on a single layout risks introducing perceptual distortions. To mitigate this issue, we adopt ensemble strategy that integrates multiple layouts, providing diverse structural views and enhancing overall performance (see Figure 2
To enable seamless visual incorpora- tion into our evolutionary framework, we use two separate functions de- signed for generating image represen- tations [10]. One function handles the initialization phase, while the other en- codes individual candidate solutions as images for their participation in ge- netic operations. The MLLM-based initialization defined as follows:
Layout
Layout 2
Layout N
@
Consensus Voting
MILMss
Timic (G, (G Visualize +)
Figure 2: Diagram of the ensemble strategy. Each layout is fed into MLLMs separately, and the final outcome is deter- mined using the consensus voting strategy.
which renders visual depiction of graph G' based on the layout parame- ters O utilizing external visualization tools. In contrast, to encode individual candidate solutions in image form, we have
I(G, θ s) Visualize O,s),
(2)
which produces visual representation of graph G arranged according layout θ, with the nodes belonging to solution s distinctly marked in color, as illustrated in Figure 3 Note that in practice, during crossover, only the seed nodes are visualized to reduce inference difficulty, whereas in mutation all nodes are visualized.
3.4.2 MLLM-based Initialization
Given graph G, initialization strategy and layout style θ, the initial population P generated as: P= MMLLLM init(Zmir(e) (G, θ); Np),
(3)
where N₂ denotes the population size.
Original Network
Solution Image
Solution Image
Mutation
Examine network image where seed nodes are colored and non-seed nodes labeled white. Carefully analyze the seed nodes present in the image.
Identify the current seed node that contributes maximization.
Crossover
influence
Examine two network images where seed nodes are colored and non-seed nodes labeled white. Carefully analyze the seed nodes each network image suggest optimal advantages of both parent networks to maximize influence spread.
Propose non-seed node that could significantly influence spread.
network's
nodes
Figure 3:
Diagram of the MLLM-based evolutionary optimization influence maximization. The solution and its underlying network are jointly represented in the image format.
3.4.3 Ensemble MLLM-based Crossover and Mutation
Given two candidate solutions 8j P where P denotes the population, the crossover operation guided by MILMS generates an offspring as:
S MILM (r
sy):
where P. denotes the crossover probability. To mitigate layout-specific bias, we propose an ensemble strategy that aggregates decisions across multiple layouts (0:)/= via:
Consensus VonsengusVoting [MLLLM_crossover]]] 0,, 0,,)),
=

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where Consensus Votting aggregates candidate solutions by evaluating the support each node receives in different layouts. Each node accumulates votes according to its frequency of occurrence, and nodes that meet a predefined threshold are chosen first. If the number of qualified nodes is insufficient, additional nodes are selected greedily based on their influence in the graph, measured by the number of uncovered neighbors.
Similarly, the ensemble mutation is defined as:
MLLM mutate mutate ():
Sy
VessensusVotinng (MLLLM (MMLLM_ mutate (I, +1)))))=)>>>
m
where IP,, denotes the mutation probability.
4 Experimental Study
The experiments are conducted on eight real-world networks, as shown in Table I nodes and edges in the simplified networks is fixed 50 and 100, respectively, remain manageable for MLLMs. The probabilities of crossover and mutation are set to 0.2 and 0.1. The number of individuals in the initialized population is set 20. The reported results averaged from 10 independent simulations unless otherwise specified. The backbone MLLM is gp1-4o-2024-11-2.
The number of
Table 1: Structural details about the networks: [V] is the number of nodes; E), the number of edges.
Network
USAir
Netscience
Polblogs
Facebook
WikiVote
Rutgers89
MSU24
Texas84
332
379
1,222
4,039
7,066
24,568
32,361
36,365
E
2,126
914
16,717
88,234
100,736
784,596
1,118,767
1,590,651
4.1 Effectiveness Examination of Cooperative Evolutionary Optimization
In this section, we compare the proposed cooperative optimization (multi-domain) with other evolu- tionary frameworks Isingle-domann Spars-D-EO and Spars-C-EO apply the vanilla evolutionary optimization to degree-bass and community- based sparsified networks without involving any ad- vanced techniques, respectively. Co-EO combines both sparsification strategies with knowledge transfer mechanism. As shown in Table [ Co-MLLM(KK) integrating MLLM with the KK layout and cooperative optimization achieves the highest fitness values across most networks. For fairness, SAEP [22. CoeCo (38, and SSR [399 optimize the population over the degree- based sparsified network. These methods highlight the advantages of self-adaptive control, wovidd-anddconeen composition, and search-space reduction in addressing complex network optimization tasks. The results indicate the potential for evolutionary optimization strategies, originally developed for the original domain, to be effectively transferred and integrated into frameworks operating on sparsified domains.
de-
Table 2: Performance comparison (Mean and Standard Deviation (SD)) of different evolutionary optimization methods across various networks.
Networks
Single
Multi-Demain
SAEP
CueCo
SSR 10)
Spars---EO
Span-C-BO
Co-EO
Co-MLLM(KK)
USAB
45.- 484-100444 (4(4)
4131132200)
219-216-11
44.9F2110)
2C14339
Netscience
14..10799)
16.810224))
162-0.26.)
14.110.782)
14.210.656-)
15.310.46-)
17610..0
Polbiogs
189.866.99-)
2215=11.66))
1986=7866-
192.246.86-3
188.5=10.59.))
208.729.2109)
48-.055
Facebook:
387.91:30.5400
378.823.288.)
283.7226.98-)
326.5214288/
337:2230993--
367.2425
WikiVote
492.2+13.166
502.0+8 366-)
504 1-12.7777
18011+23.666)
478.6-25 04(-)
1033+11.300 760
534.9
MSU24
1118,9144.50 56C-
1111.6146.10 196-
(0.7.0..4..86)
1133.11
1146.7.18.
Texast4
192.55+8800000
11157=14816610
2198 1-178229-)
298.442049986 (8(-)
1969 5-2229.24--
2459.0 109.000119000000
2512.9:111.22
Ruigers89
7044 1843
742.0+277044
712.0 20 20.39
707. 707.3-441.00-3
105.0=18.966)
736.7428.216
765.9-
ranking
The strength of the cooperative mechanism can also be observed in Figure which shows the opti- mization process of Spars-D, Spars-C and the cooperative way. Across all networks, the cooperative approach consistently outperforms the other two methods with higher fitness values, suggesting that the cooperative optimization strategy effectively leverages information from both sparsified domains enhance optimization performance.
7

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Spars-C-EO
Spars-D-EO
Cooperative
USAir
Netscience
Polblogs
Facebook
352
200
305
242
24.
13.5
272
370
Iterations WikiVote
Remations
Iterations
Itenations
MSU24
Texas84
Rutgers89
1154
1300
649
Itarations
Figure 4: Comparison of cooperative and vanilla evolutionary optimization (EO). Shaded regions represent the Standard Error of the Mean (SEM). The number of solutions in each Co-EO subprocessor population is set to half that of the single-domain configuration, since two sparsified networks are employed.
Table 3: Comparison of cooperative optimization for different transfer intervals. The results are obtained based the vanilla EO with 30 runs.
To further evaluate the effectiveness of our cooperative framework, we compare
different configurations of the knowledge transfer interval. As shown in Table [ more frequent knowledge transfer (Inter- val 2) generally results in improved opti- mization performance, demonstrating importance of timely information sharing: the elite solutions discovered in one spar- sified domain can be effectively used to guide the search in other domains, and these cross cross-ddomin insights can jointly im- prove overall optimization outcomes.
Graph USAír
Interval =
Interval 5
44.4 1 2.42
42.9 + 2.24
Netscience
1 0.77
14.9 0.86
Polblogs
199.9 +: 7.88
192.4 6.31
Facebook
3
3
WikiVote
527.5
510.8 15.37
MSU24
1154.6 = 14.60
1140.2 17.12
Texas84
2332.2 + 179.56
2235.9 + 174.71
Rutgers89
753.1 + 17.67
743.2 + 21.81
4.2 Effectiveness Examination of Ensemble Approach
To show the strength the ensemble method, we compare the ensemble variant MLLM with its single-layout counterpart. Vanilla EO refers to the probability-based evolutionary optimization. Symbols indicate statistical significance from the Wilcoxon test (95% confidence interval): '+'
for better, "z' for no difference, and for worse performance than MLLM -LLM--Eebbbe..... Table provides comparative analysis of different evolutionary optimization modes in optimizing influence maximization. As seen, The ensemble approach benefits from combining multiple layout perspectives and employing consensus voting strategy, making it more robust. In contrast, the other MLLM-bassed methods (KK, FR, and GraphOpt) show mixed performance, especially in Facebook and Texas84 networks, demonstrating the influence of layout on the performance of MLLM-basee optimization.
achieves the highest influence scores across most networks.
Figure presents the fitness progression of various MLLM-based optimization methods. Single- layout methods (KK, FR, and GraphOpt) exhibit network-depende FR layout performs significantly worse than the others in Polblogs and Facebook, while KK or GraphOpt underperform in different cases. These results highlight that single layout consistently dominates across networks. By integrating multiple layouts, the ensemble method achieves steady improvement and avoids stagnation, thereby demonstrating superior robust optimization through diverse visual inputs, Additional evidence supporting the effectiveness of the ensemble strategy presented in Figures 7 and where two tasks employed for further validation.
performance, for example, the

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Table 4: Performance comparison (Mean and Standard Deviation (SD)) of different reproduction
modes across various networks. The domain is built upon community-based sparsification method.
Reproduction modes
Networks
Vanilla
MLLM-KK
MLLM-FR
MLLM--rraphppt
MLLM-Ensemble
USAír
42.1+:2.24()
44.4:22.8((*)
44.9422.70(s)
44.7:22033s.
45.1::2.10
Nefscience
14.0++034((3))
14.6+0.67(()
15.0+-0.99(=)
14.7+0..2((=)
14,9+-063
Polblogs
190.0=6..111)
1918:16.166 16(-)
187.3:26.23()
191.0=7.22(-)
196,3:16.92
Facebook
337.44337883))
356.8=27.300/
357.1+330.70--
375.2=24.91(s))
387.3+-31.15
WikiVote
466.24-27.28()
483.3-24.08(-)
474.7+13.73()
475.2442.54()
510.0=10.30
MSU24
1053.2:1466858
1099,61441,44C)
1117 16.46-))6640(0)
1126.744-3.39(cs)
1129,2:
Texas84
2075.0#=297.49
2104.2+ 189.680)
2269.2 2269.2+1711. (7170(
2205.8+229.820)
2285.1+95..2
Rutgers89
705.6236.87())
14.09117.00((2))) 142))=
742.9420.42()
757.9=7.99
Avg ranking
4.63
3,25
3,00
3.00
1.13
KK
FR
GraphOpt
Ensemble
USAir
Netscience
Polblogs
Facebook
34.2
33.5
112
159
275
283
Iterations WikiVate
Rerations
Iberations
Iterations
MSU24
Texass84
Rutgers89
1144
102
1136
3350
1150
2200
3350
123
1010
1500
(49
1000
1900
Iterations
Reeations
Iterations
Iterations
Figure 5: Comparison of MLLM- based optimization performance across different graph layouts. Shaded regions represent the Standard Error of the Mean (SEM).
4.3 Examination of Network Layout
Initialization can be viewed as a one-time optimization of the network to a certain extent, since the MLLMs are required to select the most promising candidate. By prompting the MLLM to initialize the population, we further investigate its structural sensitivity to network images generated under different layouts and sparsification strategies, illustrated in Figure Each point represents single solution in the population, which is directly generated by the MLLMS as the initial population for optimization. Note that the sparsification strategy also influences the visualization, as the layout algorithm adaptively positions nodes based on structural properties.
As demonstrated in several cases, the choice of layout has clear impact on MILMs' performance. For instance, initial candidates generated with KK and GraphOpt exhibit higher overall fitness compared to those produced with FR in Polblogs and Facebook (Spars-C). Moreover, the influence of the sparsification strategy on MLLMS can also be observed. networks such as USAir and Texas84, the same layout yields distinct distributions, indicating that the sparsification strategy is related to the layout.
4.4 Generalizability Analysis
To examine the generalizability of MLLM-bbsse structure-aware optimization, we evaluate on tasks with different characteristics. We first study a sequential decision- decision-making making problem in which nodes are removed to disrupt network's connectivity [40), and effectiveness is measured using the area under the curve (AUC) of the largest connected component (LCC). We compare the single-layout and multi- -ayti-layout approaches on two non-sparsified networks (Dolphins and Lesmis) Figure and observe that the ensemble strategy consistently outperforms any single layout.

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KK
FR
GraphOpt
USAI
Netsienne
Polblogg
Fecebook
200
440
1.00
3.00
228
200
1:00
154
106
Strategy
Stratiggy
Stratiogy
Strategy Putgersé9
12.00
2200
Fitr
2.00
200
100
Spari
Spars
Spain
SpersO
Spatnc
Spin
SparsC
Strategy
Strategy
Strategy
Figure 6: Distribution of MLLM-based initialization with different layouts and sparsified networks.
We then consider classical permutation-bastt based combinatorial problem, the Traveling Salesman Prob- lem (TSP). In TSP, the layout is fixed by the coordinates of the cities, which makes the multi-layout ensemble inapplicable. Thus, compare the vanilla evolutionary optimization with our MLLM- based evolutionary optimization, as shown in Figure 8 The results demonstrate that incorporating MLLM-guided evolutionary operators significantly improves search effectiveness. Together, these tasks, ranging from subset selection (one additional example can be found in Figure (4) in graphs to permutation optimization as well as sequential decision-making problem, illustrate the breadth of our framework, demonstrating comprehensive evidence of the generalizability and flexibility of the structure-aware optimization approach.
Dolphing
Lesmis
15-Mode TSP Instance
20-Midde TSP instance
M.LMaaus Opdricationn
"L.-kdored Optiicaton
1AU0-12221
Nirmal 2od Remioved Rato
No modk Revevvs Rail
Aoratiors
Haradora
Figure 7: Comparison of different layout strate- gies in the network dismantling problem.
Figure 8: Comparison of different evolutionary optimization strategies the TSP.
5 Conclusion
This paper investigates the methods to facilitate the integration between evolutionary optimization and multimodal large language models (MLLMs), aiming to harness their combined strengths. This synergy is significant as it opens new avenues for solving complex problems more effectively by leveraging the deep contextual understanding of MLLMs. By utilizing MLLMs as evolutionary opera- tors, we demonstrate that incorporating graph sparsification and diverse network layouts significantly enhances the optimization process. Our findings also highlight the superiority of knowledge transfer across different sparsified networks in the cooperative evolutionary framework.
Acknowledgement
This work was supported by the Ministry of Education (MOE) Singapore, under the Academic Research Fund (AcRF) Tier Grant No. RS01/24. Kang Hao Cheong acknowledges additional support from the **STAR Human Potential Programme Grant No. H23P1M0006.
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A Discussion
A.1 Representation Comparison of Natural Language and Image
In addition the advantages of image-based encoding for graph-structured combinatorial optimiza- tion discussed in [0], we extend the discussion by presenting two additional aspects.
Token usage: In conventional LLM-based evolutionary optimization, the graph structure and candi- date solution must be described separately in text. This inflates the token count, as both structural information and solution details have to be serialized into lengthy textual descriptions. As the problem size grows, the token cost quickly becomes prohibitive. By contrast, in image-based representation, the graph and solution merged into single visual object. This compact encoding substantially reduces token usage, since the MLLM processes the image as one unified input rather than parsing long text sequences.
Reasoning: When structural information and solution are separated into different textual descriptions, the LLM must internally re-align two distinct streams of data, i.e., the abstract topology of the network and the numerical value of the solution before performing inference. This disintegration increases reasoning difficulty and introduces unnecessary cognitive overhead. In image-based encoding, however, the network and solution are presented together a single representation. This integration allows the model reason about their interactions more directly and coherently, without the burden of reconstructing connections across fragmented inputs.
A.2 Limitation and Mitigation
A.2.1 Limitation
Although graph sparsification improves the scalability by reducing the size of large networks, it also has inherent constraints that limit its applicability.
Subset selection problems: In practice, plotting beyond roughly 200 nodes produces highly cluttered images, which in turn makes MILM reasoning unreliable. This imposes a hard ceiling on the number of nodes that can be included in the visual representation. If the number of important or required selected nodes exceeds this threshold, the candidate node will also necessarily far exceed it, rendering the image-based approach invalid.
Permutation problems: In tasks such as permutation optimization, sparsification cannot be applied. These problems require complete output over all nodes in the original network. meaning node can be safely removed. Any reduction of the graph would distort the problem definition, making the method inapplicable.
A.2.2 Mitigation
To mitigate these limitations, plan to explore divide soride--nd-oon -ad--adddooo- mon-conquer strategies in the future work.
Subset selection problems: For subset selection tasks, one promising approach to employ community detection to partition the large graph into smaller subgraphs. Each subgraph remains within the visualization threshold, allowing MILMSs to reason effectively over structure and candidate solutions. The results from individual subgraphs can then be aggregated, ensuring that the final solution considers both local community-level optimization and global consistency.
For permutation-type tasks, the graph can be partitioned into geographical or structural regions (e.g.. clustering cities into local areas for TSP). The MLLM can then solve the permutation subproblem within each region before combining the partial tours into a global route, This regional -ndiid--en-cooueer strategy preserves all nodes while reducing the reasoning complexity faced by the model.
Permutation problems:
A.3 Injection Strategy
Elite candidates identified in the master domain must be adapted before they can be injected into the subprocessors' local search spaces as the sparsified networks may differ in size and structure. After projection, some candidates may fall short of the required size k. To address this, we employ
14

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Table 5: Comparison of optimization performance for different injection strategies. The result is obtained based on the vanilla evolutionary optimization with 30 runs.
Graph
Random filling
Heuristic-based filling
USAír
44.2 + 2.30
44.4 2,42
Netscience
14.8 + 0.86
14.9 - 0.77
Polblogs
198.5 + 8.03
199.9 7.88
Facebook
352.4 25.83
=
WikiVote
521.2 17.75
527.51 14.77
MSU24
1146.8 25.21
1154.6 = 14.60
Texas84
2312.1 + 156.01
2332.18 = 179.56
Rutgers89
732.9 + 27.02
753.12 + 17.67
Adjacency (LLM)
400x400 px (MLLM)
Incident (LLM)
768x768 px (MLLM)
Number of Nodes [[V] 50)
Number of Edges (|E) 400)
4500
3500
Token Count
2000
3000
1500
2.000
1000
1500
500
500
Number of Edges (E)
Number of Nodes (V)
Figure 9: Comparison of token usage across different network depiction methods and settings [0].
filling step to ensure that each injected solution is feasible by supplementing it with additional nodes. Specifically, we explore two strategies for the filling step:
Heuristic-based filling (default): prioritizes nodes with high betweenness centrality.
Random filling: selects nodes randomly from available nodes.
The results in Table 5 show that heuristic-based filling achieves slightly better performance than random filling across all tested graphs, though the differences small. This suggests that while heuristics provide a marginal advantage, the cooperative framework itself is robust and consistently delivers good optimization results regardless of the injection strategy employed. Moreover, when compared to the single-domain results in Table 2 these findings further demonstrate the effectiveness of our cooperative framework.
B Computational Cost Analysis
While large language models (LLMs) have proven effective across wide range of domains 41 42 43,, they often struggle with discrete graph traph-structured problems 44 45). To address these challenges, researchers have explored the use of MLLMs, which leverage visual representations of graphs to enhance reasoning capabilities [46] 47). For instance, recent work has demonstrated success in solving the traveling salesman problem by combining textual and visual modalities (48). Building this trend, our approach applies HILLM-based reasoning to the domain of discrete network optimization.
The scalability of the input representations is critical factor when working with LLMMs and MILMs 10. Traditional textual encodings, such as adjacency and incidence representations 49), incur token cost that grows lincarly with the size of the network, making them increasingly inefficient
15

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as the number of nodes and edges expands. In contrast, visual inputs used in MLLMs maintain constant token count, independent of the graph's 's scale (see Figure
Table Average running time (in seconds) per API call for different MLLMs performing crossover
(with 2 input images) and mutation (with I input image) during evolutionary optimization.
Kamada-Kawai
Fruchterman-Reingole
GraphOpt
Networks
Crossover
Mutation
Crossover
Mutation
Crossover
Mutation
gpt-4o-2024-11-22
3.5.0.66
2.410.744
3.410.68
2.3.10.73
3.610.81
2.3.10.99
Gemeni-2.0-flash-il
2.2.10444
2.2.0.49
2.310.51
2.110.53
2.3.10455
2.2.10522
Qwen-vl-max
3.8+1.15
2.3+1.25
3.721.588
1.3+
3.8+1.15
2.3+1.17
We compare the running time across different models in performing crossover and mutation under three layouts in Table which presents the average API call running time, revealing several key insights. First, the running time clearly depends on the foundation model, with Gemeni-2.0-flass 2-- consistently being the fastest and Qwen-vl-man generally incurring higher latency. Second, consistent with findings from [10), crossover operations requiring two input images are more time-consuming than mutation operations, which involve only one image. This trend holds across all models and layout strategies. Third, despite using the same foundation models, the running times reported here differ from those in [00), potentially due to variability in server load, highlighting the importance of deploying these models in practical environments.
VV--0, 1B-1000
IV/=100, |E|-200
Seed Size [5]
Figure 10: Average running time (in seconds) per API call for MLLMS performing initialization across different settings of network and seed size.
Figure TO shows the average running time per API call for MILMS (got-4o-0024-11-20) during initialization under different network and seed size settings. The results indicate that the running time influenced by both context difficulty, represented by network size (VV) [ED, and task difficulty, represented by solution size ((S)). Larger networks and required seed sizes consistently lead to longer running times, reflecting the increased computational demands of more complex inputs.
C Fidelity Validation of MILMs Outputs
While MLLMs demonstrate promising capabilities in reasoning over visualized network structures, their outputs may still violate task-specific constraints. Therefore, we implement a set of fidelity checks at each evolutionary stage (initialization, crossover, and mutation) to detect and correct such issues. These checks are designed to maintain the structural integrity, constraint compliance, and overall quality of the evolving population. The validation criteria applied in each stage detailed below.
The outputs of MLLMs are validated at different evolutionary stages to ensure solution quality and adherence to constraints:
Initialization Phase:
T Valid Node Check: Ensures that all nodes belong to the predefined valid set.
T Initialization Size Check: Verifies that the initial solutions meet the required seed size.
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Table 7: Evaluation of MLL.--geeenaae outputs initialization under the Spars-D setting.
Spars-D (KK)
Spars-D (FR)
Spars-D (GraphOpt)
Networks
T
T?
T7
T?
T
T7
T?
T
USAir
100%
100%
99.8%
100%
100%
100%
100%
100%
99.9%
Netscience
100%
100%
96.8%
100%
100%
94.8%
100%
100%
99.9%
Polblogs
100%
100%
99.7%
100%
100%
98.9%
100%
100%
99.6%
Facebook
100%
100%
99.3%
99.9%
100%
96.3%
99.9%
100%
96.8%
WikiVote
100%
100%
97.8%
100%
99.5%
97.2%
100%
100%
98.3%
MSU24
100%
100%
97.7%
100%
100%
91.6%
99.9%
100%
99.3%
Texas84
100%
100%
98.7%
100%
100%
99.1%
100%
100%
98.8%
Rutgers89
100%
100%
98.1%
100%
100%
96.3%
100%
100%
96.8%
Table 8: Evaluation of MLLL--geeeeee outputs at initialization under the Spars-C setting.
Spars- C (KK)
Spars- C (FR)
Spars- C (GraphOpt)
Networks
T1
T7
T
T+
T?
T
TT
Tf
T
USAir
100%
100%
99.2%
100%
100%
99.
100%
100%
99.0%
Netscience
100%
100%
96.2%
100%
100%
96.0%
100%
100%
96.1% 1%
Polblogs
100%
100%
99.7%
100%
100%
98.8%
100%
100%
99.5%
Facebook
100%
100%
93.0%
99.9%
100%
89.7%
100%
100%
90.2%
WikiVote
100%
100%
98.0%
100%
100%
97.5%
100%
100%
98.7%
MSU24
100%
100%
98.9%
100%
100%
98.9%
100%
100%
99.1% %
Texas84
100%
100%
98.3%
100%
100%
96.8%
100%
99.5%
97.5%
Rutgers89
100%
100%
95.1%
100%
100%
97.5%
100%
100%
98.8%
TÝ Low Degree Node Check: Identifies nodes with low degrees that may affect solution quality.
Crossover Phase:
T Crossover Size Check: Ensures that the offssring solutions adhere to size constraints.
TV Duplicate Node Check: Detects repeated nodes within the solution.
T Parent Node Source Check: Confirms that all nodes in the offspring inherited from parent solutions.
Mutation Phase:
Tl Node Presence Check: Ensures that nodes targeted for removal actually exist in the solution.
Th Mutation Valid Node Check: Verifies that newly added nodes are part of the valid node set. T Mutation Repetitive Node Check: Checks whether added nodes are already present the solution.
Due the concern about hallucination, we examine the correctness of MLLM outputs during initialization in Tables 7nd 8 The results indicate that T7 and T7 achieve 100% accuracy across all networks and strategies, confirming that the MLLM-bbased evolutionary process consistently selects nodes from the valid set and adheres to the required seed size constraints. This demonstrates that the initialization process is robust across different sparsification and layout strategics. However, T (Low-Degree Node Check) exhibits slight variations depending the sparsification and layout used. The degree-bacc based sparsification approach generally achieves higher T accuracy, with values close to sparsification shows slightly
or exceeding 99% across all networks. In contrast, community-basse lower T scores, especially in layouts like FR and GraphOpt, where percentages sometimes drop below 90% (e.g., Facebook: 89.7% in FR, 90.2% in GraphOpt). The differences in I scores across layouts further suggest that graph layout sensitivity should carefully considered when designing optimization strategies for influence maximization.
Figure [I illustrates the impact of graph layout choices on the validation rates of MILM- M-geeerrted solutions across various networks during evolutionary optimization. Overall, while most validation

---

tests maintain high fidelity, noticeable discrepancies arise during the mutation phase especially for test T (Mutation Repetitive Node Check). Networks such as Facebook, Polblogs, WikiVote, and Texas84 show significant variation in this metric depending on the layout used, with darker shades indicating more frequent validation failures. Notably, layouts like GraphOpt and KK are more susceptible to producing invalid outputs in this stage, particularly for complex networks (e.g.. Polblogs and Facebook). These results highlight the sensitivity MLLMs to visual layout encodings, and further justify the use of layout ensembles to enhance robustness and consistency in MLLM-based network optimization.
USAir
Nelsdence
Polblogs
Facebook
FR
FR.
CraphOpp
Couphippe
Ceashdpe
GraphiOpe
0.85
12
14
7A
75
T6
7%
74
WikVVate
RutgersB9
M5122
Toxas84
10.
KXC
XX.
10C
0.95
FR
FR
Copphhppt
Greph
18
16
12
18
Figure 11: Examination of layouts' influence validation rates for MLLMs across various networks. The sparsification strategy the community-base graph sparsification. Higher validation rates indicate greater reliability of reproduction operations, with darker colors corresponding to lower validation rates.
Figure 12 highlights the impact of different sparsification strategies, Spars-C (community-based) and Spars-D (degree-based), on the validation rates of MLLM outputs across various networks. Consistent with findings in Figure most validation checks yield high accuracy across strategies, except for T3 (Mutation Repetitive Node Check), where darker shades indicate more frequent violations. Degree-based sparsification (Spars-D) tends to result in lower validation rates in this mutation phase, especially networks like Netscience, WikiVote, Rutgers89, and Texas84 while Spars- C produces more eπrors in Polblogs and Facebook. This is because sparsification alters the structure and density of the graph, inherently affects how layouts are rendered, potentially exacerbating the sensitivity of MLLMS to visual encoding. Thus, the interplay between sparsification and layout further underscores importance of using ensemble and cooperative framework when applying MLLMs to discrete graph optimization tasks.
Moreover, evaluate the structural sensitivity of MLLMs in network analysis by examining their mutation behavior. Ideally, mutations replace low-impact seeds with high-influence candidates. As shown in Figure [3, added nodes consistently exhibit higher degrees than removed ones across networks and layouts, indicating effective refinement and the excellent structural awareness of MLLMs.
D Prompt Engineering
The prompts used in this work are presented Tables 12 The prompt consists two main components: (1) the context-setting prompt, which introduces the input information and specifies the roles of the agents, and (2) the output-directive prompt, which defines the desired output format and provides the necessary guidance restrictions. Owing to the distinct characteristics of each problem, the prompts differ as follows:
The prompt is structured around three phases of evolutionary optimiza- tion: initialization, crossover, and mutation. The mutation phase further divided into two steps: identifying the least influential node for removal and selecting the most promising node for addition.
Influence Maximization:
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Netscience
Polblogs
Farebook
Spars
Spars
Spars
Spars D
3.75
0.
Spara-
SparaeC
Spars-C
12
12
15
12
15
12
13
WikiWote
Rutgers89
MSUZA
Toxas
Soars-C
Spars
Sears
Spars-C
12
12
12
75
Figure 12: Examination sparsification strategy's influence on validation rates for MLLMs across various networks. Higher validation rates indicate greater reliability of reproduction operations, with darker colors corresponding to lower validation rates.
Suggested removed seed
Suggested added seed
USAir
Netscience
Polblogs
Fac cocebook
Dog so
GraphOpt
GraphOpt
GrophOpt
GraphOpt
Wikivatee
MSU24
TexasB4
Rutgers89
Dosiees
10C
GraphOpt
10K
FR
Graphapt
10c
FR
GraphOpt
KK
FRR
BraphhOpt
Figure 13: Comparison of degree distribution for removed and added seed nodes during MLLM-based mutation. The simplified graph is obtained by the community- based sparsification strategy.
19

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Immunization: Given its similarity to influence maximization as subset selection problem, the prompt design largely follows the same structure.
Graph Dismantling: As this is a sequential decision-making problem (used to validate the ensemble strategy), evolutionary optimization is not applied. Instead, the prompt directs the MILM identify the node most likely to cause network collapse.
Traveling Salesman Problem (TSP): Since this task requires outputting all nodes in the network, the prompt additionally provides the solution in textual form. This design reduces reasoning complexity and enhances solution feasibility.
Table 9: Structure of prompts for MLLM-based evolutionary operators of different phases.
Task
Context-setting prompt
Output directive prompt
You are an expert in network science and will be provided with one network in the form of an image. Please help me intelligently select nodes as the diffusion seeds this network to achieve influence maximization.
Only provide list of node in- dices separated by commas.
Initialization
Focus on selecting high- degree nodes nodes in strategic positions that signif- icantly enhance network con- nectivity. Provide your an- swer as list of node indices, separated by commas.
Examine a network image where seed nodes distinctly labeled. Carefully analyze the seed nodes present in each network image and suggest optimal set of seed nodes that har- ness the advantages of both parent networks to maximize influence spread.
Crossover
Examine a network image where seed nodes colored and non-seed nodes are labeled in white. Identify the current seed node that con- tributes the least influence maximization.
Focus on nodes that appear trivial or less connected. Pro- vide the index of this node.
Mutation (Removal)
Examine a network image where seed nodes are colored and non-seed nodes labeled in white. Propose non-seed node that could significantly increase the network's 's influence spread.
Focus on nodes with higher degrees or strategically criti- cal positions in the network. Provide the index of this node.
Mutation (Addition)
Vanilla
KK
FR
GraphOpt
Ensemble
Dolphins
Lesmis
35
52-
48-
35
Fitness
75
70-
Iteration
Iteration
Figure 14: Comparison of different optimization methods in the network immunization problem.
20

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Table 10:
Structure of prompts for MLLM-based evolutionary operators of different phases for network immunization.
Task
Context-setting prompt
Output directive prompt Avoid clustering immunized nodes together. Your solu- tion must include the same number of non-repetitive im- munized nodes as the parents. Provide your answer as a list of node indices, separated by commas.
You are given two parent immunization strate- gies shown as graphs with highlighted nodes. Each highlighted node is selected for immu- nization. Generate a child solution by com- bining high-impact nodes from both parents. Prioritize nodes that connect to many non- immunized nodes, i.e., these create more cut edges, helping to stop virus spread.
Crossover
You given an immunization solution visu- alized as a graph with highlighted nodes. To refine the solution, remove one node that tributes few connections to the rest of the net- work (i.e., nodes mostly surrounded by other immunized nodes or on the periphery).
The goal is improve effi- ciency: keep only the most impactful nodes for maxi- mizing the number of cut edges. Provide the index of this node.
Mutation (Removal)
Focus on nodes that, when immunized, will connect to many still-vulnerable nodes and increase the total number of cut edges. Provide the in- dex of this node.
You are given an immunization solution visu- alized as graph with highlighted nodes. Im- prove the solution by adding one new node to immunize.
Mutation (Addition)
Table 11: Structure of prompts for MLLM-based network dismantling.
Task
Context-setting prompt
Output directive prompt
You are an expert network science and you will be provided with network in the form of image. Each node is labeled with its node in black text. Your task is to help me dismantle this network.
Please tell me which node to remove to most likely col- lapse this network, i.e., make the largest connected compo- nent as small as possible.
Network Dismantling
E
Visualization
The visualization of network is supported by the external library and the details of visualization are presented T3] In this work, we adopt an ensemble approach by visualizing each network using three distinct layout algorithms before processing with MLLMs. This design choice is motivated by the observation that MLLMs are highly sensitive to spatial structure in visual inputs: Small variations in node positioning and edge arrangement can significantly influence the model's perception and output quality. To mitigate this layout-induced variability and enhance robustness, we incorporate multiple layouts during optimization. The layouts used in this study are follows:
Fruchterman-Reinne Reingold (FR): A force-directed algorithm where vertices repel each other and edges act like springs pulling connected nodes together. It tends to produce aesthetically pleasing layouts for medium-sized graphs.
Kamada-Kawai (KK): Focuses on preserving the craph-theoretical distances between nodes. minimizes energy function to position nodes such that their Euclidean distances reflect their shortest path distances.
GraphOpt: A force-directed method that uses simulated annealing techniques to optimize node placement. tailored for larger networks and emphasizes computational efficiency while aiming for visually distinct clusters.
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Table 12: Structure of prompts for MLLM-based evolutionary operators of different phases for Traveling Salesman Problem (TSP).
Task
Context-setting prompt
Output directive prompt
Use these to create a new child tour that combines parts from both parents. Try to keep the path smooth and avoid long jumps. The child tour must visit every node ex- actly once and return to the start. Return the complete visiting sequence as an or- dered list of node IDs.
You are given two parent TSP tours, and each tour shown image with numbered nodes. You are also given their visiting orders as lists of node IDs below.
Crossover
Parent tour: {solution}
Parent tour: {solution]
Make a small change the tour to try and shorten the overall path. You swap two nodes, move one node to different place, or reverse small segment. The new tour must still visit every node once and return to the start. Return the complete visiting sequence as an ordered list of node IDs.
You are given TSP tour: it is shown as image with node numbers. You are also given its visiting order as list of node IDs below.
Mutation
Current tour: (solution}
Table 13: Visualization settings summary.
Setting
Value
Graph Library
plotly gra.y.gbaaphojje
Node Size
35
Label Size
22
Canvas Size
1200 x 1200 px
Init. & Crossover: #2F7FC1
Color
Mutation:
Solution#2F7FC1,
Non-solutionnFFFFFF
Init. & Mutation: All nodes
Node Labels
Crossover: Solution only
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29


Reviews:
[{"review_id": "2QfnsHxdHW", "reviewer": "Reviewer_B64q", "summary": "The paper proposes a new evolutionary optimization approach to graph influence maximization by leveraging the reasoning capabilities of vision language models. More specifically, the proposed approach uses vision language models in an evolutionary loop and in a pointwise / pairwise fashion to update the selected nodes in a graph that might improve the overall fitness function. The authors conduct a variety of experiments to ablate various design choices in their approach over eight small/medium-sized graph datasets.", "questions": "Following up from the listed weaknesses above:\n\n1. Why have the experiments been restricted to only influence maximization? Section 2.1 of the paper lists numerous other tasks, including which would have increased the robustness of the empirical results.\n2. How does the proposed approach scale as a function of the number of nodes / edges in a dataset (e.g., in terms of # of LLM calls).\n3. Any reasons to restrict to only small-medium scale datasets for this paper?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper proposes a novel cross-disciplinary approach for influence maximization\n2. The paper is well written and easy to follow along\n\nWeaknesses:\n\n1. The paper might have a narrow scope with regards to the NeurIPS community / line of work\n2. Though the paper ablates the various design choices of their framework to a good extent, it is only limited to one task (influence maximization) when the technique is trivially generalizable to other tasks\n3. Only small datasets are used in experiments which might limit the generality of the technique in larger & complex scenarios\n4. Inference calls to large vision language models is expensive and the proposed approach will scale poorly to large graphs / datasets", "was_revised": true, "final_justification": "The authors have added more experiments on new datasets greatly improving the diversity of the empirical analyses of the paper and have answered my other concerns to a decent extent. I'm happy to increase my score and recommend acceptance.", "rebuttal": "### **Response to # Reviewer B64q**\n\nWe sincerely thank you for your constructive feedback and we are glad to hear that you found our approach novel and the paper clearly written. We have carefully addressed all of your concerns via adding new domain experiments (e.g., TSP, network immunization, dismantling), illustrating task generalization, scalability, and dataset size.\n\n---\n\n### **W1: Concern on the scope within NeurIPS community**\n**Response:**  \nThank you very much for this comment. Regarding the alignment with the NeurIPS community: our work builds on a growing interest in combining LLMs with combinatorial and graph reasoning (e.g., [1,2]), and also connects to active lines of research in evolutionary optimization [3] and graph combinatorial optimization [4]. Thus, we believe our contribution is timely and relevant to the NeurIPS audience.\n\n---\n\n### **W2 & Q1: Concern on limited task diversity/generalization**\n**Response:**  \nWe appreciate the your comment regarding the generazibility of our work. While our primary case study is influence maximization, we would like to emphasize that this was chosen not due to any task-specific limitation, but rather because it serves as a representative task (being well-known in network science) in graph-based combinatorial optimization. We fully acknowledge the importance of demonstrating broader applicability. To address this concern, we have substantially expanded our evaluation to include three additional tasks that vary widely in structure, decision space, and domain characteristics:\n- **Network Immunization**: A critical problem in network science involving node subset removal to minimize epidemic spread [5].  \n\n- **Traveling Salesman Problem (TSP)**: A classical permutation-based combinatorial problem to find the shortest route.  \n\n- **Network Dismantling**: A sequential decision-making problem where nodes are removed to break down a network's connectivity [6].\n\nThese tasks span diverse types of combinatorial optimization (subset selection, permutation-based ordering, and sequential decision-making), demonstrating that our MLLM-based evolutionary framework is not limited to a narrow task type. Furthermore, our ensemble strategy was evaluated on the network immunization and dismantling tasks, and continues to validate the benefit of layout diversity beyond a single setting.\n\nDue to word limitations, the results for these three additional tasks have been provided in our response to **# Reviewer 11CR**. We kindly refer you there for full details.\n\nTo ensure methodological consistency across different combinatorial tasks, we carefully re-implemented our approach using a unified configuration for all experiments and have updated the results in **Table 2 (and Figure 4)** of the original submission accordingly (due to the word limit, please refer to the last section of the response to **# Reviewer Mddl**). The revised results, obtained under this consistent framework, continue to support our main claims regarding the method’s advantages. We will reflect these updates in the final version accordingly for completeness and reproducibility.\n\n---\n\n### **W3 & Q2: Concern on limited dataset scale / scalability**\n**Response:**  \nWe appreciate the reviewer’s concern regarding scalability. We would like to respectfully clarify that our current study focuses on networks of up to **36,000+ nodes**, which is considered large-scale in the context of evolutionary optimization-based graph optimization. This is corroborated by recent literature, such as [7], where the tested networks are of comparable size. Even highly efficient heuristics, like the H-index-based method [8], also operate on networks within this scale.\n\nAs supported by **Reviewer #Etgw** who acknowledged the large scale of the datasets utilized in this paper, we sincerely hope that the Reviewer can re-evaluate our results, particularly when integrating models like MLLMs which introduce additional complexity.\nImportantly, one of the core contributions of our work is to improve scalability through **graph sparsification**, motivated by the scale-free nature of real-world networks [9]. This property implies that a small subset of nodes (hubs) play a disproportionately large role, allowing for intelligent reduction strategies without losing essential structural information.\n\nOur framework exploits this through **structured sparsification and cooperative optimization**, enabling MLLMs to operate effectively even in the presence of high complexity.\nFor a more detailed discussion on computational costs, please refer to our response to **W4 & Q3**.\n\n---\n\n### **W4 & Q3: Concern on MLLM inference cost / scalability**\n**Response:**  \nThank you very much for this constructive comment. We fully agree that inference with MLLMs can be computationally intensive, particularly on large graphs. Scalability is a central challenge and addressing it is a key motivation behind our framework design.\n1. **Fixed Input Size via Visual Encoding**  \n   Compared to traditional token-based LLM encodings (please refer to Figure 7 in the original submission), where input size grows linearly (or worse) with the number of nodes and edges, our **visual encoding approach decouples input size from graph scale**. Regardless of whether a graph has 1,000 or 100,000 nodes, the rendered image size remains fixed (e.g., 1200×1200 pixels, downscaled to 768×768 during GPT-4o inference). This results in constant token cost per inference, independent of network size, which offers a crucial advantage in scaling to larger graphs.\n---\n2. **Reducing Visual Clutter through Sparsification**  \n   We recognize the **visual clutter problem** that arises when plotting very large graphs. To address this, we incorporate **graph sparsification** as a core component of our pipeline. This allows us to retain essential structural features while significantly reducing the visual and computational complexity. The resulting sparsified views are more interpretable and manageable for MLLMs without sacrificing optimization quality.\n---\n3. **Scalable Modular Pipeline Design**  \n   Our framework is explicitly designed for scalability through the modular pipeline:  \n   ```\n   Graph Sparsification → Visualization → MLLM-based Optimization\n   ```\n   This design leverages the **scale-free nature of real-world networks [9]**, which implies that a small fraction of influential nodes often govern the network's behavior.  By focusing on those structurally critical components, our method remains effective even on networks of much larger size.\n---\n4. **Future Extensions with Divide-and-Conquer**  \n   For future work, our framework can be extended with a **divide-and-conquer strategy**, where large networks are partitioned into overlapping subgraphs that are optimized independently and then merged. This aligns with distributed optimization principles and further enhances scalability without increasing MLLM inference cost linearly with graph size.\nTo conclude, our work carefully considers both computational cost by using image-based representations and effectiveness and efficiency by applying graph sparsification.\n---\nWe hope that our responses and the extended experiments have addressed your concerns. We kindly hope you will consider raising your score based on the revisions and additional evidence provided.\n\n### **Reference**\n\n[1]\"Can language models solve graph problems in natural language?\" **NeurIPS** (2023).\n\n[2]\"Gita: Graph to visual and textual integration for vision-language graph reasoning.\" **NeurIPS** (2024).\n\n[3]\"Neuroevobench: Benchmarking evolutionary optimizers for deep learning applications.\" **NeurIPS** (2023).\n\n[4]\"Erdos goes neural: an unsupervised learning framework for combinatorial optimization on graphs.\" **NeurIPS** (2020).\n\n[5]“The hidden geometry of complex, network-driven contagion phenomena,” **Science** (2013).\n\n[6] \"Network dismantling.\" **PNAS** (2016).\n\n[7] \"Evolutionary Hypergraph Learning: A Multi-Objective Influence Maximization Approach Considering Seed Selection Costs.\" **IEEE TEVC** (2025).\n\n[8] \"The H-index of a network node and its relation to degree and coreness.\" **Nature Communications** (2016).\n\n[9] \"Emergence of scaling in random networks.\" **Science** (1999)."}, {"review_id": "gau2ATFi0H", "reviewer": "Reviewer_11CR", "summary": "This paper proposes an ensemble framework for structure-aware optimization on graph data, where multimodal large language models (MLLMs) are employed as evolutionary operators. The authors further incorporate the proposed framework with the following techniques for improvements:\n\n-  Applying graph sparsification techniques to prune complex networks while preserving essential structural properties. \n-  Utilizing a master-slave framework to distribute networks across multiple sparsified views, mitigating the biases produced by only viewing a single sparsified graph.\n\nThe experimental results on real-world networks validate the effectiveness of the proposed framework.", "questions": "- The paper applies a consensus voting mechanism among MLLMs. Have the authors considered a weighted voting variant, where the weights are proportional to the fitness scores of the best solutions produced by each MLLM?\n\n- While the ensemble approach with image-based MLLMs helps mitigate layout sensitivity, it would be interesting to explore whether a single graph-based MLLM could achieve similar robustness without requiring multiple layout representations. Have the authors considered this alternative in the study?\n\n- See Weaknesses for further comparisons on different combinatorial graph optimization tasks and SOTA methods with different modalities.", "limitations": "Yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper proposes an innovative algorithm that leverages MLLMs as evolutionary operators in the optimization process on graph data.\n- The framework, which utilizes multiple sparsification views and various layouts to improve solution quality, is theoretically sound and effective, as evidenced by its better performance in ablation studies.\n- The paper is well-organized, and the formulations of the entire framework are comprehensive and easy to follow.\n\nWeaknesses:\n- While the method is designed for combinatorial graph optimization, the experiments focus exclusively on the influence maximization task, which limits the insights into its generalizability. Including additional combinatorial optimization tasks in the experimental analysis would strengthen the evidence for its generalizability.\n- The paper lacks comparisons with other state-of-the-art methods involving different modalities, such as the GNN-based methods GLIE [1] and AutoGNP [2], which are designed for combinatorial graph optimization.\n\nReferences:\n\n[1] Maximizing Influence with Graph Neural Networks\n\n[2] Combinatorial Optimization with Automated Graph Neural Networks", "was_revised": true, "final_justification": "Increase the rating score to 5.\n- The additional experiments addressing the generalizability concerns raised in W1/Q3 are clear and well-executed.\n- The authors have provided a clear clarification regarding their future work in response to Q1.", "rebuttal": "### **Response to # Reviewer 11CR**\n\nWe sincerely thank you for the encouraging comments and for recognizing the novelty and clarity of our framework. In response, we have conducted additional experiments on new varied combinatorial tasks (e.g., TSP, network immunization, dismantling) and addressed your suggestions regarding comparison with other methods.\n\n---\n\n### **W1/Q3: Concern on lack of diverse combinatorial task evaluation**\n\n**Response:**  \nWe highly appreciate your suggestion to demonstrate the broader applicability of our framework beyond influence maximization. Please be assured that we have now taken concrete steps to address it by conducting additional experiments on three diverse combinatorial optimization tasks.\n\n---\n\n**Network Immunization**  \nA critical problem in network science involving node removal to minimize epidemic spread [5]. We compared the optimization process across different evolutionary methods with Dolphins and Lesmis (The data can be found in “Network data” collected by Mark Newman). The optimization goal is to maximize the number of immune edges. The results further test our framework’s ability to handle structural node selection problems.\n\n**Dolphins:**\n\n| Dolphins         | 1st           | 3rd           | 5th           | 7th           | 10th          |\n|------------------|---------------|---------------|---------------|---------------|---------------|\n| **Normal**        | **42.9 ± 2.8** | 44.7 ± 2.6    | 45.4 ± 2.8    | 46.6 ± 1.8    | 47.0 ± 1.7    |\n| **MLLM-FR**       | **42.9 ± 2.8** | 45.9 ± 3.6    | 48.3 ± 2.8    | 48.4 ± 2.8    | 49.2 ± 2.5    |\n| **MLLM-KK**       | **42.9 ± 2.8** | 43.8 ± 2.2    | 46.0 ± 3.7    | 46.7 ± 3.7    | 48.2 ± 3.2    |\n| **MLLM-GraphOpt** | **42.9 ± 2.8** | 44.4 ± 2.7    | 46.8 ± 4.5    | 48.0 ± 4.0    | 48.9 ± 4.2    |\n| **MLLM-Ensemble** | **42.9 ± 2.8** | **45.9 ± 3.1** | **48.8 ± 4.1** | **49.6 ± 4.2** | **51.2 ± 3.9** |\n\n**Lesmis:**\n\n| Lesmis           | 1st           | 3rd           | 5th           | 7th           | 10th          |\n|------------------|---------------|---------------|---------------|---------------|---------------|\n| **Normal**        | **68.7 ± 7.9**    | 69.8 ± 7.3    | 73.0 ± 9.6    | 73.5 ± 9.3    | 74.2 ± 8.6    |\n| **MLLM-FR**       | **68.7 ± 7.9**    | 72.7 ± 7.2    | 83.9 ± 7.0    | 86.6 ± 6.9    | 90.8 ± 9.6    |\n| **MLLM-KK**       | **68.7 ± 7.9**    | 75.1 ± 10.5   | 83.6 ± 6.7    | 85.0 ± 7.4    | 88.9 ± 8.4    |\n| **MLLM-GraphOpt**          | **68.7 ± 7.9**    | 76.0 ± 7.8    | 83.0 ± 7.5    | 88.3 ± 9.7    | 90.6 ± 7.2    |\n| **MLLM-Ensemble** | **68.7 ± 7.9**    | **77.4 ± 13.8**   | **85.9 ± 13.2**   | **90.7 ± 9.3**    | **95.4 ± 8.0**    |\n\n---\n\n **Traveling Salesman Problem (TSP)**  \nA classical permutation-based combinatorial problem. Note that in TSP, the layout is fixed by coordinates, rendering the multi-layout ensemble inapplicable. Therefore, we compare the optimization process of normal evolutionary optimization and MLLM-based structure-aware evolutionary optimization to demonstrate that MLLM-guided evolutionary operators can significantly improve search effectiveness. The optimization goal is to minimize the route distance.\n\n| Instance 15 | 1st           | 3rd           | 5th           | 7th           | 10th          |\n|-------------|---------------|---------------|---------------|---------------|---------------|\n| **Normal**  | **742.90 ± 39.40** | 714.20 ± 48.60 | 698.40 ± 45.80 | 684.70 ± 54.10 | 676.10 ± 48.90 |\n| **MLLM**    | **742.90 ± 39.40** | **683.60 ± 66.20** | **649.30 ± 69.40** | **635.80 ± 69.00** | **611.20 ± 66.80** |\n\n| Instance 20 | 1st            | 3rd            | 5th            | 7th            | 10th           |\n|-------------|----------------|----------------|----------------|----------------|----------------|\n| **Normal**  | **995.42 ± 46.93** | 957.56 ± 58.59 | 894.91 ± 75.14 | 874.44 ± 78.86 | 848.25 ± 53.82 |\n| **MLLM**    | **995.42 ± 46.93** | **939.34 ± 90.82** | **891.93 ± 87.98** | **872.28 ± 75.98** | **795.83 ± 53.34** |\n\n---\n\n**Network Dismantling**  \nA sequential decision-making problem where nodes are removed to break down a network's connectivity [6]. Here we use this problem to examine the influence of single-layout and multi-layout to demonstrate the advantage of our ensemble framework. The presented results show the AUC of the largest network component, with the removal process terminating after 25% of the nodes were removed.\n\n| Model         | FR     | KK     | GraphOpt | Ensemble |\n|---------------|--------|--------|----------|----------|\n| **Dolphins**  | 726.95 | 714.60 | 723.50   | **712.15**   |\n| **Lesmis**    | 494.60 | 534.40 | 626.10   | **488.80**   |\n\nThese tasks differ greatly in structure and computational nature, ranging from subset selection to permutation optimization to sequential strategies, thus providing a comprehensive validation of our structure-aware optimization framework’s generality and flexibility.\n\nMoreover, our ensemble layout strategy has also been evaluated in both the network immunization and dismantling tasks, where it consistently demonstrated superior robustness and solution quality compared to single-layout baselines. Collectively, these new experiments reinforce that our proposed MLLM-based evolutionary optimization framework is broadly applicable across a wide spectrum of combinatorial graph problems, not limited to influence maximization.\n\nTo ensure methodological consistency across different combinatorial tasks, we carefully re-implemented our approach using a unified configuration for all experiments and have updated the results in  **Table 2 ** (and  **Figure 4 **) of the original submission accordingly (due to the word limit, please refer to the ending section of the response to  **#Reviewer Mddl **). The revised results, obtained under this consistent framework, continue to support our main claims regarding the method’s advantages. We will reflect these updates in the final version accordingly for completeness and reproducibility.\n\n---\n\n### **W2/Q3: Concern on missing comparison with SOTA GNN-based methods**\n\n**Response:**  \nThank you very much for this thoughtful comment. As stated at the end of the introduction, the primary focus of this paper is not about advancing the state-of-the-art in influence maximization itself, but rather on using it (being well-known in network science) as a representative benchmark to demonstrate how MLLMs can serve as evolutionary optimization operators in a novel structure-aware optimization framework.\n\nOur main goal is to explore the integration of MLLMs into evolutionary algorithms, a direction that is applicable to a wide range of combinatorial optimization problems. Due to time constraints and the nature of our work, we seek for your kind understanding that we chose to prioritize demonstrating the generalizability of our framework across multiple tasks as shown in the response to W1.\n\nMoreover, comparing fairly with GNN-based approaches such as GLIE or AutoGNP is non-trivial, as their performance can vary significantly depending on implementation details and task-specific tuning. Similarly, the performance of any evolutionary algorithm including ours can be further improved by tuning hyperparameters like crossover/mutation rates, generation count, and population size. To ensure a controlled evaluation using consistent evolutionary parameters, we compared our method with other evolutionary optimization frameworks under the same settings, as shown in **Table 3**. The results highlight the superiority of our structure-aware evolutionary operators, in particular because we did not incorporate advanced techniques such as multi-task learning, and instead followed a standard evolutionary optimization pipeline.\n\n\n---\n\n### **Q1: Suggestion for weighted voting instead of uniform consensus**\n\n**Response:**  \nThank you for this insightful suggestion. Indeed, incorporating a weighted voting mechanism where weights are derived from the fitness of the best solutions generated by each layout could potentially enhance decision quality in our ensemble framework. This aligns with the broader direction of developing self-adaptive strategies to dynamically balance contributions from different views.\n\nIn the current work, our primary goal was to establish the effectiveness of multi-layout ensemble optimization, and to isolate its impact, we deliberately adopted the most straightforward aggregation method, i.e., majority voting. This allowed us to clearly demonstrate that layout diversity alone can significantly improve robustness and performance.\n\nWe fully agree that introducing a more sophisticated, adaptive weighting scheme could further enhance our method. New design choices (e.g., how to normalize fitness across layouts, address fitness noise, and ensure consistency during evolution) motivates future work in which a dedicated follow-up study will be conducted. Due to time constraints during the rebuttal phase, we could not include new experimental validation, but we genuinely appreciate this suggestion and plan to explore it in future work.\n\n---\n\n### **Q2: Suggestion to compare with single graph-based MLLM for robustness**\n\n**Response:**  \nThank you for this valuable question. As shown in  **Table 2** of the original manuscript, our ensemble method significantly outperforms any single-layout variant (e.g., KK, FR, GraphOpt) across multiple networks. To further strengthen this finding, we also applied the same single-vs-multi layout comparison to two additional tasks, i.e., network immunization and network dismantling as described in our response to W1. The results consistently demonstrate the advantage of layout diversity through ensemble voting.\n\n---\n\nWe hope these updates and clarifications adequately address your comments. Please let us know if any further clarification is needed. We would be grateful if you would consider adjusting your score in light of the improvements provided."}, {"review_id": "cSFIoCtsvb", "reviewer": "Reviewer_Etgw", "summary": "Traditional encoding schemes challenge in capturing the intricate structural properties of networks. This paper propose using MLLMs as evolutionary operators to facilitate structure-aware optimization over graph data. The author demonstrates how MLLMs can function as evolutionary optimization operator and  explore the potential of this integration and how this integration can be further extended.\n\nExperiments show that MLLM integration significantly outperforms single-layout approaches, and the collaborative mechanism further improves performance, achieving the best EDV scores on 8 real networks. The overall approach replaces traditional heuristics with visual large models, combining subgraph parallelization with knowledge transfer to balance accuracy and scalability.", "questions": "1. Your cooperative framework introduces several additional hyperparameters. Could you provide a sensitivity analysis or justify the choices of the default values?\n2. While the method is claimed to be applicable to full-scale graphs, the experiments are conducted only on subgraphs. Could you include or comment on results using the complete network to support the scalability claim?\n3. ANOVA is applied only during the initialisation stage. Have you considered performing a quantitative analysis of layout sensitivity for the crossover and mutation phases as well?\n4. How does the overall optimisation time of your method compare to a traditional evolutionary optimization (EO) baseline without MLLM? Could you provide a runtime comparison?", "limitations": "See weaknesses and questions above.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. Pioneers feeding multi-layout graph visualisations into a multimodal LLM, using the model itself as the generator of evolutionary operators.\n    \n2. The end-to-end architecture, from sparsified multi-view subgraphs to cross-domain knowledge transfer, is conceptually sound and tightly aligned with the authors’ stated motivation.\n    \n3. Benchmarked on eight large real-world networks against diverse baselines, with rigorous statistical tests that convincingly show the added value of both the ensemble and cooperative components.\n\nWeaknesses:\n\n1. The cooperative framework adds several hyper-parameters, yet offers no sensitivity study or justification of the chosen defaults.\n    \n2. Although the method is claimed to work on full-scale graphs, all experiments use only subgraphs; at least one evaluation on the complete network is needed.\n\n3. ANOVA is performed solely for the initialisation phase, with no quantitative layout-sensitivity analysis for crossover and mutation.\n\n4. The overall optimisation time is not compared against a traditional EO baseline without MLLM.", "was_revised": true, "final_justification": "The authors have already addressed most of my concerns. I would like to suggest an acceptance of this paper.", "rebuttal": "### **Response to #Reviewer Etgw**\n\nWe sincerely thank you for the comprehensive assessment. We are especially grateful for your recognition of the novelty, motivation alignment, and empirical rigor of our proposed framework. In response to your suggestions, we have conducted a sensitivity analysis on key hyperparameters, added a layout-sensitivity study for the reproduction phase, discussed the running time, and clarified the scalability of our framework.\n\n---\n\n### **W1/Q1: Concern on lack of hyperparameter sensitivity analysis in cooperative framework**\n\n**Response:**  \nThank you for raising this important point and we appreciate the suggestion to include parameter sensitivity to further improve our work. Among these parameters, the knowledge transfer frequency, i.e., how often elite solutions are exchanged across domains, is the most critical. In our default setting, this parameter is set to every 2 generations, which reflects a balance between sufficient cross-domain communication and preserving local evolutionary stability.\n\nIn response, we have conducted a new sensitivity experiment by increasing the transfer interval to every 4 generations. The comparison results are provided in the table below. Our findings indicate that more frequent knowledge transfer (Interval = 2) leads to better optimization performance in most cases, supporting the effectiveness of our proposed cooperative optimization framework.\n\n| Graph      | Interval = 2       | Interval = 4       |\n|------------|--------------------|--------------------|\n| **Netscience** | **15.3 ± 0.46**    | **15.3 ± 0.47**    |\n| **USAir**      | 44.5 ± 0.17        | **44.7 ± 2.44**    |\n| **Polblogs**   | **203.7 ± 9.21**   | 193.2 ± 4.50       |\n| **Facebook**   | **385.6 ± 24.53**  | 376.3 ± 25.93      |\n| **WikiVote**   | **518.3 ± 17.76**  | 498.7 ± 19.28      |\n| **Rutgers89**  | **736.7 ± 28.21**  | 730.8 ± 16.87      |\n| **MSU24**      | **1133.1 ± 28.38** | 1108.1 ± 27.81     |\n| **Texas84**    | **2459.0 ± 109.70**| 2365.3 ± 184.24    |\n\n---\n\n### **W2 & Q2: Concern on evaluation using subgraphs instead of full-scale networks**\n\n**Response:**  \nThank you for this insightful comment. We would like to clarify that our method does not work on subgraphs (an independent part of the original network). Instead, we apply graph sparsification to the full network, preserving its core structural skeleton, particularly high-impact hub nodes based on the well-researched scale-free property of real-world networks. Importantly, this property ensures that even a reduced representation can retain most of the essential structural and functional characteristics of the original network.\n\nOur sparsified networks are thus not disconnected subgraphs, but rather purposefully constructed simplifications that retain high-fidelity approximations of influence, connectivity, and topology, enabling effective optimization while remaining computationally tractable for MLLMs. This design is fundamental to addressing the scalability challenge posed by large networks when visualized in their entirety. Rendering an entire network with tens of thousands of nodes on a single canvas results in severe visual clutter and degraded MLLM performance, something we explicitly seek to avoid.\n\nTo support our scalability claim, we include a comparative experiment where candidate solutions are randomly initialized on the full, unsparsified network, and the sparsified network (as used in our framework). Using the same optimization framework in both cases, we observe that the sparsified version consistently yields better initial solution quality, reinforcing that sparsification not only improves scalability but also enhances optimization performance.\n\n| Graph      | Original          | Sparsified        |\n|------------|-------------------|--------------------|\n| **Netscience** | **16.24 ± 0.56**  | 15.89 ± 0.80       |\n| **USAir**      | 48.84 ± 2.23      | **49.27 ± 1.57**   |\n| **Polblogs**   | 183.32 ± 13.11    | **213.77 ± 5.94**  |\n| **Facebook**   | 236.46 ± 49.69    | **407.45 ± 8.86**  |\n| **WikiVote**   | 315.58 ± 28.19    | **545.13 ± 15.67** |\n| **Rutgers89**  | 327.80 ± 72.36    | **778.96 ± 25.65** |\n| **MSU24**      | 363.87 ± 102.34   | **1152.17 ± 27.81**|\n| **Texas84**    | 543.29 ± 147.81   | **2594.51 ± 144.18**|\n\nFurthermore, our utilised networks are up to **36,000+ nodes**, which are larger than those used in existing LLM/MLLM-based optimization works typically operating on small to medium-sized graphs.\n \nOur method’s scalability stems from its design: a modular pipeline of  \n`Graph Sparsification → Visualization → MLLM-based Optimization`,  \nwhich we believe is a practical and effective approach for handling large real-world graphs.\n\n---\n\n### **W3 & Q3: Concern on missing layout-sensitivity analysis for crossover and mutation**\n\n**Response:**  \nThank you for raising this important point. We acknowledge that our original layout-sensitivity analysis focused on the initialization phase, which is a one-time operation and more straightforward to evaluate statistically.\n\nIn contrast, crossover and mutation are iterative operations that occur across multiple generations, and their effects are inherently accumulative and interdependent. This makes it challenging to isolate and assess the sensitivity of each reproduction step to layout choice using standard statistical methods. For example, the fitness outcome of a solution after the 1st generation may differ substantially from that after the 10th generation even under the same layout due to the influence of prior evolutionary history. As a result, comparing layouts at the same generation across independent runs does not guarantee a fair or controlled assessment.\n\nAs a solution, we have conducted a statistical comparison between single-layout and ensemble approaches across the full optimization process, as shown in **Table 2**. These results demonstrate that layout differences significantly impact final optimization performance.\n\n---\n\n### **W4 & Q4: Concern on missing runtime comparison with traditional EO**\n\n**Response:**  \nThank you for raising this important question regarding runtime overhead. As shown in  **Table 5** of the appendix, the runtime per MLLM-based operation (crossover and mutation) depends on factors such as the model architecture (e.g., GPT-4o) and server load at inference time. These operations are inherently slower than traditional, code-based EOs, which can be executed in milliseconds due to their lightweight nature.\n\nHowever, it is important to note that traditional EO approaches are not structure-aware unless supplemented with manual, task-specific heuristics or engineered operators. In contrast, our MLLM-based method leverages multimodal perception to guide the search more intelligently, resulting in significantly improved optimization performance, as shown across multiple tasks and datasets.\n\nWhile we acknowledge the higher computational cost, we seek your kind understanding that this is not just a limitation of our framework, but a general challenge shared by all LLM-based optimization approaches. As multimodal models continue to improve in efficiency, we expect this gap to diminish over time.\n\n---\n\nWe hope these additions and clarifications have addressed your concerns. Please let us know if any further clarification is needed. We kindly request you to consider a score adjustment based on these improvements."}, {"review_id": "9sDPAArUAs", "reviewer": "Reviewer_MddL", "summary": "This paper proposes a novel Structure-Aware Cooperative Ensemble Evolutionary Optimization framework that integrates Multimodal Large Language Models (MLLMs) into evolutionary optimization for solving combinatorial problems on complex networks. The key idea is to use image-based encodings of network structures, allowing MLLMs to serve as evolutionary operators (crossover and mutation), while incorporating graph sparsification , ensemble strategies, and cooperative frameworks to enhance robustness, reduce bias, and improve performance.", "questions": "(1)How generalizable is your framework to other combinatorial graph optimization tasks such as node cover, traveling salesman problem, or community detection?\n(2)Table 5 shows significant differences in running time and performance across GPT-4o, Gemini, and Qwen. How does the choice of MLLM fundamentally affect the search behavior and final solution quality?\n(3)The simplified networks used have only 50 nodes and 100 edges. How does the method scale to graphs with thousands or millions of nodes? What architectural changes would be required?\n(4)How does the proposed MLLM-driven framework compare with traditional EA methods that use learned or hand-crafted mutation/crossover operators on graph data?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Main strengths\n(1)\tNovel Integration of MLLMs with Evolutionary Algorithms: The paper presents a creative and promising direction by using MLLMs not just as tools for reasoning or generation, but as core components in an evolutionary optimization pipeline.\n(2)\tEnsemble and Cooperative Framework Design: The proposed cooperative framework with multiple sparsified views and layout ensembles shows a strong understanding of how to mitigate MLLM limitations such as sensitivity to input presentation and structural bias.\n(3)\tComprehensive Experimental Evaluation : The evaluation on eight real-world networks, along with ablation studies and statistical significance tests, provides solid empirical support for the claims made.\n\nMain weakness\n(1)\tLimited Theoretical Contribution: While the framework is empirically sound, there is little theoretical analysis or formal guarantees about convergence, optimality, or generalization beyond the specific task of influence maximization.\n(2)\tLack of Broader Applicability Discussion: The experiments focus solely on influence maximization; it’s unclear how easily the framework can be adapted to other combinatorial optimization tasks on graphs.\n(3)\tComputational Cost and Scalability Concerns: Although the paper discusses computational efficiency, the cost of repeated MLLM API calls and the scalability to much larger graphs are underexplored.", "was_revised": true, "final_justification": "The reviewer has provided a rigorous proof in theory, and the work is based on solid foundations.", "rebuttal": "### **Response to # Reviewer Mddl**\n\nWe sincerely thank you for the detailed evaluation and for highlighting the novelty, architectural soundness, and thorough empirical validation of our proposed framework. We have taken your comments seriously and addressed the theoretical limitations, scalability, and generalization concerns. Specifically, we have added results on new combinatorial tasks (e.g., TSP, network dismantling, network immunization), clarified the impact of MLLM selection, discussed architectural scalability strategies, as well as comparisons with hand-crafted evolutionary operators.\n\n---\n\n### **W1: Concern on limited theoretical analysis or formal guarantees**\n\n**Response:**  \nThank you for this important comment. Combinatorial optimization on graphs is inherently NP-hard, especially in the context of large-scale and non-linear networks. As such, no existing method including traditional or learning-based heuristics can provide general guarantees of optimality or convergence in these settings. Our primary focus, therefore, is on designing a practical and empirically effective framework that leverages the strengths of MLLMs for structure-aware search, and demonstrates consistent performance improvements across diverse problem domains. The experimental results in our work show that the proposed approach significantly enhances optimization quality compared to both traditional and baseline methods.\n\nWe seek your kind understanding that this limitation is not specific to our framework, but reflects a broader challenge shared by all existing work operating on NP-hard problems due to inherent complexity of such kind of problem.\n\nRegarding generalization beyond influence maximization, we have conducted additional experiments on three different tasks (network immunization, TSP, and network dismantling), as detailed in our response to  **W2 & Q1**. These results further support the adaptability of our framework across a wide range of combinatorial problems.\n\n---\n\n### **W2 & Q1: Concern on unclear generalizability to other combinatorial graph tasks**\n\n**Response:**  \nThank you for your valuable comment. We fully recognize the importance of demonstrating the generalizability of our method. We have added more experiments to show the generalizability of our method as below:\n\n- **Network Immunization**: A critical problem in network science involving node removal to minimize epidemic spread [5].  \n- **Traveling Salesman Problem (TSP)**: A classical permutation-based combinatorial problem.  \n- **Network Dismantling**: A sequential decision-making problem where nodes are removed to break down a network's connectivity [6].\n\nDue to word limitations, the results for the three additional tasks have been provided in our response to **#Reviewer 11CR**. We kindly refer you there for full details.\n\nThese tasks differ in structure and computational nature, ranging from subset selection to permutation optimization to sequential strategies, thus providing a comprehensive validation of our structure-aware optimization framework’s generality. Moreover, our ensemble layout strategy has also been evaluated in both the network immunization and dismantling tasks, where it consistently demonstrated superior robustness and solution quality compared to single-layout baselines.\n\nCollectively, these new experiments reinforce that our proposed MLLM-based evolutionary optimization framework is broadly applicable across a wide spectrum of combinatorial graph problems, not limited to influence maximization.\n\n---\n\n### **W3: Concern on computational cost and scalability of repeated MLLM calls**\n\n**Response:**  \nThank you for raising this concern. To date, there are already many interests using LLM to do optimization. As shown in **Figure 7**, scalability is one of advantages of our methods in term of API call cost.\n\nFirst, compared to traditional token-based LLM encodings, where input size grows linearly (or worse) with the number of nodes and edges, our visual encoding approach decouples input size from graph scale. Regardless of whether the graph contains 1,000 or 100,000 nodes, the number of tokens used during inference is determined solely by the size of the rendered image, which remains fixed. **This leads to a constant token cost per inference, independent of the graph’s scale.**\n\nTo ensure effective and interpretable visualizations for larger graphs, our framework incorporates a **graph sparsification** step that selectively retains the most structurally important components. This approach preserves the essential features needed for optimization while keeping the visual input compact and suitable for MLLM processing. As a result, **our method achieves both efficient cost and scalability, even when applied to large graphs.**\n\n---\n\n### **Q2: Question on impact of MLLM choice on search behavior and performance**\n\n**Response:**  \nThank you for this insightful question. The choice of multimodal large language model (MLLM) indeed affects both the runtime and the quality of optimization outcomes. As shown in Table 5 in the original manuscript, we observed differences across GPT-4o, Gemini, and Qwen in terms of response latency and solution effectiveness.\n\n- **Runtime**: Variability mainly stems from differences in external factors such as API server load, rate limits, or backend infrastructure. These are more about system-level variations.  \n- **Search behavior**: Each MLLM differs in its visual-textual reasoning ability, prompt understanding, and response precision. These differences fundamentally influence the evolutionary search trajectory, as MLLMs are responsible for mutation and crossover decisions. A more capable model can better interpret graph structure, leading to more effective candidate generation and faster convergence.\n\n---\n\n### **Q3: Question on scalability from simplified to large-scale networks**\n\n**Response:**  \nThank you for this question. While the simplified networks we visualize may contain as few as 50 nodes and 100 edges, they are not arbitrary samples, but rather structurally representative miniatures of the original networks. This design choice is grounded in the scale-free property of real-world networks [1], where a small subset of high-degree hub nodes governs much of the network’s dynamics, and the majority of nodes contribute marginally. This allows us to sparsify the network while preserving its functional backbone, which is a key contribution of our framework.\n\nIndeed, regardless of the original graph size, whether thousands or millions of nodes, our pipeline can scale by selectively retaining the structurally important components through graph sparsification, keeping the input size manageable for MLLMs while retaining optimization effectiveness.\n\nIn our experiments, we work with networks containing up to **36,000+ nodes**, which is considered large-scale in the context of evolutionary optimization on graphs, and particularly significant for LLM/MLLM-based methods, where computational constraints are more pronounced.\n\nIf scaled further to networks with millions of nodes, the same architectural design  \n`Graph Sparsification → Visualization → MLLM-based Optimization`  \nremains valid. For additional scalability, our framework can be extended with divide-and-conquer techniques, where the network is partitioned into overlapping subgraphs that are optimized independently and later integrated.\n\n---\n\n### **Q4: Concern on comparison with traditional or learned EA operators**\n\n**Response:**  \nThank you for this insightful question. We have compared our MLLM-driven framework with several state-of-the-art hand-crafted evolutionary optimization methods specifically designed for graph data. The results of this comparison are presented in Table 1 of the main paper and include the following baselines:\n\n- **SAEP**: IEEE TSMC (2023)  \n- **CoeCo**: IEEE TEVC (2023)  \n- **SSR**: IEEE TCSS (2022)  \n\nAcross multiple benchmark networks, our MLLM-based optimization consistently outperforms these hand-crafted methods in terms of solution quality. This result demonstrates the potential of leveraging multimodal large language models as a powerful and generalizable alternative to manually engineered or domain-specific evolutionary operators.\n\nCritically, our current implementation uses a simple and generic EO pipeline, without incorporating advanced mechanisms such as self-adaptive parameter tuning, multi-task optimization, or co-evolution strategies. Thus, our framework is extremely novel in its own rights and fully compatible with these enhancements. We expect that integrating them would further improve performance.\n\nTo ensure methodological consistency across different combinatorial tasks, we carefully re-implemented our approach using a unified configuration for all experiments and have updated the results in  **Table 2** (and  **Figure 4**) of the original submission accordingly, i.e.,\n\n| Graph      | MLLM-Ensemble       |\n|------------|---------------------|\n| **USAir**      | 45.07 ± 2.10        |\n| **Netscience** | 14.97 ± 0.63        |\n| **Polblogs**   | 196.29 ± 6.92       |\n| **Facebook**   | 383.37 ± 31.15      |\n| **WikiVote**   | 509.99 ± 10.30      |\n| **MSU24**      | 1129.24 ± 14.04     |\n| **Texas84**    | 2285.11 ± 95.72     |\n| **Rutgers89**  | 757.94 ± 7.99       |\n\nThe revised results, obtained under this consistent framework, continue to support our main claims regarding the method’s advantages. We will reflect these updates in the final version accordingly for completeness and reproducibility.\n\n---\n\nWe hope these updates and clarifications address your concerns. Please let us know if any further clarification is needed. We would be grateful if you would consider adjusting your score in light of the extended experiments and improvements.\n\n---\n\n### **Reference**\n\n[1] \"Emergence of scaling in random networks.\" *Science* (1999)."}]

Author rebuttal:

Meta-review: (a) Scientific claims and findings
The paper proposes a framework that combines multimodal LLMs with evolutionary optimization for solving combinatorial optimization problems (COPs).
Key claims:
- Standard LLM-based optimization struggles with combinatorial search spaces.
- Introduces a Structure-Aware Cooperative Ensemble Evolutionary Optimization (SACEEO) framework, where multiple agents explore candidate solutions using diverse strategies, and a cooperative mechanism aggregates and evolves these candidates. Finally, structural information from multimodal inputs guides solution generation.
- Demonstrates improvements on benchmark COPs (e.g., TSP, Knapsack, Graph Coloring), with higher-quality solutions and faster convergence compared to baselines.
- Claims novelty in leveraging multimodal reasoning + ensemble evolutionary dynamics for optimization.

(b) Strengths

- Novel integration: Combining multimodal LLMs with evolutionary optimization is original and timely.
-  Outperforms LLM-only and classical baselines on multiple COPs.
-  Method well-described and easy to follow.
-  Evaluation spans several standard combinatorial tasks.

(c) Weaknesses 
- Limited theoretical depth: Contributions are mostly empirical; lacks formal guarantees or deeper analysis.
- Evaluation scope: Benchmarks are standard toy COPs; unclear scalability to larger industrial problems.
- Complexity: Cooperative ensemble framework increases computational overhead; not clear if efficiency is practical.
- Novelty modest: Core components (ensemble, evolutionary search) are known techniques.

(d) Key reasons for decision (accept/reject)

- Novel combination of multimodal LLMs and evolutionary optimization.
- Strong experimental performance on multiple COPs.
- Clear and well-presented framework.
- Potentially impactful direction for optimization with agentic LLMs.

(e) Discussion & rebuttal period

Reviewer points raised:
- Novelty concerns: Is SACEEO just ensemble + evolutionary optimization applied to LLMs?
           -Author rebuttal: Argued that integration with multimodal reasoning and cooperative mechanism is non-trivial.
           - Effect: Somewhat alleviated, but novelty still seen as modest.

- Scalability: Can this handle industrial-scale COPs?
               -Author rebuttal: Claimed framework generalizes, but lacked large-scale experiments.
               - Effect: Reviewers remained cautious.

- Experimental scope: Benchmarks too small/simple.
                 -Rebuttal: Authors added clarification but no new large-scale results.
                -  Effect: Concerns persisted.

Outcome: Rebuttal clarified design, but did not resolve novelty and scalability criticisms.
Decision: Accept (poster)