Aug 2026· International journal of computer information systems and industrial management applications· Vol 18, pp. 161-169· 0 citations
TL;DR
The paper concludes that intelligent system design increasingly depends on selecting a graph-theoretic representation and analysis method whose structural assumptions match the target network's actual generative structure, rather than applying a single default graph model universally.
Abstract
Graph theory has evolved from a branch of discrete mathematics concerned with abstract vertex-edge structures into the principal mathematical language for representing and analyzing complex networks, systems whose components interact through relationships that themselves carry structural, and often computationally exploitable, information. This paper reviews the foundational and applied graph-theoretic literature underlying complex network analysis and intelligent system design, tracing the field's development from classical random graph theory through the small-world and scale-free network models that reshaped network science at the turn of the century, and into the more recent graph representation learning and graph neural network (GNN) literature that has integrated graph-theoretic structure directly into machine learning architectures. The review synthesizes foundational random-graph and preferential-attachment models, structural analysis techniques including centrality measures and community detection, and the graph embedding and graph neural network methods that now underlie intelligent system design across recommendation, molecular modelling, and physical simulation domains. Distinct comparative tables map generative network model families onto their structural signature and generating mechanism, cross-reference classical structural analysis measures against their intelligent-system design application, and set graph neural network architecture families against the computational mechanism and task type each is best suited to address. The paper concludes that intelligent system design increasingly depends on selecting a graph-theoretic representation and analysis method whose structural assumptions match the target network's actual generative structure, rather than applying a single default graph model universally, and identifies the theoretical understanding of graph neural network expressiveness limits as the central future research prospect.
This paper conducts a case study on Zachary's Karate Club network and a synthetically generated scale-free network, computing centrality measures, detecting communities using the Louvain algorithm, and analysing degree-distribution behavior.
S. Sharma· Iconic research and engineer...· 0 citations
Networks provide the structural foundation for transportation systems, communication infrastructures, supply chains, power grids, information systems, social interactions, and many other complex technological and socioeconomic processes. The growing scale and interdependence of such systems have made network optimization an important area of mathematical and computational research. Graph theory offers a rigorous framework in which network entities can be represented as vertices and their relationships as edges, allowing questions of routing, connectivity, allocation, capacity, resilience, and structural efficiency to be formulated as optimization problems. This paper examines graph-theoretic models for network optimization through an integrated review of classical algorithms and contemporary network-science approaches. It discusses shortest-path models, minimum spanning trees, network flows, cuts, matching, centrality, community structure, spectral connectivity, complex networks, and multilayer representations. A generalized optimization framework is proposed in which edge selection, flow allocation, operational cost, capacity, connectivity, and robustness can be considered within a common mathematical structure. The analysis demonstrates that no single graph model is sufficient for every network-design problem. Classical polynomial-time algorithms remain highly effective for well-structured problems, while large, uncertain, multilayer, and combinatorial systems increasingly require approximation, decomposition, robust optimization, and hybrid computational strategies. The study argues that the principal value of graph-theoretic optimization lies not only in identifying minimum-cost paths or maximum flows but also in representing structural dependencies that influence system-wide efficiency and resilience. Future research should therefore integrate dynamic graphs, uncertainty-aware optimization, multilayer modeling, spectral methods, and data-driven decision support while preserving mathematical interpretability.
S. Anantharaman, V. Vishnupriya· Stanzaleaf International Jou...· 0 citations
Community detection is a key problem in complex-network analysis: densely connected groups may correspond to social circles, scientific fields, biological modules, or functional subsystems. This review considers how spectral graph methods translate a network into matrix form and then use eigenvalues and eigenvectors to uncover this structure. After introducing complex networks, graph matrices, community structure, and evaluation criteria, it develops the main ideas behind unnormalized and normalized graph Laplacians, the Fiedler vector, spectral embedding, normalized cut, and modularity-based eigenvector methods. Evidence from benchmark and real-world studies is then used to compare these classical techniques with regularized, non-backtracking, local, overlapping, neural-embedding, and randomized multi-layer extensions. Spectral methods remain competitive and mathematically interpretable. Their performance, however, depends on sparsity, degree heterogeneity, community overlap, network scale, and the choice of evaluation criteria. Future work is likely to combine sparse linear algebra and randomized eigensolvers with dynamic, multi-layer, and interpretable graph-learning models. By connecting core linear algebra with practical structure discovery, the review clarifies both the continuing value and the limits of the spectral perspective.
Ji Li· Theoretical and Natural Scie...· 0 citations
A diffusion-enhanced inductive link prediction framework that combines Graph Diffusion Convolution (GDC), structural node descriptors, and neighborhood aggregation from GraphSAGE is proposed that achieves higher accuracy than the other models on the benchmark datasets.
Time-evolving networks, or temporal networks, play a crucial role in modeling dynamic interactions across various domains, including biology, social sciences, and information technology. Unlike static networks, these systems undergo continuous changes in topology and edge weights, influencing processes such as information flow, transportation efficiency, and neural activity. Understanding and controlling these networks are essential for predicting future behavior and optimizing dynamic processes. This work focuses on the problem of dynamic centrality, a measure of node importance in time-dependent networks. Specifically, we address how to steer network centrality to a desired state by making minimal modifications to the network structure. This problem is formulated as an optimal control problem for an ordinary differential equation, either matrix- or vector-based, where the control acts on network edges. The proposed framework generalizes centrality control problems studied in static networks and leverages the Pontryagin Maximum Principle for efficient solutions. For large-scale problems, the required matrix-function actions are approximated by Krylov-type techniques, avoiding the explicit formation of dense matrix functions. Numerical experiments on synthetic and real temporal networks show that the proposed framework can effectively steer receive centrality under prescribed control constraints.
Abstract For more than 50 years, the linear sequence and the multiple sequence alignment have been the foundational data structures of protein science, and they remain central to homology search, phylogenetic inference, covariance-based contact prediction, and modern protein language models. However, relational and graph-based representations are increasingly being adopted alongside sequence-based methods to capture biological relationships that linear data structures express only implicitly. Proteins fold as three-dimensional residue interaction networks, evolve through high-dimensional genotype networks defined by mutational connectivity, and operate within cellular protein–protein interaction graphs. Here, we review how graph theory is being used to describe and understand these relationships across protein science, with an emphasis on what these methods offer biochemists working on enzyme superfamilies, protein engineering, drug targets, and functional annotation. We trace the development of these ideas from early theoretical topologies, through statistical coupling and the structural network analyses, to the geometric and graph-like representations used in recent machine-learning-driven advances. Throughout, we emphasise that graphs do not replace sequences or MSAs but provide a complementary representation for biochemical relationships that are difficult to express in one dimension.
Dana S. Matthews, Sacha B. Pulsford, Anthony Brancewicz et al.· Biochemical Journal· 0 citations
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