Computational Complex Networks, 2nd Edition

Topic Information

Dear Colleagues,

Computational methods and models in complex networks have recently proved useful in investigating a variety of networked systems, where diverse network properties can be analyzed. They are widely applied in areas such as network communication, control, prediction, estimation, and security. Recently, fruitful achievements in the research of computation in complex networks have been reported in the literature. At the same time, with the deepening of research, many new problems and challenges have emerged. In particular, fractal theory and fractional calculus have provided powerful tools for characterizing the structural complexity, scaling behaviors, and dynamic processes of complex networks. Incorporating these perspectives can enhance the modeling of network heterogeneity, long‑range dependence, and memory effects. This Topic aims to provide the latest theoretical methods or practical algorithms for complex networks and their applications, including those that utilize fractal or fractional‑order approaches. It is hoped that the contents of this Topic can provide useful information and technical references for readers interested in this area to promote progress in network computation. This Topic has a wide scope, covering contributions from theoretical advances to practical applications.

Dr. Alexandre G. Evsukoff
Dr. Yilun Shang
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Algorithms algorithms	2.6	5.4	2008	17.6 Days	CHF 1800	Submit
Complexities complexities	-	-	2025	15.0 days *	CHF 1000	Submit
Entropy entropy	2.1	4.9	1999	20.9 Days	CHF 2600	Submit
Fractal and Fractional fractalfract	3.5	6.8	2017	17.2 Days	CHF 2700	Submit
Information information	4.3	8.2	2010	18.7 Days	CHF 1800	Submit
Physics physics	1.4	3.5	2019	34.9 Days	CHF 1400	Submit

* Median value for all MDPI journals in the first half of 2026.

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Published Papers (5 papers)

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All Algorithms Complexities Entropy Fractal Fract Information Physics

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34 pages, 13900 KB  

Open AccessArticle

A Multi-Criteria Framework for Comparative Topological Regime Characterization in Complex Networks

by Fabiane de Fatima Carvalho, Ivan Bergier, Silvia Maria Fonseca Silveira Massruhá and Jayme Garcia Arnal Barbedo

Complexities 2026, 2(3), 21; https://doi.org/10.3390/complexities2030021 - 11 Sep 2026

Viewed by 125

Abstract

Collaboration networks are frequently studied as empirical instances of complex social systems, yet standardized methodological frameworks for consistently identifying heterogeneous mesoscopic structural regimes remain limited. This study proposes an integrated multi-criteria classification framework and demonstrates its application to the structural characterization of communities [...] Read more.

Collaboration networks are frequently studied as empirical instances of complex social systems, yet standardized methodological frameworks for consistently identifying heterogeneous mesoscopic structural regimes remain limited. This study proposes an integrated multi-criteria classification framework and demonstrates its application to the structural characterization of communities extracted from a large-scale scientific collaboration network. The framework combines community detection, classical network metrics, statistical modeling of weighted degree tails, small-world diagnostics, information-entropy measures, and fractal analysis based on the Song–Havlin–Makse box-covering renormalization framework. As an empirical application, the methodology is applied to the giant coauthorship component of Embrapa’s scientific production (1974–2024), derived from the Brazilian Agricultural Research Database (BDPA), comprising 60,636 nodes. The weighted Louvain algorithm partitions the network into 25 major communities, which are evaluated through an integrated classification protocol combining the Akaike Information Criterion model selection, Kolmogorov–Smirnov goodness-of-fit tests, small-worldness diagnostics, and fractal scaling analysis. The proposed framework identifies three network families, namely Barabási–Albert (BA-like)/scale-free small-world, scale-free fractal (non-BA) and small-world (non-scale-free), while explicitly distinguishing supported and ambiguous classifications according to the overall consistency of the statistical and structural evidence. The results demonstrate that distinct mesoscopic structural regimes coexist within the same connected collaboration system, highlighting the usefulness of the proposed reproducible multi-criteria framework for comparative topological characterization across complex collaboration networks. Full article

(This article belongs to the Topic Computational Complex Networks, 2nd Edition)

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23 pages, 2912 KB  

Open AccessArticle

ECPD-SG: An Emotion-Aware Contrastive Prototype Algorithm for Change Point Detection in Dynamic Social Graphs

by Yingjie Xie, Yinbo Liu, Yanfei Liu, Junfang Li and Wenjun Wang

Algorithms 2026, 19(7), 536; https://doi.org/10.3390/a19070536 - 2 Jul 2026

Viewed by 272

Abstract

Change point detection in dynamic social graphs aims to identify significant transitions in evolving interaction patterns. The task is particularly challenging due to sparse and noisy interactions and frequent user turnover, while the existing methods largely overlook the semantic and emotional signals in [...] Read more.

Change point detection in dynamic social graphs aims to identify significant transitions in evolving interaction patterns. The task is particularly challenging due to sparse and noisy interactions and frequent user turnover, while the existing methods largely overlook the semantic and emotional signals in user-generated content by focusing primarily on structural changes. To address these limitations, this paper proposes ECPD-SG, an emotion-aware contrastive prototype learning algorithm for unsupervised change point detection in dynamic social graphs. ECPD-SG constructs emotion-aware graph snapshots by integrating textual and affective features into node representations and recalibrating interaction weights through emotion-aware attention. It then summarizes temporal node representations into adaptive prototypes and models their evolution using optimal-transport-based alignment and contrastive learning. Change points are detected from prototype-level shift scores with an adaptive CUSUM decision rule. Experiments on real-world dynamic social graph datasets show that ECPD-SG achieves competitive or superior performance over representative baselines, while ablation and sensitivity analyses verify the effectiveness of its key components. Full article

(This article belongs to the Topic Computational Complex Networks, 2nd Edition)

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31 pages, 17875 KB  

Open AccessArticle

GCR-Net: Stable Reinforcement Learning for Community Detection with Unknown Community Count in Attributed Networks

by Wencai He, Zhijie Peng, Yuanbin He, Ziyu Zhang, Mingshen Zhang and He Zhu

Algorithms 2026, 19(6), 484; https://doi.org/10.3390/a19060484 - 16 Jun 2026

Viewed by 368

Abstract

Community detection in attributed networks becomes considerably more challenging when the number of communities is unknown in advance. Most existing deep community detection methods assume a fixed community count, whereas reinforcement learning (RL)-based alternatives often suffer from overestimated action values and unstable target [...] Read more.

Community detection in attributed networks becomes considerably more challenging when the number of communities is unknown in advance. Most existing deep community detection methods assume a fixed community count, whereas reinforcement learning (RL)-based alternatives often suffer from overestimated action values and unstable target updates. To address these limitations, we propose GCR-Net (Graph Community Recognition Network), an RL-guided framework that combines representation learning with adaptive community-count selection. The method adapts decoupled value estimation and gradual anchor-network updates from established deep RL techniques to a formal MDP over candidate community counts. Experiments on citation, social, biomedical, and proteininteraction benchmarks, together with synthetic graphs with more than ten communities, show that GCR-Net delivers competitive NMI and ARI scores with lower variance and more stable optimization than conventional RL baselines. Statistical tests indicate that the clearest gains concern training stability rather than large accuracy margins. Full article

(This article belongs to the Topic Computational Complex Networks, 2nd Edition)

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23 pages, 2933 KB  

Open AccessArticle

Iterative Generation and Generalized Degree Distribution of Higher-Order Fractal Scale-Free Networks

by Lin Qi, Jiaxin Zhang, Ying Fan and Feiyan Guo

Fractal Fract. 2026, 10(5), 306; https://doi.org/10.3390/fractalfract10050306 - 30 Apr 2026

Viewed by 502

Abstract

Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have emerged as a research hotspot due to their ability to express interactions among [...] Read more.

Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have emerged as a research hotspot due to their ability to express interactions among multiple nodes. This study proposes an iterative generation model for higher-order fractal networks. The iteration is controlled by three parameters: the dimension $K$ of the simplicial complex, the multiplier $m$, and the iteration count $t$. The constructed network is a pure simplicial complex. Theoretical analysis using the similarity dimension and experimental verification using the box-counting dimension demonstrate that the generated networks exhibit fractal characteristics. When the multiplier $m$ is large, the generalized degree distribution of the generated networks exhibits scale-free properties. Full article

(This article belongs to the Topic Computational Complex Networks, 2nd Edition)

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32 pages, 617 KB  

Open AccessArticle

Analyzing Late Antiquity Shifts of Trade Regime in the Iberian Peninsula and Their Causes via Change Point Detection Methods

by Juan Julián Merelo-Guervós

Complexities 2026, 2(2), 12; https://doi.org/10.3390/complexities2020012 - 16 Apr 2026

Viewed by 873

Abstract

History attempts to make sense of disparate information by trying to create discourse that lays a series of events with crisp cause–effect relationships in a sequence. Epochal shifts, such as the change from Antiquity to the Middle Ages, are especially complex since they [...] Read more.

History attempts to make sense of disparate information by trying to create discourse that lays a series of events with crisp cause–effect relationships in a sequence. Epochal shifts, such as the change from Antiquity to the Middle Ages, are especially complex since they involve a large number of economic, political and even religious factors which occur over long periods and that might overlap and interact through reciprocal feedback mechanisms, making this cause–effects sequence difficult to establish. In this research we adopt a data-driven and well-established methodology to identify, with quantifiable statistical precision, the moment when this shift happened, and from there arrive at its possible causes. We will use historical coin hoard data to find out whether such a shift is detected in a peripheral part of the Roman Empire, the Iberian Peninsula. To do so, we will apply different changepoint analysis methods to a time series of trade links created from that data, and conduct a retrospective analysis based on that result, analyzing the structure of the trade networks before and after the link. Thus, we progress from identifying when the shift happened to identifying where it took place, which in turn allows us to get to investigate why it happened, namely, historical events that could have caused it. This methodology can be used to analyze epochal changes in several steps using time-stamped network data, possibly finding disregarded causes or cause–effect links that could have been overlooked by qualitative methods; in this case, we have applied it to a dataset of coin hoards either found in the Iberian Peninsula or including coins minted there, finding a changepoint in the early 5th century, which, through network analysis, has been linked to a loss of trade with the area of Britannia. Full article

(This article belongs to the Topic Computational Complex Networks, 2nd Edition)

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