Volume 35, Number 2 (2026)

---

Second-Order Three-Neighbor Number-Conserving Cellular Automata
Akiko Fukuda, Sennosuke Watanabe, Yuki Nishida and Junta Matsukidaira

In this paper, the number-conserving rules for second-order three-neighbor binary cellular automata (SOCAs) are discussed. The SOCAs include the well-known elementary cellular automata and elementary reversible cellular automata and have a total of $2^{64}$ local rules. We show that the number of number-conserving SOCAs is limited to 74 through a polynomial representation of the local rules. A concrete description of the polynomial representation for 15 representative rules is provided. In addition, we establish a necessary and sufficient condition for each SOCA to be number conserving. This condition can be regarded as a discrete version of the flux form in fluid dynamics.

Keywords: second-order cellular automata; number conservation; polynomial representation; flux form

Cite this publication as:
A. Fukuda, S. Watanabe, Y. Nishida and J. Matsukidaira, “Second-Order Three-Neighbor Number-Conserving Cellular Automata,” Complex Systems, 35(2), 2026 pp. 119–140.
https://doi.org/10.25088/ComplexSystems.35.2.119

---

Characterization of Maximal Length Cellular Automata Using NSRTD
Som Banerjee and Mamata Dalui

Maximal length cellular automata (CAs) have gained significant attention from researchers due to their applications in different areas like random number generation, cryptography and test pattern generation. This paper reports the theoretical framework of the next state rule min term transition diagram (NSRTD) for the characterization of maximal length CAs in null-boundary condition. The proposed solution helps in verifying whether a given cellular automaton (CA) is a maximal length CA or not in $O\left(2^n\right)$ time. Also, the proposed solution can identify and eliminate a CA candidate that fails to configure a maximal length CA in linear time.

Keywords: cellular automata; maximal length CA; fixed-point attractor; NSRTD; null boundary CA

Cite this publication as:
S. Banerjee and M. Dalui, “Characterization of Maximal Length Cellular Automata Using NSRTD,” Complex Systems, 35(2), 2026 pp. 141–160.
https://doi.org/10.25088/ComplexSystems.35.2.141

---

Elementary Cellular Automata for Reservoir Computing with Modest Resources
Kazuhiro Yokota

Several elementary cellular automaton (ECA) rules have been found useful for reservoir computing (RC). In this paper, we investigate how effectively ECA rules operate with reduced computational resources. We examine successful rules with the 5-bit memory task and nonlinear autoregressive moving-average (NARMA) benchmarks. Our model is a slightly modified version of one previously reported that is optimized for modest resources. We find that the features produced by a cellular automaton (CA) at each timestep vary greatly in their contributions to the computation. The performance on the NARMA task is improved, especially in a rule from Wolfram’s class 2. The result demonstrates the feasibility of implementing RC with modest resources, such as those in embedded systems, and helps elucidate the role of cellular automata in RC.

Keywords: reservoir computing; elementary cellular automaton

Cite this publication as:
K. Yokota, “Elementary Cellular Automata for Reservoir Computing with Modest Resources,” Complex Systems, 35(2), 2026 pp. 161–180.
https://doi.org/10.25088/ComplexSystems.35.2.161

---

Strengthening the Common Neighbor Algorithm by the Degree of Importance of Nodes for Predicting Missing Links in Complex Networks
Mourad Charikhi

Social and complex networks analyses are important areas of research. Link prediction is a discipline in this field that attempts to find missing links and predict new links in networks. Node proximity approaches are widely used in link prediction problems; however, they suffer from weak performance. We propose a new method based on node proximity that exploits common neighbors and the degree of importance of nodes using the well-known PageRank algorithm. Our method improves the performance of existing local methods with a low computation time. Experiments conducted on nine datasets show the advantage of our new method compared to the basic methods of common neighbors, Adamic–Adar and resource allocation. We compare our approach and 14 state-of-the-art techniques, including path and local information approaches. The results indicate that our new approach provides a significant improvement in terms of area under the receiver operating characteristic curve (AUC) score with linear computational complexity.

Keywords: link prediction; similarity metrics; PageRank algorithm; network evolution; social network

Cite this publication as:
M. Charikhi, “Strengthening the Common Neighbor Algorithm by the Degree of Importance of Nodes for Predicting Missing Links in Complex Networks,” Complex Systems, 35(2), 2026 pp. 181–202.
https://doi.org/10.25088/ComplexSystems.35.2.181