Probability and complex quantum trajectories: Finding the missing links

Moncy V. John

Wen Liang, Fei Yan, Abdullah M. Iliyasu +2 more

Prediction of missing links is an important part of many applications, such as friends' recommendations on social media, reduction of economic cost of protein functional modular mining, and implementation of accurate recommendations in the shopping platform. However, the existing algorithms for predicting missing links fall short in the accuracy and the efficiency. To ameliorate these, we propose a simplified quantum walk model whose Hilbert space dimension is only twice the number of nodes in a complex network. This property facilitates simultaneous consideration of the self-loop of each node and the common neighbour information between arbitrary pair of nodes. These effects decrease the negative effect generated by the interference effect in quantum walks while also recording the similarity between nodes and its neighbours. Consequently, the observed probability after the two-step walk is utilised to represent the score of each link as a missing link, by which extensive computations are omitted. Using the AUC index as a performance metric, the proposed model records the highest average accuracy in the prediction of missing links compared to 14 competing algorithms in nine real complex networks. Furthermore, experiments using the precision index show that our proposed model ranks in the first echelon in predicting missing links. These performances indicate the potential of our simplified quantum walk model for applications in network alignment and functional modular mining of protein-protein networks.

E. C. G. Sudarshan

A. R. Nexus

This work (O3 — Part II) completes the derivation of the quantum mechanical formalism within the Emergent Pre-Quantizable (EPPQ) framework by establishing the necessary emergence of the complex Hilbert space structure from purely relational dynamics. Building upon O3 — Part I, where the Schrödinger equation and the Born rule were derived conditionally from coarse-grained Markov dynamics on metastable basins, this paper addresses the remaining foundational gap: the origin of complex amplitudes and quantum phases. We demonstrate that a discrete U(1) gauge structure naturally arises from the contextual layer of the EPPQ vacuum, formalized as a principal bundle over the vacuum graph. The associated holonomies (Wilson lines) induce geometric phases along microscopic trajectories. By analyzing the interference of these trajectories, we prove that: • Transition amplitudes must take values in the complex field ℂ • The Born rule P = |ψ|² is the unique probability measure compatible with superposition and gauge invariance • The effective Hamiltonian emerges from the skew-symmetric component of the coarse-grained Markov generator and is directly determined by contextual holonomies Furthermore, we show that the same structure yields, in a specific spectral sector, the Yukawa matrices of the Standard Model, including nontrivial complex phases responsible for CP violation. The results establish that the complex structure of quantum mechanics is not an independent postulate, but a necessary consequence of relational holonomy and trajectory interference. This provides the missing link between the geometric constructions of O1/O4 and the emergent quantum dynamics of O3. This work is part of a broader program (EPPQ) aimed at deriving spacetime, quantum theory, and particle physics from a minimal relational ontology.

Wen Liang, Fei Yan, Abdullah M. Iliyasu +1 more

Link prediction remains a challenging pursuit in existing complex networks. Our study proposes a Grover coin driven quantum walk (GCQW) model for prediction of missing edges on complex networks. The GCQW model uses observed probabilities of common neighbours of two nodes as similarity between the nodes. Furthermore, each walk step of the proposed model is determined by a three degree of influence rule. Results of experiments based on the area under the receiver operating characteristic curve (AUC) index demonstrate the proposed model’s performance in eight real complex networks outperforms nine conventional comparison algorithms. Outcomes show that even when the ratio of testing to training is set in the range 0.1∼0.5, our GCQW model maintained a stable and competitive performance in terms of the AUC index. The proposed GCQW model will be expectedly applied in function modular mining of protein-protein interaction networks and friend recommendation of social media.

Abdo Abou Jaoudé