Fractal elements and topologies as a solution to the scalability of photonic neuromorphic processors

Background. Classical network topologies, such as the two-dimensional mesh, impose constraints on the scaling of photonic neuromorphic processors, leading to exponential growth of optical losses, a quadratic increase in occupied area, and a linear increase in latency with the number of nodes. Biological neural networks demonstrate an alternative, evolutionarily optimized principle for constructing their architecture, utilizing self-similar and hierarchical laws to ensure high connectivity with a short average path length. Therefore, the study and adaptation of such fractal topologies represent a relevant task for overcoming the scaling barriers of photonic neuromorphic computing systems. Aim. To perform a theoretical analysis and quantitative comparison of the main parameters of self-similar topologies to substantiate the boundaries of their applicability in scaling photonic neuromorphic processors; to formulate recommendations for the selection and adaptation of interconnection architectures. Methods. A systematic comparative analysis of 13 types of network topologies was conducted, including stochastic (fractal dendrites, neural clusters, small-world network, scale-free networks) and deterministic fractal structures (trees, Sierpinski carpets, Hilbert curve, Koch snowflake), as well as traditional ones. For each topology, five key parameters were calculated and analyzed at scales up to N=107 nodes: diameter, average path length, clustering coefficient, critical failure probability, and maximum optical losses. Based on multi-factor comparison, a final classification of topologies according to their practical applicability was performed. Results. It was established that topologies with logarithmic diameter scaling (small-world network, fractal trees) provide minimal latency and optical losses. However, many deterministic fractals have zero local clustering, while stochastic models are technologically difficult to implement. Based on a scoring assessment across five criteria (scalability, clustering, reliability, maximum optical losses, technological implementability), the «small-world network» topology was recognized as optimal. To optimize the balance of key characteristics, a concept of hybrid «engineered fractals» was proposed – deterministic truncated hierarchies combining a fractal backbone for global connections with regular clusters at the local level. Conclusion. It is shown that the potential of fractal topologies for scaling photonic neuromorphic systems is specific and realizable primarily through their adapted, hybrid forms. The main recommendation for designing large-scale processors is to use a deterministic hierarchical implementation of the «small-world network» or «engineered fractals», which allows combining logarithmic latency scaling, high fault tolerance, and technological implementability in planar integrated photonics.