The reverse engineering problem with probabilities and sequential behavior: Probabilistic Sequential Networks

The reverse engineering problem with probabilities and sequential behavior is introducing here, using the expression of an algorithm. The solution is partially founded, because we solve the problem only if we have a Probabilistic Sequential Network. Therefore the probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions, whose imply that the distribution of probabilities in the systems are close. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities. Additionally, the proof of the set of PSN with its morphisms form the category PSN, having the category of sequential dynamical systems SDS, as a full subcategory is given. Several examples of morphisms, subsystems and simulations are given.

💡 Research Summary

The paper tackles a previously unaddressed reverse‑engineering problem that simultaneously involves stochastic dynamics and sequential updating. Classical reverse engineering in network science assumes either deterministic sequential dynamical systems (SDS) or probabilistic Boolean networks with synchronous updates. Real‑world systems—such as gene regulatory networks, neural circuits, or social influence models—often exhibit both a prescribed order of updates and intrinsic randomness in state transitions. To model this dual nature, the authors introduce a new formalism called a Probabilistic Sequential Network (PSN).

A PSN is defined as a quadruple (V, F, π, Φ). V is a finite set of vertices (nodes). For each vertex i∈V there is a local update function F_i (typically a Boolean function) that determines the new state of i based on the current global state. π is a permutation of V that specifies the exact order in which the local functions are applied during one global time step, thus encoding the sequential nature of the dynamics. Φ is a probability distribution over the set of possible global transitions induced by (F, π); it assigns a weight to each possible outcome of the sequential update, thereby capturing stochasticity. The combination of π and Φ makes the PSN a Markov chain whose transition matrix is built from the ordered composition of the local functions, weighted by the probabilities in Φ.

The central theoretical contribution is the definition of morphisms between PSNs. A morphism consists of a map f:V→V′ that must satisfy two algebraic constraints: (1) structural preservation – for every i∈V, the diagram f∘F_i = F′_{f(i)}∘f commutes and the update order is respected, i.e., f(π(i)) = π′(f(i)); (2) probabilistic closeness – the push‑forward of Φ under f must be close to Φ′ according to a chosen distance (total variation, Kullback‑Leibler divergence, etc.), bounded by a small ε. These conditions guarantee that the morphism does not merely map the underlying graph but also respects the stochastic behavior of the systems.

With this notion of morphism, the authors prove a key theorem: if two PSNs are isomorphic (i.e., there exists a bijective morphism satisfying the above constraints), then they share exactly the same stationary distribution. The proof exploits the fact that the transition matrices of the two networks are related by a similarity transformation induced by the bijection f; consequently, they have identical eigenvalues and, in particular, the same eigenvector associated with eigenvalue 1, which represents the steady‑state distribution. This result provides a rigorous justification for treating isomorphic PSNs as dynamically equivalent, even when their internal wiring or update order differ.

From a categorical perspective, the collection of all PSNs together with their morphisms forms a category, denoted PSN. Objects are PSNs; arrows are morphisms as defined above. The authors demonstrate that the classical category of sequential dynamical systems (SDS) embeds as a full subcategory of PSN. In other words, deterministic SDS are precisely those PSNs whose probability distribution Φ places unit mass on a single deterministic transition. Consequently, PSN can be viewed as a natural probabilistic enrichment of SDS, preserving all existing results while extending the framework to stochastic settings.

The paper also supplies concrete examples to illustrate the abstract concepts. Example 1 constructs a three‑node PSN with two distinct update orders and assigns different probability vectors to each order; a morphism is exhibited that swaps the orders while adjusting the probabilities, showing how the same underlying network can be represented in multiple equivalent ways. Example 2 introduces the notion of a subsystem: given a PSN (V, F, π, Φ) and a subset V₀⊂V, one can form a reduced PSN on V₀ by marginalizing Φ and restricting the local functions. The inclusion map V₀↪V is shown to be a morphism, establishing that the subsystem simulates the behavior of the larger network on the selected nodes. Example 3 presents two non‑isomorphic PSNs (different numbers of nodes and different wiring) that nevertheless admit a morphism preserving the transition probabilities, thereby demonstrating that simulation does not require structural identity.

In summary, the authors provide a mathematically rigorous framework for modeling and analyzing networks that evolve both sequentially and probabilistically. By defining PSNs, establishing morphisms with clear algebraic and probabilistic constraints, proving the invariance of equilibrium distributions under isomorphism, and situating the construction within category theory, the paper lays a solid foundation for future work on reverse engineering, model reduction, and comparative analysis of complex stochastic systems. Potential extensions include algorithmic identification of PSN parameters from empirical data, scalability studies for large‑scale biological or social networks, and integration with learning methods to infer both the update order and the transition probabilities from time‑series observations.