The category of directed graphs is isomorphic to a particular category whose objects are labeled undirected bipartite graphs and whose morphisms are undirected graph morphisms that respect the labeling. Based on this isomorphism, we begin by showing that the class of all directed graphs is a Graph Isomorphism Complete class. Afterwards, by extending this categorical framework to weighted prime graphs, we prove that the categories of multidirected graphs with and without self-loops are each isomorphic to a particular category of weighted prime graphs. Consequently, we prove that these classes of multidirected graphs are also Graph Isomorphism Complete.

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering.

We develop an information-theoretic, cost-first framework for discrete dynamics in which the primitive operation is ratio-based comparison. Given two quantities compared via their ratio $x=a/b$, we assign a cost $F\left(x\right)$ measuring deviation from equilibrium ($x=1$). Adopting a reciprocal d’Alembert composition law motivated by coherent chaining, together with quadratic calibration at unity, uniquely determines a reciprocal comparison cost $J\left(x\right)=\frac{1}{2}\left(x+x^{-1}\right)-1$. Taking J as input, we model recognition events as deterministic updates on directed graphs recorded in a minimal ledger. Minimality (no intra-tick ordering metadata) together with non-commutativity of events implies atomic ticks: at most one event per tick. With conservation, pairwise locality, and quantization in $\delta \mathbb{Z}$, each event is recorded as a balanced double-entry posting. For graphs with cycles, assuming time-aggregated cycle closure over a finite clearing horizon, we show that cleared cycle closure is equivalent to path-independence and that the cumulative flow admits a scalar potential on each connected component (unique up to additive constant) via a discrete Poincaré lemma. On hypercube graphs $Q_d$, atomic single-edge updates impose a $2^d$-tick minimal period for timestamp-unique coverage, realized by cyclic Gray codes (explicitly for $d=3$). The framework links ratio-based cost functions, conservative graph flows, and discrete potential theory through explicitly stated axioms and structural assumptions.

We present a formal framework for modeling neural network dynamics using Category Theory, specifically through Markov categories. In this setting, neural states are represented as objects and state transitions as Markov kernels, i.e., morphisms in the category. This categorical perspective offers an algebraic alternative to traditional approaches based on stochastic differential equations, enabling a rigorous and structured approach to studying neural dynamics as a stochastic process with topological insights. By abstracting neural states as submeasurable spaces and transitions as kernels, our framework bridges biological complexity with formal mathematical structure, providing a foundation for analyzing emergent behavior. As part of this approach, we incorporate concepts from Interacting Particle Systems and employ mean-field approximations to construct Markov kernels, which are then used to simulate neural dynamics via the Ising model. Our simulations reveal a shift from unimodal to multimodal transition distributions near critical temperatures, reinforcing the connection between emergent behavior and abrupt changes in system dynamics.

Emergent effect is crucial to understanding the properties of complex systems that do not appear in their basic units, but there has been a lack of theories to measure and understand its mechanisms. In this letter, we consider emergence as a kind of structural nonlinearity, discuss a framework based on homological algebra that encodes emergence as the mathematical structure of cohomologies, and then apply it to network models to develop a computational measure of emergence. This framework ties the potential for emergent effects of a system to its network topology and local structures, paving the way to predict and understand the cause of emergent effects. We show in our numerical experiment that our measure of emergence correlates with the existing information-theoretic measure of emergence.

We develop an information-theoretic framework for discrete dynamics grounded in a comparison-cost functional on ratios. Given two quantities compared via their ratio \(x=a/b\), we assign a cost \(F(x)\) measuring deviation from equilibrium (\(x=1\)). Requiring coherent composition under multiplicative chaining imposes a d'Alembert functional equation; together with normalization (\(F(1)=0\)) and quadratic calibration at unity, this yields a unique reciprocal cost functional (proved in a companion paper): \[ J(x) = \tfrac{1}{2}\bigl(x + x^{-1}\bigr) - 1. \] This cost exhibits reciprocity \(J(x)=J(x^{-1})\), vanishes only at \(x=1\), and diverges at boundary regimes \(x\to 0^+\) and \(x\to\infty\), excluding ``nothingness'' configurations. Using \(J\) as input, we introduce a discrete ledger as a minimal lossless encoding of recognition events on directed graphs. Under deterministic update semantics and minimality (no intra-tick ordering metadata), we derive atomic ticks (at most one event per tick). Explicit structural assumptions (conservation, no sources/sinks, pairwise locality, quantization in \(δ\mathbb{Z}\)) force balanced double-entry postings and discrete ledger units. To obtain scalar potentials on graphs with cycles while retaining single-edge impulses per tick, we impose time-aggregated cycle closure (no-arbitrage/clearing over finite windows). Under this hypothesis, cycle closure is equivalent to path-independence, and the cleared cumulative flow admits a unique scalar potential on each connected component (up to additive constant), via a discrete Poincaré lemma. On hypercube graphs \(Q_d\), atomicity imposes a \(2^d\)-tick minimal period, with explicit Gray-code realization at \(d=3\). The framework connects ratio-based divergences, conservative graph flows, and discrete potential theory through a coherence-forced cost structure.