Bitopological Spaces: Listening to the Direction of Time When It Matters
How two topologies — built from the same data — can hear the difference between past and future
Most of the tools we use to analyze sequences of data — averages, correlations, spectral analyses — treat time as a label that could run in either direction without changing the answer. Reverse the order of your data points and many standard methods give you identical results. But in the real world, the direction of time matters profoundly. Economies expand slowly and crash suddenly. Heartbeats rise smoothly and fall steeply. A method blind to direction is a method blind to one of the most fundamental features of how systems change.
A recent paper by independent researcher José Mauricio Gómez Julián introduces a construction that addresses this gap. Taking a known method from graph theory and extending it to directed graphs, the paper produces a pair of topologies — mathematical frameworks for understanding structure and connectivity — whose divergence is a topological fingerprint of temporal irreversibility. Applied to three decades of American economic data, the method recovers a picture that is both mathematically precise and economically interpretable. Here is a walk through the main ideas.
Seeing and Being Seen
The starting point is a beautifully simple idea introduced by Lucas Lacasa and collaborators in 2008. Imagine plotting a time series — say, 129 consecutive quarterly growth rates of U.S. GDP — as points above a timeline. Now connect two points with a line if they can “see” each other: the straight segment between them passes above every intermediate data point, as if you stood at one point and shone a flashlight toward the other with no obstacles in the way.
The result is a visibility graph: a network whose nodes are time points and whose edges encode a geometric relationship. Visibility graphs have been used to classify chaotic systems, detect heartbeat anomalies, and distinguish between types of economic regimes. They translate the shape of a time series into the structure of a graph, opening the door to the vast toolkit of network science.
But standard visibility graphs are undirected: an edge between two points does not record which one came first. If you orient each edge from the earlier time point to the later one, you obtain a directed visibility graph — a directed acyclic graph in which the arrows always point forward in time. This orientation carries information about temporal asymmetry that the undirected graph throws away entirely.
From Networks to Structure
Here is where the paper’s contribution begins.
In 2018, Huda Nada and collaborators introduced a procedure for turning any undirected graph into a topological space. For those unfamiliar with the term, a topology is a mathematical framework that defines what it means for groups of points to be “open,” for sets to be “connected,” and for spaces to have “structure.” It operates at a level more abstract than distances or coordinates — it captures the pattern of how sets overlap and separate.
The Nada construction works as follows. For each vertex of the graph, compute its closed neighborhood: the vertex itself plus all of its immediate neighbors. Take this family of neighborhoods and generate a topology by closing it under two operations: finite intersections (combine neighborhoods by overlapping them) and arbitrary unions (combine neighborhoods by collecting them). The result is a topology on the vertex set, and its invariants — connected components, separation properties, component counts — capture structural features of the graph.
This procedure is universal: it works for any family of subsets of any set. The mathematical content lies in identifying the right family to use.
Gómez Julián’s key observation is that for a directed graph, you do not get one family of neighborhoods — you get two. For each vertex:
The forward closed neighborhood includes the vertex itself and all the vertices it points to — the later time points it can see. The backward closed neighborhood includes the vertex itself and all the vertices that point to it — the earlier time points from which it is visible.
Apply the Nada procedure to the forward neighborhoods and you get a topology τ+. Apply it to the backward neighborhoods and you get a topology τ−. The resulting triple (V, τ+, τ−) is what mathematicians call a bitopological space: a set equipped with two topologies simultaneously, a concept introduced by John Kelly in 1963.
The extension is, in a precise mathematical sense, trivial — the topology axioms do not care where the generating family came from. But recognizing this trivial extension as the right thing to do, and showing that the resulting bitopological structure captures something real about temporal asymmetry, is the paper’s central insight.
When Two Topologies Disagree
If the process generating your data is symmetric — equally likely to go up as down, at the same speed — then the forward and backward neighborhoods are statistically exchangeable. The two topologies τ+ and τ− look the same, and the bitopological structure adds nothing beyond the undirected construction.
But if the process is asymmetric, the two topologies diverge. Consider the prototypical asymmetry of economic and physical systems: gradual expansion followed by sudden contraction. Forward visibility through a gradual rise connects many points — each point can see far ahead through the gentle slope. Backward visibility through an abrupt drop connects few — the sharp fall blocks the line of sight. The forward topology ends up more connected (fewer separate components) than the backward topology.
This divergence is the topological fingerprint of temporal irreversibility. The paper defines three quantitative measures of it:
The asymmetry direction Δ = C− − C+, where C+ and C− are the numbers of connected components in the forward and backward topologies. Positive Δ means the forward topology is more connected. The component-count irreversibility index IC, which normalizes the difference to lie between 0 and 1. And the base-size irreversibility index IB, which measures the analogous difference in the sizes of the generating bases.
These are pure numbers — no calibration, no free parameters, no training data. They emerge from the structure of the data and the construction itself.
Peeling the Onion: Three Layers of Structure
One of the paper’s most clarifying contributions is the identification of three nested layers of topological structure on the same time series, each revealing different information.
Layer 1 — The Alexandrov topology (reachability). For a directed acyclic graph, the most basic topology treats as “open” any set that is closed under forward reachability: if a node is in the set, all its descendants are too. This topology has exactly one connected component for any weakly connected graph, because every node can reach every later node through some directed path. At this level, the system is globally indecomposable. It tells us what we already know: the economy is a single connected process in which each quarter influences every subsequent quarter through chains of causation.
Layer 2 — The undirected Nada topology (local fragmentation). When you apply the Nada construction to the undirected shadow of the visibility graph, the topology fragments dramatically. The intersection closure of the neighborhoods reveals clusters of time points that share local structural similarity — groups of observations linked by overlapping visibility neighborhoods — that go beyond mere reachability. This layer uncovers genuine structure that the reachability topology hides entirely.
Layer 3 — The bitopological layer (temporal asymmetry). When you split the construction into forward and backward, a further distinction emerges. The forward and backward topologies have different component counts — and the difference is invisible to the undirected construction and invisible to the Alexandrov construction. It lives only in the gap between the two directed topologies.
Each layer is contained within the next: the Alexandrov topology is a subtopology of the Nada topology (a theorem proved in the paper), which in turn underlies the bitopological structure. But each coarser layer hides information that the finer layer reveals.
What the American Economy Looks Like Through a Topological Lens
The paper applies the full pipeline to the quarterly growth rate of U.S. real GDP from Q1 1992 to Q1 2024 — 129 observations spanning the dot-com bust, the Global Financial Crisis, and the COVID-19 shock. Two graph constructions are used: the Horizontal Visibility Graph and the Natural Visibility Graph, both in their directed forms.
The headline finding is that Δ = +4 in both constructions. The forward topology has 4 fewer connected components than the backward topology, regardless of which visibility-graph variant you use. This positive value is consistent with the well-documented asymmetry of the American business cycle over this period: expansions are gradual and sustained (1992–2000, 2001–2007, 2009–2020), while contractions are sharp and short-lived (2001, 2008–2009, 2020). Forward visibility through a gradual expansion is unobstructed; backward visibility through an abrupt contraction is fragmentary.
What makes this finding compelling is its invariance. The HVG and NVG produce very different graphs — 248 vs. 406 edges, different base sizes, different absolute component counts — yet they agree on the sign and magnitude of Δ. The signal appears robust: a feature of the underlying data, not an artifact of how you choose to draw the graph.
Another detail worth noting: the base-size irreversibility index IB is exactly zero in both constructions. The forward and backward topologies are generated by bases of equal size (250 and 250 for the HVG, 239 and 239 for the NVG). The asymmetry lives entirely in the structure of those base elements and how their intersections distribute — not in how many there are. The two topologies are built from the same number of building blocks, but those blocks fit together differently depending on whether you are looking forward or backward through time.
A Single Shock
Perhaps the most striking empirical finding is what the topology says about the COVID-19 shock.
The second quarter of 2020 recorded the sharpest contraction in U.S. GDP on record — an annualized rate of roughly −31%. The third quarter recorded the sharpest rebound — roughly +33%. These are the two most extreme observations in the entire 129-quarter series, opposite in sign and opposite in economic interpretation.
A naive analysis would naturally separate them: one is the worst crash, the other the best recovery. They sit at opposite ends of the value spectrum.
But the Nada topology classifies them together. Under both the undirected and directed topologies, under both the HVG and NVG, these two observations belong to the same connected component.
Why? Because the topology is not a proximity measure. It does not group points by how close their values are. It groups them by the structure of their visibility neighborhoods — which other points they can see, and how those visibility patterns intersect. Despite their extreme and opposite values, the two quarters share neighborhoods that overlap substantially. The intersection closure, which drives the Nada construction, puts them in the same cluster.
This matches the interpretation most economists give to the event: the contraction and the rebound are two phases of a single exogenous shock, driven by the same underlying cause — the pandemic and the policy response to it. The topology recovers this interpretation from the geometry of the data alone, without any economic priors built in.
Certifying the Construction
The paper takes reliability seriously at three levels.
Machine-checked proofs. The central theorem and related core results have been formalized in Lean 4, a proof assistant, against the Mathlib mathematical library. A computer has verified that the proofs are logically correct, with no gaps or hidden assumptions. The formalization is archived alongside the paper as part of a reproducibility bundle on Zenodo.
Polynomial-time algorithms. Every step of the construction has an explicit algorithm with proven complexity bounds. The connected components of each topology can be computed in polynomial time without enumerating the full topology, which can be exponentially large. The key trick is to work through a combinatorial proxy for the topology called the specialization preorder, using bitset operations that are highly efficient in practice.
Honest uncertainty. A three-valued decision procedure for pairwise connectedness reports “pairwise connected,” “pairwise disconnected,” or “undecided” — the last when the computation exhausts its resource budget. Rather than guessing, the algorithm honestly reports that it has not finished the work. This is a methodological commitment as much as a technical one: a topological statement counts as established only when the computation has completed the work that establishes it.
No free parameters. The construction has no tuning knobs. The topological invariants — component counts, base sizes, irreversibility indices — are determined entirely by the data and the definitions. There is nothing to calibrate, nothing to overfit.
A New Lens
The paper does not propose to replace existing methods of time series analysis. Correlation, spectral analysis, regime-switching models, and the many other tools of econometrics and statistics capture information that topology cannot see: amplitude, frequency, distributional shape. The paper is explicit about this complementarity.
What the topological construction offers is a new lens — one that responds to the relational structure of a time series rather than its metric structure. It asks not “how big is this change?” but “what does this change connect to, and what does it disconnect from, and is the answer different depending on which direction in time you are looking?”
For systems where temporal asymmetry is a defining feature — business cycles, climate dynamics, physiological signals, causal event sequences — this lens may reveal structure that traditional tools, by their very construction, cannot see.
The application to U.S. GDP is a proof of concept. The construction is general: it applies to any time series that can be turned into a directed visibility graph, which is to say, any time series at all. Whether the invariants it produces are useful features for classification, prediction, or interpretation in broader contexts is an empirical question that the paper opens but does not close.
What it does establish is this: there exists a construction that takes a time series, produces two topologies from it, and quantifies the gap between them as a measure of temporal irreversibility. The construction is mathematically sound, mechanically verified, algorithmically tractable, parameter-free, and — when applied to the American economy across three turbulent decades — gives answers that make economic sense.
That is a foundation worth building on.
