Ordinal Networks: Methods & Applications

• Ordinal networks are methods that map data sequences to permutations, encoding the relative ordering of observations rather than their absolute values.
• They are constructed by converting scalar or image data into graphs where nodes represent ordinal patterns and edges capture temporal or spatial transitions.
• Applications span complex systems analysis and deep learning, where order-aware models improve prediction accuracy and reveal underlying dynamical structures.

Searching arXiv for the cited works and related uses of “ordinal networks”. arxiv_search query: "ordinal networks arXiv" Ordinal networks are methods that encode relative order rather than absolute magnitude. In one established usage, a time series or image is transformed into a weighted, directed graph whose vertices are ordinal patterns and whose edges represent temporal or spatial transitions among those patterns. In another, neural network models are modified so that outputs, losses, or pooling operators explicitly respect ordered targets or ranked activations. Across these usages, the common principle is that ordering information is structurally meaningful and should not be discarded by nominal classification losses, value-only summaries, or standard max-pooling rules (Pessa et al., 2020, Cao et al., 2019, Kumar, 2018).

1. Multiple meanings of the term and the common role of order

Ordinal time-series analysis maps observations to ordinal patterns, that is, permutations describing order relations among delayed samples rather than the samples themselves. A recent review describes ordinal time-series analysis as based on mapping time series to ordinal patterns, “i.e., order relations between the values of a time series and not the values themselves,” and notes that this approach was introduced in 2002 by C. Bandt and B. Pompe (Lehnertz, 2023). In this setting, an ordinal network is typically a graph over permutations.

In supervised learning, the same order-centric principle appears in ordinal regression. If labels belong to an ordered set $\{r_1 \prec r_2 \prec \dots \prec r_K\}$, then treating them as unrelated categories with standard multi-class cross-entropy ignores that neighboring labels are closer than distant ones. The resulting models can be converted into $K-1$ binary subtasks of the form “is $y>r_k$?”, but naive formulations may violate rank monotonicity across those subtasks (Cao et al., 2019).

A third usage occurs inside convolutional architectures. The Ordinal Pooling Network replaces fixed max- or average-pooling by sorting activations inside each pooling region and learning weights by rank, so that aggregation depends on ordinal position rather than spatial position alone (Kumar, 2018).

Related, but conceptually distinct, work on Bayesian networks from ordinal data assumes that observed ordinal variables arise by marginally discretizing latent Gaussian variables whose structural dependence follows a directed acyclic graph. Here, ordinality constrains the data-generating model rather than defining a transition network over permutations (Luo et al., 2020).

2. Construction of ordinal-pattern transition networks

For a scalar time series $\{x_t\}_{t=1}^T$, the standard construction begins by choosing an embedding dimension $d$ or $D$ and a delay $\tau$. One then forms delay vectors such as

$$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$

or equivalently

$$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$

Each vector is mapped to the unique permutation that sorts its components in ascending order, with ties broken by temporal index order when needed. The resulting symbolic sequence $(\pi_t)$ takes values in the $K-1$0 or $K-1$1 possible ordinal patterns (Almendral et al., 2023, Flores et al., 12 Jul 2025).

The network is obtained by interpreting ordinal patterns as nodes and successive symbols as directed transitions. If $K-1$2 counts how often pattern $K-1$3 is followed by pattern $K-1$4, then a common row-stochastic representation is

$K-1$5

so that $K-1$6 for each $K-1$7. Other formulations normalize by the total number of transitions, yielding a weighted adjacency matrix $K-1$8 or $K-1$9 whose entries represent empirical transition frequencies over the whole sequence rather than conditional successor probabilities (Almendral et al., 2023, Pessa et al., 2019). Forbidden transitions are simply those with zero count; some arise from dynamical structure, and others from overlap constraints in the embedding itself (Flores et al., 12 Jul 2025).

The same logic extends to images. For a grayscale image, one extracts overlapping subblocks of size $y>r_k$0, flattens each block into a vector of length $y>r_k$1, maps that vector to its ordinal pattern, and then links neighboring patterns according to horizontal and vertical adjacency. In this case, the network reflects spatial transitions between local orderings rather than temporal ones (Pessa et al., 2020, Pessa et al., 2022).

Multivariate generalizations use coupled embeddings. Cross ordinal partition transition networks can be built either from difference vectors between two series or from Cartesian products of separate ordinal symbols, while joint ordinal partition transition networks use a single $y>r_k$2-dimensional embedding. These constructions were proposed to study synchronization in coupled systems and to expose missing patterns that are not apparent from scalar permutation entropy alone (Guo et al., 2018).

3. Entropic, graph-theoretic, and topological summaries

Once an ordinal network has been constructed, its analysis usually combines information-theoretic and graph-theoretic descriptors. Standard permutation entropy uses only pattern frequencies,

$y>r_k$3

where $y>r_k$4 is the empirical frequency of pattern $y>r_k$5. Transition-based entropy instead uses the ordinal-transition matrix,

$y>r_k$6

thereby incorporating temporal ordering among patterns. Local conditional entropies such as

$y>r_k$7

measure successor unpredictability at individual nodes, while global node entropy, degree distributions, weighted clustering coefficients, average path length, and Gini coefficients summarize the network as a whole (Almendral et al., 2023, Pessa et al., 2022, Pessa et al., 2020).

A notable analytic result concerns random series. For continuous IID white noise, ordinal networks have an exact asymptotic transition structure: all $y>r_k$8 patterns occur with equal frequency, each pattern has exactly $y>r_k$9 feasible successors because consecutive delay vectors share $\{x_t\}_{t=1}^T$0 entries, and one successor has probability $\{x_t\}_{t=1}^T$1 while the remaining $\{x_t\}_{t=1}^T$2 feasible successors each have probability $\{x_t\}_{t=1}^T$3. This exact baseline makes it possible to detect non-random behavior through deviations in transition structure or truly missing transitions (Pessa et al., 2019).

For images, overlap constraints likewise induce nontrivial combinatorial structure in the network. The two-dimensional construction yields exact connectivity restrictions inherited from shared pixels between neighboring windows. For $\{x_t\}_{t=1}^T$4, the full network has 24 nodes and at most 416 edges, and the random-image baseline can be derived exactly by counting admissible inequalities between overlapping blocks (Pessa et al., 2020).

Topological Data Analysis has also been applied to ordinal networks. For the driven magnetic pendulum, ordinal partition networks were converted into unweighted, undirected graphs, shortest-path distances were used to build Vietoris–Rips filtrations, and persistent homology in dimension one was summarized by a periodicity score $\{x_t\}_{t=1}^T$5 and a normalized persistent entropy $\{x_t\}_{t=1}^T$6. Periodic regimes produced a small number of large, regular loops, whereas chaotic regimes produced many short, irregular loops. In that study, periodic windows satisfied $\{x_t\}_{t=1}^T$7 and $\{x_t\}_{t=1}^T$8, while chaotic windows yielded values near unity (Myers et al., 2020).

4. Scientific applications of ordinal-pattern networks

In coupled dynamical systems, ordinal-transition methods have been used to infer structural role from local dynamics alone. For networks of coupled Rössler oscillators, transition entropy $\{x_t\}_{t=1}^T$9 separated hub and leaf nodes in star networks at the onset of phase-locking, whereas marginal permutation entropy $d$0 separated them only near full synchronization and with a smaller gap. In a scale-free network with $d$1, the degree-averaged $d$2 varied monotonically with degree under weak coupling and reversed slope under stronger coupling, and the reported difference $d$3 between hub and leaf nodes was up to $d$4 larger than $d$5. On experimental networks of $d$6 nonlinear electronic circuits across 20 topologies, calibration on one network and blind interpolation on the other 19 yielded correct degree assignments satisfying $d$7 for 91% of the $d$8 nodes (Almendral et al., 2023).

Multivariate ordinal partition transition networks have been used to study synchronization boundaries directly. In unidirectionally coupled Rössler systems, the cross- and joint-network entropies $d$9 and $D$0 decreased sharply near the phase-synchronization threshold, and two-parameter plots reproduced the Arnold tongue. Missing probabilities of individual patterns revealed thin periodic windows inside the tongue that the global entropies did not resolve cleanly (Guo et al., 2018).

For stochastic signals, ordinal networks support both discrimination and estimation. In fractional Brownian motion, the average weighted shortest-path length of the ordinal network was found to vary monotonically with the Hurst exponent, and a $D$1-nearest-neighbors regressor based on this feature achieved $D$2, compared with $D$3 for DFA and $D$4 for the QuantileGraph method. In earthquake magnitude series from Southern California, the average weighted shortest-path length decreased after the Landers, Hector Mine, and Baja California mainshocks, consistent with stronger persistence in aftershock-dominated regimes (Pessa et al., 2019).

In image analysis and materials science, ordinal networks have been generalized to spatial data. For randomized ornaments, fractional Brownian surfaces, Ising-model accumulation surfaces, and natural textures, measures such as global node entropy, average weighted shortest-path length, and the Gini index tracked roughness, symmetry, and criticality. In supervised tasks, the best ordinal-network feature achieved about 70% accuracy versus about 40% for the best gray-level co-occurrence matrix feature in Hurst-exponent classification, and the top four features in binary Ising classification were ordinal-network measures at about 95% accuracy versus about 80% for the best GLCM descriptor (Pessa et al., 2020). In liquid-crystal microscopy, ordinal networks formed by only 24 nodes were reported to encode crucial information about mesophase transitions, doping concentration, and temperature, enabling simple statistical learning models with “outstanding accuracy” (Pessa et al., 2022).

Time-resolved functional brain networks provide another application domain. A sliding-window protocol computes pairwise ordinal interactions—such as permutation mutual information, symbolic transfer entropy, or ordinal synchronization—between channels and yields evolving weighted or directed adjacency matrices. Reported applications include epilepsy, anesthesia, disorders of consciousness, stroke, and coma, alongside explicit caveats concerning volume conduction, common-source effects, finite-sample bias, embedding choice, and difficulty separating direct from indirect coupling (Lehnertz, 2023).

A recent extension compares whole ordinal networks rather than single summary measures. In falling-paper experiments, each trajectory was converted into an ordinal network of ordinal-pattern transitions derived from observed area time series, then embedded as a node in a higher-level similarity network whose edges represented pairwise dynamical similarity. Community detection with Infomap clustered trajectories into tumbling and chaotic falls in excellent agreement with expert visual classification, outperformed previous approaches based on classical physical features from complete three-dimensional trajectories, especially for cross-shaped papers, and indicated that trajectories disagreeing with expert labels occupied more central positions in the similarity network (Flores et al., 12 Jul 2025).

5. Ordinal neural networks for ordered prediction

The ordinal-regression literature in deep learning starts from the observation that ordered labels should not be treated as unrelated categories. A standard transformation maps each target $D$5 to $D$6 binary indicators

$D$7

so that the $D$8-th subtask asks whether the true label exceeds rank $D$9. This extended-binary formulation is flexible but can produce logically inconsistent outputs if the subtasks are trained independently (Cao et al., 2019).

CORAL addresses this problem by using a shared weight vector $\tau$0 and learned scalar biases $\tau$1 on top of any feature extractor $\tau$2, with logits

$\tau$3

Training minimizes the sum of binary cross-entropies across subtasks. The key theoretical result is that any optimum satisfies $\tau$4, which implies monotone probabilities

$\tau$5

for every $\tau$6. On age estimation with a ResNet-34 backbone, CORAL-CNN achieved MAE values of 2.64 on MORPH-2, 3.47 on AFAD, and 5.25 on CACD, compared with 2.83, 3.51, and 5.38 for the reference ordinal regression network. In an ablation on consistency, OR-CNN exhibited on average 1–2 rank inconsistencies per test example, whereas CORAL-CNN had zero inconsistencies across all examples (Cao et al., 2019).

Later methods removed the weight-sharing restriction while retaining guaranteed rank consistency. CORN models conditional probabilities

$\tau$7

so that unconditional rank probabilities satisfy

$\tau$8

Because the product is nonincreasing in $\tau$9, consistency follows from the chain rule rather than from shared linear structure. On balanced benchmarks with a ResNet34 backbone, CORN achieved MAE 2.98 on MORPH-2, 2.81 on AFAD, and 0.76 on Fireman, matching or improving upon CORAL and non-consistent baselines (Shi et al., 2021). CONDOR formalized the same Markov-chain view, proved “universal rank consistency,” derived a maximum-likelihood loss, and reported best WBCE, MAE, and EMD on ordinal MNIST, strong results on Amazon reviews, and a GRU-D case study for COVID-19 severity in which CONDOR-WBCE reached mortality-only AUROC .9038 (Jenkinson et al., 2021).

Cost-aware extensions incorporate asymmetric clinical risk. Ordinal Cross-Entropy defines an ordinal cost matrix $$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$0, converts it into a normalized penalty-reward matrix $$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$1, and retains a softmax output layer while weighting the logarithmic terms by ordinal cost. The method was evaluated on the APTOS 2019 diabetic retinopathy benchmark with five ordinal levels and pronounced class imbalance. Under symmetric costs with DenseNet121, OCE achieved the lowest Cost Error, 1.081, versus 1.095–1.259 for competitors, while also reporting accuracy 0.784, AUC 0.897, MAE 0.324, and QWK 0.803 (Dvora et al., 24 Jun 2026).

Ordinal regression has also been applied to text. In sex-trafficking detection, an ordinal regression neural network used $$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$2 parallel sigmoid outputs $$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$3 on top of a 3-layer gated-feedback recurrent neural network, together with an order-violation penalty

$$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$4

On Trafficking-10K under 10-fold cross-validation, ORNN achieved MAE 0.769, macro-averaged MAE 1.238, and binary accuracy 0.818, improving on four classical ordinal models and the prior HTDN baseline in the reported comparison (Wang et al., 2019).

A common misconception is that any multi-sigmoid network for ordered labels is automatically ordinal. The literature shows that this is false: monotonicity must be enforced either architecturally, as in CORAL, or probabilistically, as in CORN and CONDOR, or softened by an explicit violation penalty, as in ORNN. Another misconception is that ordinal modeling requires specialized backbones. In these studies, the dominant pattern is instead architecture-agnostic replacement of the final layer, loss, or training loop (Cao et al., 2019, Shi et al., 2021).

6. Architectural variants, adjacent models, and methodological limits

Ordinality can also enter a network at the feature-aggregation level. In the Ordinal Pooling Network, the activations $$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$5 inside a pooling region are sorted in descending order,

$$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$6

and the pooled output is

$$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$7

If $$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$8, the operation reduces to max-pooling; if $$v_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(d-1)\tau}),$$9, it reduces to average-pooling. For a layer with $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$0 channels and pooling region size $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$1, OPN adds $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$2 scalar weights. In the published MNIST experiment, max-pooling achieved 0.85% validation and 0.89% test error, LBPN 0.86% and 1.00%, and OPN 0.74% and 0.80%, with faster convergence in validation-accuracy curves (Kumar, 2018).

A separate line of work concerns learning Bayesian networks from ordinal observations. OSEM assumes latent Gaussian variables $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$3 whose dependencies follow a directed acyclic graph, with observed ordinal categories generated by cut-points

$$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$4

Structural EM is then used with a Monte Carlo E-step over truncated normal regions and a decomposable BIC-style score in the M-step. In simulations over network sizes $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$5, sample sizes $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$6, and $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$7 ordinal levels, OSEM “uniformly dominates” the compared alternatives in recovering the true CPDAG pattern, and on real data it was applied to psychological survey data from 408 patients with co-morbid symptoms of obsessive-compulsive disorder and depression (Luo et al., 2020).

Across the different meanings of ordinal networks, several limitations recur. In ordinal-pattern networks, the number of patterns and transitions grows factorially with embedding dimension; the falling-paper study states that in practice one must keep $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$8 small, for example $$\mathbf{X}_t = (x_t, x_{t+\tau}, x_{t+2\tau}, \dots, x_{t+(D-1)\tau}).$$9, unless very long series are available, while the Rössler-network study notes that reliable estimation of $(\pi_t)$0 requires series lengths $(\pi_t)$1 (Flores et al., 12 Jul 2025, Almendral et al., 2023). Because only order relations are retained, amplitude information may be lost, and parameter choices such as $(\pi_t)$2, $(\pi_t)$3, $(\pi_t)$4, and $(\pi_t)$5 can materially affect results (Pessa et al., 2020, Pessa et al., 2022). In deep ordinal regression, high thresholds may induce small conditional training sets, decision thresholding at 0.5 remains a design choice, and even methods with consistent probabilities may still benefit from post-hoc calibration when precise probability estimates are needed (Shi et al., 2021, Cao et al., 2019).

Taken together, these strands define ordinal networks not as a single algorithm but as a research program built around order structure. In symbolic dynamics and complex systems, ordinal networks provide compact graph representations of temporal or spatial ordering. In machine learning, they motivate rank-consistent predictors, cost-aware losses, and pooling schemes that preserve more than the single largest activation. The unifying claim is modest but consequential: order is often informative in its own right, and network-based models can exploit that information in ways that nominal or value-only formulations do not.