Towards reproducible edge inference with formal guarantees

3.1. Gene expression dataset

The gene expression dataset we analyzed was from [27], which employed RNA-seq to profile gene expression patterns in peripheral blood mononuclear cells (PBMCs) from 17 HCC patients and 17 age-matched healthy controls. We normalized raw per-gene read counts using Transcripts Per Million (TPM) [28] to account for transcript length and sequencing depth, followed by log₂ transformation to stabilize variance. Normalization and transformation were performed using the edgeR [29] R package (v 3.42.4, R version 4.3.0). For more detailed information on the RNA-seq data processing, we refer readers to our previous work [30].

3.2. Biological pathways and network data

We retrieved 251 metabolic and signaling pathways and their corresponding gene sets from the KEGG database [25] using the R packages AnnotationDbi v1.66.0 [31] and org.Hs.eg.db v3.19.1 [32] with R (v4.4.1). Pathway interactions (e.g., activations, inhibitions) were parsed from KEGG Markup Language (KGML) files downloaded from the KEGG website (https://www.genome.jp/kegg/pathway.html). The Bio.KEGG.REST module in Biopython v1.1.76 [33] for Python (v3.12.6) was used to convert KEGG gene IDs to NCBI gene symbols, ensuring consistency with the HCC gene expression dataset.

For each processed pathway, the obtained network contained one node for every gene involved in the pathway, and directed edges between genes were placed for activations, inhibitions, or other interactions defined in KEGG. In cases where a KEGG pathway defined nodes containing multiple genes (e.g., a group of genes performing a shared function), these were separated into individual nodes, while maintaining for each of the separated nodes all edges associated with the original group of genes. This step ensured that the final network accurately captured all individual gene interactions and maintained a one-to-one correspondence with the gene list used for gene expression analysis. Finally, each pathway network was described in the form of an adjacency matrix.

3.3. Representation of gene expression data

For each pathway and each health condition (HCC or control), the gene expression profiles (expression levels measured in the 17 corresponding PBMC samples) of all genes involved in the pathway were mapped to a 17-dimensional space to obtain a condition-specific point cloud. In this point cloud, each point corresponds to a gene node in the pathway network, and each dimension in the space reflects the gene expression levels in one of the 17 samples for that health condition.

Each obtained point cloud can be used for a PH analysis, or—jointly with the corresponding pathway network for which it was constructed—for a PPH analysis with GenPath-PPH. In both analyses, the topological properties of the two point clouds of a pathway, one for HCC and one for healthy controls, can be compared.

3.4. Distance metric and filtration

To prioritize biologically meaningful gene relationships, we replaced the Euclidean distance originally used by PathHom [22] with a correlation-based distance metric $d=1-|\rho |$, where $\rho$ is the Pearson correlation coefficient. This coefficient is frequently used by the scientific community to determine the strength of co-expression between two genes [34]. This correlation-based distance metric ranges from 0 (perfect correlation, regardless of direction) to 1 (no correlation), helping to emphasize strong gene interactions while filtering noise from weak correlations.

Using the absolute value $|\rho |$ aligns with the biological intuition that coordinated activity—whether positive or negative—can jointly drive pathway behavior. A negative correlation can be as biologically relevant as positive correlation, e.g., when one gene inhibits the activity of another. Thus, this formulation captures both co-activation and inverse regulatory patterns within pathway networks.

Alternative similarity measures, such as Spearman or partial correlation, or mutual information, can also be applied depending on the data characteristics and the type of dependencies of interest. In this study, we selected the Pearson-based metric for its interpretability and its established use in co-expression network analysis.

This distance metric was then used as the correlation threshold for filtrations (see Section 2). At very low filtration values $f$ (close to zero), only the edges between the closest genes ($d\le f$, i.e., highly correlated) are taken into account to determine topological features in terms of Betti numbers. As $f$ increases, edges between more distant genes (those representing weaker correlations) are progressively included, until at $f=1$ the entire network is considered. In this study, we increase the filtration values/thresholds stepwise from 0 to 1 with a step size of 0.01.

3.5. Global differences between health conditions

For each pathway, two series of Betti numbers (one for the control group and one for the disease group)—as a function of the filtration value, i.e., distance threshold ($f\in \{0.01k|k\in {\mathbb{N}}_0,k\le 100\}$)—were obtained from PPH computations for both dimension 0 (connected components) and dimension 1 (cycles).

These series of Betti numbers were then converted to persistence diagrams (PDs; see Fig. 8 in Section 5 for examples), which are graphical representations of the topological features (such as connected components or cycles) and their persistence across the filtration scale. Each point in a PD represents a topological feature, with the $x$-coordinate indicating its birth time (when the feature first appears) and the $y$-coordinate indicating its death time (when the feature disappears or ceases to exist) during the filtration process.

The PDs were then transformed into persistence landscapes (PLs), which hierarchically rank topological features based on their prominence across filtrations. Each landscape function ${\lambda }_k$ represents the $k$-th most persistent feature, where ${\lambda }_1$ captures the most dominant structure (e.g., stable gene cluster or cycle), ${\lambda }_2$ the second most dominant, and so on.

PLs provide a computationally convenient way to summarize the persistence of topological features. PLs are sequences of piecewise-linear functions, making them faster to compute compared to PDs, and allowing arithmetic comparisons (such as addition, subtraction, and averaging) that are not possible with PDs. By using functional norms (sup-norm, 1-norm, and 2-norm) we can measure the maximum deviation, total absolute difference, and Euclidean distance of a PL, respectively. This makes it possible to perform meaningful comparisons between the PLs of different conditions (for example, disease versus control).

To evaluate whether the observed differences in PLs ($\mathrm{\Delta }PL$) were statistically significant, a permutation test was conducted separately for each homological dimension $n$ (connected components and cycles). PLs from all pathways across both conditions were pooled, randomly reassigned into two groups matching the original number of PLs for each group, and used to recalculate the average PLs ($\mathrm{\Delta }P{L}_0$ for connected components and $\mathrm{\Delta }P{L}_1$ for cycles). This process was repeated 1000 times to generate a null distribution of randomized differences. For each iteration, the sup-norm (maximum deviation), the 1-norm (total absolute difference), and the 2-norm (Euclidean distance) were computed for $\mathrm{\Delta }P{L}_0$ and $\mathrm{\Delta }P{L}_1$. The differences observed from the true data were compared to this null distribution, and $p$-values were calculated as the proportion of permutations with differences greater than or equal to the observed values from the true data. Finally, $p$-values from all three norms were adjusted using the Benjamini–Hochberg method (BH [35]) to control the false discovery rate (FDR $<0.05$).

3.6. Pathway-level differences

The two series of Betti numbers obtained for each pathway, health condition and dimension of topological features (see Section 3.5) can also be used to explore pathway-level differences and identify specific pathways where topological differences distinguish between conditions, i.e., where the disease condition significantly affects the topological structure, for example, by increasing fragmentation (e.g., more isolated components indicating less correlation between genes) or enhancing cyclic interactions (e.g., stronger correlation between genes involved in directed cycles).

3.7. Baseline methods for comparison

As a comparison to our new GenPath-PPH framework, we used three alternative pathway analysis approaches: the PH analysis based on topological descriptors that we presented in a previous study and two topology-unaware methods that are frequently employed by the scientific community.

5.1. GenPath-PPH workflow

4.3. Implementation

The workflow is implemented in a modified version of PathHom [22] (available at https://github.com/WeilabMSU/PathHom, accessed on 13 August 2024), with custom functions developed for connectivity tracking, visualization, and statistical analysis. Our code can be found at https://github.com/DrMSAbdullahi/GenPath-PPH. Users can specify pathways in the form of adjacency matrices, which can in principle be taken from databases other than KEGG.

In GenPath-PPH, the filtration step size is fixed at 0.01, providing a balance between detecting relevant topological features and computational efficiency. Larger step sizes may skip features due to having too few filtration levels, whereas smaller step sizes generate many intermediate features, increasing computation time. The filtration scale is also defined based on the distance type (e.g., 1 for (1 - $|\rho |$), 2 for (1-$\rho$), or user-defined for Euclidean distance).

Other parameters are flexible and user-configurable, including: the number of permutations, the norm used in PLs, the homology dimensions to track (e.g., ${\beta }_0$, ${\beta }_1$), and the number of samples per condition group. These can be adjusted depending on the dataset and analysis goals.

4.3. Illustrative example

Apart from these structures, the toy pathway contains two additional genes, a gene connected to cycle 1 ($g5$) and a completely disconnected one ($g10$). For all ten genes, we generated simulated gene expression profiles for four samples.

Fig. 2(c) illustrates how for PPH we select only those pairwise co-expression measurements (Pearson correlation coefficients in Fig. 2(b)) that correspond to the directed edges in the pathway network, and Fig. 2(d) shows their conversion to distances $1-|\rho |$.

How the topological features of a directed network can be tracked across increasing filtration values is exemplified in Fig. 3. Filtration values $f$, which are used as distance thresholds, were increased stepwise from 0 to 1, with a step size of 0.01. At every step, only edges with correlation-derived distances of $d=1-|\rho |\le f$ were considered, starting from the initially disconnected point cloud for $f=0$ (see Fig. 3(a)). Hence, initially the number of connected components is equivalent to the number of genes in the network (${\beta }_0$=10) and consequently no 1-cycles can be observed (${\beta }_1$=0).

Figs. 3(b)–3(f) depict several of the filtration steps and the network edges that are considered at these steps. At filtration $f=0.2$ (Fig. 3(b)), only the edge $g9\mapsto g7$ is observed ($d=0.184\le f$, see Fig. 2(d)), reducing the number of connected components to ${\beta }_0=9$ with still no cycles (${\beta }_1=0$). At filtration $f=0.4$ (Fig. 3(c)), one of the two cycles has emerged (${\beta }_1=1$) because all involved edges have $d\le f$, and only ${\beta }_0=7$ connected components remain. At $f=0.95$, finally, all edges of the network are included and the topological summaries stabilize at ${\beta }_0=3$ and ${\beta }_1=2$ Fig. 3(f). Generally, since the maximum possible edge distance $d$ is 1, at the last filtration step $f=1$, a pathway network will always be observed in its entirety.

An approach based on classical PH, in contrast, would not take the detailed gene interaction information of a pathway into account and would instead use all theoretically possible edges simply according to the distances derived from all the pairwise correlations illustrated in Fig. 2(b). For a description of this approach, we refer to our previous study [30].

To highlight the conceptual and computational differences between classical PH and direction-aware PPH, we provide a side-by-side illustration in Fig. 4 and explain how each detects topological features.

The barcode plots in Fig. 4 display the persistence of connected components (dimension 0) and cycles (dimension 1) across the filtration procedure for both PH and PPH. For PPH Fig. 4(b), the barplot shows at which filtration thresholds two disconnected components merge because a network edge appears (loss of a red bar, ${\beta }_0$ decreases by one), and at which filtration thresholds cycles become observable because their last edge appears (gain of a blue bar).

In contrast, for PH Fig. 4(a), merging of two components does not depend on the existing network edges; any sufficiently strong correlation (low enough distance) between any two genes is enough for merging. Here, the filtration value $f$ is interpreted as the radius of spheres centered at each data point (gene); components merge when these spheres touch, i.e., when $d/2\le f$. For these reasons, the number of connected components tends to decrease faster and at $f=d_{\text{max}}/2$ only one large component remains, where $d_{\text{max}}$ is the maximum pairwise distance among genes.

Structural features of dimension 1 also behave differently. In PH, cycles or “holes” appear when the filtration value $f$ is sufficient to let the overlapping filtration spheres of multiple genes or data points form closed bounded structures (e.g., a triangle or a square) but disappear once all spheres overlap completely (see Fig. 4(a)). Because PH does not consider pathway directionality and edge constraints, it can introduce spurious cycles that do not represent true network topology. PPH, by considering connectivity only along directed pathway edges, correctly identifies two cycles in the simulated pathway (see Fig. 2(a) and the blue bars in Fig. 4(b)), whereas PH produces three transient “holes” (see Fig. 4(a)) that do not reflect the true interaction structure.

5.1. Global differences across pathways

To assess global topological differences between the control and the disease groups, we constructed PLs (see Section 3.5) and computed condition-specific averages over all pathways for dimensions 0 (connected components) and 1 (1-cycles). These averaged PLs are shown in Figs. 5(a) (dimension 0) and 5(b) (dimension 1).

The left panels show PLs for the healthy control samples, where peaks in the landscape functions ${\lambda }_k$ reflect baseline connectivity and cyclic interactions; the central panels display PLs for the samples from HCC patients, revealing altered topology (e.g., fragmentation of connected components or dysregulation of regulatory cycles); and the right panels plot the difference $\mathrm{\Delta }PL$, highlighting intervals where HCC-associated ${\lambda }_k$ values exceed or fall below those in controls.

For dimension 0 (connected components; Fig. 5(a)), the negative difference indicates that on average, the control group has more pronounced connectivity (higher landscape peaks) than the disease group, which suggests greater fragmentation of connected components in the latter.

In contrast, for dimension 1 (cycles; Fig. 5(b)), the difference is mixed. At high filtration values (distance thresholds) the difference is slightly negative, but on average and for most of the filtration range, the difference is positive, meaning that the disease group has a stronger co-regulation of cyclic interactions (e.g., regulatory cycles) than the control group.

Permutation tests using different norms (sup-norm for maximum deviation, 1-norm for total absolute difference, and 2-norm for Euclidean distance) confirmed statistically significant differences (FDR $<0.05$ for all norms), validating PLs as a powerful tool for detecting global topological differences between conditions. The 1-norm captures the overall (global) difference across landscapes, the 2-norm emphasizes larger but distributed deviations, and the sup-norm detects the maximum local deviation at any point. When all norms are significant, as in our case, it implies that differences between control and disease are both globally consistent and locally pronounced. Collectively, these findings demonstrate that PLs are a powerful tool for detecting global topological differences between conditions.

5.2. Identification of pathway-level differences

As demonstrated in the previous subsection, the global analysis demonstrates overall class separability, whereas this pathway-level analysis focuses on identifying specific pathways that contribute to (or are altered by) the disease condition.

We used two metrics (KS test statistic and Cohen’s $d$ effect size) to identify significant pathway-level differences. For both metrics, we applied permutation testing to calculate $p$-values and corrected them for multiple testing (see Section 3.6). Both metrics were independently applied to topological features of dimension 0 (connected components) and those of dimension 1 (cycles).

A pathway was considered significant if its FDR was below 0.05 for both dimensions (0 and 1) and for both metrics. This restrictive approach helped us pinpoint pathways with the most consistent and relevant topological differences in their gene expression networks.

In total, 31 pathways were highlighted by GenPath-PPH due to their consistent relevance regarding both metrics and both dimensions (see Table 1). For the overlap of results from the four individual tests, see Fig. 6. Additional details about the pathways identified by each test are provided in the Supplementary Document S1.

To validate these findings, we performed principal component analysis (PCA) on multiple pathway-level metrics–KS statistics, mean Betti number differences, and Cohen’s $d$. PCA reduced these metrics to principal components that capture the main variation across pathways, with the first two principal components explaining 84.1 % of the total variance, while the third component accounted for the remaining 15.9 %. As shown in the 3D plot (Fig. 7), pathways formed two partially distinct clusters that correspond to significant and non-significant groups identified by GenPath-PPH. Clustering robustness was assessed using the silhouette coefficient (0.25), indicating moderate separation. This suggests that the PCA structure reflects meaningful but partially overlapping topological differences between conditions. The multivariate PCA analysis therefore supports the reliability of the pathway-level topological insights provided by GenPath-PPH.

As an illustrative example, the p53 signaling pathway demonstrates how GenPath-PPH captures topological differences between control and disease groups. The barcodes and persistence diagrams in Fig. 8 highlight distinct features in both dimensions. For dimension 0 Fig. 8(a), connected components (red bars) merged earlier in the disease group than in controls. This suggests altered gene connectivity, with a higher fragmentation, i.e., fewer persistent clusters in the disease group. For dimension 1 Fig. 8(b), two 1-cycles (blue bars) emerged at filtration values $f=0.58$ and $f=0.98$ in the disease group, compared to $f=0.92$ and $f=0.96$ in the control group. The persistence intervals (birth-to-death distances) for these cycles were 0.42 and 0.02 in the disease group, compared to 0.08 and 0.04 in the control group. This suggests a considerably strengthened coordination of one of the cycles in the disease group, i.e., the genes involved in that cycle remain connected over a wider range of similarity thresholds, which indicates a more stable and dominant interaction pattern. These contrasting patterns illustrate how GenPath-PPH detects meaningful differences in both local connectivity and higher-order interactions in pathway structures between conditions.

5.3. Comparisons with alternative pathway analysis methods

We compared the pathways identified by GenPath-PPH with those detected by three other methods (PH-TD [30], GSEA, and HGEA). GSEA and HGEA are widely used enrichment-based methods in biological research, while PH-TD is a topological method we previously introduced. PH-TD applies persistent homology to point clouds formed from gene expression similarity but does not account for the structural information of biological pathway networks.

Notably, GenPath-PPH identified 31 pathways with significant topological differences between conditions, compared to 23 detected by PH-TD, 27 by GSEA, and 16 by HGEA (see Table 2). Additionally, GenPath-PPH shared 4 pathways with PH-TD, 5 with GSEA, and 3 with HGEA (see Table 3 for more details). The UpSet plot in Fig. 9 shows limited overlap across the four methods, with no single pathway identified by all of them. This lack of shared pathways mainly results from the three baseline methods (PH-TD, GSEA, and HGEA) not detecting a common set of pathways. These results demonstrate the distinct detection patterns of each method and the complementary insights they provide. To understand these differences, it is important to consider the strengths and limitations of each method.

GSEA and HGEA depend on gene-level significance and assume gene independence, which may overlook pathway-level structural organization of genes. PH-TD adopts a topological perspective but is limited to undirected representations of gene expression similarities, and so does not account for the directional nature of biological interactions in the pathway networks. In contrast, GenPath-PPH captures both topological and directional features of gene similarities and pathway structure by integrating the expression data with the directed KEGG pathway graphs using persistent path homology.

One advantage of GenPath-PPH is its ability to detect small changes in topological features across scales that occur within biologically meaningful, directed gene interactions. By restricting analysis to genes with known biological relationships, it avoids noisy signals from unrelated expression patterns. This likely explains why it uniquely identified 21 pathways that were not detected by the other methods. Among these, NF-$\kappa$B signaling, prolactin signaling, and sphingolipid signaling are well-known to be associated with HCC [39–41]. These pathways contain regulatory motifs such as feedback loops and directionally constrained cascades, which are preserved in KEGG but are often missed by undirected or enrichment-based approaches.

For example, the NF-$\kappa$B signaling pathway contains several directed cyclic-like structures, including cycles with asymmetric edge orientations, which are detectable using path homology. In both control and disease groups, GenPath-PPH identified three such 1-dimensional cycles. Among them, the first appeared at a lower filtration threshold of 0.5 in the disease group compared to 0.85 in the control group. Since the filtration progresses from 0 (perfect correlation) to 1 (no correlation), this earlier appearance suggests stronger correlations among the genes forming the cycle in the disease condition. Furthermore, all three cycles in the disease group appeared earlier and persisted longer than their counterparts in the control group. This indicates that under disease conditions, coordinated gene interactions not only emerge earlier but also remain stable over a wider range of similarity thresholds, suggesting the presence of stronger and more dominant regulatory motifs. These persistent cyclic structures may reflect sustained or rigid signaling activity. Notably, the NF-$\kappa$B pathway regulates numerous genes involved in inflammation, cell growth, and apoptosis through direct promoter binding in cancers including HCC [42,43], which may contribute to the formation and stability of these features.

Apart from the well known JAK-STAT, NF-$\kappa$B, and p53 signaling pathways, many other pathways highlighted by our approach have previously been linked to HCC—either through experimental validation or through indirect evidence, such as the construction of pathway-based risk prognosis models (see Supplementary Document S2). Those pathways that have not yet been directly linked to HCC, or have only indirect support, can therefore be considered as potential candidates.

One of the most promising candidate pathways is aminoacyl-tRNA bio-synthesis. This pathway has been associated with various cancer types, and several aminoacyl-tRNA synthetases are differentially expressed in liver cancer as well [44,45] and show expression levels that correlate with patient survival [45]. Moreover, a recent proteomics study has found the pathway to be enriched in differentially expressed proteins [46]. Although these findings do not directly provide evidence of its involvement in HCC initiation or progression, they are consistent with the importance of the pathway as suggested by our results.

Another interesting candidate is the complement and coagulation cascades pathway. Previous studies have constructed prognostic models based on pathway-related gene signatures that can successfully stratify patients into high- and low-risk groups with distinct survival outcomes [47,48]. Moreover, HCC is found to be associated with a hypercoagulable state driven by liver dysfunction, systemic inflammation, and tumor-related factors [49], which further supports the biological plausibility of this finding.

Altogether, these results suggest that while enrichment-based methods like GSEA and HGEA provide general insight into gene set overrepresentation, and PH-TD offers a complementary topological view based on undirected similarity structures, GenPath-PPH introduces a direction-aware and structure-informed approach. A complete comparison of all the 251 pathways and the detection results from each method is given in Supplementary Document S1 (Table S1).

By integrating pathway structure with gene expression data and focusing only on biologically interacting genes, GenPath-PPH reduces misleading associations from unrelated gene pairs. This approach depends on the quality of the pathway annotations and may miss novel or unrecognized associations, potentially introducing bias toward the pathways represented in the chosen database (e.g., KEGG). The framework can also accommodate other pathway databases, allowing broader application while maintaining sensitivity to biologically relevant pathways shaped by their network structure.

A.1 Path complexes

Let $G=(N,E)$ be a directed network (digraph) with a vertex set $N$ and directed edge set $E\subseteq N\times N$. A path complex $\mathcal{P}$ on $N$ is a collection of ordered vertex sequences (paths)

such that

• (i) every vertex $v\in N$ yields the 0-path $(v)\in \mathcal{P}$;

• (ii) whenever $(v_0,\dots ,v_n)\in \mathcal{P}$, any truncation obtained by removing the first or last vertex is again in $\mathcal{P}$.

Each subset ${\mathcal{P}}_n\subset \mathcal{P}$ denotes the collection of all $n$-paths in $\mathcal{P}.$ Hence ${\mathcal{P}}_0=N$ and ${\mathcal{P}}_1=E.$

We denote by ${\mathcal{P}}_n$ the set of allowed $n$-paths, thus ${\mathcal{P}}_0=N$ and ${\mathcal{P}}_1=E$. The elements of $\mathcal{P}$ are called allowed elementary paths.

An elementary $n$-path is an ordered sequence $(v_0,\dots ,v_n)\in {\mathcal{P}}_n$. Concatenation of paths (when compatible) and truncation preserve membership in the path complex under the admissibility rules above; these algebraic closure properties are the structural foundation for path homology.

Fig. A.10 illustrates the basic structures forming a path complex. Specifically, panel A.10(a) shows the elementary building blocks of path complexes, including 0-paths (individual vertices), 1-paths (edges), and 2-paths (higher-order structures—sequences of consecutive edges). Panel A.10(b) provides examples of triangular configurations. The left structure (T1) forms a simple directed loop or a 1-cycle (from $g_1\to g_2\to g_3\to g_1$, with ${\beta }_1=1$), whereas the right structure (T2) forms a directed graph without a conventional closed loop (edges $g_1\to g_2\to g_3$ and $g_1\to g_3$ with ${\beta }_1=0$) (e.g., feedback-like or sequential interactions among genes). Panel A.10(c) extends this concept to four-node configurations, where some arrangements create closed cycles (with ${\beta }_1=1$) while others remain acyclic (i.e., ${\beta }_1=0$). These examples highlight the variety of connectivity patterns that can exist in directed networks (e.g., cyclic or non-cyclic signaling relationships) and set the stage for defining path homology in subsequent sections.

A.2 Chain spaces and boundary operator

A.3 Path homology groups and betti numbers

A.4 Filtrations and persistent path homology (PPH)

Over a field, finite persistence modules decompose into interval modules, and this decomposition is represented visually by barcodes or persistence diagrams; the collection of intervals over all $n$ is the persistent path homology of $G$.

A.5 Relation to classical (undirected) persistence and computational remarks

Chowdhury and Mémoli show that PPH captures directed asymmetry: in the symmetric (undirected) case the PPH construction reduces to the usual PH of the corresponding symmetric complex. In practice, this means that PPH is sensitive to edge direction and can distinguish directed cycles that classical (undirected) PH may treat as spurious or may not detect correctly.

From a computational perspective, when coefficients are taken over a field, the computation of PPH follows the same algebraic-reduction principles used for standard PH; the worst-case computational complexity remains comparable to that of classical PH algorithms (see [19] for details).

A.6 Application note

In GenPath-PPH we realize filtrations by thresholding a similarity measure (e.g., absolute Pearson correlation transformed into a distance). Edges that meet the threshold at level ${ϵ}_i$ are included in $G_{{ϵ}_i}$; the resulting nested graphs produce induced path complexes whose Betti numbers ${\beta }_0^{({ϵ}_i)},{\beta }_1^{({ϵ}_i)}$ are then tracked across ${ϵ}_i$ to obtain Betti curves and persistence summaries.

A.7 Intuitive summary

Classical PH studies how clusters and loops form as connections gradually appear in an undirected network. PPH extends this idea to directed networks by keeping track of the direction of each edge. It detects structures such as feedback loops, signal flows, and other asymmetric relationships that ordinary persistence treats as equivalent. Hence, PPH provides a mathematically consistent way to describe how direction-dependent interactions (for instance, regulatory effects in gene networks) behave across scales.