key: cord-0193897-7b5pipq4
authors: Klimm, Florian
title: Topological data analysis of truncated contagion maps
date: 2022-03-03
journal: nan
DOI: nan
sha: cb4048943b8974f5ce2d73d2f5a822cb83347810
doc_id: 193897
cord_uid: 7b5pipq4

The investigation of dynamical processes on networks has been one focus for the study of contagion processes. It has been demonstrated that contagions can be used to obtain information about the embedding of nodes in a Euclidean space. Specifically, one can use the activation times of threshold contagions to construct contagion maps as a manifold-learning approach. One drawback of contagion maps is their high computational cost. Here, we demonstrate that a truncation of the threshold contagions may considerably speed up the construction of contagion maps. Finally, we show that contagion maps may be used to find an insightful low-dimensional embedding for single-cell RNA-sequencing data in the form of cell-similarity networks and so reveal biological manifolds. Overall, our work makes the use of contagion maps as manifold-learning approaches on empirical network data more viable.

The investigation of dynamical processes on networks has been one focus for the study of contagion processes. It has been demonstrated that contagions can be used to obtain information about the embedding of nodes in a Euclidean space. Specifically, one can use the activation times of threshold contagions to construct contagion maps as a manifoldlearning approach. One drawback of contagion maps is their high computational cost. Here, we demonstrate that a truncation of the threshold contagions may considerably speed up the construction of contagion maps. Finally, we show that contagion maps may be used to find an insightful low-dimensional embedding for single-cell RNA-sequencing data in the form of cell-similarity networks and so reveal biological manifolds. Overall, our work makes the use of contagion maps as manifold-learning approaches on empirical network data more viable.

It is known that the analysis of spreading processes on networks may reveal their hidden geometric structures. These techniques, called contagion maps, are computationally expensive, which raises the question of whether they can be methodologically improved. Here, we demonstrate that a truncation (i.e., early stoppage) of the spreading processes leads to a substantial speed-up in the computation of contagion maps. For synthetic networks, we find that a carefully chosen truncation may also improve the recovery of hidden geometric structures. We quantify this improvement by comparing the topological properties of the original network with the constructed contagion maps by computing their persistent homology. Lastly, we explore the embedding of single-cell transcriptomics data and show that contagion maps can help to distinguish different cell types.

The study of spreading processes has a rich tradition in epidemiology 1-3 , the social sciences [4] [5] [6] [7] [8] [9] , and applied mathematics [10] [11] [12] . Due to improved availability of interaction data, the description of contagion processes on networks has been a focus in recent years 13 . It is known that spreading phenomena are influenced by the geometry of a networks's underlying embedding 14 , for example for the spread of pandemics 15 and the adaption of technologies 16 . By constructing contagion maps, the interplay between dynamical processes and underlying geometry can be used to detect manifold structure in networks 17 . Contagion maps may reliable detect manifold structure in noisy data, even if standard approaches, such as ISOMAP fail 18 . Generalisation of these approaches, for example to contagions on simplicial complexes 19 and Kleinberglike networks 20 , is an active field of research. The construction of contagion maps, however, is computationally expensive, which makes the exploration of strategies for their speedup a pressing issue.

In this work, we introduce a generalisation of contagion maps that we call truncated contagion maps. The core idea is that an early stoppage of the contagion processes that are used to detect the manifold may drastically reduce the computational cost of the embedding algorithm. Furthermore, as we will show, the activations in the early stages of a contagion often follow the underlying manifold of the networks more clearly than the later stages. Omitting these late-stage dynamics results in an improved manifold detection, which we verify with techniques from topological data analysis.

Topological data analysis is a field that aims to develop tools that allow the extraction of qualitative features in data that are hidden to standard approaches. A wide range of tools and methods for the topological analysis of data have been developed [21] [22] [23] and applied to problems in biology 24, 25 , social sciences [26] [27] [28] , and many others [29] [30] [31] . Persistent homology 32, 33 in particular is a topological data analysis method that received a lot of interest from practitioners because it enables the extraction of topological features in high-dimensional point-cloud data. In particular, topological data analysis is considering higher-order interactions between data points by constructing simplicial complexes.

Advances in single-cell transcriptomics enable the measurement and analysis of gene expression at a single-cell resolution 34 . One challenge in the analysis of single-cell transcriptomics data is that experimental advances lead to an explosion in data set sizes 35 . As complex organisms have thousands to ten thousands of genes, their gene-expression space is high-dimensional, making single-cell transcriptomics data notoriously hard to visualise. Yet, the vast majority of this space is empty and cells follow few low-dimensional manifolds that represent, for example, cell types or differentiation trajectories. The presence of low-dimensional structures embedded in high-dimensional gene-expression spaces makes single-cell transcriptomics a promising field for the application of manifold-learning approaches 36 . In this work, we apply truncated contagion maps to single-cell transciptomics data and show that the constructed embeddings are fruitful in representing the complex biological structure, which repre- sents cell types and developmental trajectories. The remainder of this article is organised as follows. First, we summarise the existing methods and introduce truncated contagion maps. Second, we investigate the temporal development of the contagion processes. Third, we demonstrate that truncated contagion maps may improve the recovery of topological features in contagion maps. Fourth, we discuss the improved computational complexity of truncated contagion maps versus full contagion maps. Fifth, we apply truncated contagion maps to single-cell transcriptomics data of developing mouse oocytes. Finally, we discuss our findings.

Noisy geometric networks 17 are geometric networks 37 to which 'noisy' non-geometric edges have been added. They are defined as follows. Consider a manifold M that is embedded in an ambient space A such that M ⊂ A . First, we construct nodes V that are embedded in the manifold M such that w (i) ∈ M for nodes i ∈ V . Second, we construct edges which are of one of two types, either geometric or nongeometric. The geometric edges E (G) follow the manifold such that i, j ∈ E (G) if the distance along the manifold between nodes i and j is below some distance threshold. The non-geometric edges E (NG) are constructed in a random process that does not consider the manifold. We obtain the noisy geometric network as G = (V, E (G) ∪ E (NG) ).

In this manuscript, we focus on noisy ring lattices NRL(N, d (G) , d (NG) ) as prototypical examples of noisy geometric networks. Their parameters are the number N of nodes, the geometric degree d (G) , and the non-geometric degree d (NG) . The NRL networks are defined as follows. First, we place N nodes uniformly spaced along the unit circle M = {(a, b)|a 2 + b 2 = 1} ⊂ R 2 such that w (i) = (cos(2πi/N), sin(2πi/N)) is the location of node i. Second, we construct geometric edges such that each node is connected to its d (G) nearest-neighbour nodes. Third, we construct N × d (NG) non-geometric edges uniformly at random such that each node is has exactly d (NG) non-geometric edges. In practice, the non-geometric edges are constructed with stub-matching: We create a list of stubs that contains each node exactly d (NG) times. Then we construct edges iteratively by drawing without replacement uniformly at random from this list while rejecting self-edges or parallel edges. In rare cases this procedure might fail to terminate (e.g., if only stubs of the same node are remaining) but this problem can be overcome by restarting the procedure. The ratio α = d (NG) /d (G) of non-geometric to geometric edges quantifies the 'noisiness' of a noisy ring lattice with α = 0 indicating that there are exclusively geometric edges, and α = 1 indicating that there are as many non-geometric edges as geometric edges. In Fig. 2 , we show two example noisy geometric lattices with α = 0 and α = 1, respectively.

The Watts threshold model (WTM) is a well-established deterministic binary-state discrete-time model of social contagions on networks 38 and is a modification of Mark Granovetter's threshold model 7 . The WTM is defined as follows. Each node is in one of two states, either active or inactive. At time t = 0 all nodes but a set of seed nodes S is inactive. All nodes have the same activation threshold T ∈ [0, 1] (homogenous threshold), which determines how easily a node adapts their neighbours' activation. An active node stays active but an inactive node is activated if more than a threshold fraction T of its d neighbours are active. We update all node states synchronously at each time point and stop the contagion once a steady state is reached (for a discussion of synchronous vs asynchronous update procedures, see 39 ).

As introduced by Taylor et al., we use the WTM to construct deterministic embeddings for networks. These embeddings are based on the nodes' activation times under different initial conditions. Specifically, we define a contagion map as

j indicating the activation time of node i for contagion j. For the initial conditions, we use cluster seeding such that the neighbourhood N ( j) of node j is active at T = 0. After a steady state of the WTM is reached some nodes might still be inactive. The infinite activation time of these inactive nodes is set to 2N ∞ because 2N is much larger than the largest possible activation time of N − 1 (An activation time of N − 1 would be reached by a single node if in each time step exactly one additional node is activated, for example, along a line graph.)

The obtained matrix X of activation times is not necessarily manifold geometric lattice noisy geometric lattice

Noisy geometric networks contain geometric edges that follow the underlying manifold and non-geometric edges that ignore the underlying manifold. Here, we show noisy geometric lattices of size N = 50, whose underlying manifold M is the unit circle

The geometric lattice contains exclusively geometric edges and has d (G) = 2. After adding d (NG) = 2 non-geometric edges per node, we obtain a noisy geometric lattice with α = 1.

symmetric. To study an embedding that defines a semimetric, we investigate the symmetric contagion map X symmetric = X + X . See Algorithms 1 and 2 for pseudocode for the contagion maps and contagion dynamics, respectively. For clarity, we will refer to the contagion maps as defined by Taylor et al. as 'full contagion maps' in contrast to the 'truncated contagion maps' that we will introduce now.

Algorithm 1: Computing (truncated) contagion maps

Threshold T Number of steps s // Ignored for full contagion maps output: Node embedding X symmetric ∈ N N×N begin for each node j ∈ V = {1, . . . , N} do Initialise cluster seeding by S infecting j and its neighbours; In this work, we introduce truncated contagion maps as a generalisation of full contagion contagion maps. Intuitively, they represent full contagion maps in which contagions are not run until a steady state is reached. Rather, we stop the contagion after s ∈ N >0 steps. Specifically, we define a truncated contagion map as V → {x (i) } i∈V , where

(1)

where a (i) j indicates the activation time of node i for contagion j. For the initial conditions, we use cluster seeding such that the neighbourhood N ( j) of node j is active at T = 0. Note that full contagion maps are a special case of truncated contagion maps for s = N. In practice, however, we anticipate an approximately similar behaviour of truncated and full contagion maps for s N because most contagions reach a steady state after a small number s of steps (in comparison to the network size N). As for full contagion maps, we study exclusively the symmetric contagion maps X symmetric = X + X such that we obtain a semimetric.

Persistent homology is a widely adapted method from topological data analysis. It aims to identify stable (i.e., persistent) topological features of point cloud data across a range of resolutions. To obtain these topological features, we must construct simplicial complexes from the point cloud data. Specifically, we employ an Vietoris-Rips simplicial filtration, which is a sequence of simplicial complexes that constructs topological structure based on the data. Given a data set X, a metric d : Y × Z → R, and a scale parameter α ∈ R, we can construct a Vietoris-Rips complex as

Computing the Vietoris-Rips complex for a selection of scale parameters α 1 ≤ α 2 ≤ · · · ≤ α a , then yields a Vietoris-Rips simplicial filtration

which is a sequence of embedded simplicial complexes. The most common way to visualise the persistence of topological features across a filtration is a barcode, which indicates in which filtration step a topological feature is born and in which step it ceases to exist. We use the RIPSER implementation 40 for the computation of Vietoris-Rips persistence barcodes and use the Euclidean norm. To summarise the 1D persistence barcodes (i.e., the presence of loops in the data), we compute ring stability as ∆ = l 1 − l 2 , where l 1 and l 2 are the persistence of the longest and second longest 1ifetime of loops in the data 17 .

In Fig. 1 , we show two example Vietoris-Rips persistence barcodes constructed from contagion maps. 

In this section, we will describe the spread of WTM contagions on noisy ring lattices. In particular, we will demonstrate that in certain parameter regimes, the initial spread is mainly occurring along the manifold (called 'wave front propagation', WFP) and at later stages of the contagion the spread is less restricted to the manifold and leads to new contagion clusters at distant parts of the manifold ('appearance of new clusters', ANC).

Taylor et al. proved two critical thresholds for noisy ring lattices of noisiness α:

which indicate the thresholds above which no WFP and no ANC, respectively, occur. We show these critical thresholds, which intersect at (α, T ) = (1/2, 1/3), in Fig. 3 and observe four qualitatively different contagion regimes: exclusively WFP, exclusively ANC, WFP and ANC, and neither. As we will show, these regimes generalise from full contagion maps to truncated contagion maps. In the following, we investigate four parameter choices in more detail. Specifically, we summarise the Watts threshold model contagions for thresholds T ∈ {0.05, 0.2, 0.3, 0.45} on a noisy ring lattice with N = 400 nodes and noisiness of α = d (NG) /d (G) = 2/6 = 1/3 (see Fig. 4 ). In Fig. 4a , we show the size q(t) of the contagion over time (i.e., the number of nodes that are active). First, we note that for T ∈ {0.05, 0.2, 0.3} a global cascade is triggered as q(t) → 1, whereas for T = 0.45 the contagion size remains constant. Second, a smaller threshold T results in a quicker contagion spread which in turn yields a smaller time until a global cascade is reached. By comparing the chosen thresholds with the critical thresholds of WFP (Eq. 4) and ANC (Eq. 5) we observe that T = 0.05 and T = 0.2 fall in the regime in which WFP and ANC occur, T = 0.3 falls in the regime in which exclusively WFP occurs, and T = 0.45 results in no contagion (see coloured crossed in Fig. 3) .

To study whether the contagion spreads along the manifold or independent of it, we investigate along which type of edges the contagion spreads predominantly. In Fig. 4b , we show the number E(t) of activating edges (i.e., edges incident to a node that is active at t and a node that becomes active at t + 1). We distinguish between geometric edges (solid lines) and non-geometric edges (dashed lines). We observe that for thresholds T ∈ {0.05, 0.2} the contagion spreads quickly and to similar extent along geometric and non-geometric edges. For a threshold of T = 0.3 in contrast, we find a slow contagion process. We also find that initially the contagion process is strongly dominated by contagion along geometric edges (see inlay). After time t ≈ 20, however, the influence of non-geometric edges increases. Our observation indicates that for slow contagion processes, the contagion initially spreads along the manifold as WFP. Yet, a spreading along non-geometric edges becomes more important over time. This highlights that the derived critical thresholds (Eqs. 4 and 5) are strictly valid only for the early contagion steps and become less appropriate the longer the contagion progresses. As we will show in Sec. IV, this behaviour will allow us to improve the recovery of topological features with truncated contagion maps in comparison with full contagion maps.

In this section, we analyse the topological properties of contagion maps on noisy ring lattices. This allows us to identify parameter regimes in which the contagion processes follow the underlying manifold. In particular, we compare the behaviour of truncated contagion maps with that of full contagion maps.

In Fig. 6 , we study contagion maps for noisy ring lattices with N = 400 nodes and noisiness of α = d (NG) /d (G) = 2/6 = 1/3. Each column represents a distinct threshold T ∈ {0.05, 0.2, 0.3, 0.45}, which correspond to the example thresholds in Fig. 4 and distinct regimes in the bifucation analysis of the critical thresholds critical thresholds (Eqs. 4 and 5). In the top row, we show the noisy ring lattice with nodes placed at their locations w (i) = (cos(2πi/N), sin(2πi/N). The colour of each node indicates the node's activation time during one realisation of the WTM in which the green nodes are the seed nodes. For all other nodes, the colour map ranges from blue (small activation time) to yellow (large activation time). Grey nodes never adopt the contagion. As expected, we observe that for two smallest thresholds T ∈ {0.05, 0.2} the contagion quickly spreads to the whole network and that for the largest threshold T = 0.45 no contagion beyond the seed nodes occurs. For intermediary treshold T = 0.3, however, we observe that the contagion spreads approximately along the circular manifold.

In the remaining rows of Fig. 6 , we show the point clouds x (i) obtained from contagion maps. Specifically, rows two, three, and four show truncated contagion maps for s = 10 steps, s = 40 steps, and s = 60 steps and the last row shows the full contagion map, which is equivalent to s = N. To visualise the N-dimensional point clouds, we use a two-dimensional projection through principal component analysis (PCA). Similar to the networks in the top row, we colour each point with the activation time under one realisation of the WTM. Yet, as we truncate the contagion after s steps a node might be grey (i.e., never activated) either because a steady state is reached or because the contagion stopped earlier. Therefore, the number of grey (i.e., never activated) nodes varies with the number s of considered contagion steps although the seed nodes were identical.

In general, we find that the contagion maps x (i) for threshold T = 0.3, which is in a regime in which there is exclusively WFP predicted, most closely resemble the two-dimensional ring topology of the original network (up to rotation). Further-more, in this regime, the truncation parameter s has a strong influence on the obtained embedding. For a small number s = 10 of contagion steps, the ring structure is perturbed and appears folded. Increasing the number of contagion steps to s = 20 makes the ring structure more clearly visible. Increasing the number s of contagion steps further, however, results in a 'broadening' of the circular point clouds. Our observation suggest that for the threshold T = 0.3 a truncated contagion map might better recover the underlying circular manifold than the full contagion map.

After the bifurcation analysis in Sec. III and the qualitative investigation in Fig. 5 , we now quantify the recovery of the manifolds topological features with persistent homology. In particular, we know that the underlying manifold M of a noisy ring lattice is the unit circle. By comparing how closely the persistent homology of the contagion maps resemble the topological properties of the unit circle, we can quantify the recovery of its topological properties. A unit circle has one circular hole and thus its first Betti number is b 1 = 1. Therefore, we compute the ring stability ∆ as a quantification of how 'ring-like' a contagion map is. In Fig. 6a , we show the ring stability ∆ of contagion maps on noisy ring lattices of size N = 400 and noisiness α = 1/3. We vary the threshold T ∈ [0, 0.6] and highlight the critical thresholds T (ANC) = 0.25 and T (WFP) = 0.375 (Eqs. 4 and 5) as vertical dotted lines. All contagion maps have their largest ring stability ∆ in the regime between these thresholds (i.e., when only WFP is possible). Furthermore, the truncated contagion maps with s = 10 and s = 20 have a higher ring stability ∆ than the full contagion map, which matches our observations in Fig. 5 that an appropriate truncation of the contagion map may improve the recovery of the ring-like topology. In other threshold regimes, the ring stability ∆ is almost indistinguishable between truncated and full contagion maps.

In Fig. 6b , we investigate the influence of the number s of contagion steps on the ring stability ∆ of contagion maps with a threshold of T = 0.3. We find that for a large number s → N of steps, the ring stability is the same as for the full contagion map (dashed line), which matches our theoretical expectation (see 'Truncated contagion maps' in the Methods section). For a small number s < 60 of steps, the truncated contagion map has a larger ring stability ∆ than the full contagion map, whereas for s > 60 steps, the full contagion map outperforms the truncated contagion map. We explain this behaviour as a trade-off between data quality and data quantity: The early-stage contagions more closely follow the underlying manifold, yielding better embeddings for the truncated contagion maps. Yet, full contagions have more data as less entries in the contagion map have infinite activation times.

In Fig. 3 , we compare the analytically predicted bifurcation diagram with the ring stability ∆ of truncated contagion maps with s = 20 steps for various thresholds T ∈ [0, 0.6] for noisy ring lattices with N = 400 and various noisiness values α ∈ [0, 1]. Parameter combinations (α, T ) with large ring stability ∆ closely match the regime in which the bifurcation analysis predicts exclusively WFP. This indicates that in regimes which are dominated by WFP, the truncated contagion map consistently recovers the underlying topology of the unit circle, whereas in other parameter regimes this is not possible.

It has been established that the typical computational cost of full contagion maps is O (N 2 d) , where N is the number of nodes in the input network and d the average degree of its nodes 17 . As truncated contagion maps omit some activation steps, we anticipate that less computational time is needed than for the full contagion map.

We empirically verify the reduction of computational cost for contagion maps with thresholds T = 0.3 on noisy ring lattices with noisiness α = 1/3 and consider s = 20 steps in the truncated contagion maps, which is a parameter combination which yields a good recovery of the ring manifold (see Sec. IV). We vary the network sizes N ∈ {32, . . . , 10 4 } to identify the computational complexity (see Fig. 7 ). We find that for small network sizes N ≤ 100, the computation time δt of truncated and full contagion maps are indistinguishable because a steady state is reached before a truncation after s = 20 influences the contagions. Yet, for networks of sizes N > 100, the truncated contagion map is quicker to compute than the full contagion map. Furthermore, we identify that the gap between computational costs widens with increasing network size N. To obtain estimates of the computational complexity of the form δt = ζ N γ , we fit its linearised form log(δt(log(N))) = log ζ + γ log(N) to the logarithm of the data. We find that the computational time of the full contagion map and the truncated contagion map scale superquadratic (δt ∝ N 2.24 ) and subquadratic (δt ∝ N 1.79 ) with the problem size N, respectively, yielding an improved scaling of almost √ N for the truncated contagion maps. The reduced computational complexity of the truncated contagion maps can be understood by investigating the time until a steady state is reached. If comparing networks of the same class (i.e., similar characteristics, such as characteristic path length and mean degree), in larger networks the time until a global cascade has been reached tends to be longer than in smaller networks. Thus, a restriction to a fixed number s of steps by the truncated contagion maps, omits a larger number of steps for larger net-works. This increasing omission reduces the computational time for larger networks, leading to a substantial reduction of the computation time for truncated contagion maps.

In this section, we investigate a single-cell transcriptomics dataset of growing mouse oocytes 41, 42 , in which the oocyte undergo several key stages to generate mature oocytes. Specifically, Gu et al. measure gene expression in five successive stages: primordial follicle, primary oocytes, secondary oocytes, tertiary oocytes, and antral oocytes. The raw gene expression matrix Count ∈ N (c×g) in which entry Count i, j indicates the strength of expression of gene i in cell j is available from GEO under GSE114822. For further analysis, we use SCANPY 43 and follow current best practices 44 . For example, we filter out 5937 genes that are detected in less than 3 cells (for details, see online material). We construct a cellsimilarity graph as a k-nearest-neighbour graph G = (V, E) in which the N = 223 nodes represent the cells V ∈ {1, . . . , c} and each node is connected to its k = 20 nearest neighbours in gene-expression space.

Then, we construct a truncated contagion map with threshold T = 0.3 and truncation s = 10 from the cell-similarity graph (see Fig. 8 ). The contagion map in Fig. 8a has a blockdiagonal structure in which the the non-growing oocytes (i.e., primordial) are clearly separated from the growing oocyte (primary, secondary, and tertiary) and the fully-grown oocyte (antral). A two-dimensional PCA projection of the contagion map in Fig. 8b revels the oocyte development trajectory in gene-expression space. The primary oocytes are developing into secondary and tertiary oocytes and finally in antral oocytes. The separation of the developmental stages is not sharp but rather a continuum, yet a pseudo-temporal ordering that matches the cell identities is observable.

Our study of the single-cell transcriptomics data shows that contagion maps can reveal a manifold structure in a highdimensional gene expression space by constructing a cellsimilarity graph. In particular, we observe that primordial oocytes are clearly separated from developing oocytes and within the developing oocytes, we observe an ordering in gene-expression space.

In this work, we introduced truncated contagion maps, an extension of contagion maps that has improved computational complexity and is thus computationally less expensive. This is achieved by truncating (i.e., stopping) the contagion processes before they reach a steady state. Furthermore, we demonstrate by quantifying the persistent homology that, for appropriate parameter choices, truncated contagion maps might improve the recovery of the embedding manifold of noisy geometric networks. We explain this behaviour with the dominance of spreading along the manifold in early steps of a contagion, whereas at later steps the spread along non-geometric edges increases. Lastly, we demonstrated that contagion maps are a manifold-learning technique that can reveal low-dimensional structure in a cell similarity network constructed from singlecell transcriptomics data.

Our findings suggest various avenues for further studies. For example, it is an open question whether these truncation approaches can also be extended to the simplical complex variant of the contagion maps 19 , which is particularly relevant as higher-order interactions are important in cellular interactions 45, 46 . Furthermore, we hypothesise that a subsampling approach with selected seeding, as used in the k-means++ clustering algorithm 47 , or landmark selection, as used in L-Isomap 48 , might further reduce the computational complexity of contagion maps.

FK is supported as an Add-on Fellow for Interdisciplinary Life Science by the Joachim Herz Stiftung. We thank Martin Vingron for fruitful discussions.

Data openly available in a public repository that does not issue DOIs. PYTHON code to reproduce the results in this paper are available under https://github.com/floklimm/ contagionMap.

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