- Research
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Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge
- Vsevolod Salnikov^{1},
- Daniele Cassese^{1, 2, 3}Email authorView ORCID ID profile,
- Renaud Lambiotte^{3} and
- Nick S. Jones^{4}
- Received: 19 March 2018
- Accepted: 18 June 2018
- Published: 28 August 2018
Abstract
In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. k-dimensional holes die when every concept in the hole appears in an article together with other k+1 concepts in the hole, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the size of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We provide further description of the conceptual space by looking for the simplicial analogs of stars and explore the likelihood of edges in a star to be also part of a homological cycle. We also show that authors’ conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.
Keywords
- Co-occurrence
- Topological data analysis
- Persistent homology
Introduction
Co-occurrence networks capture relationships between words appearing in the same unit of text: each node is a word, or a group of words, and an edge is defined between two nodes if they appear in the same unit of text. Co-occurrence networks have been used, among other things, to study the structure of human languages (Ferrer-i-Cancho and Solé 2001), to detect influential text segments (Garg and Kumar 2018) and to identify authorship signature in temporal evolving networks (Akimushkin et al. 2017). Other applications include the study of co-citations of patents (Wang et al. 2011), articles (Lazer et al. 2009) and genes (Jenssen et al. 2001; Mullen and et al. 2014). Here we focus on the co-occurrences of concepts (theorems, lemmas, equations) in scientific articles to gain understanding in the structure of knowledge in Mathematics. Similar problems have been considered in scientometrics, even if previous works have limited their analysis to keywords, or words appearing in abstracts (Radhakrishnan et al. 2017; Zhang et al. 2012; Su and Lee 2010), and focused only on binary relations between words, as we clarify below.
The main novelty of our work is to study co-occurrences in a simplicial complex framework, using persistent homology to understand the conceptual landscape of mathematics. The adoption of a simplicial complex framework is motivated by the fact that concepts are inherently hierarchical, so simplicial complexes might seem a natural representation: often elementary conceptual units connect together to form nested sequences of higher-order concepts. A simplicial complex approach to model the semantic space of concepts was already suggested by (Chiang 2007), even if not in a topological data analysis framework (Patania et al. 2017b), while application of topological data analysis tools to visualisation of natural language can be found in (Jo et al. 2011; Wagner and et al. 2012; Sami and Farrahi 2017). Several reasons motivate the use of higher-order methods in this context. First, co-occurrence networks tend to be extremely dense in practice and require additional tools to filter the relations and sparsify the network to extract information (Serrano et al. 2009; Slater 2009). Second, in the original dataset, interactions are not pairwise and it is unclear if the constraints induced by a network framework, in terms of nodes and pairwise edges, do not obscure important structures in the system. By modelling co-occurrence relations as a simplicial complex, we thus go beyond the network description that reduces all the structural properties to pairwise interactions and their combinations, explicitly introducing higher-order relations. Note that this modelling approach, in particular the use of simplicial persistent homology, has found uses when the data is inherently multidimensional (Petri et al. 2013), with applications in neuroscience (Petri et al. 2014; Stolz et al. 2017), biology (Chan et al. 2013; Mamuye et al. 2016) to the study of contagion (Taylor and et al. 2015) and to coauthorship networks (Patania et al. 2017a; Carstens and Horadam 2013).
A second contribution of our work is the analysis of the full text of a large corpus of articles, which allows us to bypass the high-level categorisation provided by keywords but also to identify the use of methodological tools and to gain insight into mathematical praxis. However, the main purpose of this article is to use the resulting dataset of concepts and articles as a testbed in which to apply methods from topological data analysis, and to go beyond a standard network analysis.
Dataset
The dataset analysed has been scraped from arXiv, and includes a total of 54177 articles from 01/1994 to 03/2007, of which 48240 in mathematics (math) and 5937 in mathematical physics (math-ph). We have limited the timeframe due to naming conventions in arXiv: since 03/2007 subject is not a part of the article identifier, thus if one wants to export it additional queries to metadata are needed. That is easily expandable, but we decided to limit the dataset at this moment for computational speed. The date is extracted from article id, hence it refers to the submission date. Notice that some of the articles in the first years may have been written some years before 1991 (arXiv first article’s date). In order to describe the mathematical content of articles from the LATE X file we look at different concepts occurrences in the text. Clearly the choice of the concepts set can influence the outcome: choosing them manually by a small group of people would result in a strong bias towards the understanding and priorities of the individuals in the group. Thus we wanted to have something either globally accepted by scientific community or at least created by a sufficiently large group of people. Another point in the selection of a good concepts list is the possibility to make a similar research for other disciplines, thus we chose to get it from some general, easily accessible source. Our strategy consisted in parsing a concepts list from Wikipedia, which includes 1612 equations, theorems, lemmas. Clearly these concepts are not homogeneous, meaning that some of them might represent extremely specific theorems, while others can be very general, like differential equation, but similar holds for any text processing with different words having different frequencies. Our position on that is still to minimize the manually introduced bias: we consider that all concepts have similar weight and try to have as complete set as possible. Moreover it is possible that two different names represent for example the same theorem due to historical reasons. For the moment we consider such synonyms as distinct entities as the usage of one of them but not the other may reflect structural properties: for example a lemma might have different names depending on a (sub-)field of mathematics and manually merging them is not correct.
Simplicial complexes
A simplicial complex is a space obtained as the union of simple elements (nodes, edges, triangles, tetrahedra and higher dimensional polytopes). Its elements are called simplices, where a k-simplex is a set of k+1 distinct nodes and its subsets of cardinality d≤k are called its d-faces. We say that two simplices intersect if they share a common face. More formally:
Definition
Let V be a set of vertices, then a n-dimensional simplex is a set of cardinality n+1 of distinct elements of V, {v_{0},v_{1},…,v_{n}}, v_{i}∈V. A simplicial complex is a collection K of simplices such that if σ∈K and τ⊂σ then τ∈K, so for every simplex in K all its faces are also in K. The k-skeleton of K is the union of all simplices in K up to dimension k.
Simplicial complexes can be seen as generalisation of a network beyond pairwise interactions, that differ from hypergraphs as all subsets of a simplicial complex must also be simplices. As an illustrative example of how simplicial complexes capture higher-order interactions where networks fail to do so, consider that in a co-occurrence network it is not possible to distinguish between three concepts appearing in the same paper and three concepts appearing in three papers each containing two concepts: in a network both cases are represented by a triangle, while in a simplicial complex the first is a 2-simplex (a filled triangle) and the second is a cycle made of three 1-simplices (an empty triangle).
As for networks, also for simplicial complexes we can define simplicial measures that are the higher-order analogs of networks ones, for example (Estrada and Ross 2018) defines several simplicial centrality measures, providing also the characterisation of some families of simplicial complexes. In this paper we use the simplicial analogs of stars to provide a further description of the concept space.
Definition
A simplicial star \(S_{l}^{k}\) consists of a central (k−1)-simplex that is a face of lk-simplices, and there is no other simplex but their subsimplices.
Persistent homology
Dataset
Years | Papers | Concepts | Authors | |
---|---|---|---|---|
Total | 1994 - 2007 | 35018 | 1612 | 23471 |
Included | 1994 - 2007 | 8375 | 1067 | 8852 |
The homological features of complexes are usually studied on a filtration of the complex, that is a sequence of simplicial complexes starting at the empty complex and ending with the full complex, so that the complex at step n<m is embedded in the complex at m for all the steps. In this way it is possible to focus on the persistency of homological features: as the filtration evolves the shape of data changes, so birth and death of holes can be recorded. A hole is born at step s if it appears for the first time in the corresponding step of the filtration, and dies at t if after step t the hole disappears. The difference between birth and death of homological features is called persistence, and can be recorded by a barcode, a multiset of intervals bounded below (Carlsson et al. 2005) visualizing the lifetime of the feature and its location across the filtration: the endpoints of each interval are the steps of the filtration where the homological feature is born and dies (Horak et al. 2009). An alternative visualisation is provided by the persistence diagrams, which are built by constructing a peak function for each barcode, proportional to its length (Edelsbrunner et al. 2002; Stolz et al. 2017).
The way the filtration is built depends on the analysis that one wants to do on the data, a very common method on a weighted network is the weighted rank clique filtration (Petri et al. 2013). This is done by filtering for weights: after listing all edge weights w_{t} in descending order, at every step t one takes the graph obtained keeping all the edges which weight is greater than or equal to w_{t}. The simplicial complex at that step of the filtration is built by including all the maximal k-cliques of the graphs as k-simplices. The obtained simplicial complex is called clique complex.
Reducing the computational burden
Computing persistent homology is very costly if there are large simplices in the simplicial complex, as for each simplex the computation requires to list all the possible subsimplices. For instance in our (rather small) dataset, there are already simplices with 37 vertices and the number of its (k-1)-subsimplices is \(\frac {37!} {(37-k!)k!}\), making it impossible to finish the task in reasonable time with standard tools. In order to reduce the computational burden, we put an upper bound on the dimension of simplices, that is we take the subcomplex that only includes simplices up to a maximum dimension d_{M}=5. In other words, we compute the homology of the d_{M}-skeleton of our simplicial complex K.
In the d_{M}-skeleton of K all simplices of dimension d>d_{M} are replaced by collections of their d_{M}-faces, that is a complex of dimension d_{M} made by glueing together \(\binom {d+1}{d_{M}+1} {d_{M}}\)-simplices along their d_{M}−1 faces, such that for each d_{M}−1 face there are d−d_{M} simplices sharing that face. To make an illustrative example if d_{M}=2, the 2-skeleton of a 3-simplex is the collection of triangles in the boundary of the tetrahedron.
As a trivial example consider the 2-skeleton of the tetrahedron, this contains a homological cycle of dimension 2, as there is a void bounded by triangles inside the tetrahedron. But no homological cycle of dimension 1 nor 0, as all its edges are in the boundary of some 2-simplex. An illustration of the 2-skeleton of a complex can be seen in Fig. 4.
Experimental results
Looking at the distribution of killers’ size, the largest simplex in H_{0} is made of 27 concepts, while both in H_{1} and H_{2} has 38 concepts (it is actually the same article in both). Interestingly and not surprisingly, these two articles are both surveys, the largest for H_{0} regards open questions in number theory (Waldschmidt 2004) and the largest for H_{1} is a survey on differential geometry (Yau 2006). The most frequent killers’ sizes for H_{0}, H_{1} and H_{2} are respectively 4,7,11.
To give a better idea of the meaning of holes death, let us consider some examples. The smallest cycle in H_{1} is an empty triangle, one of such cycles is given by the three simplices {(Schur’s lemma, Stone-Von Neumann Theorem), (Schur’s lemma, Spectral Theorem), (Spectral Theorem, Stone-Von Neumann Theorem)} which is killed by a 2-simplex when the three concepts appear together in the paper (Mantoiu et al. 2004). Another interesting example is a 5-step long cycle in H_{1}, {(Boltzmann equation, Alternate Interior Angles Theorem), (Boltzmann equation, Vlasov equation), (Inverse function theorem, Vlasov equation), (Arzelá - Ascoli theorem, Alternate Interior Angles Theorem), (Arzelá - Ascoli theorem, Inverse function theorem)}, that is killed by the 10-simplex which nodes are: { Blum’s speedup theorem, Boltzmann equation, Alternate Interior Angles Theorem, Kramers theorem, Perpendicular axis theorem, Ordinary differential equation, Kronecker’s theorem, Arzelá - Ascoli theorem, Navier - Stokes equations, Vlasov equation, Moreau’s theorem} (Gottlieb 2000). The article, classified in arXiv as Probability, establishes the conditions for a family of n-particles Markov processes to propagate chaos, and shows its application to kinetic theory. We think this is a another possible interpretation of killing holes: a theoretical result that has several applications, hence bridges related areas and closes a homological cycle.
Simplicial stars represent potentially interesting structures in the conceptual space, which can be visualised as small substructures attached to the ‘surface’ of a densely connected cluster, like receptors on the membrane of a cell. To grasp the intuition consider that if we partition the concepts in a S^{k} star between those in the core (the 0-faces of the core k-simplex) and those in the periphery (the zero faces of the (k+1) simplices that have the core as common face, without the core faces) by definition there can’t be any edge (or higher-order simplex) between any of the concepts in the periphery. This means that periphery nodes ‘touch’ the surface of a densely connected area, and each of them belongs to a different simplex lying on the surface, while nodes in the core are one step far from the surface. Considering only those with at least two simplices we count 567 S^{2} stars and 284 S^{3} stars. We do not check for higher-order stars for computational reasons.
Probabilities and standard errors for edges in the cores and peripheries of stars and of a random edge to be in H_{k}
H _{1} | H _{2} | H _{3} | |
---|---|---|---|
S^{2} (cores) | 0.794 (0.005) | 0.806 (0.004) | 0.785 (0.003) |
S^{2} (peripheries) | 0.571 (0.006) | 0.51 (0.005) | 0.547 (0.003) |
S^{3} (cores) | 0.743 (0.005) | 0.905 (0.003) | 0.91 (0.002) |
S^{3} (peripheries) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) |
Random Edge | 0.240 (0.002) | 0.300 (0.002) | 0.641 (0.003) |
Authors analysis
Discussion
This work is a first attempt to explore the importance of homological holes in mathematics, and it is important to issue certain caveats here. It seems clear that our observations would not have been possible within a classical network analysis of co-occurrences, and we also think that holes deaths can be informative in capturing advances in the discipline. However, we do not claim that we are describing the essence of mathematical practice here. If on one hand, by extracting concepts from the whole text of the article instead of just focusing on the keywords we avoid the bias of authors’ own classification of their work, on the other hand it is possible that for some articles the conceptual content is not well captured by our approach: we cannot exclude that the set of concepts that better identify the content of the article do not find an exact match in our list, while those finding a match are poorly representative of the article. We believe that such cases are a minority as our list is very comprehensive, still this is an aspect to take in consideration when analysing our results, even if this is an issue that is potentially arising every time one does text analysis, irrespective of the tools used to explore the data. In this direction, an important step would be to assess the robustness of our observations by purposely introducing errors in our data analysis, for instance by focusing only on a fraction of our list of identified concepts, and keeping the other unknown.
Overall, this paper suggests new directions for the study of co-occurrences by focusing on their higher-order properties and there is substantial space for further development. The first and most urgent point, in our opinion, regards the necessity to validate the findings from the data by having well-founded null models for comparison. Recently some null models for simplicial complexes have been developed (Young et al. 2017; Courtney and Bianconi 2016), which we did not use here for computational reasons, as producing samples from our relatively large dataset proved too challenging to manage with our resources. Another very important point is to devise methods and find algorithms to localise homological holes with precision, as this could be very useful for some specific applications, and it can still be considered an open problem. In this regard we think that it would be pivotal to unify the study of the geometry and the topology of the structure. A geometrical approach has been used in (Eckmann and Moses 2002) where they find that curvature is a good measure of thematic cohesion in the WWW. More recently (Wu et al. 2015; Bianconi and Rahmede 2017) studied the emergence of geometry in growing simplicial complexes, without a pre-existing embedding space and metric. This approach could be naturally extended to our case, as we have no natural embedding space for concepts, and also because the co-occurrence simplicial complex that we study is dynamic by nature, as new simplices appear at each time step eventually glueing to existing simplices. Embedding concepts in a space could facilitate the localisation of holes, in this regard (Bianconi and Rahmede 2017) find that the natural embedding for their growing simplicial complexes is hyperbolic, and in their case the position of the incoming nodes depends on the position of the nodes of the face of the simplex to which the new simplex glues. Such a framework could be ideal to the purpose of precise holes identification in our case. Moreover adopting a growing simplicial complex model (generalized to allow for simplices of different size at each time step) and fitting it to our data we could be able to predict the future evolution of the conceptual space of mathematics.
Conclusions
In this paper, we have studied the topological structure of conceptual co-occurrences in mathematics articles, using data from arXiv. We modelled co-occurrences in a simplicial framework, focusing on higher-order relations between concepts and applying topological data analysis tools to explore the evolution of research in mathematics. We find that homological holes are ubiquitous in mathematics, appearing to show an intrinsic characteristic of how research evolves in the field: holes are likely to represent groups of concepts that are closely related but do not belong to a unitary subfield, and the death of a hole is either a sign that anticipates a potential advance in that conceptual area (for example a review trying to bridge the concepts and suggesting research lines), or an actual advance, that is an article (or a set of articles) that unifies a subgroup of concepts in the cycle, for example a theoretical result with application to different areas. Less interesting, but we cannot exclude it as we have no other way to verify than reading each of the papers killing a hole, a hole-killer (especially if of very large size) could be a scarcely relevant article mentioning many concepts without providing any true contribution.
We also find that the higher the number of concepts in a hole, the longer it takes to die, hence the length of a hole is a good proxy of how distant these concepts are, in terms of their likelihood to appear together in an article. So in this sense large holes could be seen as potential spaces for important advances in mathematics. Moreover we further explore the structure of co-occurrences by looking at the simplicial analogs of stars in higher dimension, which represent groups of concept (those in the core of the star) that supports and connects many otherwise unrelated concepts, and we find that concepts appearing in stars tend also to appear in holes more often than they would do at random, suggesting that both structures lie at the frontier of mathematical research.
We also explore authors’ conceptual profile by ordering them on the basis of their conceptual entropy, so that we can differentiate between those authors who tend to specialise and publish mostly about few concepts, and others that do research on a broad range of topics, that we call polymaths. Comparing authors’ profiles with a random model, we find that authors’ entropy as a function of how may concepts authors use across different publications, is bounded above, while in the null model, entropy is always increasing for larger set of random concepts. This is reasonable, and means that even the more prolific polymaths, even if they publish a large number of articles, will still tend to specialise to some extent, instead of doing research always on new topics. Moreover we find that polymaths contribute to homological holes more than specialists, so polymaths are often at the frontier of research.
Further work could be done by using larger datasets, as it would be very interesting to explore the birth and death of holes in a larger time-span, and to study simplicial co-occurrences in other disciplines, in order to see if any difference appears in the way research evolves in different fields. Furthermore, conceptual spaces emerging from co-occurrences relations could be explored adding a further dimension to the filtration: in our case we focus on a temporal filtration, disregarding the weights of simplices, this could be extended by filtering along time and weight using multidimensional persistence (Carlsson and Zomorodian 2009).
Declarations
Acknowledgements
The authors thank two anonymous reviewers for their constructive comments and Henry Adams, Oliver Vipond and Alexey Medvedev for useful discussions and suggestions.
Funding
Daniele Cassese wishes to thank support from FNRS (Belgium).
Availability of data and materials
Data available upon request.
Authors’ contributions
All authors conceived the study; VS and DC performed the numerical simulations and created the Figures; All authors wrote and reviewed the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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