The why, how, and when of representations for complex systems

Leo Torres; Ann S. Blevins; Danielle S. Bassett; Tina Eliassi-Rad

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The why, how, and when of representations for complex systems

Leo Torres, Ann S. Blevins, Danielle S. Bassett, Tina Eliassi-Rad

arXiv:2006.02870v1Jun 4, 2020cs.SIcs.DMq-bio.QM

TL;DR

Complex-systems research is fragmented by diverse formalisms and inconsistent vocabulary, limiting communication across disciplines. The paper introduces a common language, examines dependencies and representations across the analysis pipeline, and uses real-world examples to show that representation choices influence results. It concludes that dependencies, research questions, and careful representation and analysis choices should be considered together.

Problem
The proliferation of domain-specific formalisms and inconsistent language hinders communication, collaboration, and insight across complex-systems research.
Method
The paper aligns complex-system analysis pipelines using a common vocabulary, examines dependencies across graph, simplicial-complex, and hypergraph formalisms, and applies the framework to two datasets.
Results
The coauthorship and email examples show that the system, dependencies, research question, and mathematical representation influence analysis results.
Takeaways & Limitations
Researchers should consider dependencies, representation assumptions, research questions, and analysis techniques when studying complex systems.
Takeaways & Limitations
Available methods may support only one formalism, even when another formalism better represents the system under study.

Abstract

from arXiv · show

Complex systems thinking is applied to a wide variety of domains, from neuroscience to computer science and economics. The wide variety of implementations has resulted in two key challenges: the progenation of many domain-specific strategies that are seldom revisited or questioned, and the siloing of ideas within a domain due to inconsistency of complex systems language. In this work we offer basic, domain-agnostic language in order to advance towards a more cohesive vocabulary. We use this language to evaluate each step of the complex systems analysis pipeline, beginning with the system and data collected, then moving through different mathematical formalisms for encoding the observed data (i.e. graphs, simplicial complexes, and hypergraphs), and relevant computational methods for each formalism. At each step we consider different types of \emph{dependencies}; these are properties of the system that describe how the existence of one relation among the parts of a system may influence the existence of another relation. We discuss how dependencies may arise and how they may alter interpretation of results or the entirety of the analysis pipeline. We close with two real-world examples using coauthorship data and email communications data that illustrate how the system under study, the dependencies therein, the research question, and choice of mathematical representation influence the results. We hope this work can serve as an opportunity of reflection for experienced complexity scientists, as well as an introductory resource for new researchers.

1 Introduction

Complex systems span scales, domains, and behaviors, but their study is hindered by fragmented formalisms and inconsistent vocabulary. The paper proposes a shared language and pipeline for representing and analyzing such systems.

Motivation: Complex systems contain many interacting units whose collective behavior is not immediately predictable from the individual parts.Examples range from neuronal populations and cellular reactions to markets, the Internet, and power grids.
Motivation: Network science provides mathematical and computational frameworks for representing complex systems across disciplines.These frameworks support methodologies drawn from discrete mathematics and computational data structures.
Motivation: The proliferation of frameworks hinders interdisciplinary communication and makes it difficult to gather insight across systems.Different approaches can also obstruct collective insight within a single subfield.
Paper aim: The paper aligns complex-system analysis pipelines and develops a common vocabulary centered on dependencies, representations, formalisms, and computations.It follows a simplified pipeline from system properties through representation and computation to interpretation, using coauthorship and Enron email examples.
Definitions: Dependencies describe how the existence of one relation can provide information about the existence of another relation.The paper distinguishes these system properties from units, relations, and attributes used in the analysis pipeline.
Definitions: A formalism is a mathematical framework for representing and studying a complex system, while a representation is one concrete encoding produced from it.The paper discusses graph, simplicial-complex, and hypergraph formalisms.

2 Dependencies by the system, for the system

Complex-system analysis should identify dependencies in the system and preserve them through representation and downstream analysis. Subset, temporal, and spatial dependencies can arise intrinsically or from external data constraints, and overlooking them can bias results.

Dependency-aware analysis: Dependencies describe how one relation’s existence provides information about another relation, guiding which system properties a model should encode.The paper treats dependency selection as a fundamental design decision in complex-system analysis.
Subset dependencies: A subset dependency exists when a relation among nodes implies that some or all subsets are also relations, but this implication depends on how relations are defined.Shared features can produce subset dependencies, whereas chemical reactions need not make every subset independently reactive.
Subset dependencies: Whether to represent subset dependencies depends on available data, the research question, and relation definitions, because the choice changes usable computations, formalisms, and downstream results.Path questions may be answerable with simpler dyadic relations, while polyadic data may be needed to encode the dependency explicitly.
Temporal dependencies: Ignoring temporal dependency can make a pairwise graph suggest impossible routes, whereas preserving sequence distinguishes train arrivals and reveals required transfers.The subway example shows that observed paths may be non-Markovian: future movement can depend on prior states, not only the current station.
Spatial dependencies: Combining spatial positions with topology helps assess costly long-distance connections, while omitting spatial dependency can misattribute structural patterns to the wrong cause.Examples include confusing geographically driven disease spread or friendship counts with effects of food consumption or personal charisma.
External sources of dependencies: Dependencies may be imposed by data availability or processing, so observed subset structure may reflect sparse sampling or induced correlations rather than the system’s true structure.Changing the correlation or similarity measure can change inferred node relations.
External sources of dependencies: Recognizing subset, temporal, and spatial dependencies early helps preserve them through the analysis pipeline and affects the conclusions that downstream analyses can support.The paper emphasizes that dependencies may originate within the system or from external factors.

3 Formal representations of complex systems

Complex-system research uses multiple mathematical formalisms to represent different system aspects and dependencies. That variety supports specialized modeling but complicates communication, collaboration, and the analysis pipeline.

Formalism overview: Different formalisms support modeling of unique system aspects and dependencies, but their multiplicity creates challenges for communication, collaboration, and progress across complexity science.The same system may therefore be represented and analyzed through different formal frameworks.
Formalism overview: Graphs, simplicial complexes, and hypergraphs are three prevalent formalisms used to represent complex systems.They are selected as common examples among many possible mathematical formalisms in the literature.
Formalism overview: All three discussed formalisms represent system units as nodes, while graphs encode pairwise relations as edges.The paper frames representations as mirrors of a complex system’s units and relations.

3.1 Graphs

Graph theory provides a common formalism for representing complex systems as vertices connected by pairwise edges. Its broad use spans mathematics, social network analysis, computer science, and contemporary complex-systems research.

Graph definition: A graph G=(V,E) consists of vertices V and edges E, with each edge connecting exactly two vertices.Edges are unordered sets of two nodes, while vertices are the graph’s main units.
Graph definition: Graphs model how units fit together through pairwise relations represented by edges.This formalism represents dyadic structure rather than relations involving three or more nodes directly.
History and use: Graph theory’s use beyond mathematics includes sociograms, social network analysis, and graph-like computer-science data structures.Its application to general complex systems expanded rapidly in recent decades.
History and use: The spread of graph-based complex-system modeling was driven largely by discoveries such as small-world structure and heavy-tailed degree distributions.The passage identifies these discoveries as major drivers of recent adoption.

3.2 Simplicial Complexes

Simplicial complexes represent polyadic relations while enforcing downward closure: every subset of a related node set must also be related.

A simplicial complex consists of vertices and subsets of vertices called relations, with every subset of each relation included.This condition is called downward closure.
A k-simplex represents k + 1 collectively interacting nodes, and all node subsets within it must also form simplices.
Simplicial complexes capture polyadic relations among units through simplices rather than pairwise edges.

3.3 Hypergraphs

Hypergraphs generalize graphs by allowing hyperedges to connect any number of vertices, without requiring the downward-closure property of simplicial complexes.

A hypergraph contains vertices and hyperedges, with each hyperedge able to connect an arbitrary number of vertices.Unlike graph edges, hyperedges may connect three, four, five, or more nodes.
Hypergraphs do not require every subset of a hyperedge to be another hyperedge.A hyperedge can connect v1, v2, and v3 even when no hyperedge connects v1 and v2 exclusively.
This flexibility makes hypergraphs suitable for representing polyadic relations in systems previously modeled with graph theory.Applications described include social networks, Enron email data, yeast protein interactions, and neurodevelopmental motifs.

3.4 Variations

Complex-system formalisms can be extended to encode attributes such as direction, weight, time, relation type, and higher-order temporal structure.

Directionality: Directed formalisms encode an ordered relation between two nodes, distinguishing vA →vB, vB →vA, and vA ↔vB.Directionality is relevant in systems including brains, transportation networks, and metabolic pathways.
Weights: Weighted formalisms assign real-valued weights to encoded relations to represent differing interaction strengths or magnitudes.
Time: Dynamic graphs represent evolving systems as sequences of graphs on the same node set, preserving node identities across time.The sequence is written G1, . . . , GT.
Relation types: Multilayer graphs separate relation types into layers, with each layer containing relations of exactly one type.Layers may or may not involve the same nodes.
Higher-order temporal structure: Higher Order Networks encode frequent paths or transitions as nodes so standard Markovian assumptions can represent temporal dependencies.In the transportation example, this yields a more faithful representation because observed continuations depend on prior station sequences.
Combinations: These variations can be combined into composite representations such as edge-weighted dynamic or directed multilayer networks.

3.5 Encoding system dependencies

Encoding dependencies requires defining relations and choosing a formalism that matches the system’s subset, spatial, and temporal structure. Because real systems can have multiple dependencies, representation choices remain interdependent and incomplete.

Defining relations: Relation definitions determine whether dependencies appear in the resulting representation or arise as artifacts absent from the actual system.
Defining relations: In the brain-activity example, defining relations by co-firing produces a subset dependency, unlike definitions based on jointly required activation or pairwise similarity.
Defining relations: Bus-route relations can encode no subset or temporal dependency when defined by complete routes, whereas shared-line relations introduce subset dependency.
Choosing a formalism: Modeling choices for relations, representations, and dependencies are unavoidably interdependent, requiring researchers to distinguish encoded, neglected, and externally introduced dependencies.
Subset dependencies: Simplicial complexes suit systems where every subset of a related set is related, while hypergraphs allow only some subsets to be related.
Temporal dependencies: Higher Order Networks encode temporal dependencies by representing ordered sets of original-system units as graph nodes and connecting them according to temporal dependence.
Spatial dependencies: Spatial dependencies can be represented with edge weights, but using weights for another information type prevents simultaneously encoding spatial proximity there.
Multiple dependencies: Real-world systems often contain multiple dependencies, motivating methods that systematically represent and encode them together.Examples include subway systems with temporal and spatial dependencies and coauthor systems with temporal and subset dependencies.

4 Mathematical relationships between formalisms

The paper treats transitions among graphs, simplicial complexes, and hypergraphs as optional transformations that can enable additional methods but may lose information or require assumptions. These transformations are not generally reversible, so representation choice constrains what can be inferred.

4 Mathematical relationships between formalisms: Moving between formalisms is an optional pipeline step that should follow choosing an appropriate representation for the data.The authors emphasize that transformations should be used cautiously rather than treated as automatic replacements for representation choice.
From hypergraph to simplicial complex: Forgetting independent sets: Mapping a hyperedge to a simplex imposes downward closure, adding subset relations that may not have existed independently in the hypergraph.A hyperedge on k + 1 nodes becomes a k-simplex, which requires every subset to be a simplex.
From hypergraph to simplicial complex: Forgetting independent sets: Hypergraphs preserve whether subset relations exist independently, whereas simplicial complexes can be specified by their maximal simplices alone.Thus, converting a hypergraph to a simplicial complex can erase information about absent or independently arising subset hyperedges.
From simplicial complex to graph: Forgetting polyadic relations: Converting a simplicial complex to a graph keeps its 1-simplices as edges and forgets polyadic relations among nodes.Nodes sharing a simplex become pairwise adjacent in the graph, even though the original higher-order relation is no longer represented explicitly.
From simplicial complex to graph: Forgetting polyadic relations: A hypergraph can also be converted directly to a graph by connecting node pairs that belong to a common hyperedge, yielding the same graph as the two-step conversion through a simplicial complex.Both procedures retain pairwise co-membership while discarding the original higher-order organization.
From graph to simplicial complex: Assuming polyadic relations: Constructing a simplicial complex from a graph assumes that every clique represents a polyadic relation, producing the clique or flag complex.The construction fills each (k + 1)-clique with a k-simplex.
From simplicial complex to hypergraph: Assuming that only maximal simplices are independent: Converting a simplicial complex to a hypergraph can either preserve all subset relations or retain only maximal simplices, depending on the chosen protocol.These alternatives encode different assumptions about which relations are independently meaningful.
From hypergraph to graph, and from graph to hypergraph: A graph can be translated through the other formalisms and recovered, but translating a hypergraph downward and back produces a different hypergraph because independent higher-order information is lost.The reverse transformation cannot reconstruct whether subset hyperedges originally existed independently.

5 Methods suitable for each representation

Each representation supports distinct computational views of a complex system, and method availability should not determine the representation by itself. Graphs emphasize pairwise structure, simplicial complexes expose higher-dimensional topology, and hypergraphs retain polyadic and absent-substructure information.

Methods and dependencies: A computational method that accepts only one formalism does not justify representing data in that formalism if it is ill-suited to the system.The paper uses epidemic-threshold calculations as an example: graph-based methods exist, while corresponding polyadic theory remains nascent.
Methods for graphs: Graph analyses span local, meso-scale, and global organization, using measures such as degree, clustering, path length, centrality, modularity, and small-worldness.These methods benefit from the long history and broad development of graph analysis.
Methods for graphs: A node’s graph clustering coefficient uses the number of edges between its neighbors relative to the number of possible such connections.The numerator counts triangles containing the node, while the denominator normalizes by the maximum possible number of triangles.
Methods for simplicial complexes: Simplicial-complex analyses extend graph concepts through node participation across simplex dimensions and can identify higher-dimensional topological voids.Downward closure makes it possible to determine which lower-dimensional simplices belong to higher-dimensional simplices and where expected simplices are absent.
Methods and dependencies: Different formalisms can yield different structural interpretations of the same system: core-periphery in a graph, a circular organization in a simplicial complex, and two communities in a hypergraph.The comparison illustrates that representation choice changes the perspective produced by downstream analysis.
Methods for hypergraphs: Hypergraphs can represent absent substructures, such as missing three-node relations inside larger relations, and support statistics such as the fill coefficient.Graphs lack polyadic relations, while simplicial complexes impose downward inclusion, preventing the same question from being posed in those formalisms.
Methods and dependencies: The paper concludes that proper representation and computational analysis should be chosen with awareness that each pipeline and method offers a distinct perspective on the underlying system.The updated pipeline therefore includes both representation choice and consideration of complementary computational methods.

6 Examples

The coauthorship and email examples show that dependencies, research questions, and mathematical representations jointly shape analysis, and that different representations can produce conflicting results.

Coauthorship: Coauthorship analysis must distinguish dependencies arising from the system, data collection, and the chosen research question.A pairwise coauthorship question can impose subset dependencies even when the underlying data may not record them.
Coauthorship: The graph, simplicial complex, and hypergraph representations encode different perspectives on the same coauthor data.Moving between formalisms after starting from a graph can also produce inaccurate representations of the original data.
Coauthorship: Hypergraph degree identifies author a2 as the most collaborative, whereas graph degree and simplicial participation treat authors a2 and a3 as equally collaborative.For authors a1 and a4, simplicial complexes and hypergraphs distinguish project sizes that the graph does not.
Coauthorship: Spearman degree correlations are highest between simplicial-complex and hypergraph representations, while graph–hypergraph correlation falls below 0.5 in some years.The stronger correlation between simplicial complexes and hypergraphs is attributed to their shared encoding of polyadic relations.
Email communications: Email-team rankings show little correlation between graph and hypergraph clustering coefficients, with Spearman r = 0.1 and p = 0.16 for teams of at least five.The result persists for teams of at least 6, 7, or 8 people, so intervention priorities would depend on the chosen representation.
Email communications: Even with a fixed research question, complete data, and few interactions among dependency types, representation choice alone can yield different insights.The example attributes the divergence to different assumptions about dyadic versus polyadic interactions.

7 Applications

The paper applies its framework to brain, transportation, and cellular systems, identifying underexplored dependencies and representation challenges. These applications motivate richer representations and probabilistic alternatives where observations are incomplete.

Cross-domain perspective: Alternative analysis pipelines may provide perspectives that are more revealing or more faithful to the system than commonly used pipelines.The applications revisit dependencies, formalisms, and data challenges as dimensions shaping those alternatives.
Brain: Brain research has incorporated spatial structure, but temporal dependency remains particularly little understood within the representation of brain activity.The paper suggests higher-order network representations to investigate how past relations affect future activity.
Transportation: Transportation studies have used multilayer, spatial, and temporal representations, while subset dependency has received comparatively little attention.The paper proposes examining subset dependencies and polyadic relations to address system failures, predict activity, and reveal hidden properties.
Cellular systems: Cellular-system interactions are rarely directly observable, making system representation especially difficult.The paper presents probabilistic relation weights and ensembles of graphs, simplicial complexes, or hypergraphs as possible alternatives.

8 Discussion and Conclusion

The paper argues that complex-system analyses require pipelines tailored to each system, dataset, and dependency structure rather than routinely reused field-specific workflows. It presents a dependency-aware sequence from system and data assessment through formalism selection and analysis methods, aiming to support more accurate insight and interdisciplinary collaboration.

Discussion and Conclusion: The paper examines complex-system analysis as a pipeline spanning dependencies, mathematical formalisms, conversions between representations, and analysis techniques.It emphasizes that representation changes can lose or impute information.
Discussion and Conclusion: Different systems, and even different datasets from the same system, may require entirely different analysis pipelines.The authors caution that modeling decisions do not necessarily transfer across systems or datasets.
Discussion and Conclusion: Figure 18 orders the workflow from research question, data availability, and dependencies to formalism selection and then analysis methods.The proposed sequence is intended to preserve dependencies before computational analysis begins.
Discussion and Conclusion: Pipeline decisions should be based on the system, its dependencies, data-induced dependencies, and the limits of the system fragment under study.The authors recommend evaluating each new system and dataset individually before selecting a formalism.
Discussion and Conclusion: The authors frame their vocabulary and dependency-aware representations as a first attempt toward more accurate analyses, collaborations, and future techniques.They hope continued discussion will identify gaps in current formalisms and generate novel methods.

10 Citation diversity statement

The paper describes a proactive effort to consider citation diversity and reports the gender-name categories used to evaluate its references. It notes that this measurement approach has limits in representing gender diversity.

Citation diversity statement: The authors sought to proactively choose references that reflect diversity and evaluated one aspect using gendered-name probabilities.The passage describes citation-bias concerns involving women and other minorities.
Citation diversity statement: The gendered-name method cannot account for intersex, non-binary, or transgender people.The authors identify this as a limitation and call for improved approaches.