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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:2006.02870v1cs.SIcs.DMq-bio.QM

TL;DR

Complex-systems research uses varied frameworks and vocabularies, making interdisciplinary communication and shared insight difficult. This paper aligns analysis pipelines with domain-agnostic language and examples, illustrating that dependencies and representation choices shape results.

  • Problem

    Varied frameworks and analysis pipelines make interdisciplinary communication and collective insight across complex-systems research difficult.

    Source: §1 · p. 4

  • Method

    The paper aligns complex-systems analysis pipelines with a common vocabulary, examining dependencies, mathematical formalisms, their relationships, and analysis methods.

    Source: §1 · p. 4, §8 · p. 36

  • Results

    The coauthorship and email examples underscore that dependencies, representation choices, and analysis techniques influence complex-systems results.

    Source: §8 · p. 36

  • Takeaways & Limitations

    Researchers should account for system dependencies and formalism assumptions when studying complex systems.

    Source: §8 · p. 36

    Limited data may make an observed and encoded subset dependency reflect data sparsity rather than the system’s true structure and function.

    Source: §2.4 · p. 11

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

The paper argues that complex-systems research needs shared language and deliberate choices about how systems are represented and analyzed. It defines core terms, traces an analysis pipeline, and uses coauthorship and email data to illustrate how modeling choices shape results.

1 Introduction

  • A complex system has many interacting parts whose collective behavior is not immediately predictable from the parts’ individual behavior.The paper notes that such systems span domains and scales, from neuronal populations to the Internet and power grids.Source: §1 · p. 4
  • Multiple representation frameworks and analysis pipelines hinder interdisciplinary communication and make it difficult to gather insight across systems.The authors argue that researchers should understand the assumptions of each formalism and how formalism choices and relationships affect analysis and interpretation.Source: §1 · p. 4
  • The paper aligns complex-systems analysis pipelines and vocabulary, examines dependencies, graphs, simplicial complexes, hypergraphs and their computations, and illustrates modeling choices with coauthorship and Enron email data.It recommends making each modeling decision case by case, considering dependencies, formalisms, relationships between formalisms, and the research question.Source: §1 · p. 4

1.1 Definitions

  • The definitions aim to make communication across disciplines more precise and operationalize the study of complex-system structure and behavior.The authors provide a consistent vocabulary for terms used throughout the paper.Source: §1.1 · p. 5
  • Units are system parts; relations are sets of one or more units, and their patterns form system structure, while system activity is sometimes called function.A relation is dyadic when it contains two units and polyadic when it contains three or more; a complex system’s units and relations have functionality different from the sum of its units acting individually.Source: §1.1 · p. 5
  • A system fragment is a subset of a system’s nodes and relations, reflecting researchers’ often-limited access; a dependency is a property linking the existence of one relation to information about another.Fragments may be limited by scale, measurement constraints, or data access, and need not function autonomously as systems.Source: §1.1 · pp. 5–6
  • A formalism is a mathematical framework, a representation is a specific encoding using that framework, and encoding is the process of creating it; attributes are information attached to nodes or relations.The paper focuses on graphs, simplicial complexes, and hypergraphs, while noting that node and relation attributes can extend these base formalisms.Source: §1.1 · p. 6

2 Dependencies by the system, for the system

The section defines dependencies as system properties linking the existence of one relation to another and examines how subset, temporal, and spatial dependencies—and external influences—affect representations and analyses.

2 Dependencies by the system, for the system

  • Dependencies are system properties in which one relation informs another; this section distinguishes subset, temporal, and spatial dependencies.Other dependency types also exist, and researchers continue to develop ways to identify and incorporate them.Source: §2 · p. 6

2.1 Subset dependencies

  • A subset dependency holds when a relation implies that some or all subsets are also relations, subject to a predicate defining which subsets qualify.Whether to encode it depends on available polyadic data, the research question, and relation definitions; counting Marta’s conversation partners without distinguishing separate from group conversations can misestimate her popularity.Source: §2.1 · p. 7

2.2 Temporal dependencies

  • Pairwise station transitions can erase temporal dependencies, making a commute appear possible even when its observed sequence never occurs.This representation assumes Markovian transitions; mismanaging temporal dependencies can create nonexistent connections or overstate the importance of rarely used edges.Source: §2.2 · pp. 8, 10, Figure 3 · p. 9

2.3 Spatial dependencies

  • Spatial dependencies arise when distance influences whether nodes relate, and ignoring them can bias models and attribute structural patterns to the wrong causes.Combining topology with spatial information can reveal costly, potentially important long-distance links; omitting location and mobility in an outbreak analysis could wrongly implicate a widely consumed food rather than person-to-person contact.Source: §2.3 · pp. 10–11

2.4 External sources of dependencies

  • External factors—including data limits, acquisition or processing, and research questions—can shape observed dependencies or obscure dependencies essential to interpretation.Sparse data may conceal subset rules; correlation processing can induce transitivity absent from the system; and a protein-complex question can change whether relations exhibit subset dependency, while a food-focused outbreak question may omit spatial effects.Source: §2.4 · pp. 11–12

3 Formal representations of complex systems

The section introduces graphs, simplicial complexes, and hypergraphs as formalisms for representing systems of units and relations. Their differing capabilities can model distinct system aspects and dependencies, while their variety complicates communication and analysis choices.

3 Formal representations of complex systems

  • The multiplicity of formalisms allows modeling distinct system aspects and dependencies but complicates communication, collaboration, scientific progress, and analysis-pipeline choices.Source: §3 · p. 12
  • The section discusses graphs, simplicial complexes, and hypergraphs as commonly used formalisms for representing systems composed of units and relations.The units in all three formalisms are called nodes.Source: §3 · pp. 12–13
  • Graphs encode dyadic node relations as edges, whereas simplicial complexes and hypergraphs encode polyadic relations as simplices and hyperedges, respectively [20].Graph representations have also supported discoveries such as the prevalence of small-worldness in real-world networks.Source: §3 · p. 13

3.1 Graphs

Graphs represent systems as vertices connected by edges, with each edge linking exactly two vertices. This formalism supports analyses of connected groups and paths, while assuming relations are pairwise.

Graphs

  • A graph G = (V, E) represents units as vertices and their pairwise relations as edges, each connecting exactly two vertices.Edges are unordered pairs of nodes.Source: §3.1 · p. 13
  • Graph analyses commonly seek densely connected node sets, such as cliques or communities, or trace paths through the structure.These questions concern how objects traverse the graph or how nodes are connected.Source: §3.1 · p. 13
  • Graphs benefit from extensive mathematical theory, generally simple computations, and insightful visualization, but assume relations occur exclusively pairwise.Some systems, including transportation networks, may have only pairwise relations, whereas many others—especially in biology—have polyadic relations.Source: §3.1 · p. 13

3.2 Simplicial Complexes

Simplicial complexes represent polyadic interactions while requiring every subset of a relation to also be a relation. Their use spans point-cloud analysis and applications to biological, physical, and image data.

3.2 Simplicial Complexes

  • A simplicial complex represents polyadic relations as simplices, with downward closure requiring every subset of a simplex to also be a simplex.A k-simplex comprises k + 1 vertices that may function in unison; motifs of interest include topological cavities and maximal simplices.Source: §3.2 · p. 13, Figure 6 · p. 14
  • A simplicial complex can be encoded as a binary matrix with one row per maximal simplex and one column per vertex, marking vertex participation.A maximal simplex is not contained in any larger simplex.Source: §3.2 · p. 13
  • Applied algebraic topology emerged in the early 2000s; simplicial complexes have since represented point clouds, neural recordings [82], images [195], and brain-network architecture [194].Recent work has defined generative models for simplicial complexes with given topological features [47].Source: §3.2 · p. 15

3.3 Hypergraphs

Hypergraphs encode polyadic relations as arbitrary-sized vertex subsets without requiring downward closure. This flexibility broadens what can be represented, but limits formal analysis methods.

3.3 Hypergraphs

  • Hypergraphs generalize graphs by letting each hyperedge connect any number of vertices; they can be encoded as a binary matrix of hyperedges by vertices [204, 25].A matrix entry of 1 indicates that a vertex participates in a hyperedge.Source: §3.3 · p. 15
  • Unlike simplicial complexes, hypergraphs need not include every subset of a hyperedge, so a smaller relation’s place within larger relations can be ambiguous [187], trading flexibility for fewer formal analysis methods.For example, a pairwise relation within two larger hyperedges does not reveal whether it is a sub-relation of either, both, or neither.Source: §3.3 · p. 15
  • Applications include social networks, Enron email, yeast interactions, neurodevelopmental motifs [84], evolving systems [17] [52] [53], and biochemical failure; many methods mimic graph analysis.The flexibility of hypergraphs for modeling polyadic relations helped make them appealing in systems originally studied with graph theory.Source: §3.3 · p. 15

3.4 Variations

Researchers extend graphs, simplicial complexes, and hypergraphs in varied ways to encode more information about a system. The useful variations depend on the system and research question.

Directed

  • Directed formalisms represent ordered relations: edges in graphs, orientations in simplicial complexes, and source-to-sink hyperarcs in hypergraphs.The passages describe applications ranging from workflows and cardiac excitation to directed synapses, migration, and biological pathways.Source: Directed · p. 16

Weighted

  • Weighted representations assign interaction values to relations; unlike weighted hypergraphs, weighted simplicial complexes require each face to weigh at least as much as its superset.Graph weights can represent interaction strength or frequency, while weighted simplicial complexes are often studied using persistent homology.Source: Weighted · p. 16

Dynamic

  • Dynamic graphs, simplicial complexes, and hypergraphs represent changes over time on a shared vertex set, with tools for evolving topology and graph communities.The authors found few applied dynamic hypergraphs at the time of writing, but suggest them for evolving gene interactions, brain-region relations, and social groups.Source: Dynamic · p. 17

Higher Order Networks

  • Higher-order networks encode frequent paths or transitions as nodes, capturing temporal dependencies that a physical graph can miss.In the subway example, paths through D and E split into distinct nodes because observed continuations differ; the representation changes if the observed routes change.Source: Higher Order Networks · pp. 17–18

Further variations

  • Variations can be combined, and representations can also include weighted nodes, internally structured nodes, or coupled dynamical systems.Examples include edge-weighted dynamic networks and directed multilayer networks.Source: Further variations · p. 18

Other Formalisms

  • Other formalisms include graphons, metapopulation models, random sequences of sets, and sheaves, each with distinct complex-systems applications.The cited examples include estimating large, noisy systems with graphons and framing network coding or sensor-network consensus with sheaves.Source: Other Formalisms · p. 18

3.5 Encoding system dependencies

Choosing how to define relations is essential to whether a representation captures the system’s dependencies or introduces properties from data externalities. Representation choices must account for the dependencies present, omitted, or externally induced, and for interactions among multiple dependencies.

3.5 Encoding system dependencies

  • Relation definitions determine whether representations capture intended dependencies or introduce artifacts, making relations, formalisms, and dependencies interdependent.For example, brain-region co-activation can encode a subset dependency, while alternative relation definitions may not; bus-route relations likewise yield different representations, and may omit temporal dependencies.Source: §3.5 · pp. 18–19, Figure 7 · p. 19

Encoding subset dependencies

  • Use simplicial complexes when every subset of a relation is also related, and hypergraphs when only some subsets are related; their different sub-relation semantics affect interpretation.In simplicial complexes, a face is necessarily a sub-relation; in hypergraphs, whether one hyperedge is a sub-edge of another is not specified. [187]Source: Encoding subset dependencies · p. 20

Encoding temporal dependencies

  • Higher Order Networks encode temporal dependencies by making nodes from ordered sets of units, but extending them to simplicial complexes or hypergraphs remains an open question.HONs are defined for paths on graphs; extensions that also capture arbitrary subset or spatial dependencies have yet to be established.Source: Encoding temporal dependencies · p. 20

Encoding spatial dependencies

  • Edge weights can encode spatial proximity, but when they also carry traffic or road-length data, researchers must aggregate information, use multilayer networks, or seek a holistic representation.Systems may combine multiple dependencies, such as temporal and spatial constraints in subway networks; methods for systematically encoding multiple dependencies are still needed.Source: Encoding spatial dependencies · p. 21

4 Mathematical relationships between formalisms

Graphs, simplicial complexes, and hypergraphs can be translated into one another, but each translation may add assumptions or discard information. Moving up from a graph through a simplicial complex to a hypergraph and back recovers the graph, whereas translating an independently specified hypergraph through the formalisms does not recover the original hypergraph.

From hypergraph to simplicial complex: Forgetting independent sets

  • A hyperedge on k + 1 nodes becomes a k-simplex, imposing downward closure and treating subset hyperedges as redundant.The simplicial complex retains maximal hyperedges but can lose independently observed subset relations and the total number of relations involving a node; hypergraphs allow methods unavailable for simplicial complexes.Source: From hypergraph to simplicial complex:… · pp. 21–22, Figure 9 · p. 22

From simplicial complex to graph: Forgetting polyadic relations

  • A simplicial complex becomes a graph by connecting nodes that share a simplex, which erases whether a fully connected triple also had a 2-simplex.Directly connecting two nodes whenever they share a hyperedge produces the same graph as translating through a simplicial complex first.Source: From simplicial complex to graph:… · pp. 22–23

From graph to simplicial complex: Assuming polyadic relations

  • A graph's cliques can be filled with simplices to form a clique or flag complex, assuming every clique is a jointly related functional unit.That assumption may fail, as when three people converse only in pairs and never together as a group.Source: Figure 9 · p. 22, From graph to simplicial complex:… · p. 23

From simplicial complex to hypergraph: Assuming that only maximal simplices are independent

  • A simplicial complex can yield a hypergraph with hyperedges for every subset of each simplex, or only for maximal simplices as a conservative choice.The maximal-simplices approach assumes the fewest hyperedges needed to preserve polyadic relations with the most nodes.Source: Figure 9 · p. 22, From simplicial complex to hypergraph:… · p. 23

From hypergraph to graph, and from graph to hypergraph

  • A graph survives translation to a simplicial complex, then a hypergraph, and back, but an independently specified hypergraph cannot be recovered after translating down to a graph and back up.Independent hyperedges can be lost, including those between subsets of a larger hyperedge; the reverse translation therefore yields a different hypergraph.Source: Figure 9 · p. 22, From hypergraph to graph, and from… · p. 23
  • Translating between formalisms is optional and requires caution because translations add assumptions or forget relations and independencies.Other protocols are possible, including keeping only a graph's edges in a simplicial complex or assigning graph edge weights from hypergraph structure.Source: §4 · p. 21, From hypergraph to graph, and from… · p. 23, Figure 10 · p. 24

5 Methods suitable for each representation

Each representation supports distinct analyses and perspectives, so methods should be selected to suit the system rather than letting available methods dictate its representation. Interpretation should also account for system dependencies and the distinct structures each formalism encodes.

5 Methods suitable for each representation

  • Methods that currently use only one formalism do not justify choosing it when that representation is ill-suited to the system.For example, existing epidemic-threshold methods use graphs, while methods for polyadic formalisms remain nascent; that gap alone does not warrant a graph representation [94] [96].Source: §5 · p. 24

5.1 Methods for graphs

  • Graph methods span local, meso-scale, and global analyses, including degree, clustering, paths, centrality, modularity, small-worldness, and core-periphery structure.Degree distributions can constrain large-scale organization, while degree and clustering describe local structure; many network statistics are also correlated across graph-ensemble instances and can distinguish system types.Source: §5.1 · pp. 24–25

5.2 Methods for simplicial complexes

  • Simplicial-complex methods extend graph measures through simplex participation and simplex-level clustering, and use homology to detect topological voids.Downward closure identifies which simplices belong to higher-dimensional simplices, making homology especially suited to this formalism.Source: §5.2 · pp. 25–26

5.4 Methods and dependencies

  • Graph, simplicial-complex, and hypergraph analyses respectively suggest core-periphery structure, a global circular organization, and two communities in the same example system.The paper emphasizes that formalism choice shapes structural interpretation and recommends careful representation and analysis choices.Source: Figure 12 · p. 26, §5.4 · pp. 28–29

5.2 Methods for simplicial complexes

  • A Dowker dual swaps nodes and relations in a concurrence matrix, potentially reducing computational burden while preserving isomorphic homology groups.This can be more tractable when the original complex has many nodes but relatively few relations, provided the representation still suits the scientific question.Source: §5.2 · pp. 26–27

5.3 Methods for hypergraphs

  • Hypergraph methods define vertex hyperdegree and hyperedge cardinality, treating each hyperedge as an independent relation rather than stratifying vertex degree by cardinality.Consequently, related computations on hypergraphs and simplicial complexes must be interpreted differently.Source: §5.3 · p. 27
  • Hypergraphs can represent absent substructures, such as missing three-node relations within a four-node hyperedge, and the fill coefficient measures the fraction of smaller hyperedges present.Graphs and simplicial complexes cannot support the same question because they lack polyadic relations or require downward inclusion, respectively.Source: §5.3 · p. 28

5.4 Methods and dependencies

  • Analyses and interpretations should account for dependencies, including subset dependencies or interactions between structure and geography.The paper cites Rentian scaling and modularity maximization with spatial null models as examples of methods that intertwine dependencies.Source: §5.4 · p. 28

6 Examples

The examples show that system dependencies, available data, research questions, and representation choices shape what can be measured and concluded. Even with the same data and research question, different formalisms can produce conflicting results.

6.1 Coauthorship

  • Coauthorship data may obscure spatial dependencies, and author-set data alone supports graph or simplicial-complex construction but not a hypergraph that records each paper as one relation.A hypergraph is inappropriate when the author-only system has a subset dependency, since a polyadic relation implies its dyadic sub-relations.Source: §6.1 · pp. 29–30
  • In the coauthorship toy example, graph, simplicial-complex, and hypergraph analyses gave different accounts of author collaboration, with the hypergraph revealing exclusive groups not recoverable from the other representations.For example, it distinguishes a1’s individual project with a2 and team project including a3 from a4’s two-person papers; graph degree also makes a2 and a3 appear equally collaborative, unlike hypergraph degree.Source: §6.1 · pp. 30–31, Figure 15 · p. 31
  • Converting a coauthorship graph into a clique complex and then a hypergraph can create a different structure from one built from the raw data, so translations must preserve the original data’s assumptions.In the toy example, this conversion made author pairs appear to contribute equivalently across representations.Source: §6.1 · pp. 31–32, Figure 15 · p. 31
  • In the DBLP dataset, simplicial-complex and hypergraph degree rankings correlated most, while graph–hypergraph correlation fell below 0.5 in some years.The dataset contains 3,700,681 articles from 2000–2016 and 1,930,378 authors; agreement across representations was higher for high-degree than low-degree nodes, warranting caution when interpreting small-degree nodes.Source: §6.1 · p. 32, Figure 16 · p. 33

6.2 Email communications

  • For Ana’s email analysis, a team comprises 5–25 people receiving the same email simultaneously; graph- and hypergraph-based average clustering measures might correlate, but subtle definitional differences lead to varying results.This concerns past aggregate communication, with no spatial dependencies and no need to incorporate temporal dependency.Source: §6.2 · pp. 32–33, Figure 17 · p. 34
  • r = 0.1 (p = 0.16) was the Spearman rank correlation coefficient between team rankings from graph and hypergraph clustering, and the rankings differed across teams of at least 5, 6, 7, or 8 people.Choosing either representation could therefore lead Ana to allocate resources to different teams.Source: §6.2 · pp. 33, 35, Figure 17 · p. 34

7 Applications

Applications in brain, transportation, and cellular systems show how system-specific dependencies and data constraints shape representation choices. The section identifies underrepresented dependencies and opportunities for richer encodings.

Brain

  • Brain analyses span graphs, simplicial complexes, and hypergraphs across scales, while temporal and spatial dependencies merit more direct representation.Temporal dependencies may constrain future activity based on past activity; multilayer representations or sheaves may encode spatial position directly.Source: §7 · p. 35

Transportation

  • Transportation studies often encode spatial and temporal effects as distance or travel-time weights, while higher-order representations can capture temporal dependencies.Subset dependencies are less studied; investigating their identification or inclusion may improve failure prevention or activity prediction, and simplicial complexes or hypergraphs may elucidate polyadic relations.Source: §7 · p. 36

Cellular systems

  • Cellular interaction networks are usually reconstructed from indirect measurements, and uncertain relations could be represented probabilistically and compared across ensembles of graphs, simplicial complexes, or hypergraphs.Temporal and spatial information can aid reconstruction; polyadic relations are known to matter but are less often encoded with simplicial complexes or hypergraphs, while increasing access to temporal data offers opportunities to study temporal dependencies.Source: §7 · p. 36

8 Discussion and Conclusion

The paper surveys how dependencies, representations, and analysis methods shape complex-systems pipelines. It argues that pipelines should be tailored to each system and dataset rather than adopted simply because they are customary.

Discussion and Conclusion

  • The paper examines system dependencies, formalism assumptions and relationships, information loss during conversion, and analysis examples to show why representation and methods matter.Dependencies include temporal, subset, and spatial forms; conversion between formalisms can lose or impute information.Source: §8 · p. 36
  • There is no perfect analysis pipeline: modeling decisions for one system or dataset may not carry over even to another dataset from the same system.The authors caution against choosing pipelines merely because they are common within a field.Source: §8 · pp. 36, 38
  • Design pipelines around the system, dependencies, data-collection effects, and study fragment, then choose fitting formalisms and methods because earlier choices can bias results.The authors recommend respecting dependencies and system qualities through appropriate representation and analysis to uncover insight.Source: Figure 18 · p. 37, §8 · p. 38
  • The authors hope continued collaboration will bring complex-systems fields closer together, identify gaps in current formalisms, and enable more insightful analyses and novel techniques.They envision researchers using multiple representations and carefully chosen computations that respect system dependencies.Source: §8 · p. 38

10 Citation diversity statement

The authors sought to choose references reflecting diversity in thought, contribution, gender, and other factors, against evidence that women and other minorities are under-cited. They assessed reference-author gender using pronouns and gendered-name probabilities, while noting limitations of both methods.

10 Citation diversity statement

  • Prior work identified under-citation of papers by women and other minorities relative to their representation in fields [63] [34].The authors say they sought to choose references reflecting diversity in thought, contribution, gender, and other factors.Source: §10 · p. 38
  • Pronoun-based estimates found 45% man(first)/man(last), 12% man/woman, 11% woman/man, 13% woman/woman, 0% non-binary, and 19% unknown categorization.The method used pronouns affiliated with authors online or known through personal friendships, excluding self-citations to the current paper’s first and last authors; pronouns may not indicate gender identity or remain consistent across contexts or time.Source: §10 · p. 38
  • Gendered-name estimates found 60% man/man names, 12% man/woman names, 11% woman/man names, 10% woman/woman names, and 7% unknown categorization [63].The method used databases estimating the probability that a name is carried by a woman, but cannot account for intersex, non-binary, or transgender people; the authors call for future work supporting equitable scientific practices.Source: §10 · p. 39

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