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Cited by 160Complex Network Analysis TechniquesComplex Systems and DynamicsBioinformatics and Genomic Networks

On the origins of hierarchy in complex networks

Open access
Corominas-Murtra, Bernat, Goñi, Joaquín, Solé, Ricard V., Rodríguez-Caso, Carlos·2013·Proceedings of the National Academy of Sciences
DOI 10.1073/pnas.1300832110PMC PMC3746874
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Abstract

Hierarchy seems to pervade complexity in both living and artificial systems. Despite its relevance, no general theory that captures all features of hierarchy and its origins has been proposed yet. Here we present a formal approach resulting from the convergence of theoretical morphology and network theory that allows constructing a 3D morphospace of hierarchies and hence comparing the hierarchical organization of ecological, cellular, technological, and social networks. Embedded within large voids in the morphospace of all possible hierarchies, four major groups are identified. Two of them match the expected from random networks with similar connectivity, thus suggesting that nonadaptive factors are at work. Ecological and gene networks define the other two, indicating that their topological order is the result of functional constraints. These results are consistent with an exploration of the morphospace, using in silico evolved networks.

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AI snapshot

Written from the paper’s full text.

TL;DR

The paper defines a 3D morphospace of hierarchy using treeness, feedforwardness, and orderability, then maps 125 real networks and null models into it. It finds four major clusters, with two matching random-network expectations and two—ecological and gene networks—apparently shaped by functional constraints.

Key findings

  • The morphospace Ω is defined by coordinates \(u(G)=(T,F,O)\), with perfect hierarchy at \((1,1,1)\) and a totally cyclic network at \((0,0,0)\); random homogeneous and broad networks occupy basically the same region, within the bow-tie plane \(T(G)=0\).
  • Among 125 real networks across 13 classes, most fall into four clusters: metabolic, neural, linguistic, and some social networks lie in the lower bow-tie domain within the random-graph cloud; electronic circuits and software graphs form a feedforward band with \(-0.2<T(G)<0.2\); gene regulatory networks plus a kinase network show slightly positive \(T(G)\) and high \(O\); ecological flow graphs cluster around \(u(G)=(0.35,0.45,0.25)\).
  • Isolated extremes include a cell lineage at \(u(G)=(0.95,1,1)\) and a small social network at \(u(G)=(-1,0.06,0.09)\).
  • In silico evolution over 300 target points, three \(O\) values \(\{0.15,0.5,0.85\}\), and \(10^3\) generations showed some regions are accessible while others are inaccessible, suggesting non-adaptive processes may play a major role in shaping hierarchies.

Methods

  • Directed graphs are condensed by collapsing strongly connected components into a node-weighted condensed graph \(G_C\); hierarchy is quantified by treeness \(T\) from forward/backward entropies over DAGs and leaf-removal subgraphs, feedforwardness \(F\) from path weights of cyclic modules, and orderability \(O\) as the fraction of nodes not in any cycle.
  • The study analyzes 125 real networks from 13 classes and compares them with directed Erdős–Rényi, Barabási–Albert, and Callaway model ensembles; randomization uses two methods preserving either the undirected degree sequence and component structure or the directed degree sequence and component structure, with 100 replicas per network and \(4|E|\) link switches or \(20|E|\) trials.
  • An evolutionary search starts from small random graphs and selects graphs minimizing Euclidean distance to target points in Ω via random link addition and deletion, with 250 replicas per target point and up to 1000 generations.

Limitations

  • The paper states that the study uses a static view dominated by causal relations and does not include functionality, dynamics, or weighted structures; these are left for further work.
  • The paper notes that the evolutionary algorithm’s parameters may shape the observed fitness landscape, that the experiment has high computational cost, and that small network sizes provide only a coarse-grained picture of reachability.
  • The formalism assumes connected directed graphs; the Appendix states that unconnected components have no causal relation and that hierarchy loses meaning across them. Exact analytical estimates are not derived for \(F\), and the \(T\) and \(O\) random-graph estimates rely on simplifying assumptions.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1073/pnas.1300832110

Cites

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Statistics

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Claims the paper makes

0 of 48 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § IV. NULL MODELS AND REAL NETWORK ANALYSIS · p. 4

    Metabolic networks display a large central cycle, much larger than that observed in their randomized counterparts.

    “They display a large central cycle, much larger than that observed in their randomized counterparts.”
    Quote in the paperNot a substantial claim
  • Findingown data
    § IV. NULL MODELS AND REAL NETWORK ANALYSIS · p. 4

    A first group consisting of metabolic, neural, linguistic and some social networks is found at the lower part of the bow-tie domain, embedded within the cloud of random graphs.

    “First, a group consisting of metabolic, neural, linguistic and some social networks is found at the lower part of the bow-tie domain, clearly embedded within the cloud of random graphs (Fig. 2d) .”
    Quote in the paperSubstantial claim
  • Findingown data
    § On the origins of Hierarchy in Complex Networks · p. 1

    Ecological and gene networks define the other two of the four major groups.

    “Ecological and gene networks define the other two, indicating that their topological order is the result of functional constraints.”
    Quote in the paperSubstantial claim
  • Findingown data
    § IV. NULL MODELS AND REAL NETWORK ANALYSIS · p. 4

    Most real networks fall into four clusters.

    “However, most networks fall into four clusters.”
    Quote in the paperSubstantial claim
  • Findingown data
    § On the origins of Hierarchy in Complex Networks · p. 1

    Two of the four major groups match the expected from random networks with similar connectivity.

    “Two of them match the expected from random networks with similar connectivity, thus suggesting that non-adaptive factors are at work.”
    Quote in the paperSubstantial claim
  • May combine two claims. 31
  • Classified as probably not a substantial claim. 44
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    Findingown data
    § IV. NULL MODELS AND REAL NETWORK ANALYSIS · p. 6

    A third group displays slightly positive values of T(G) and is composed of graphs with cycles of small size but with a predominant position at the feedforward structure giving rise to a very high O with variable F(G); these are gene regulatory networks plus a protein kinase network.

    “The third group displays slightly positive values of T ( G ) and is composed of graphs with cycles of small size but with a predominant position at the feedforward structure giving rise to a very high O with variable F ( G ). These are gene regulatory networks plus a protein kinase network.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given
    § On the origins of Hierarchy in Complex Networks · p. 1

    The results are consistent with an exploration of the morphospace using in silico evolved networks.

    “These results are consistent with an exploration of the morphospace using in silico evolved networks.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § IV. NULL MODELS AND REAL NETWORK ANALYSIS · p. 4

    A second group placed at the O(G)≈1 plane shows a narrow band of feedforward nets including electronic circuits with -0.2<T(G)<0.2 and software graphs slightly biased to negative T(G) values.

    “The second group placed at the O ( G ) ≈ 1 plane shows a narrow band of feedforward nets including electronic circuits with -0 . 2 &lt; T ( G ) &lt; 0 . 2 and software graphs slightly biased to negative T ( G ) values.”
    Quote in the paperNot a substantial claim

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    Findingown data
    § V. MORPHOSPACE ACCESSIBILITY BY DRIVEN EVOLUTION · p. 7

    The most interesting results concern the presence of inaccessible regions, labeled in dark red color.

    “However, the most interesting results here concerns the presence of inaccessible regions, labeled in dark red color.”
    Quote in the paperSubstantial claim

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    Findingown data
    § V. MORPHOSPACE ACCESSIBILITY BY DRIVEN EVOLUTION · p. 7

    At O=0.5, a large region of high reachability is observed for extreme values of T but is not occupied by real networks.

    “Here, a large region of high reachability is observed for extreme values of T but, not occupied by real networks.”
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    Findingnone given
    § 3. The accessibility of the Morphospace: Evolution · p. 28

    It is easier for small networks to display extreme configurations.

    “since, for example it is easier for small networks to display extreme configurations.”
    Quote in the paperNot a substantial claim

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    Findingown data
    § c. Confronting real data with their randomized counterparts · p. 27

    In most cases, real networks avoid the general cyclic character observed in randomized networks; randomizations present a larger cyclic character than real networks.

    “In most cases real data appears to avoid the general cyclic character observed in randomized networks.”
    • Also stated in: § c. Confronting real data with their randomized counterparts · p. 27 · hedged
      “In general, such a trend presents a larger cyclic character in randomizations than in real networks.”
    Quote in the paperSubstantial claim

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    Findingown data
    § c. Confronting real data with their randomized counterparts · p. 27

    Food webs and most GRNs show positive biased T values.

    “Positive biased values are observed in food webs and most of GRNs.”
    Quote in the paperSubstantial claim

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    Findingown data
    § c. Confronting real data with their randomized counterparts · p. 27

    Electronic circuits show an anti-hierarchical configuration (T < 0), with a tendency to occupy slightly negative, but significant, values of T.

    “On the contrary, electronic circuits show an anti hierarchical configuration ( T &lt; 0), with a tendency to occupy slightly negative, but significant, values of T .”
    Quote in the paperNot a substantial claim

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    Findingown data
    § c. Confronting real data with their randomized counterparts · p. 27

    When compared to randomized ensembles obtained using Method B, the trend for metabolisms and word corpora is similar to the common values observed in randomized versions.

    “However, when compared to randomized ensembles obtained using method b , the trend is similar to the common values observed in randomized versions.”
    Quote in the paperNot a substantial claim

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    Findingown data
    § 1. Model Networks · p. 23

    As connectivity decreases from high values, O starts to increase but networks still aggregate around T approximately 0 and F approximately 0.

    “Departing from the value O ≈ 0 displayed at high connectivities, as connectivity decreases, O starts to increase but nets still aggregate around T ≈ 0 and F ≈ 0.”
    Quote in the paperNot a substantial claim

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    Findingown data
    § 3. The accessibility of the Morphospace: Evolution · p. 28

    Not all regions of Ω are equally accessible; some regions are fairly improbable to be achieved.

    “And indeed, thanks to this approximation, we showed that not all regions are equally accessible but, instead, some regions are fairly improbable to be achieved.”
    Quote in the paperSubstantial claim

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    Findingreasoning
    § 1. Model Networks · p. 23

    For low connectivities, T values tend to be closer to T = 0, but both F and O increase to reach the region around T = 0, F = 1, O = 1.

    “For low connectivities, T values tend to be closer to T = 0 but both F and O change their behavior, increasing their values to reach the region around the point T = 0 , F = 1 , O = 1.”
    Quote in the paperSubstantial claim

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    Findingown data
    § b. O in random directed graphs · p. 22

    For directed Erdos-Renyi graphs, the theoretical prediction from equation (D18) fits real data well and the behavior of the orderability coordinate O can be clearly predicted from that equation.

    “In figure (16) we plot estimates of equation (D18) and real values from an ensemble of Erdos-Renyi graph and we see that, despite the strong assumptions made, the behavior of O can be clearly predicted from the mentioned equation, providing a good insight to the behavior of such coordinate.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: Small SCCs that can also be present within the graph represent a negligible fraction of it when compared to the size of the GSCC.
    • The authors state this as an assumption: There is no correlation between in-degree and out-degree, p(k_in, k_out) approximately equals p(k_in)p(k_out).

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Premisecitation
    § I. INTRODUCTION · p. 1

    A morphospace is a phenotype space where a small set of quantitative traits can be defined as axes.

    “Awell established concept where such questions are addressed involves the use of a morphospace [32-36] namely a phenotype space where a small set of quantitative traits can be defined as the axes.”
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    Premisecitation
    § I. INTRODUCTION · p. 1

    Hierarchy pervades a coherent form of organization that allows reducing costs associated to reliable information transmission and supports efficient genetic and metabolic control in cellular networks.

    “Hierarchy seems to pervade a coherent form of organization that allows reducing the costs associated to reliable information transmission [39] and to support efficient genetic and metabolic control in cellular networks [40].”
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    § On the origins of Hierarchy in Complex Networks · p. 1

    Hierarchy is a pervasive feature of the organization of natural and artificial systems.

    “Hierarchy seems to pervade complexity in both living and artificial systems.”
    • Also stated in: § I. INTRODUCTION · p. 1 · hedged
      “Hierarchy seems a pervasive feature of the organization of natural and artificial systems [2, 3].”
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    Interpretationreasoning
    § IV. NULL MODELS AND REAL NETWORK ANALYSIS · p. 4

    Random graphs account for both bow-tie structures and feed-forward sparse webs.

    “That implies, among other things, that they account for both bow-tie structures and feed-forward sparse webs.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    Hierarchy is a polysemous word involving order, levels, inclusion or control as possible descriptors, none of which captures its complexity or the problem of its measure and origins.

    “But hierarchy is a polysemous word, involving order, levels, inclusion or control as possible descriptors [21], none of which captures either its complexity or the problem of its measure and origins.”
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    Premisecitation
    § I. INTRODUCTION · p. 1

    Previous work using complex networks theory has quantitatively tackled the problem of hierarchy.

    “Although previous work using complex networks theory has quantitatively tackled the problem [3, 4, 22-25, 27-31], some questions remain:”
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    Interpretationnone given
    § I. INTRODUCTION · p. 1

    Deviations from the ideal tree picture of hierarchy occur because several elements are on the top, downstream elements interact horizontally, or feedback loops are present.

    “Such deviations occur because: (a) several elements are on the top, (b) downstream elements interact horizontally or (c) feedback loops are present.”
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    Premisecitation
    § II. THE COORDINATES OF HIERARCHY · p. 1

    SCC detection has been shown to be a powerful approach for subsystem identification, unraveling the presence of nested organizations.

    “It is worth to mention that SCC detection has been shown as a powerful approach for subsystem identification [15, 42, 43] unraveling the presence of nested organizations.”
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    Interpretationreasoning
    § III. THE DEFINITION OF THE MORPHOSPACE Ω · p. 4

    While O(G) and F(G) converge in their upper bound in a single value that defines the region of feedforward networks, they differ when O goes to zero.

    “It is worth to stress that, while O ( G ) and F ( G ) converge in their upper bound in a single value that defines the region of feedforward networks, they differ when O goes to zero.”
    Quote in the paperSubstantial claim

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    Premisecitation
    § I. INTRODUCTION · p. 1

    In most real systems, signal integration requires gathering inputs from different sources, while robust processing and control requires crosstalk and feedbacks, often organized in a modular fashion.

    “In most real systems, signal integration requires gathering inputs from different sources, while robust processing and control requires crosstalk and feedbacks, as represented by Fig. (1d), which are often organized in a modular fashion [38].”
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    § I. INTRODUCTION · p. 1

    Herbert Simon defined complex systems as nested hierarchical networks of components organized as interconnected modules.

    “Fifty years ago [1], Herbert Simon defined complex systems as nested hierarchical networks of components organized as interconnected modules.”
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    Interpretationreasoning
    § 1. Orderability , O · p. 14

    The Orderability O of a graph G is a raw indicator of the fraction of the net that can be ordered.

    “This is a raw indicator of the fraction of the net that can be ordered.”
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    Interpretationnone given
    § VI. CONCLUDING REMARKS · p. 7

    Some of these constraints result from the underlying rules of network growth and change, which strongly limit the repertoire of potential designs.

    “Some of them result from the underlying rules of network growth and change, which strongly limit the repertoire of potential designs.”
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    Study design and conductnone given
    § 1. Directed Graphs · p. 11

    The randomization methods used respect the structure of components of the graph (i.e., connectedness if present), although connectedness may be lost during a randomization process.

    “It is worth to note that, connectedness may be lost during a randomization process but, as we shall see in section E 2, the randomization methods we use respect the structure of components of the graph -i.e., connectedness if it be.”
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    Interpretationreasoning
    § Appendix B: The fundamentals of hierarchy (Postulates of Hierarchy) · p. 10

    The perfect hierarchical system is represented by a directed tree with a single root in which all leafs have the same length.

    “It is straightforward to realize that the directed, inverted tree in which all the arrows go downwards starting from a single root naturally emerges as the graphical description of the perfect hierarchical system -see fig. (5).”
    • Also stated in: § 2. Directed Acyclic Graphs · p. 12 · plain
      “Lemma : Under the hallmark described (see Postulates, section B), the kind of graph G depicting the perfect hierarchy is a directed tree in which all the leafs have the same length.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A perfectly hierarchical system has the properties of order, reversibility, and pyramidal structure.

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    Interpretationnone given
    § b. O in random directed graphs · p. 21

    The giant strongly connected component represents the largest cyclic region of the graph.

    “The GSCC will represent the largest cyclic region of the graph.”
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    Interpretationreasoning
    § 2. Directed Acyclic Graphs · p. 11

    A directed acyclic graph can represent a more or less entangled structure of interconnected causal processes.

    “This sorting implies that a DAG can represent a more or less entangled structure of interconnected causal processes.”
    Quote in the paperSubstantial claim

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    Interpretationnone given
    § Proof · p. 18

    Meaningful information can only be obtained from the study of a single connected component because it represents a unique causal structure.

    “However, meaningful information can only be obtained from the study of a single component since it represents a unique causal structure.”
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    Interpretationnone given
    § 2. Feedforwardness , F · p. 15

    The combination of the orderability coordinate O(G) and the feedforwardness coordinate F(G) indicates how cycles are located within the network and what impact they have on the order of the structure.

    “The combination of O ( G ) and F ( G ) tells us how are cycles located within the net and the impact they have in the order of the structure.”
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    Premisecitation
    § c. Dissection of the layers of the graph · p. 13

    Layers of a directed acyclic graph can be identified through a backward or bottom-up leaf removal (LR_b) algorithm or through a forward or top-down leaf removal (LR_f) algorithm.

    “Layers can be identified through a backward or bottom up leaf removal ( LR b ) algorithm or through a forward or top down leaf removal ( LR f ) algorithm [33].”
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    Interpretationreasoning
    § 2. Directed Acyclic Graphs · p. 12

    The directed tree with all leafs the same length defined above satisfies all the postulates of hierarchy and is the graphical representation of a perfect hierarchical system.

    “It turns out that the type of graph defined above satisfies all the postulates defined in B is the graphical representation of a perfect hierarchical system.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A perfectly hierarchical system has the properties of order, reversibility, and pyramidal structure.

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    Premisecitation
    § 3. Treeness, T · p. 15

    Deviations from reversibility can be captured by measuring the uncertainty in reversing a given path, a measure called topological reversibility.

    “Intuitively, we first observe that deviations from reversibility can be properly captured measuring the uncertainty in reversing a given path, a measure called topological reveresibility [3].”
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    Interpretationown data
    § VI. CONCLUDING REMARKS · p. 7

    The matching of random and real webs in the first two clusters suggests that their hierarchical features can be accounted for from the spontaneous correlations associated to random graphs of a given degree, indicating that the observed webs are simply the most probable ones.

    “The matching of random and real webs in the first two clusters suggests that their hierarchical features can be accounted for from the spontaneous correlations associated to random graphs of a given degree, indicating that the observed webs are simply the most probable ones.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The region occupied by null models indicates where no selection pressure is at work.

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    Interpretationreasoning
    § 1. Orderability , O · p. 14

    A tree or, more generally, a feed-forward network will display O(G) = 1.

    “A tree or, more generally, a feed-forward network will display O ( G ) = 1, in agreement to their orderable nature.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § a. Condensation: Obtaining a DAG from any directed graph · p. 12

    The condensed graph G_C of any directed graph G is a directed acyclic graph.

    “It turns out that, by definition, G C is a DAG .”
    • Also stated in: § a. Condensation: Obtaining a DAG from any directed graph · p. 12 · plain
      “Notice, as illustrated in figure (6) that G C is a DAG , as expected.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § VI. CONCLUDING REMARKS · p. 7

    The computational nature of regulatory networks and the combination of layers and cycles common to energy flows in food webs separate them from the large cluster.

    “The computational nature of regulatory networks and the combination of layers and cycles common to energy flows in food webs separate them from this large cluster.”
    Quote in the paperSubstantial claim

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    Interpretationnone given
    § c. Dissection of the layers of the graph · p. 13

    Although the LR_f and LR_b algorithms identify the same number of layers, the specific composition of every layer is not necessarily the same.

    “Although LR f and LR b algorithms identify the same number of layers, the specific composition of every layer is not necessarily the same.”
    Quote in the paperSubstantial claim

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    Interpretationreasoning
    § 1. Orderability , O · p. 14

    A network which is totally cyclical would display O(G) = 0.

    “In contrast, a network which is totally cyclical would display O ( G ) = 0.”
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    Interpretationcitation
    § 2. Directed Acyclic Graphs · p. 12

    The length of the longest path of a finite DAG is always bounded.

    “We observe that the length of the longest path of a finite DAG is always bounded.”
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  • Classified as probably not a substantial claim.

    Interpretationnone given
    § c. Dissection of the layers of the graph · p. 13

    The non-cyclic nature of a directed acyclic graph enables the definition of a finite set of layers.

    “The non-cyclic nature of such graph structures enables us to define a finite set of layers.”
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    Premisecitation
    § VI. CONCLUDING REMARKS · p. 7

    Both biological and cultural evolution operate under a number of deep constraints.

    “Both biological and cultural evolution operate under a number of deep constraints [60].”
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    Interpretationnone given
    § Proof · p. 18

    It is mathematically possible to apply the hierarchy formalism to networks containing more than one connected component.

    “It is worth to note that it is mathematically possible to apply this formalism to networks containing more than one connected component.”
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    Interpretationreasoning
    § Appendix D: Detailed derivation of the Coordinates of Hierarchy · p. 14

    The values of the three hierarchy coordinates allow identification and quantification of deviations from ideal hierarchical behavior.

    “Through the values of these coordinates we must be able to identify and properly quantify deviations from such an ideal behavior.”
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  • Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

    Interpretationnone given
    § c. Dissection of the layers of the graph · p. 13

    The dissection of G_C in the two sequences of subgraphs will enable exhaustive exploration of the role of all layers in the further hierarchy measure.

    “The dissection of G C in the above described two sequences of subgraphs will enable us to exhaustively explore the role of all layers in the further hierarchy measure.”
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    Interpretationreasoning
    § Appendix D: Detailed derivation of the Coordinates of Hierarchy · p. 14

    The three hierarchy coordinates allow extraction of information about the structure of a network under the conceptual background of hierarchy based on order, reversibility and pyramidal structure.

    “From the coordinates of u ( G ) we can extract information about the structure of the net under the conceptual background of hierarchy presented in section B based on the concepts of order , reversibility and pyramidal structure .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A perfectly hierarchical system has the properties of order, reversibility, and pyramidal structure.

  • May combine two claims.

    Interpretationown data
    § IV. NULL MODELS AND REAL NETWORK ANALYSIS · p. 6

    The T(G)>0 values of the fourth group indicate a certain degree of pyramidal structure, and the low O(G) values are consistent with an important role played by loops.

    “Their T ( G ) &gt; 0 values indicate a certain degree of pyramidal structure and the low O ( G )'s are consistent with an important role played by loops.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § 3. Treeness, T · p. 15

    If the quantification of the uncertainty in reversing paths is lower than the quantification of alternatives, the network qualitatively shows a pyramidal structure.

    “If the quantification of the uncertainty in reversing the paths displays a lower value than the quantification of alternatives, we say that, qualitatively, the net shows a pyramidal structure.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § a. T in random directed graphs · p. 20

    Bidirectional links introduce a symmetry within the adjacency matrix under the transpose operation, so their presence would have no impact on equation (D13) and the reasoning for the random-graph result still holds.

    “However, we observe that these bidirectional links already introduce a symmetry within the adjacency matrix under the transpose operation and, consequently, their presence would have no impact in equation (D13) and thus the reasoning still holds.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

    Study design and conductnone given
    § b. Randomization Methods · p. 25

    Randomization Method B preserves the directed degree sequence and the component structure.

    “Method B: Preserving the directed degree sequence and the component structure.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). May combine two claims.

    Study design and conductreasoning
    § b. O in random directed graphs · p. 21

    The independence condition among in-degree and out-degree is very strong, so the results must be seen as a rough estimation of the qualitative behavior of the GSCC.

    “We emphasize that this independence condition among k in and k out is a very strong one, consequently, results must be seen as a rough estimation of the qualitative behavior of the GSCC .”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: There is no correlation between in-degree and out-degree, p(k_in, k_out) approximately equals p(k_in)p(k_out).

  • May combine two claims.

    Interpretationown data
    § c. Confronting real data with their randomized counterparts · p. 27

    Food webs and most GRNs are hierarchic (T > 0) in terms of how the causal flow is organized.

    “Such nets are thus hierarchic ( T &gt; 0), in terms of how the causal flow is organized.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

    Study design and conductnone given
    § c. Confronting real data with their randomized counterparts · p. 26

    Using percentiles in box plots allows eluding assumptions on the statistical behavior of the ensembles of randomized graphs.

    “This chart allows us to elude any assumption on the statistical behavior of the ensembles of randomized graphs.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § 5. Analytical estimates for random graphs · p. 19

    The analysis focuses on T and O because the active presence of paths and cycles in the computation of F makes any rough estimation extraordinarily complex and of no practical use.

    “We will focus on T and O , since the active presence of paths and cycles in the computation of F makes any rough estimation extraordinarily complex and of no practical use.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § 4. The coordinates of hierarchy for directed graphs · p. 17

    The relation between O and F enables estimation of how many cyclic regions are in the graph and at what point of the causal flow they are located.

    “For example, the relation between O and F enables us to estimate how many cyclic regions are in the graph and at what point of the causal flow they are located.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductnone given
    § b. Randomization Methods · p. 25

    Randomization Method A preserves the undirected degree sequence and the component structure.

    “Method A: Preserving the undirected degree sequence and the component structure.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § c. F in random directed graphs · p. 22

    The drastic reduction of F by increasing average degree is tied to the impact of SCC size affecting the length of pathways in the condensed graph.

    “It is not ignored by the authors that the drastic reduction of F by increasing 〈 k 〉 is tied to the impact of SCC size affecting to the length of pathways in the condensed graph.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4. The coordinates of hierarchy for directed graphs · p. 17

    Low values of O and F indicate that the network is mainly cyclical and that the feed-forward-like analysis provided by T is less relevant than when O and F are close to 1.

    “In addition, they provide an estimator of relevance of coordinate T , since low values of O,F indicate us that the net is mainly cyclical and that the feed-forward-like analysis provided by T is less relevant than in the case where O,F are close to 1.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). The quote only partly states this claim.

    Interpretationnone given
    § Proof · p. 18

    The reversed directed tree case belongs to the perfect antihierarchical system.

    “This latter case would belong to the perfect antihierachical system.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductreasoning
    § 5. Analytical estimates for random graphs · p. 19

    Exact calculations of the hierarchy coordinates in random directed graphs are hard and far from the scope of the work, so the analytic results are presented starting from simplifying assumptions to obtain indications for large, sparse random networks.

    “Exact calculations of the above presented values are hard and far away from the scope of this work. We therefore will present the analytic results starting from some simplifying assumptions to obtain indications on what we should expect in the case of large, sparse random networks.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § b. Randomization Methods · p. 25

    The conservation of the directed degree sequence makes Method B slightly more restrictive than Method A.

    “The conservation of σ ( G ) makes this randomization method slightly more restrictive than the other presented above.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § 1. Model Networks · p. 23

    A network with a handful of nodes is more likely to give extreme values than a large network.

    “It is easy to see that a network with a handful of nodes is more likely to give extreme values than large networks.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Study design and conductnone given
    § 3. The accessibility of the Morphospace: Evolution · p. 28

    Parameters of the evolutionary algorithm may be determinant in shaping the observed fitness landscape, thereby accelerating or slowing the convergence process to the desired target point.

    “It is worth to note that parameters of the evolutionary algorithm may be determinant in shaping the observed fitness landscape, thereby accelerating or slowing the convergence process to the desired target point.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 1. Model Networks · p. 23

    For large |V|, the directed Erdos-Renyi model does not ensure connectedness until p approximately equals log(|V|)/|V|.

    “For | V | ≫ 0, this model does not ensure connectedness until values of p ≈ log( | V | ) / | V | [7].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [7]

      Not checked in the free run.

67 of 67

  • The paper relies on this without support.

    No general theory that captures all features of hierarchy and its origins has been proposed yet.

    “Despite its relevance, no general theory that captures all features of hierarchy and its origins has been proposed yet.”

  • The paper relies on this without support.

    Null models do not consider optimal designs or functional constraints.

    “Since null models do not consider optimal designs or functional constraints, they provide good insights about the part of the morphospace where no selection pressure is at work.”

  • The paper relies on this without support.

    The region occupied by null models indicates where no selection pressure is at work.

    “Since null models do not consider optimal designs or functional constraints, they provide good insights about the part of the morphospace where no selection pressure is at work.”

  • The paper relies on this without support.

    In general, neither biological nor technological webs match the feedforward pattern.

    “But, in general, neither biological nor technological webs match the feedforward pattern.”

  • The paper relies on this without support.

    The 125 real networks used are representative of natural and artificial systems.

    “Here we use N n = 125 real networks encompassing 13 classes of natural and artificial systems”

  • The authors state this as an assumption.

    Unconnected components have no causal relation of any kind, and therefore the concept of hierarchy loses any meaning for them.

    “we assume that unconnected components have no causal relation of any kind and, therefore, the concept of hierarchy looses any meaning.”

  • The authors state this as an assumption.

    A perfectly hierarchical system has the properties of order, reversibility, and pyramidal structure.

    “The perfectly hierarchical system will have the following three properties: - order , - reversibility and - pyramidal structure .”

  • The paper relies on this without support.

    Links in a directed graph describe some kind of causal relation between two elements of the set which is considered relevant for the system.

    “In any directed graph, links describe some kind of causal relation between two elements of the set which is considered relevant for the system.”

  • The paper relies on this without support.

    Rigorous and systematic confrontation of real data with their randomized counterparts elucidates much of the possible origins of the observed patterns.

    “It is important to stress that the rigorous and systematic confrontation of real data with their randomized counterparts elucidates much of the possible origins of the observed patterns.”

  • The authors state this as an assumption.

    The ensemble G_C(G) is well behaved, meaning that the averages collapse to the most probable value of a given observable.

    “The first one concerns the assumption that the ensemble G C ( G ) is well behaved , which means that the averages collapse to the most probable value of a given observable.”

  • The authors state this as an assumption.

    Small SCCs that can also be present within the graph represent a negligible fraction of it when compared to the size of the GSCC.

    “if we assume that small SCC s that can be also present within the graph represent a negligible fraction of it when compared to the size of the GSCC”

  • The authors state this as an assumption.

    There is no correlation between in-degree and out-degree, p(k_in, k_out) approximately equals p(k_in)p(k_out).

    “Following [16], we know that, if there is no correlation between in and out -degrees p ( k in , k out ) ≈ p ( k in ) p ( k out ), we can estimate the size of the GSCC as:”
12 of 12

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Evolutionary experiment uses two different network sizes

    Major

    The main text says every graph in the evolutionary search has 50 nodes, while the appendix specifies 25 nodes for each graph in that experiment. Network size is relevant to the reported morphospace-accessibility results.

  • Word-corpora count disagrees with the listed texts

    Minor

    The dataset description calls these four lexical graphs but names six source texts.

  • Method A is described as preserving the wrong degree sequence

    Minor

    The results discussion says method a preserves the directed degree sequence, whereas its definition says it preserves the undirected degree sequence. The paper assigns directed-degree preservation to method B.

3 of 3

Cites

Flagged first, then by citations

2 of the 79 references couldn’t be matched to a known paper.

Cited by

Sorted by citations

Related

Authors
Corominas-Murtra, B., Goñi, J., Solé, R. V., & Rodríguez-Caso, C.
Affiliations
  • Institució Catalana per a la Recerca i Estudis Avancats–Complex Systems Laboratory, Universitat Pompeu Fabra, 08003 Barcelona, Spain;
  • Indiana University Bloomington (US)
  • Institució Catalana de Recerca i Estudis Avançats (ES)
ORCID
Venue
Proceedings of the National Academy of Sciences
Publisher
National Academy of Sciences
Published
2013-07-29
Volume
110
Issue
33
Pages
13316-13321
ISSN
  • 0027-8424 (ISSN-L)
  • 1091-6490
DOI
10.1073/pnas.1300832110
PMC
PMC3746874
OpenAlex
W2141141714
Cited by
160
References
79
Citations by year
2026: 7 · 2025: 10 · 2024: 9 · 2023: 13 · 2022: 12 · 2021: 15 · 2020: 12 · 2019: 12 · 2018: 13 · 2017: 12 · 2016: 23 · 2015: 10 · 2014: 9 · 2013: 3
FWCI
10.67
Type
journal article
Language
en
Open access
Locations
  • Proceedings of the National Academy of Sciences (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • Europe PMC (PubMed Central) (publishedVersion)
  • PubMed Central (publishedVersion)
  • PubMed (publishedVersion)
  • DIGITAL.CSIC (Spanish National Research Council (CSIC)) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Systems and Dynamics
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Bioinformatics and Genomic Networks
Keywords
network hierarchy, complex networks, morphospace, theoretical morphology, network topology, ecological networks, gene regulatory networks, social networks, technological networks, random networks, functional constraints
MeSH terms
Cell Physiological Phenomena, Biological Evolution, Models, Theoretical, Social Support, Ecosystem, Gene Regulatory Networks

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-07.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-07.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

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