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On the origins of hierarchy in complex networks
Open accessAbstract
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.
Checked 2026-10-07 · 1 reference not checked yet
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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.
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69 Own interpretation · 38 Own result · 27 Not checked yet · 13 No basis given · 7 Own method
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“They display a large central cycle, much larger than that observed in their randomized counterparts.”
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“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) .”
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“Ecological and gene networks define the other two, indicating that their topological order is the result of functional constraints.”
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“However, most networks fall into four clusters.”
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“Two of them match the expected from random networks with similar connectivity, thus suggesting that non-adaptive factors are at work.”
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“As shown in Fig. (2a), the larger the SCC , the lower are the values of F and O .”
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“Embedded within large voids in the morphospace of all possible hierarchies, four major groups are identified.”
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“Interestingly, randomized nets of this group show a similar behavior although with a more central position within the cloud of random nets (see Fig. (3).”
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“This co-occupation, within the bow-tie plane with T ( G ) = 0, shows that random graphs appear located right in the middle between hierarchical and antihierarchical structures, independently of the type of degree distribution of the studied models.”
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“As Fig. (2d) shows, a few isolated systems reach the boundaries of Ω: a cell lineage located at u ( G ) = (0 . 95 , 1 , 1) and a small social network located at u ( G ) = ( -1 , 0 . 06 , 0 . 09).”
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“Attending to F and O , we find an interesting region, defined by O ( G ) = 0 and F ( G ) > 0.”
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“An interesting case is given by the presence of bow-tie patterns in metabolic networks [53].”
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- Not checked yetcites [53] source
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“Fig. (2b-c) shows that both homogeneous and broad random networks occupy (basically) the same region in Ω.”
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“Close to O = 1, there only exists graph configurations for F = 1, coinciding with feedforward networks encompassing, electronic circuits and software networks.”
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“As expected hierarchical ( T > 0) and anti-hierarchical regions ( T < 0) are quite symmetric.”
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“This tells us that 〈 H f 〉 - 〈 H b 〉 = 0.”
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“Low levels of O seem to reduce the space of possible conformations.”
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“We remark that this is what is observed in random graphs, independently of their degree distributions, as we see in figure (15) and in the forthcoming sections.”
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“The majority of webs display a balance between integration of multiple signals and control over multiple targets under a bow-tie structural pattern.”
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“Such a behavior is relaxed at O = 0 . 5”
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“The large voids surrounding these clusters of webs (defining four major groups) are partially inaccessible and partially reachable, as shown by means of a directed evolution algorithm.”
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“One key observation is that most data sets are found within the envelope predicted by the ensembles of random graphs (see Fig. (3)).”
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“We are going to show that the expected value 〈 T 〉 over an ensemble of random directed graphs is 0.”
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“Finally, the fourth group is defined by an isolated cluster of ecological flow graphs, located around u ( G ) = (0 . 35 , 0 . 45 , 0 . 25).”
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“The larger O , the more reduced is the range of the possible of F .”
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“This is the reason why, at O = 0 . 15, reachability is rather heterogeneous.”
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“At O = 0 . 15 the extremes of T and F are inaccessible under our in silico evolutionary experiments.”
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“Fig. (4) reveals that the cloud of null models represented in Fig. (2b) is easily accessible, as indicated by the dark blue color of the region.”
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“Such a trend is less dramatic when O increases (see O = 0 . 5 and O = 0 . 85 of Fig. 4) since the fraction of nodes belonging to cycles is small enough to produce a rich combination of configurations in the resulting node-weighted condensed graphs.”
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“In all cases, model networks having high connectivities aggregate around the region defined by the rectangle T = ( -1 , 1), F ≈ (0 , 0 . 5) O ≈ 0.”
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“In relation to T coordinate, graphs tend to show T values close to zero.”
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“Looking at F and O values, real networks are generally far from the whiskers of the ensemble, when confronting real data against the ensemble obtained using the randomization method a -where directed degree sequence is preserved.”
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“Looking at the O for randomized ensembles obtained through method a , metabolisms and word corpora show a higher cyclic character than their respective randomized ensembles.”
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“Analysis of model networks obtained using the directed versions of ER, BA and Callaway models show similar results.”
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“This trend seems to be reduced when the directed degree sequence is conserved (randomization method b ).”
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“This difference is more accentuated for method a (the directed degree sequence is not conserved) than for method b , mainly due to the fact that the latter imposes a more dramatic restriction in the graphical configuration.”
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“Figure (17) shows the dramatic impact of 〈 k 〉 in F .”
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“In figure (25) looking at the position of real networks, they fall out from the percentile 50 th of their randomized ensembles in the Ω space.”
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“This likely reflects the advantage of reusing and recycling molecules.”
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“Ecological and gene networks define the other two, indicating that their topological order is the result of functional constraints.”
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“Here too the dispersal seems consistent with what is expected from very diluted random graphs (see Fig. (2c)).”
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“In this particular structure, a large SCC occupies a central position in a quite symmetrized feedforward structure of inputs and outputs [45].”
What the sources say
- Not checked yetcites [45] source
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“This little difference allows to unravel a family of rare networks highlighted by its extreme configuration -as shown by green icons in Fig. (1a). Essentially they are formed by chains of small SCC s disposed in a feedforward structure.”
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“Two of them match the expected from random networks with similar connectivity, thus suggesting that non-adaptive factors are at work.”
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- Rests on an assumption: The region occupied by null models indicates where no selection pressure is at work.
In the paper
“This sparseness is a consequence of engineering practices focused on reducing the wiring costs while keeping the system connected [19].”
What the sources say
- Not checked yetcites [19]
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“Interestingly, when cycles are incorporated to this structure at T = 0, the resulting structure is a bow-tie organization.”
What the sources say
- Not checked yetcites [45] source
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“Since F ( G ) = 1 is only possible when O ( G ) = 1, feedforward networks belong to the F ( G ) = O ( G ) = 1 line.”
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“T ( G ) has been shown to be a very good indicator of the deviations of a given acyclic graph from the ideal hierarchical, tree-like picture [3].”
What the sources say
- Not checked yetcites [3]
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“A completely random DAG displays H f -H b ≈ 0 [3].”
What the sources say
- Not checked yetcites [3]
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“It turns out that deviations of the pyramidal structure can be also quantified through an information-theoretic measure, namely, from the difference between the amount of statistical entropy generated when we cross the graph according to the flow direction depicted by the arrows and the uncertainty in reversing the paths [3].”
What the sources say
- Not checked yetcites [3]
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In the paper
“The four clusters point to different scenarios pervading the origins of their hierarchical organization.”
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In the paper
“DAG s have been also referred to as ordered graphs because it is always possible to topologically sort a DAG [1].”
What the sources say
- Not checked yetcites [1]
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In the paper
“Deviations are due to the dispersion produced by the finite-size statistics, specially when the resulting G C become very little, by the imposition of a high number of cycles, as it happens for O = 0 . 15.”
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In the paper
“Bow-tie networks, which have been suggested to define a flexible, robust and evolved type of systems [38, 40], perhaps resulting from selection for robustness would be a byproduct of the generation rules responsible for network growth.”
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In the paper
“For directed graphs, in addition, we observe further the emergence of the Giant Strongly Connected Component , ( GSCC ) namely, a single SCC containing a finite fraction S of the nodes of the graph [16].”
What the sources say
- Not checked yetcites [16] source
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In the paper
“Again, we will find F ( G ) = 1 in DAG s, as well as F ( G ) = 0 in networks consisting of a single strongly connected component.”
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In the paper
“This constraint inevitably produces a G C with just a handful of nodes.”
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In the paper
“As a consequence, non- adaptive processes would have played a major role in shaping hierarchies in nature [59].”
What the sources say
- Not checked yetcites [59] source
hedgeddirect- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: The region occupied by null models indicates where no selection pressure is at work.
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In the paper
“Finally if the situation is opposite to the former one, the graph will display a funnel-like or inverted pyramid structure.”
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In the paper
“This may indicate that a part of Ω is accessible and yet not occupied, suggesting that the spontaneous correlations created by random fluctuations provide the source of order for free.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Lemma : G displays T ( G ) = F ( G ) = O ( G ) = 1 if and only if G is a directed tree in which all the leafs have the same length, as defined in section C 1.”
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In the paper
“Corollary : G displays T ( G ) = -1 , F ( G ) = O ( G ) = 1 if and only if G is a directed tree in which all the leafs have the same length, as defined in section C 1 but with changing the direction of all arrows.”
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In the paper
“Intuitively, a graph having pyramidal structure will have the balance H f -H b positive, whereas a graph having inverted pyramidal structure will display H f -H b negative.”
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In the paper
“Since these null models do not consider optimal designs nor functional traits, we conclude that, as it was reported in the context of modularity [55, 56], hierarchical order may be a byproduct of inevitable random fluctuations, which spontaneously generate graph correlations”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: Null models do not consider optimal designs or functional constraints.
- Rests on an assumption: The region occupied by null models indicates where no selection pressure is at work.
In the paper
“Notice that the probability to have 2 GCC is vanishingly small [18] and thus, with high probability”
What the sources say
- Not checked yetcites [18] source
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In the paper
“F ( G ) dispersion is due to the variable size of modules located at the top of the structure.”
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In the paper
“Then, as it happens for a broad number of topological measures, values of T and F are extremely sensible to small variations in network configurations.”
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In the paper
“The special location in Ω, far from the random cloud, is caused by a small fraction of genes, the DNA-binding elements (transcription factors), located at the top of the network, which participate in cycles.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Intuitively, a graph having pyramidal structure will have the balance H f -H b positive, whereas a graph having inverted pyramidal structure will display H f -H b negative.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If the creation of new paths in a top down exploration of the graph is quantitatively equal to the uncertainty in reversing them, then, there is no argument to attribute to the net any pyramidal shape.”
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In the paper
“The special status of these networks (not shared by other webs) is consistent with the well known picture of a trophic pyramid combined with the presence of recycling [16, 54].”
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“A graph consisting in a big cycle will be the paradigmatic example of non-hierarchical system, showing thus T ( G ) = F ( G ) = O ( G ) = 0.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Finally, a graph consisting of a chain of nodes, or a DAG in which all the layers have the same size and all nodes the same connectivity will be the paradigmatic example of non-hierarchical but ordered system , showing T ( G ) = 0 , F ( G ) = O ( G ) = 1.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Again, we will find F ( G ) = 1 in DAG s, as well as F ( G ) = 0 in networks consisting of a single strongly connected component.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The solutions for this orderability are only possible by forcing to the network population to exhibit a large fraction of the nodes within cycles.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As it happened in the previous cases, the range occupied by real and randomized data seems to be fairly justified in most cases by directed degree sequence.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“therefore, the normalization of the relative size of the GSCC with the size of the GCC provides us an estimation about the actual impact of cycles over the connected structure of the graph.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“Local swap diverges from standard random rewiring algorithms because it keeps invariant the component structure of the graph.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In these cases, we expect that more than 1000 generations are needed.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This points to the conclusion that not local correlation among connectivities but the pattern of inputs and outputs would explain part of TFO values.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Among other factors, the size of the network may play a crucial role to achieve certain regions of the morphospace, since, for example it is easier for small networks to display extreme configurations.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Therefore, we can argue that real networks of under scrutiny have not a representative graphical configuration for their degree sequence.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We observe that, if the laves have not the same length, then a 0 will emerge in some computation, lowering the value to T ( G ) < 1.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Actually, the region T = 0 , F = 1 , O = 1 is only occupied by DAG s.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The high variance on the value of T can be explained by the fact that G C is very small, since high connectivity produces a drastic process of condensation.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This indicates that, although real networks tend to live in the null model regions as we saw earlier, system's constraints confer differences in the pattern of connections that distinguish real networks from their randomized counterparts.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This latter situation can be explained by the low presence -or complete absence- of cycles in networks having small connectivities.”
plaindirect- Asserted: passes — The authors put it forward as true.
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- It may state two separate claims as one. 31
- Probably not a substantial claim. 44
- The quote only partly states this claim. 3
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“These results are consistent with an exploration of the morphospace using in silico evolved networks.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The 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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, the most interesting results here concerns the presence of inaccessible regions, labeled in dark red color.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Here, a large region of high reachability is observed for extreme values of T but, not occupied by real networks.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“since, for example it is easier for small networks to display extreme configurations.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedged- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
- Also stated in: § c. Confronting real data with their randomized counterparts · p. 27 · hedged
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Positive biased values are observed in food webs and most of GRNs.”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“On the contrary, electronic circuits show an anti hierarchical configuration ( T < 0), with a tendency to occupy slightly negative, but significant, values of T .”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, when compared to randomized ensembles obtained using method b , the trend is similar to the common values observed in randomized versions.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- 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).
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“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].”
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“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].”
What the sources say
- Not checked yetcites [2] source
- Not checked yetcites [3]
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Also stated in: § I. INTRODUCTION · p. 1 · hedged
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“That implies, among other things, that they account for both bow-tie structures and feed-forward sparse webs.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“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.”
What the sources say
- Not checked yetcites [21] source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“Although previous work using complex networks theory has quantitatively tackled the problem [3, 4, 22-25, 27-31], some questions remain:”
What the sources say
- Not checked yetcites [3]
- Not checked yetcites [4] source
- Not checked yetcites [22] source
- Not checked yetcites [23] source
- Not checked yetcites [24] source
- Not checked yetcites [25] source
- Not checked yetcites [27] source
- Not checked yetcites [28] source
- Not checked yetcites [29] source
- Not checked yetcites [30] source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Such deviations occur because: (a) several elements are on the top, (b) downstream elements interact horizontally or (c) feedback loops are present.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“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].”
What the sources say
- Not checked yetcites [38] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Fifty years ago [1], Herbert Simon defined complex systems as nested hierarchical networks of components organized as interconnected modules.”
What the sources say
- Not checked yetcites [1]
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This is a raw indicator of the fraction of the net that can be ordered.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Some of them result from the underlying rules of network growth and change, which strongly limit the repertoire of potential designs.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- The authors state this as an assumption: A perfectly hierarchical system has the properties of order, reversibility, and pyramidal structure.
- Also stated in: § 2. Directed Acyclic Graphs · p. 12 · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The GSCC will represent the largest cyclic region of the graph.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This sorting implies that a DAG can represent a more or less entangled structure of interconnected causal processes.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, meaningful information can only be obtained from the study of a single component since it represents a unique causal structure.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
In the paper
“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].”
What the sources say
- Not checked yetcites [33] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: A perfectly hierarchical system has the properties of order, reversibility, and pyramidal structure.
In the paper
“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].”
What the sources say
- Not checked yetcites [3]
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Rests on an assumption: The region occupied by null models indicates where no selection pressure is at work.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A tree or, more generally, a feed-forward network will display O ( G ) = 1, in agreement to their orderable nature.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Also stated in: § a. Condensation: Obtaining a DAG from any directed graph · p. 12 · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Although LR f and LR b algorithms identify the same number of layers, the specific composition of every layer is not necessarily the same.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In contrast, a network which is totally cyclical would display O ( G ) = 0.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We observe that the length of the longest path of a finite DAG is always bounded.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The non-cyclic nature of such graph structures enables us to define a finite set of layers.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
In the paper
“Both biological and cultural evolution operate under a number of deep constraints [60].”
What the sources say
- Not checked yetcites [60] source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is worth to note that it is mathematically possible to apply this formalism to networks containing more than one connected component.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Through the values of these coordinates we must be able to identify and properly quantify deviations from such an ideal behavior.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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 .”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- The authors state this as an assumption: A perfectly hierarchical system has the properties of order, reversibility, and pyramidal structure.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Their T ( G ) > 0 values indicate a certain degree of pyramidal structure and the low O ( G )'s are consistent with an important role played by loops.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Method B: Preserving the directed degree sequence and the component structure.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“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 .”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- 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).
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Such nets are thus hierarchic ( T > 0), in terms of how the causal flow is organized.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“This chart allows us to elude any assumption on the statistical behavior of the ensembles of randomized graphs.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Method A: Preserving the undirected degree sequence and the component structure.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This latter case would belong to the perfect antihierachical system.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. The quote only partly states this claim.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The conservation of σ ( G ) makes this randomization method slightly more restrictive than the other presented above.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is easy to see that a network with a handful of nodes is more likely to give extreme values than large networks.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“For | V | ≫ 0, this model does not ensure connectedness until values of p ≈ log( | V | ) / | V | [7].”
What the sources say
- Not checked yetcites [7]
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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:”
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.
Rated major: it would touch a result — a value a result rests on, reported and recomputed p-values on opposite sides of the significance level, two passages that conflict with no sign which is right, or a cited work retracted before this paper was published.
In the paper
§ V. MORPHOSPACE ACCESSIBILITY BY DRIVEN EVOLUTION · p. 6
“The number of nodes for every graph, | V | = 50, remains constant in this process”
§ 3. The accessibility of the Morphospace: Evolution · p. 27
“The initial condition of this algorithm is defined by a population of graphs P (always single connected components), being | P | = 25, each one made of | V | = 25 nodes”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ a. Network dataset · p. 24
“Four directed lexical graph from different texts according the description in [20]:”
§ a. Network dataset · p. 24
“'Angie's Wren Xmas Tale' (Paul Auster), 'Frankenstein or the modern prometeous' (Mary Shelley),' Ulisses' (James Joyce), 'Moby dick' (Herman Melville), 'Black cat' (Edgar Allan Poe) and a fragment of the New York Times column.”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ c. Confronting real data with their randomized counterparts · p. 27
“randomization method a -where directed degree sequence is preserved”
§ b. Randomization Methods · p. 25
“Method A: Preserving the undirected degree sequence and the component structure.”
Citations
Cites
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Bollobás, B. (1985) · Cited by 6,196
The structure of the nervous system of the nematode Caenorhabditis elegans
White, J. G., Southgate, E., Thomson, J. N., & Brenner, S. (1986) · Philosophical Transactions of the Royal Society of London. B, Biological Sciences · doi:10.1098/rstb.1986.0056 · Cited by 5,589
A General Model for the Origin of Allometric Scaling Laws in Biology
West, G. B., Brown, J. H., & Enquist, B. J. (1997) · Science · doi:10.1126/science.276.5309.122 · Cited by 5,016
The Large-Scale Organization of Metabolic Networks
Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N., & Barabási, A.-L. (2000) · Nature · doi:10.1038/35036627 · Cited by 4,972
Cytoscape 2.8: new features for data integration and network visualization
Smoot, M. E., Ono, K., Ruscheinski, J., Wang, P., & Ideker, T. (2010) · Bioinformatics · doi:10.1093/bioinformatics/btq675 · Cited by 4,770
The Architecture of Complexity
Simon, H. A. (1991) · Springer eBooks · doi:10.1007/978-1-4899-0718-9_31 · Cited by 4,479
Hierarchical Organization of Modularity in Metabolic Networks
Ravasz, E., Somera, A. L., Mongru, D. A., Oltvai, Z. N., & Barabási, A.-L. (2002) · Science · doi:10.1126/science.1073374 · Cited by 4,224
Random graphs with arbitrary degree distributions and their applications
Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2001) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.64.026118 · Cited by 3,675
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., & Wiener, J. (2000) · Computer Networks · doi:10.1016/s1389-1286(00)00083-9 · Cited by 2,810
Cited by
Sorted by citations
Colloquium : Criticality and dynamical scaling in living systems
Muñoz, M. A. (2018) · Reviews of Modern Physics · doi:10.1103/revmodphys.90.031001 · Cited by 556
Structure and dynamics of core/periphery networks
Csermely, P., London, A., Wu, L.-Y., & Uzzi, B. (2013) · Journal of Complex Networks · doi:10.1093/comnet/cnt016 · Cited by 446
Operational resilience: concepts, design and analysis
Ganin, A. A., Massaro, E., Gutfraind, A., Steen, N., Keisler, J. M., Kott, A., Mangoubi, R., & Linkov, I. (2016) · Scientific Reports · doi:10.1038/srep19540 · Cited by 308
Boguñá, M., Bonamassa, I., Domenico, M. D., Havlin, S., Krioukov, D., & Serrano, M. Á. (2021) · Nature Reviews Physics · doi:10.1038/s42254-020-00264-4 · Cited by 232
Different Views of Hierarchy and Why They Matter: Hierarchy as Inequality or as Cascading Influence
Bunderson, J. S., van der Vegt, G. S., Cantimur, Y., & Rink, F. A. (2015) · Academy of Management Journal · doi:10.5465/amj.2014.0601 · Cited by 211
Metadata
- 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)
- 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
- 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
- green
- arxiv.org
- 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.
A language model drew the claims from the paper on 2026-10-07, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-07 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
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.
What counts as a claim ›How we measure confidence ›How we check this ›
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