## Centers of complex networks (2003)

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Venue: | J Theor Biol |

Citations: | 15 - 0 self |

### BibTeX

@ARTICLE{Wuchty03centersof,

author = {Stefan Wuchty and Peter F. Stadler},

title = {Centers of complex networks},

journal = {J Theor Biol},

year = {2003},

volume = {223},

pages = {2003}

}

### OpenURL

### Abstract

Abstract. The central vertices in complex networks are of particular interest because they might play the role of organizational hubs. Here, we consider three different geometric centrality measures, eccentricity, status, and centroid value, that were originally used in the context of resource placement problems. We show that these quantities lead to useful descriptions of the centers of biological networks which often, but not always, correlate with a purely local notion of centrality such as the vertex degree. We introduce the notion of local centers as local optima of a centrality value “landscape ” on a network and discuss briefly their role. 1 S. Wuchty, P.F. Stadler: Centers of Complex Networks 2

### Citations

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Citation Context ...utions define three different notions of “central” vertices. The distance matrix D can be computed rather efficiently e.g. using Dijkstra’s algorithm with time complexity O(|V | 2 ln |V |), see e.g. (=-=Cormen et al., 1990-=-). The excentricity of a vertex x in G and the radius ρ(G), respectively, are defined as e(x) = max y∈V The center of G is the set d(x, y) and ρ(G) = min e(x) (3) x∈V C(G) = {x ∈ V |e(x) = ρ(G)} . (4)... |

2279 | Emergence of Scaling in Random Networks
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Citation Context ...graphs (Erdős & Rényi, 1960; Bollobás, 1985). (b) Scale-Free Networks with a power law distribution P (d) ∼ d −γ . A simple model for this type of networks was introduced recently by Barabási et al. (=-=Barabási & Albert, 1999-=-; Barabási et al., 1999). Metabolic networks (Wagner & Fell, 2000; Jeong et al., 2000) and food-webs (Montoya & Solé, 2002) belong to this class. (c) Broad-Scale Networks for which P (d) has a power-l... |

1956 | On Random Graphs
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Citation Context ...o mention a few examples. Starting with the seminal paper by (Watts & Strogatz, 1998), it has been recognized that these real life network differ qualitatively from the classical random graph models (=-=Erdős & Rényi, 1960-=-; Bollobás, 1985) by the so-called small-world property: while the graphs are very sparse on average, the mutual distances between their vertices are nevertheless much shorter than expected. The recen... |

1846 | Random Graphs
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Citation Context ...les. Starting with the seminal paper by (Watts & Strogatz, 1998), it has been recognized that these real life network differ qualitatively from the classical random graph models (Erdős & Rényi, 1960; =-=Bollobás, 1985-=-) by the so-called small-world property: while the graphs are very sparse on average, the mutual distances between their vertices are nevertheless much shorter than expected. The recent review by (Alb... |

1302 | Statistical mechanics of complex networks
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- 2002
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Citation Context ...985) by the so-called small-world property: while the graphs are very sparse on average, the mutual distances between their vertices are nevertheless much shorter than expected. The recent review by (=-=Albert & Barabási, 2002-=-) indicates that current research focuses on the one hand evolving graphs with various preferential attachment rules and, on the other hand, on characterizing new empirically determined graphs in term... |

392 | The large-scale organization of metabolic networks
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Citation Context ...tribution P (d) ∼ d −γ . A simple model for this type of networks was introduced recently by Barabási et al. (Barabási & Albert, 1999; Barabási et al., 1999). Metabolic networks (Wagner & Fell, 2000; =-=Jeong et al., 2000-=-) and food-webs (Montoya & Solé, 2002) belong to this class. (c) Broad-Scale Networks for which P (d) has a power-law regime followed by a sharp cut-off, e.g. exponential or Gaussian decay of the tail... |

327 |
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Citation Context ... to weighted graphs. So, simply define deg(x) as the sum of the weight of the edges that contain x and define the length of a path as the sum of the weights of each edges. 2.2. Degree Distributions. (=-=Amaral et al., 2000-=-) showed that there are (at least) three structurally different classes of networks that are distinguished by the distribution P (k) of the vertex degrees k = deg(x): (a) Single-Scale Networks with a ... |

301 |
Lethality and centrality in protein networks
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Citation Context ...ites listed in table 2 are the same for d(x) and f(x). 4.2. Protein Networks. A second class of networks that have received particular attention recently are networks of (direct) protein interaction (=-=Jeong et al., 2001-=-; Wagner, 2001). The likelihood that the elimination of a protein from the genome issS. Wuchty, P.F. Stadler: Centers of Complex Networks 10 Table 1. Central nodes in the metabolic network of E. coli.... |

230 | Mean-Field Theory for ScaleFree Random Networks
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(Show Context)
Citation Context ...60; Bollobás, 1985). (b) Scale-Free Networks with a power law distribution P (d) ∼ d −γ . A simple model for this type of networks was introduced recently by Barabási et al. (Barabási & Albert, 1999; =-=Barabási et al., 1999-=-). Metabolic networks (Wagner & Fell, 2000; Jeong et al., 2000) and food-webs (Montoya & Solé, 2002) belong to this class. (c) Broad-Scale Networks for which P (d) has a power-law regime followed by a... |

197 | Topology of Evolving Networks: Local Events and Universality
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Citation Context ... present vertex k with probability Π(k) = d(k) � � d(k) , (2) where d(k) is the degree of vertex k. A recent extension of the model allows the tuning of the scaling exponent γ in the range 2 ≤ γ ≤ 3 (=-=Albert & Barabási, 2000-=-a). ksS. Wuchty, P.F. Stadler: Centers of Complex Networks 4 The vertex degrees are an intrinsically local characterization of a graph. Consequently, they allow a meaningful interpretation only when t... |

146 | The small world inside large metabolic networks
- Wagner, Fell
- 2001
(Show Context)
Citation Context ...ath length. The vertex degree is typically used as a measure of centrality in these networks. In the graph evolution models, high vertex degrees usually indicate “old” vertices. (Fell & Wagner, 2000; =-=Wagner & Fell, 2000-=-) indeed show that the metabolites with the highest connectivity are part of the oldest “core” metabolism. It is a bit surprising, however, that classical graph-theoretical properties of such large re... |

135 |
The yeast protein interaction network evolves rapidly and contains few redundant duplicate genes
- Wagner
- 2001
(Show Context)
Citation Context ... 2 are the same for d(x) and f(x). 4.2. Protein Networks. A second class of networks that have received particular attention recently are networks of (direct) protein interaction (Jeong et al., 2001; =-=Wagner, 2001-=-). The likelihood that the elimination of a protein from the genome issS. Wuchty, P.F. Stadler: Centers of Complex Networks 10 Table 1. Central nodes in the metabolic network of E. coli. For the full ... |

120 |
Structural determination of paraffin boiling points
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- 1947
(Show Context)
Citation Context ...9). InsteadsS. Wuchty, P.F. Stadler: Centers of Complex Networks 5 of the status, one may of course use the average distance ℓ(x) of a vertex from x. Clearly, d(x) = (|V | − 1)ℓ(x). The Wiener index (=-=Wiener, 1947-=-) is W (G) = 1 � d(x) = 2 1 � � � n d(x, y) = ℓ(G) , (7) 2 2 x∈V x,y∈V where ℓ(G) is the mean path length in G. It provides an important characteristic of molecular graphs. For details, see (Gutman et... |

86 |
Small Worlds
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- 1999
(Show Context)
Citation Context ...class. (c) Broad-Scale Networks for which P (d) has a power-law regime followed by a sharp cut-off, e.g. exponential or Gaussian decay of the tail. An example is the movie-actor network described in (=-=Watts, 1999-=-) The Erdős-Rényi model (ER) (Erdős & Rényi, 1960) assumes a fixed number n = |V | of vertices and assigns edges independently with a certain probability p. For details see the book by (Bollobás, 1985... |

85 | Renormalization group analysis of the small-world network model
- Newman
- 1999
(Show Context)
Citation Context ...ministic graph, usually a circular arrangement of vertices in which each vertex is connected to k nearest neighbors on each side. Subsequently, edges are “rewired” (in the original version) or added (=-=Newman & Watts, 1999-=-; Newman et al., 2000) with probability p. Both ER and SW graphs exhibit an approximately Gaussian degree distributions. The other extreme is the scale-free model (BA) (Barabási & Albert, 1999; Barabá... |

75 |
The small world of metabolism
- Fell, Wagner
- 2000
(Show Context)
Citation Context ...icient, and average path length. The vertex degree is typically used as a measure of centrality in these networks. In the graph evolution models, high vertex degrees usually indicate “old” vertices. (=-=Fell & Wagner, 2000-=-; Wagner & Fell, 2000) indeed show that the metabolites with the highest connectivity are part of the oldest “core” metabolism. It is a bit surprising, however, that classical graph-theoretical proper... |

70 |
The Theory of Graphs
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- 1962
(Show Context)
Citation Context ...d as a graph G which consists of a set V of vertices and a set E of edges which we regard as un-ordered pairs of distinct vertices. Hence we consider only simple undirected graphs in the language of (=-=Berge, 1985-=-). A path in G is an alternating sequence (x0, e1, x1, . . . , eℓ, xℓ) of vertices and edges, where the ei = {xi−1, xi} are the edges connecting subsequent vertices. The length of a path is its number... |

66 |
On the properties of small-world network models
- Barrat, Weigt
- 2000
(Show Context)
Citation Context ...he literature on the small-world networks is the clustering coefficient that measures how close the neighborhood of a each vertex comes on average to being a complete subgraph (clique) (Herzel, 1998; =-=Barrat & Weigt, 2000-=-; Watts & Strogatz, 1998). Again, this measure is intrinsically local. A more global measure is the average length ℓ(G) of a path between two vertices, see e.g. for an extensive discussion (Newman et ... |

65 | Mean-field solution of the small-world network model,” Phys
- Newman, Moore, et al.
- 2000
(Show Context)
Citation Context ...y a circular arrangement of vertices in which each vertex is connected to k nearest neighbors on each side. Subsequently, edges are “rewired” (in the original version) or added (Newman & Watts, 1999; =-=Newman et al., 2000-=-) with probability p. Both ER and SW graphs exhibit an approximately Gaussian degree distributions. The other extreme is the scale-free model (BA) (Barabási & Albert, 1999; Barabási et al., 1999) with... |

51 | RNA folding at elementary step resolution
- Flamm, Fontana, et al.
- 2000
(Show Context)
Citation Context ...ting paths between local centers that are themselves as central as possible. The program barriers 1 that was originally designed to investigate the structure of the energy landscape of RNA molecules (=-=Flamm et al., 2000-=-; Flamm et al., 2002) can be used to solve exactly this problem. Starting from a list of vertices that is sorted by the value of the cost function g(x), barriers identifies all local minima and the sa... |

49 |
Scale-free Behavior in Protein Domain Networks
- Wuchty
- 2001
(Show Context)
Citation Context ...indicated e.g. by the position of the inflection point of the sigmoidal curves, increases significantly from unicellular organism to vertebrates. These data are in good agreement with the results of (=-=Wuchty, 2001-=-), which were obtained using vertex connectivity. In Table 3, we list the ten domains with the smallest centroid value, i.e., the ones that are most central, for six different organism. Again, the dat... |

44 |
Small-world patterns in food webs
- Montoya, Sole
- 2002
(Show Context)
Citation Context ...odel for this type of networks was introduced recently by Barabási et al. (Barabási & Albert, 1999; Barabási et al., 1999). Metabolic networks (Wagner & Fell, 2000; Jeong et al., 2000) and food-webs (=-=Montoya & Solé, 2002-=-) belong to this class. (c) Broad-Scale Networks for which P (d) has a power-law regime followed by a sharp cut-off, e.g. exponential or Gaussian decay of the tail. An example is the movie-actor netwo... |

37 |
Sur les assemblages de lignes
- Jordan
(Show Context)
Citation Context ...= min d(x) . (5) x∈V M(G) = {x ∈ V |d(x) = σ(G)} . (6) The median is the solution of the “service facility location problem” (B). Both the center and the median of a graph were already considered by (=-=Jordan, 1869-=-). InsteadsS. Wuchty, P.F. Stadler: Centers of Complex Networks 5 of the status, one may of course use the average distance ℓ(x) of a vertex from x. Clearly, d(x) = (|V | − 1)ℓ(x). The Wiener index (W... |

33 | Barrier trees of degenerate landscapes
- Flamm, Hofacker, et al.
- 2002
(Show Context)
Citation Context ...ocal centers that are themselves as central as possible. The program barriers 1 that was originally designed to investigate the structure of the energy landscape of RNA molecules (Flamm et al., 2000; =-=Flamm et al., 2002-=-) can be used to solve exactly this problem. Starting from a list of vertices that is sorted by the value of the cost function g(x), barriers identifies all local minima and the saddle points that con... |

32 |
The roles of mutation, inbreeding, crossbreeeding and selection in evolution
- Wright
- 1932
(Show Context)
Citation Context ...scale-free graphs compared to ER random graphs, we observe larger values of f(x) as well. 3.2. Landscapes on Graphs. The concept of a fitness landscape originated in the 1930s in theoretical biology (=-=Wright, 1932-=-; Wright, 1967) as a mean of visualizing evolutionary adaptation. The same abstract concept arise naturally in many other areas of scientific study, for instance the physics of disordered systems, com... |

28 |
Classification of scale-free networks
- Goh, Oh, et al.
- 2002
(Show Context)
Citation Context ...fied in terms of the number of shortest paths that run through a given vertex. Most recently, a classification of scale-free networks based on the scaling of betweenness centrality has been proposed (=-=Goh et al., 2002-=-). A comparison of this measure with resource placement centralities will be described elsewhere. 3. Properties of Centrality Measures 3.1. Correlations. Both the vertex degree and the three geometric... |

22 | Relevant cycles in chemical reaction networks. Advances in complex systems
- Gleiss, Stadler, et al.
- 2001
(Show Context)
Citation Context ...erage length ℓ(G) of a path between two vertices, see e.g. for an extensive discussion (Newman et al., 2000). The distribution of short cycles, i.e., detours, may be regarded as an intermediate case (=-=Gleiss et al., 2001-=-). 2.3. Geometric Centrality. Geometric notions of centrality are closely linked to facility location problems. Suppose, we are given a graph G representing, say, a traffic network. We may then ask qu... |

19 |
Surfaces of selective value
- Wright
- 1967
(Show Context)
Citation Context ...phs compared to ER random graphs, we observe larger values of f(x) as well. 3.2. Landscapes on Graphs. The concept of a fitness landscape originated in the 1930s in theoretical biology (Wright, 1932; =-=Wright, 1967-=-) as a mean of visualizing evolutionary adaptation. The same abstract concept arise naturally in many other areas of scientific study, for instance the physics of disordered systems, combinatorial opt... |

15 |
Prediction of protein essentiality based on genomic data
- Jeong, Oltvai, et al.
(Show Context)
Citation Context ...e other, larger peak refers to the main (giant) component of the network. a viable phenotype therefore will require a gradual answer. A promising approach in this direction was recenly undertaken by (=-=Jeong et al., 2002-=-), who found correlations of phenotypic effects of single gene deletions in Yeast with fluctuations in the corresponding mRNA expression levels, functional classification of gene products and the numb... |

14 |
Status and contrastatus
- Harary
- 1959
(Show Context)
Citation Context ...G) = {x ∈ V |e(x) = ρ(G)} . (4) C(G) is the center to the “emergency facility local problem” (A) which is always contained in a single block of G (Harary & Norman, 1953). The status d(x) of a vertex (=-=Harary, 1959-=-) and the status σ(G) of the graph G, respectively, are defined as d(x) = � y∈V The median (Slater, 1980) of G is the set d(x, y) and σ(G) = min d(x) . (5) x∈V M(G) = {x ∈ V |d(x) = σ(G)} . (6) The me... |

13 | Molecular evolution in large genetic networks: does connectivity equal constraint
- Hahn, Conant, et al.
(Show Context)
Citation Context ...akdown. A recent re-evaluation of the available data indicated, however, that lethal proteins cannot be cleanly distinguished from viable ones by their degree alone (Wuchty, 2002). In the same vein, (=-=Hahn et al., 2002-=-) show that connectivity is not related to robustness against aminoacid substitutions in protein networks. This poses the question whether the correlation of lethality and connectivity is a local or a... |

11 |
Medians of arbitrary graphs
- Slater
- 1980
(Show Context)
Citation Context ...s always contained in a single block of G (Harary & Norman, 1953). The status d(x) of a vertex (Harary, 1959) and the status σ(G) of the graph G, respectively, are defined as d(x) = � y∈V The median (=-=Slater, 1980-=-) of G is the set d(x, y) and σ(G) = min d(x) . (5) x∈V M(G) = {x ∈ V |d(x) = σ(G)} . (6) The median is the solution of the “service facility location problem” (B). Both the center and the median of a... |

10 |
Distance in graphs
- Entringer, Jackson, et al.
- 1976
(Show Context)
Citation Context ...), is then solved by the vertices x that maximize |Vxy| − |Vyx| over all possible locations of the competitor y. The following identity d(x) + |Vxy| = d(y) + |Vyx| (9) holds for all connected graphs (=-=Entringer et al., 1976-=-). Following (Slater, 1975) we define centroid value of a vertex and the graph G itself as f(x) = d(x) − min y�=x The centroid of G is the set d(y) and ϕ(G) = min f(x) . (10) x∈V Z(G) = {x ∈ V |f(x) =... |

5 |
The dissimilarity characteristic of Husimi trees
- Harary, Norman
- 1953
(Show Context)
Citation Context ...f G is the set d(x, y) and ρ(G) = min e(x) (3) x∈V C(G) = {x ∈ V |e(x) = ρ(G)} . (4) C(G) is the center to the “emergency facility local problem” (A) which is always contained in a single block of G (=-=Harary & Norman, 1953-=-). The status d(x) of a vertex (Harary, 1959) and the status σ(G) of the graph G, respectively, are defined as d(x) = � y∈V The median (Slater, 1980) of G is the set d(x, y) and σ(G) = min d(x) . (5) ... |

5 |
Interaction and domain networks of yeast. Proteomics
- Wuchty
- 2002
(Show Context)
Citation Context ...s increases the probability of breakdown. A recent re-evaluation of the available data indicated, however, that lethal proteins cannot be cleanly distinguished from viable ones by their degree alone (=-=Wuchty, 2002-=-). In the same vein, (Hahn et al., 2002) show that connectivity is not related to robustness against aminoacid substitutions in protein networks. This poses the question whether the correlation of let... |

3 |
How to quantify ‘small-world’ networks
- Herzel
- 1998
(Show Context)
Citation Context ...only used in the literature on the small-world networks is the clustering coefficient that measures how close the neighborhood of a each vertex comes on average to being a complete subgraph (clique) (=-=Herzel, 1998-=-; Barrat & Weigt, 2000; Watts & Strogatz, 1998). Again, this measure is intrinsically local. A more global measure is the average length ℓ(G) of a path between two vertices, see e.g. for an extensive ... |

3 |
Maximum facility location
- Slater
- 1975
(Show Context)
Citation Context ... u, v ∈ V , u �= v, define Vxy = {w ∈ V |d(x, w) < d(y, w)} , (8) i.e., Vuv is the set of vertices that are closer to u than to v. The competitive location problem (C), which was first considered by (=-=Slater, 1975-=-), is then solved by the vertices x that maximize |Vxy| − |Vyx| over all possible locations of the competitor y. The following identity d(x) + |Vxy| = d(y) + |Vyx| (9) holds for all connected graphs (... |

2 |
Extremal values for ratios of distance trees
- Barefoot, Entringer, et al.
- 1997
(Show Context)
Citation Context ... the centroid Z(G) are always contained in the same block of a connected graph G (Smart & Slater, 1999). Both, the center and the centroid may serve as the root of a distance preserving spaning tree (=-=Barefoot et al., 1997-=-). The centroid value f(x) may, perhaps surprisingly, become 0 or even negative. If this is the case, then d(x) = miny d(y) = σ(G). It follows that f(x) < 0 for at most one vertex x ∗ , in which case ... |

2 |
Centrality, convexity and intersections in graphs
- Nieminen
- 1984
(Show Context)
Citation Context ..., M(G), and Z(G) may be pairwise disjoint and even separated by arbitrary distances if G is large enough (Slater, 1999). A slightly different, much less studied notion of centrality is introduced in (=-=Nieminen, 1984-=-). An induced subgraph H of G is convex if it contains a shortest path (in G) between any two of its vertices. A branch of G at a vertex x is a maximal convex induced subgraph that does not contain x.... |

2 |
centroid subgraphs
- Slater
- 1999
(Show Context)
Citation Context ...(x) and e(x). The mutual location of the three types of “central” vertices is of obvious interest. The median M(G) and the centroid Z(G) are always contained in the same block of a connected graph G (=-=Smart & Slater, 1999-=-). Both, the center and the centroid may serve as the root of a distance preserving spaning tree (Barefoot et al., 1997). The centroid value f(x) may, perhaps surprisingly, become 0 or even negative. ... |

2 |
Structural determination of para#ne boiling points
- Wiener
- 1947
(Show Context)
Citation Context ...9). Instead S. Wuchty, P.F. Stadler: Centers of Complex Networks 5 of the status, one may of course use the average distance #(x) of a vertex from x. Clearly, d(x) = (|V | - 1)#(x). The Wiener index (=-=Wiener, 1947-=-) is W (G) = 1 2 # x#V d(x) = 1 2 # x,y#V d(x, y) = # n 2 # #(G) , (7) where #(G) is the mean path length in G. It provides an important characteristic of molecular graphs. For details, see (Gutman et... |

1 | Stadler: Centers of Complex Networks 15 - Wuchty, F - 1977 |

1 | Fifty Years of the Wiener index volume 36 of MATCH - Gutman, Klavzar - 1996 |

1 |
Central vertices in a graph
- Slater
- 1976
(Show Context)
Citation Context ...ng d. Again we have Z(G) = M(G). Conversely, if ϕ(G) > 0 then the minima of d(x) do not minimize f(x) and hence median and centroid are disjoint. Such graphs are called secure graphs. It is shown in (=-=Slater, 1976-=-) that there are no secure graphs with |V | < 9 vertices. An example with |V | ≥ 9 is given (Smart & Slater, 1999, Fig.4). It is shown in (Smart & Slater, 1999) that C(G), M(G), and Z(G) may be pairwi... |

1 |
A survey of sequences of central subgraphs
- Slater
- 1999
(Show Context)
Citation Context ...x) = d(x) − min y�=x The centroid of G is the set d(y) and ϕ(G) = min f(x) . (10) x∈V Z(G) = {x ∈ V |f(x) = ϕ(G)} . (11) We have inverted the sign of f(x) compared to the discussion in Slater’s work (=-=Slater, 1999-=-) as we prefer the centrality measure f(x) to be minimal at the most central vertices in analogy to d(x) and e(x). The mutual location of the three types of “central” vertices is of obvious interest. ... |

1 | Stadler: Centers of Complex Networks 16 - Wuchty, F - 1998 |

1 |
How to quantify "small world networks
- Herzel
- 1998
(Show Context)
Citation Context ...mmonly used in the literature on the small-world networks is the clustering coe#cient that measures how close the neighborhood of a each vertex comes on average to being a complete subgraph (clique) (=-=Herzel, 1998-=-; Barrat & Weigt, 2000; Watts & Strogatz, 1998). Again, this measure is intrinsically local. A more global measure is the average length #(G) of a path between two vertices, see e.g. for an extensive ... |