## A Faster Algorithm for Betweenness Centrality (2001)

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Venue: | Journal of Mathematical Sociology |

Citations: | 335 - 5 self |

### BibTeX

@ARTICLE{Brandes01afaster,

author = {Ulrik Brandes},

title = {A Faster Algorithm for Betweenness Centrality},

journal = {Journal of Mathematical Sociology},

year = {2001},

volume = {25},

pages = {163--177}

}

### Years of Citing Articles

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### Abstract

The betweenness centrality index is essential in the analysis of social networks, but costly to compute. Currently, the fastest known algorithms require #(n ) time and #(n ) space, where n is the number of actors in the network.

### Citations

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Citation Context ...th connecting s and t in G. By definition, dG(s, s) = 0 for every s ∈ V , and dG(s, t) = dG(t, s) for s, t ∈ V . We assume familiarity with standard algorithms for shortest-paths problems (see, e.=-=g., Cormen et al. 1990). Several-=- measures capture variations on the notion of a vertex’s importance in a graph. Let σst = σts denote the number of shortest paths from s ∈ V to t ∈ V , where σss = 1 by convention. Let σst(v... |

3024 | Authoritative Sources in a Hyperlinked Environment
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Citation Context ...tion of centrality indices on networks with thousands of actors. Several notions of centrality originating from social network analysis are in use to determine the structural prominence of Web pages (=-=Kleinberg, 1999; -=-Brin et al., 1998; Bharat and Henzinger, 1998). However, there is an Ω(n 3 ) bottleneck in existing implementations, due to the particularly important betweenness centrality index (Freeman, 1977; An... |

778 |
Data Structures and Algorithms
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Citation Context ...n each pair of vertices. Some superfluous work is avoided in a suitably defined instance, called geodetic semiring (Batagelj, 1994), of the closed semiring generalization for shortest paths problems (=-=Aho et al., 1974)-=-. It yields an Θ(n 3 ) algorithm for betweenness by augmenting the Floyd/Warshall algorithm for the all-pairs shortest-paths problem with path counting. To exploit the sparsity of typical instances, ... |

679 | LEDA: a platform for combinatorial and geometric computing. volume 38
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- 1995
(Show Context)
Citation Context ...; for v ∈ P [w] do δ[v] ← δ[v] + σ[v] σ[w] · (1 + δ[w]); if w �= s then CB[w] ← CB[w] + δ[w]; end end 10sand undirected graphs using the Library of Efficient Data Structures and Algorit=-=hms (LEDA, see Mehlhorn and Näher 1999-=-). Performance is compared with an implementation that uses the same code to determine the length and number of shortest paths between all pairs of vertices, but sums all pairdependencies explicitly. ... |

629 | Centrality in social networks: Conceptual clarification
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- 1979
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Citation Context ...l of Mathematical Sociology 25(2):163-177, (2001). 1sAn essential tool for the analysis of social networks are centrality indices defined on the vertices of the graph (Bavelas, 1948; Sabidussi, 1966; =-=Freeman, 1979-=-). They are designed to rank the actors according to their position in the network and interpreted as the prominence of actors embedded in a social structure. Many centrality indices are based on shor... |

600 |
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Citation Context ...d to count the number of shortest paths according to this lemma. BFS takes time O(m), and Dijkstra’s algorithm runs in time O(m + n log n), if the priority queue is implemented with a Fibonacci heap=-= (Fredman and Tarjan, 1987).-=- Corollary 4 Given a source s ∈ V , both the length and number of all shortest paths to other vertices can be determined in time O(m+n log n) for weighted, and in time O(m) for unweighted graphs. Co... |

508 | Social Network Analysis: a handbook - Scott - 2002 |

432 | Improved Algorithms for Topic Distillation in a Hyperlinked Environment
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Citation Context ...ks with thousands of actors. Several notions of centrality originating from social network analysis are in use to determine the structural prominence of Web pages (Kleinberg, 1999; Brin et al., 1998; =-=Bharat and Henzinger, 1998).-=- However, there is an Ω(n 3 ) bottleneck in existing implementations, due to the particularly important betweenness centrality index (Freeman, 1977; Anthonisse, 1971), which makes comparative centra... |

322 |
A set of measures of centrality based on betweenness
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Citation Context ... (Kleinberg, 1999; Brin et al., 1998; Bharat and Henzinger, 1998). However, there is an Ω(n 3 ) bottleneck in existing implementations, due to the particularly important betweenness centrality index=-= (Freeman, 1977-=-; Anthonisse, 1971), which makes comparative centrality analyses of networks with more than a few hundred actors prohibitive. As a remedy, network analysts are now suggesting simpler indices, for inst... |

314 | The Web as a graph: Measurements, models, and methods
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- 1999
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Citation Context ...rk is less than 29, corresponding to 0.3% density. (Clearly, density tends to zero when the average degree is fixed.) Recent experiments estimate an even lower average outdegree of 7.2 for Web pages (=-=Kleinberg et al., 1999-=-). Figure 4 gives running times for betweenness index calculations on random graphs with a fixed average vertex degree of 20. These results imply that our implementation can compute, e.g., the between... |

300 | Pajek-program for large network analysis
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Citation Context ...lable 3 Courtesy of Robert Foreman and Thomas Valente of the Epidemiology Data House at Johns Hopkins University. See Valente et al. (1998) for background on the data. 12snetwork analysis tool Pajek (=-=Batagelj and Mrvar, 1998-=-) and by researchers performing centrality analyses on networks of words extracted from electronic text. For typical instances, they obtained improvements from about 12 hours CPU time on an SGI Medusa... |

154 | Structural models: An introduction to the theory of directed graphs - Harary, Norman, et al. - 1965 |

103 |
The centrality index of a graph
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(Show Context)
Citation Context ...blished in Journal of Mathematical Sociology 25(2):163-177, (2001). 1sAn essential tool for the analysis of social networks are centrality indices defined on the vertices of the graph (Bavelas, 1948; =-=Sabidussi, 1966-=-; Freeman, 1979). They are designed to rank the actors according to their position in the network and interpreted as the prominence of actors embedded in a social structure. Many centrality indices ar... |

58 | What can you do with a web in your pocket
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(Show Context)
Citation Context ...y indices on networks with thousands of actors. Several notions of centrality originating from social network analysis are in use to determine the structural prominence of Web pages (Kleinberg, 1999; =-=Brin et al., 1998; -=-Bharat and Henzinger, 1998). However, there is an Ω(n 3 ) bottleneck in existing implementations, due to the particularly important betweenness centrality index (Freeman, 1977; Anthonisse, 1971), wh... |

58 | Theoretical foundations for centrality measures - Friedkin - 1991 |

52 |
The rush in a directed graph
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(Show Context)
Citation Context ...99; Brin et al., 1998; Bharat and Henzinger, 1998). However, there is an Ω(n 3 ) bottleneck in existing implementations, due to the particularly important betweenness centrality index (Freeman, 1977=-=; Anthonisse, 1971-=-), which makes comparative centrality analyses of networks with more than a few hundred actors prohibitive. As a remedy, network analysts are now suggesting simpler indices, for instance based only on... |

37 |
Eccentricity and centrality in networks
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(Show Context)
Citation Context ... s to t that some v ∈ V lies on. The following are standard measures of centrality: 1 CC(v) = � t∈V dG(v, t) closeness centrality (Sabidussi, 1966) 1 CG(v) = maxt∈V dG(v, t) CS(v) = graph cent=-=rality (Hage and Harary, 1995) � σst(v) stres-=-s centrality (Shimbel, 1953) s�=v�=t∈V CB(v) = � s�=v�=t∈V σst(v) σst betweenness centrality (Freeman, 1977; Anthonisse, 1971) High centrality scores thus indicate that a vertex can re... |

34 |
Structural parameters of communication networks
- Shimbel
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(Show Context)
Citation Context ...standard measures of centrality: 1 CC(v) = � t∈V dG(v, t) closeness centrality (Sabidussi, 1966) 1 CG(v) = maxt∈V dG(v, t) CS(v) = graph centrality (Hage and Harary, 1995) � σst(v) stress cen=-=trality (Shimbel, 1953) s�=v�=t∈-=-V CB(v) = � s�=v�=t∈V σst(v) σst betweenness centrality (Freeman, 1977; Anthonisse, 1971) High centrality scores thus indicate that a vertex can reach others on relatively short paths, or th... |

22 |
Integration and Radiality: Measuring the Extent of an Individual’s Connectedness and Reachability in a Network
- Valente, Foreman
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(Show Context)
Citation Context ...th based centrality measures defined in Section 2 are easily computed during the execution of single-source shortest-paths traversals. The same holds for a recently introduced index called radiality (=-=Valente and Foreman, 1998) CR(v) = �-=-�� t∈V (D(G) + 1 − dG(v, t)) , (n − 1) · D(G) where D(G) = maxs,t∈V dG(s, t), and for other potential measures as well. This is a significant practical advantage, reducing the combined time a... |

17 |
A Mathematical Model for Group Structure." Human Organization
- Bavelas
- 1948
(Show Context)
Citation Context ...ogramm III). Published in Journal of Mathematical Sociology 25(2):163-177, (2001). 1sAn essential tool for the analysis of social networks are centrality indices defined on the vertices of the graph (=-=Bavelas, 1948-=-; Sabidussi, 1966; Freeman, 1979). They are designed to rank the actors according to their position in the network and interpreted as the prominence of actors embedded in a social structure. Many cent... |

14 | The Gatekeeper, Pair-Dependency, and Structural Centrality." Quality and Quantity - Freeman - 1980 |

7 | Social Network Analysis
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(Show Context)
Citation Context ...any pair of vertices), we are only interested in the number of shortest paths between each pair of vertices. Some superfluous work is avoided in a suitably defined instance, called geodetic semiring (=-=Batagelj, 1994)-=-, of the closed semiring generalization for shortest paths problems (Aho et al., 1974). It yields an Θ(n 3 ) algorithm for betweenness by augmenting the Floyd/Warshall algorithm for the all-pairs sho... |

4 | Satellite exchange in the Baltimore needle exchange program - Valente, Junge, et al. - 1998 |

2 |
Ego-network betweenness. Paper presented at
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- 1999
(Show Context)
Citation Context ...f networks with more than a few hundred actors prohibitive. As a remedy, network analysts are now suggesting simpler indices, for instance based only on linkages between the neighbors of each actors (=-=Everett et al., 1999-=-), to at least obtain rough approximations of betweenness centrality. In this paper, we show that betweenness can be computed exactly even for fairly large networks. We introduce more efficient algori... |

1 | reported by Steven Corman - As - 2000 |