## A graph-based approach towards discerning inherent structures in a digital library of formal mathematics (2004)

Venue: | In Lecture Notes in Computer Science |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Lorigo04agraph-based,

author = {Lori Lorigo and Jon Kleinberg and Richard Eaton and Robert Constable},

title = {A graph-based approach towards discerning inherent structures in a digital library of formal mathematics},

booktitle = {In Lecture Notes in Computer Science},

year = {2004},

pages = {220--235},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. As the amount of online formal mathematical content grows, for example through active efforts such as the Mathweb [21], MOWGLI [4], Formal Digital Library, or FDL [1], and others, it becomes increasingly valuable to find automated means to manage this data and capture semantics such as relatedness and significance. We apply graph-based approaches, such as HITS, or Hyperlink-Induced Topic Search, [11] used for World Wide Web document search and analysis, to formal mathematical data collections. The nodes of the graphs we analyze are theorems and definitions, and the links are logical dependencies. By exploiting this link structure, we show how one may extract organizational and relatedness information from a collection of digital formal math. We discuss the value of the information we can extract, yielding potential applications in math search tools, theorem proving, and education.

### Citations

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Citation Context ...ernet by operating on this directed graph. Popular eigenvector-based methods that are effective in web search by finding authoritative sources include Kleinberg’s HITS algorithm and Google’s PageRank =-=[9]-=-. In addition to abilities to rank important objects, capabilities to cluster or organize data into groups based on the graph structure have been developed and exploited. Web Trawling [12] uses a grap... |

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Citation Context ...ngly valuable to find automated means to manage this data and capture semantics such as relatedness and significance. We apply graph-based approaches, such as HITS, or Hyperlink-Induced Topic Search, =-=[11]-=- used for World Wide Web document search and analysis, to formal mathematical data collections. The nodes of the graphs we analyze are theorems and definitions, and the links are logical dependencies.... |

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Citation Context ...ructure have been developed and exploited. Web Trawling [12] uses a graph-theoretic approach to enumerate communities on the web, based on the findings of densely bipartite sub-graphs. Recent work in =-=[17]-=- finds communities in networks by iteratively removing detectable edges from the network to divide the collection into related groups. Automatically categorizing or grouping related theorems in a form... |

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Citation Context ...n stratified by depth, the graph exhibits a ``wide'' region in the middle. In this way, it can potentially be viewed as an interesting analogue, for acyclic graphs, of the ``bowtie'' model of the Web =-=[10]-=-. In order to extract information from these levels, one may iteratively run the HITS algorithm, iteratively finding the hubs and authorities for the graph containing the union of level i and i+1, for... |

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Citation Context ...he World Wide Web links. The power law distribution, which appears as a straight line in log-log density plots, is prevalent in many growing real-world networks, such as power grids, and the Internet =-=[5]-=-. Our data does not closely fit a power law over the entire set of degrees, but it is nearly linear over certain intervals of degree values: after the peak in the out-link graph, and also in the earli... |

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Citation Context ...er eigenvalues. These often are semantically related in practice. While the individual non-principal eigenvectors are thus a direct way to expose further structure in the data, we note that Ng et al. =-=[19]-=- observe instabilities in some cases in the use of these eigenvectors, and recommend more generally studying subspaces spanned by the non-principal eigenvectors. Since the Nuprl Standard library has a... |

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Citation Context ...n Choices We implemented the algorithm in LISP, and ran the code inside of Cornell’s FDL on two different collections of formal mathematics that belong to that library: the Nuprl5 Standard collection =-=[2]-=-, and the Event Structures collection [8]. These two collections were easily accessible to us and also presented a good contrast for measuring and evaluating our results. Large collections of PVS cont... |

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Citation Context ...NY {lolorigo, kleinber, eaton, rc}@cs.cornell.edu http://www.cs.cornell.edu Abstract. As the amount of online formal mathematical content grows, for example through active efforts such as the Mathweb =-=[21]-=-, MOWGLI [4], Formal Digital Library, or FDL [1], and others, it becomes increasingly valuable to find automated means to manage this data and capture semantics such as relatedness and significance. W... |

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Citation Context ...e’s PageRank [9]. In addition to abilities to rank important objects, capabilities to cluster or organize data into groups based on the graph structure have been developed and exploited. Web Trawling =-=[12]-=- uses a graph-theoretic approach to enumerate communities on the web, based on the findings of densely bipartite sub-graphs. Recent work in [17] finds communities in networks by iteratively removing d... |

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Citation Context ... kleinber, eaton, rc}@cs.cornell.edu http://www.cs.cornell.edu Abstract. As the amount of online formal mathematical content grows, for example through active efforts such as the Mathweb [21], MOWGLI =-=[4]-=-, Formal Digital Library, or FDL [1], and others, it becomes increasingly valuable to find automated means to manage this data and capture semantics such as relatedness and significance. We apply grap... |

11 | A Causal Logic of Events
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Citation Context ... LISP, and ran the code inside of Cornell’s FDL on two different collections of formal mathematics that belong to that library: the Nuprl5 Standard collection [2], and the Event Structures collection =-=[8]-=-. These two collections were easily accessible to us and also presented a good contrast for measuring and evaluating our results. Large collections of PVS content also currently reside in the FDL, whi... |

8 |
social networks are different from other types of networks
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Citation Context ...d more. Information about its dependency graph follows in Table 1. Table 1. Nuprl5 Standard Dependency Graph Data. Most fields should be clear. The assortativity of a network was defined by Newman in =-=[18]-=- and is a measure of the variance of the link distribution, or degree-degree correlations often used in social-network analysis. In practice, r is positive for social networks, where the nodes represe... |

6 | The DMLF project: A new initiative in classical special functions
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Citation Context ... methods, theorem provers, information seeking, information networks, link analysis, clustering 1 Introduction Invaluable progress has been made in the development of digital libraries of mathematics =-=[1, 3, 4, 15, 21]-=-. Such progress also includes content and presentationspecific representations of mathematical knowledge [13, 14, 16], including architectures for exchanging mathematical content. These efforts are br... |

2 | Mathematical Knowledge Management in HELM
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Citation Context ... methods, theorem provers, information seeking, information networks, link analysis, clustering 1 Introduction Invaluable progress has been made in the development of digital libraries of mathematics =-=[1, 3, 4, 15, 21]-=-. Such progress also includes content and presentationspecific representations of mathematical knowledge [13, 14, 16], including architectures for exchanging mathematical content. These efforts are br... |

2 |
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(Show Context)
Citation Context ...progress has been made in the development of digital libraries of mathematics [1, 3, 4, 15, 21]. Such progress also includes content and presentationspecific representations of mathematical knowledge =-=[13, 14, 16]-=-, including architectures for exchanging mathematical content. These efforts are bringing together otherwise disparate collections of formal mathematics, and providing rich access to mathematical know... |

2 |
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Citation Context ...progress has been made in the development of digital libraries of mathematics [1, 3, 4, 15, 21]. Such progress also includes content and presentationspecific representations of mathematical knowledge =-=[13, 14, 16]-=-, including architectures for exchanging mathematical content. These efforts are bringing together otherwise disparate collections of formal mathematics, and providing rich access to mathematical know... |

1 |
Dependency graphs
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(Show Context)
Citation Context ...ght to be documented? Also, tools to visualize and interact with the digital collections surely aid us in analyzing the structures. Several theorem provers offer tree-like user interfaces and work in =-=[7]-=- presents methods for pruning the dependency graphs so that they are visually appealing. We are interested in how visualization in this context can aid a user searching a digital library of mathematic... |