A Network Structure of the Synoptic Gospels Employing Clustering Coefficients

In this presentation, we propose a graph-theory approach to
the systematization of the texts of the Synoptic Gospels.
Specifically, we apply a new graph-clustering technique for
data processing that utilizes a clustering-coefficient threshold
in creating semantic networks for the Gospels. Taking Greek
texts (Nestle-Aland, 1979) as the textual source, an adjacency
matrix for graphs is created from word co-occurrence data for
a given range of the texts. We also develop a web-based
application for the dynamic representation of the network
structures based on the relationships between words and
concepts.
Network representations are useful in modeling natural language
semantics (Stevers, & Tenenbaum, 2005). Recently, Dorow
(2006) has proposed the notion of curvature (clustering
coefficient) as a clustering tool to detect ambiguity and to
acquire semantic classes rather than using word frequencies.
The clustering coefficient represents the interconnective strength
between neighboring nodes in a graph. Following Watts and
Strogatz (1998), the clustering coefficient of the node (n) can
be defined as: (number of links among n's
neighbors)/(N(n)*(N)-1)/2), where N(n) denotes the set of n's
neighbors. The coefficient assumes values between 0 and 1.
In processing a text, word pairs are computed by a windowing
method (Takayama, Flournoy, Kaufmann, &. Peters). The
windowing method provides a relatively simple representation
of similarity levels that is suitable for clustering. The technique
involves moving a certain sized window over a text to extract
all fixed-sized word grams (Vechthomova, Roberston, & Jones,
2003). Word pairings are subsequently made by combining all
extracted words. Co-occurrence data is computed with two
window sizes that reflect syntactic and semantic considerations.
The first size is set at 2 for the nearest co-occurrence, while the
second size is set to 20 to collect words both before and after
a sentence. Morphological data for the BibleWorks Greek New
Testament Morphology (BNM) was used in a stemming process
to obtain base word forms from morphological variants.
Consequently, 2,688 word occurrences were identified.
In Figure 1, the degree distribution for the word occurrences
nodes are plotted in log-log coordinates with showing the best
fitting power law distribution (r=1.4). Following Barabasi and
Albert (1999), the feature of power-law degree distribution
indicates the small-world structure of the semantic network.
Figure 2 is a plot for the clustering coefficient as a function of
degree. There are 16 words that have a clustering coefficient
value of less than 0.1. These are articles, prepositions, pronouns,
and conjunctions, which are usually regarded as "noise" words.
We use the clustering coefficient value as a threshold in order
to eliminate such noise words and to control the datasets for
graph clustering. In applying the clustering technique, nine
datasets were created with 0.1 increments in the value of the
clustering coefficient (from 0.1 to 0.9). The datasets for each
window size were converted into adjacency matrices for
Recurrent Markov Clustering (RMCL) (Jung, Miyake, &
Akama, 2006).
RMCL is an improved form of Markov Clustering (MCL) (van
Dongen, 2000). The MCL algorithm is based on random walks
for a graph, and its model simulates flow by using the two
simple algebraic operations of expansion and inflation on the
stochastic transition matrix. The method has been applied to a
number of corpora, such as a French synonym dictionary
(Gfeller, 2005) and the British National Corpus (Dorow,
Widdows, Ling, Eckmann, Danilo, and Moses, 2005). In
contrast, RMCL allows for greater control over the sizes of
concept domains by modifying graph granularity and the
generality of concepts. The recurrent process gets feedback
from the state of overlapping clustering before the output of
MCL. This reversal procedure is a key feature of RMCL in
generating a virtual adjacency matrix for the non overlapping
clusters as a resultant state of convergence actually yielded by
the MCL process. The resultant downsized matrix provides a
simpler graph of the conceptual structures underlying similar
words.
Figure 3 shows the transitions in cluster sizes for the MCL and
RMCL processes as a function of the clustering coefficient with
a window size of 2, while Figure 4 shows similar data with a
window size of 20. The term input data refers to the initial
adjacency matrices. Figure 4 indicates that while cluster sizes
are almost identical for RMCL and MCL at clustering
coefficient values of 0.6 or more, RMCL clearly yields smaller
clusters at values of 0.5 or less. The MCL clustering results in
Figure 4 for the window size of 20 indicate the strong
connectedness between nodes, as all nodes are grouped into a
single cluster for coefficient values of 0.2 or less.
The results for both the MCL and RMCL processes are
implemented as graph network structures in a web application,
which has been developed to dynamically represent the
relationships between words and how MCL components might
correspond to concepts. Figure 5 is a screen shot of the web application, which uses Apache2 as an http server and Tomcat5
as a JSP and servlet engine. WebMathematica is used to call a
Mathematica kernal to calculate the graphs.
In conclusion, the clustering coefficient represents a useful tool
for manipulating datasets to eliminate noise words. The RMCL
methods allows for the creation of compact semantic networks,
which clearly present the relationships between words.
In future works, we will apply different clustering-coefficient
thresholds to detect ambiguous words, particularly words with
low coefficient-low degree values. We will also evaluate
systematizations of the Synoptic Gospels by biblical scholars
and explore the effectiveness of our approach in furthering
biblical studies.
Figure 1: The Degree Distributions
Figure 2: Clustering coefficient plotted as a function of degree
Figure 3: Clustering Coefficient (window size =2)
Figure 4: Clustering Coefficient (window size =20) Figure 5: Screen shot of a web application
Bibliography
Baranbasi, A.L., and R. Albert. "Emergence of Scaling in
Random Networks." Science 286 (1999): 509-512.
Dorow, B., D. Widdows, K. Ling, J. Eckmann, D. Sergi, and
E. Moses. "Using Curvature and Markov Clustering in Graphs
for Lexical Acquisition and Word Sense Discrimination."
MEANING-2005, 2nd Workshop organized by the MEANING
Project, February 3rd-4th 2005, Trento, Italy. Trento,Italy,
2005.
Dorow, K. "A Graph Model for Words and Their Meaning".
Doctoral Thesis. University of Stuttgart, 2006. <http://el
ib.uni-stuttgart.de/opus/volltexte/2007/2
985/pdf/diss_27022007.pdf>
Gfeller, D., J. C. Chappelier, and P. De Los Rios. "Synonym
Dictionary Improvement through Markov Clustering and
Clustering Stability." International Symposium on Applied
Stochastic Models and Data Analysis. . 106-113.
Jung, J., Maki Miyake, and H. Akama. "Recurrent Markov
Cluster (RMCL) & #12288; Algorithm for the Refinement of
the Semantic Network, LREC2006lity." International
Conference on Language Resources and Evaluation. .
1428-1432.
Nestle-Aland. Novum Testmentum Graece . 26th edition.
Stuttgart: German Bible Society, 1987.
Steyvers, M., and J. B. Tenenbaum. "The Large-Scale Structure
of Semantic Networks: Statistical Analyses and a Model of
Semantic Growth." Cognitive Science 29.1 (2005): 41-78.
Takayama, Y., R. Flournoy, S. Kaufmann, and S. Peters.
Information Mapping: Concept-based Information Retrieval
Based on Word Associations. . <http://www-csli.sta
nford.edu/semlab-hold/infomap/theory.html>
van Dongen, Stijn Marinus. "Graph Clustering by Flow
Simulation". PhD Thesis. University of Utrecht, 2000. <http
://igitur-archive.library.uu.nl/dissertat
ions/1895620/inhoud.htm>
Vechthomova, O., S. Robertson, and S. Jones. "Query
Expansion With Long-Span Collocate." Information Retrieval
6 (2003): 251-273.
Watts, D., and S. Strogatz. "Collective Dynamics of
‘Small-World’ Networks." Nature 393 (1998): 440-442.