Michael Bommarito | University of Michigan (original) (raw)
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Papers by Michael Bommarito
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
The United States Code (Code) is an important source of Federal law that is produced by the inter... more The United States Code (Code) is an important source of Federal law that is produced by the interactions of many heterogeneous actors in a complex, dynamic space. The Code can be represented as the union of a hierarchical network and a citation network over the vertices representing the language of the Code. In this paper, we investigate the properties of the Code's citation network by examining the directed degree distributions of the network. We find that the power-law model is a plausible fit for the outdegree distribution but not for the indegree distribution. In order to better understand this result, we construct a model with the assumption that the probability of citation is a per-word rate. We calculate the adjusted degree of each vertex under this model and study the directed adjusted degree distributions. These adjusted degree distributions indicate that both the adjusted indegree and outdegree distributions seems to follow a log-normal form, not a power-law form. Our findings indicate that the power-law is not generally applicable to degree distributions within the United States Code but that the distribution of degree per word is well-described by a log-normal model.
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
ABSTRACT In this paper, we highlight the potential of n-grams as a vehicle to explore the ‘evolut... more ABSTRACT In this paper, we highlight the potential of n-grams as a vehicle to explore the ‘evolution’ of the law and legal language. Using the full text corpus of decisions of the United States Supreme Court (1791-2005), we explore the n-gram space, offer some initial results based upon our calculations and highlight the beta version of our n-gram search interface.
ABSTRACT We present an adjusted method for calculating the eigenvalues of a time-dependent return... more ABSTRACT We present an adjusted method for calculating the eigenvalues of a time-dependent return correlation matrix that produces a more stationary distribution of eigenvalues. First, we compare the normalized maximum eigenvalue time series of the market-adjusted return correlation matrix to that of logarithmic return correlation matrix on an 18-year dataset of 310 S&P 500-listed stocks for two (small and large) window or memory sizes. We observe that the resulting new eigenvalue time series is more stationary than time series obtained through the use of existing method for each memory. Later, we perform this analysis while sweeping the window size τ ε {5, ..., 100} in order to examine the dependence on the choice of window size. We find that the three dimensional distribution of the eigenvalue time series for our market-adjusted return is significantly more stationary than that produced by classic method. Moreover, our model offers an approximate polarization domain of smooth L-shaped strip. The polarization with large amplitude is revealed, while there is persistence in agreement with small amplitude most of the time.
SSRN Electronic Journal, 2000
Procedia - Social and Behavioral Sciences, 2010
SSRN Electronic Journal, 2000
ABSTRACT
Quantitative Finance, 2011
... DOI: 10.1080/14697680903449815 Ahmet Duran a ... of Science ®] View all references) studied a... more ... DOI: 10.1080/14697680903449815 Ahmet Duran a ... of Science ®] View all references) studied an inverse problem involving a semi-unconstrained parameter optimization for a dynamical system of nonlinear asset flow differential equations to describe trader population dynamics ...
Physica A: Statistical Mechanics and its Applications, 2010
Acyclic digraphs arise in many natural and artificial processes. Among the broader set, dynamic c... more Acyclic digraphs arise in many natural and artificial processes. Among the broader set, dynamic citation networks represent a substantively important form of acyclic digraphs. For example, the study of such networks includes the spread of ideas through academic citations, the spread of innovation through patent citations, and the development of precedent in common law systems. The specific dynamics that produce such acyclic digraphs not only differentiate them from other classes of graphs, but also provide guidance for the development of meaningful distance measures. In this article, we develop and apply our sink distance measure together with the singlelinkage hierarchical clustering algorithm to both a two-dimensional directed preferential attachment model as well as empirical data drawn from the first quarter century of decisions of the United States Supreme Court. Despite applying the simplest combination of distance measures and clustering algorithms, analysis reveals that more accurate and more interpretable clusterings are produced by this scheme.
Physica A: Statistical Mechanics and its Applications, 2010
The United States Code (Code) is a document containing over 22 million words that represents a la... more The United States Code (Code) is a document containing over 22 million words that represents a large and important source of Federal statutory law. Scholars and policy advocates often discuss the direction and magnitude of changes in various aspects of the Code. However, few have mathematically formalized the notions behind these discussions or directly measured the resulting representations. This paper addresses the current state of the literature in two ways. First, we formalize a representation of the United States Code as the union of a hierarchical network and a citation network over vertices containing the language of the Code. This representation reflects the fact that the Code is a hierarchically organized document containing language and explicit citations between provisions. Second, we use this formalization to measure aspects of the Code as codified in These measurements allow for a characterization of the actual changes in the Code over time. Our findings indicate that in the recent past, the Code has grown in its amount of structure, interdependence, and language.
Environmental Science & Policy, 2013
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
The United States Code (Code) is an important source of Federal law that is produced by the inter... more The United States Code (Code) is an important source of Federal law that is produced by the interactions of many heterogeneous actors in a complex, dynamic space. The Code can be represented as the union of a hierarchical network and a citation network over the vertices representing the language of the Code. In this paper, we investigate the properties of the Code's citation network by examining the directed degree distributions of the network. We find that the power-law model is a plausible fit for the outdegree distribution but not for the indegree distribution. In order to better understand this result, we construct a model with the assumption that the probability of citation is a per-word rate. We calculate the adjusted degree of each vertex under this model and study the directed adjusted degree distributions. These adjusted degree distributions indicate that both the adjusted indegree and outdegree distributions seems to follow a log-normal form, not a power-law form. Our findings indicate that the power-law is not generally applicable to degree distributions within the United States Code but that the distribution of degree per word is well-described by a log-normal model.
SSRN Electronic Journal, 2000
SSRN Electronic Journal, 2000
ABSTRACT In this paper, we highlight the potential of n-grams as a vehicle to explore the ‘evolut... more ABSTRACT In this paper, we highlight the potential of n-grams as a vehicle to explore the ‘evolution’ of the law and legal language. Using the full text corpus of decisions of the United States Supreme Court (1791-2005), we explore the n-gram space, offer some initial results based upon our calculations and highlight the beta version of our n-gram search interface.
ABSTRACT We present an adjusted method for calculating the eigenvalues of a time-dependent return... more ABSTRACT We present an adjusted method for calculating the eigenvalues of a time-dependent return correlation matrix that produces a more stationary distribution of eigenvalues. First, we compare the normalized maximum eigenvalue time series of the market-adjusted return correlation matrix to that of logarithmic return correlation matrix on an 18-year dataset of 310 S&P 500-listed stocks for two (small and large) window or memory sizes. We observe that the resulting new eigenvalue time series is more stationary than time series obtained through the use of existing method for each memory. Later, we perform this analysis while sweeping the window size τ ε {5, ..., 100} in order to examine the dependence on the choice of window size. We find that the three dimensional distribution of the eigenvalue time series for our market-adjusted return is significantly more stationary than that produced by classic method. Moreover, our model offers an approximate polarization domain of smooth L-shaped strip. The polarization with large amplitude is revealed, while there is persistence in agreement with small amplitude most of the time.
SSRN Electronic Journal, 2000
Procedia - Social and Behavioral Sciences, 2010
SSRN Electronic Journal, 2000
ABSTRACT
Quantitative Finance, 2011
... DOI: 10.1080/14697680903449815 Ahmet Duran a ... of Science ®] View all references) studied a... more ... DOI: 10.1080/14697680903449815 Ahmet Duran a ... of Science ®] View all references) studied an inverse problem involving a semi-unconstrained parameter optimization for a dynamical system of nonlinear asset flow differential equations to describe trader population dynamics ...
Physica A: Statistical Mechanics and its Applications, 2010
Acyclic digraphs arise in many natural and artificial processes. Among the broader set, dynamic c... more Acyclic digraphs arise in many natural and artificial processes. Among the broader set, dynamic citation networks represent a substantively important form of acyclic digraphs. For example, the study of such networks includes the spread of ideas through academic citations, the spread of innovation through patent citations, and the development of precedent in common law systems. The specific dynamics that produce such acyclic digraphs not only differentiate them from other classes of graphs, but also provide guidance for the development of meaningful distance measures. In this article, we develop and apply our sink distance measure together with the singlelinkage hierarchical clustering algorithm to both a two-dimensional directed preferential attachment model as well as empirical data drawn from the first quarter century of decisions of the United States Supreme Court. Despite applying the simplest combination of distance measures and clustering algorithms, analysis reveals that more accurate and more interpretable clusterings are produced by this scheme.
Physica A: Statistical Mechanics and its Applications, 2010
The United States Code (Code) is a document containing over 22 million words that represents a la... more The United States Code (Code) is a document containing over 22 million words that represents a large and important source of Federal statutory law. Scholars and policy advocates often discuss the direction and magnitude of changes in various aspects of the Code. However, few have mathematically formalized the notions behind these discussions or directly measured the resulting representations. This paper addresses the current state of the literature in two ways. First, we formalize a representation of the United States Code as the union of a hierarchical network and a citation network over vertices containing the language of the Code. This representation reflects the fact that the Code is a hierarchically organized document containing language and explicit citations between provisions. Second, we use this formalization to measure aspects of the Code as codified in These measurements allow for a characterization of the actual changes in the Code over time. Our findings indicate that in the recent past, the Code has grown in its amount of structure, interdependence, and language.
Environmental Science & Policy, 2013