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13
Combinatorial Landscapes
 SIAM REVIEW
, 2002
"... Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, ne ..."
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Cited by 33 (2 self)
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Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, nearness, distance or accessibility. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.
Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. II. Program Implementation of the WeisfeilerLeman Algorithm
, 1997
"... The stabilization algorithm of Weisfeiler and Leman has as an input any square matrix A of order n and returns the minimal cellular (coherent) algebra W (A) which includes A. In case when A = A(\Gamma) is the adjacency matrix of a graph \Gamma the algorithm examines all configurations in \Gamma hav ..."
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Cited by 10 (4 self)
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The stabilization algorithm of Weisfeiler and Leman has as an input any square matrix A of order n and returns the minimal cellular (coherent) algebra W (A) which includes A. In case when A = A(\Gamma) is the adjacency matrix of a graph \Gamma the algorithm examines all configurations in \Gamma having three vertices and, according to this information, partitions vertices and ordered pairs of vertices into equivalence classes. The resulting construction allows to associate to each graph \Gamma a matrix algebra W (\Gamma) := W (A(\Gamma)) which is an invariant of the graph \Gamma. For many classes of graphs, in particular for most of the molecular graphs, the algebra W (\Gamma) coincides with the centralizer algebra of the automorphism group Aut(\Gamma). In such a case the partition returned by the stabilization algorithm is equal to the partition into orbits of Aut(\Gamma). We give algebraic and combinatorial descriptions of the WeisfeilerLeman algorithm and present an efficient com...
STABCOL: Graph Isomorphism Testing Based on the WeisfeilerLeman Algorithm
, 1997
"... A coherent algebra is a matrix algebra over the field of the complex numbers which is closed under conjugate transposition and elementwise multiplication of matrices and which contains the identity matrix and the all 1 matrix. This algebraic structure has a variety of important applications. Amon ..."
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Cited by 9 (3 self)
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A coherent algebra is a matrix algebra over the field of the complex numbers which is closed under conjugate transposition and elementwise multiplication of matrices and which contains the identity matrix and the all 1 matrix. This algebraic structure has a variety of important applications. Among others, coherent algebras are an appropriate tool in the design of algorithms for two notoriously hard graph theoretical problems: the problems of deciding whether two graphs are isomorphic and of finding the automorphism partition of a graph. Weisfeiler and Leman stated a polynomial algorithm which computes the coherent algebra which is generated by the adjacency matrix of a graph. However, for almost three decades, no reasonable time bound was known for this method. Very recently, one of the authors established a theoretical time bound of O(n 3 log n) with n denoting the number of vertices in the graph. The aim of this paper is to document a computer implementation of the algor...
Efficiently Indexing Shortest Paths by Exploiting Symmetry in Graphs
 In EDBT 2009
"... Shortest path queries (SPQ) are essential in many graph analysis and mining tasks. However, answering shortest path queries onthefly on large graphs is costly. To online answer shortest path queries, we may materialize and index shortest paths. However, a straightforward index of all shortest path ..."
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Cited by 9 (2 self)
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Shortest path queries (SPQ) are essential in many graph analysis and mining tasks. However, answering shortest path queries onthefly on large graphs is costly. To online answer shortest path queries, we may materialize and index shortest paths. However, a straightforward index of all shortest paths in a graph of N vertices takes O(N 2) space. In this paper, we tackle the problem of indexing shortest paths and online answering shortest path queries. As many large real graphs are shown richly symmetric, the central idea of our approach is to use graph symmetry to reduce the index size while retaining the correctness and the efficiency of shortest path query answering. Technically, we develop a framework to index a large graph at the orbit level instead of the vertex level so that the number of breadthfirst
Spectral Landscape Theory
 Evolutionary Dynamics—Exploring the Interplay of Selection, Neutrality, Accident, and Function
, 1999
"... INTRODUCTION Evolutionary change is caused by the spontaneously generated genetic variation and its subsequent fixation by drift and/or selection. Consequently, the main focus of evolutionary theory has been to understand the genetic structure and dynamics of populations, see e.g. [101]. In recent ..."
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Cited by 8 (3 self)
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INTRODUCTION Evolutionary change is caused by the spontaneously generated genetic variation and its subsequent fixation by drift and/or selection. Consequently, the main focus of evolutionary theory has been to understand the genetic structure and dynamics of populations, see e.g. [101]. In recent years, however, alternative approaches have gained increasing prominence in evolutionary theory. This development has been stimulated to some extent by the application of evolutionary models to designing evolutionary algorithms such as Genetic Al Evolutionary Dynamics edited by J.P. Crutchfield and P. Schuster, 1999 2 Spectral Landscape Theory gorithms, Evolution Strategies, and Genetic Programming, as well as by the theory of Complex Adaptive Systems [69, 79, 38]. The generic structure of an evolutionary model is x 0 = S (x; w) ffi T (x;<F
Landscapes on Spaces of Trees
, 2001
"... Combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology, for instance the problem of reconstruction a phylogeny using maximum likelihood or parsimony approaches. The collection of possible phylogenetic trees is arranged as a socalled ..."
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Cited by 6 (1 self)
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Combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology, for instance the problem of reconstruction a phylogeny using maximum likelihood or parsimony approaches. The collection of possible phylogenetic trees is arranged as a socalled Robinson graph by means of the nearest neighborhood interchange move. The coherent algebra and spectra of Robinson graphs are discussed in some detail as their knowledge is important for an understanding of the landscape structure. We consider simple model landscapes as well as landscapes arising from the maximum parsimony problem, focusing on two complementary measures of ruggedness: the amplitude spectrum arising from projecting the cost functions onto the eigenspaces of the underlying graph and the topology of local minima and their connecting saddle points.
Experimental Studies of the Universal Chemical Key (UCK) Algorithm on the NCI
 Database of Chemical Compounds, Proceedings of the 2003 IEEE Computer Society Bioinformatics Conference (CSB 2003), IEEE Computer Society, Los Alamitos
"... We have developed an algorithm called the Universal Chemical Key (UCK) algorithm that constructs a unique key for a molecular structure. The molecular structures are represented as undirected labeled graphs with the atoms representing the vertices of the graph and the bonds representing the edges. T ..."
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Cited by 4 (3 self)
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We have developed an algorithm called the Universal Chemical Key (UCK) algorithm that constructs a unique key for a molecular structure. The molecular structures are represented as undirected labeled graphs with the atoms representing the vertices of the graph and the bonds representing the edges. The algorithm was tested on 236,917 compounds obtained from the National Cancer Institute (NCI) database of chemical compounds. In this paper we present the algorithm, some examples and the experimental results on the NCI database. On the NCI database, the UCK algorithm provided distinct unique keys for chemicals with different molecular structures. 1.
Network quotients: Structural skeletons of complex systems
 Physical Review E
"... A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article w ..."
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Cited by 3 (2 self)
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A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article we utilize inherent network symmetry to collapse all redundant information from a network, resulting in a coarsegraining which we show to carry the essential structural information of the ‘parent ’ network. In the context of algebraic combinatorics, this coarsegraining is known as the quotient. We systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of ‘realworld ’ quotients with respect to those of their parent networks. In particular, we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity (heterogeneity and hubs vertices) and communication (diameter and mean geodesic distance), suggesting that quotients constitute the essential structural skeleton of their parent network. We summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms. PACS numbers: 89.75.k 89.75.Fb 05.40.a 02.20.a
The Ring of Graph Invariants  Upper and Lower Bounds for Minimal Generators, Graphs and Combinatorics
"... In this paper we study the ring of graph invariants, focusing mainly on the invariants of simple graphs. We show that all other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this ring. In fact, every graph invariant is a linear combination of the basi ..."
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Cited by 1 (1 self)
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In this paper we study the ring of graph invariants, focusing mainly on the invariants of simple graphs. We show that all other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this ring. In fact, every graph invariant is a linear combination of the basic graph invariants which we study in this paper. To prove that two graphs are isomorphic, a number of invariants are required, which are called separator invariants. The minimal set of separator invariants is also the minimal generator set for the ring of graph invariants. We find lower and upper bounds for the minimal number of generator/separator invariants needed for proving graph isomorphism. The minimal number of generators/separators is the transcendence degree of the ring of graph invariants. Finally we find a sufficient condition for Ulam’s conjecture to be true based on Redfield’s enumeration formula. 1.
STABCOL: An Efficient Implementation of the WeisfeilerLeman Algorithm
"... A coherent algebra is a matrix algebra over the field of the complex numbers which is closed under conjugate transposition and elementwise multiplication of matrices and which contains the identity matrix and the all 1 matrix. This algebraic structure has a variety of important applications. Among o ..."
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Cited by 1 (0 self)
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A coherent algebra is a matrix algebra over the field of the complex numbers which is closed under conjugate transposition and elementwise multiplication of matrices and which contains the identity matrix and the all 1 matrix. This algebraic structure has a variety of important applications. Among others, coherent algebras are an appropriate tool in the design of algorithms for two notoriously hard graph theoretical problems: the problems of deciding whether two graphs are isomorphic and of finding the automorphism partition of a graph. Weisfeiler and Leman stated a polynomial algorithm which computes the coherent algebra which is generated by the adjacency matrix of a graph. However, for almost three decades, no reasonable time bound was known for this method. Very recently, one of the authors established a theoretical time bound of O(n 3 log n) with n denoting the number of vertices in the graph. The aim of this paper is to document a computer implementation of the algorithm of Wei...