Chapter 1. Introduction

Transcription

1 Chapter 1 Introduction The starting point for the work represented here was the second-named author s doctoral dissertation. In that work, combinatorial properties of Clifford algebras were used to develop a graph-theoretic construction of the iterated stochastic integral of a process defined on a Clifford algebra of arbitrary signature using adjacency matrices with entries in a commutative subalgebra of a Clifford algebra. That initial work led directly to the publications [114], [115], [116], and [117]. Since then, the authors have extended the work to the study of random graphs [101], graph processes [109, 110], random walks on Clifford algebras [102, 104], Appell systems on Clifford algebras [100], homology and cohomology theories of Clifford operator calculus [107], combinatorics of partitions [106], and partition-dependent stochastic measures [108]. The principal motivation for pursuing problems in graph theory is the abundance of real-world applications, notably in computer science, where new methods of tackling computationally difficult problems are needed. Graphs provide natural models for wireless networks, traffic sensors, and the world wide web; moreover, these graphs evolve in real time. Random walks on graphs are of interest as models of internet searches, data transmission, and even error propagation. Clifford algebras have natural connections with graph theory. The group structure underlying the Clifford algebra Cl p,q of dimension 2 p+q has a Cayley graph resembling the (p + q)-dimensional hypercube. Their natural connections with spinors and applications to problems in theoretical physics make them an excellent bridge between areas of pure mathematics and theoretical physics. The material contained herein touches on a number of areas of research, including algebra, combinatorics, analysis, probability, and operator theory. 3

2 4 Operator Calculus on Graphs: Theory and Applications in Computer Science Essential background such as terminology and basic definitions is included where appropriate and practical. The reader is also guided to a number of references for more details on broader topics. Generally speaking, the book s flow begins with algebraic preliminaries including operator calculus on a number of group and semigroup algebras. From there, graph theory is introduced and a number of results are obtained using operator calculus methods. Graph theory gives way to probability, from classical to quantum. After several chapters of algebraic probability, the work turns toward a more applied direction with chapters on Appell systems and homology. Finally, the discussion turns to symbolic computations and complexity. More specifically, the work is organized as follows. Algebraic preliminaries appear in Chapter 2. To begin, six group and semigroup algebras with interesting combinatorial properties are constructed. These algebras actually fall within the realm of Clifford algebras, which have well-known and long-standing applications in physics, engineering, and geometry. The group and semigroup algebra approach to Clifford algebras and subalgebras distills the combinatorial aspects to their purest forms. Included in Chapter 2 are the details of Clifford algebras and their combinatorially interesting subalgebras, zeon algebras and their generalizations, and fundamental notions of operator calculus. Operator calculus methods developed in the chapter are inherent throughout the rest of the book. Chapter 3 develops some analytical machinery for working with Clifford algebras and matrices whose entries are elements of Clifford algebras. Some attention is paid to the concept of generating functions in the Clifford algebra sense, as these are of great interest in combinatorics. Essential terminology and standard results from graph theory appear at the beginning of Chapter 4, where the operator calculus approach to graph theory begins. In particular, a number of special adjacency matrices are constructed for finite graphs. While powers of the adjacency matrix of a finite graph reveal information about walks on the graph, they fail to distinguish closed walks from cycles. Using elements of an appropriate commutative, null-square generated algebra, these new adjacency matrices are well suited for symbolic computations and allow one to sieve out self-avoiding structures in graphs simply by computing matrix powers. Chapter 5 extends the adjacency matrix results from finite graphs to probabilistic models of random graphs. In particular, expected numbers and probabilities of occurrence of self-avoiding structures are considered. Letting X k denote the number of k-cycles occurring in a random graph, this

3 Introduction 5 algebra, together with a probability mapping, allows E(X k ) to be recovered in terms of matrix traces. Higher moments of X k can also be computed, and conditions are given for the existence of higher moments in growing sequences of random graphs by considering infinite-dimensional algebras. In Chapter 6, an algebraic probability space of nilpotent adjacency matrices associated with finite graphs is defined. Each nilpotent adjacency matrix is a quantum random variable whose m th moment corresponds to the number of m-cycles in the graph. Each matrix admits a canonical quantum decomposition into a sum of three algebraic random variables: a = a + a Υ + a Λ, where a is classical while a Υ and a Λ are quantum. Connected components in graph processes are also considered by encoding the relevant information from graph processes into a second quantization operator. Using tools of quantum probability and infinite-dimensional analysis, it is thereby possible to derive formulas for exact values of quantities that otherwise could only be approximated. Such quantities include the expected size of a maximal connected component, the probability of existence of a component of particular size, and the expected number of spanning trees in a random graph. The chapter includes a method of encoding the relevant information from graph processes into a second quantization operator and using tools of quantum probability and infinite-dimensional analysis to derive formulas that reveal the exact values of quantities that otherwise can only be approximated. In particular, the expected size of a maximal connected component, the probability of existence of a component of particular size, and the expected number of spanning trees in a random graph are obtained. In Chapter 7, nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, d- dimensional unit cube [0, 1] d. A random geometric graph is constructed by randomly choosing a set of points in the unit cube [0, 1] d and connecting two points by an edge if their Euclidean distance is at most some fixed distance r > 0. Graphs of this type are of particular interest as models of wireless networks. Using operator calculus methods, cycles are counted, sizes of maximal connected components are computed, and closed formulas are obtained for graph circumference and girth. In Chapter 8, combinatorial properties of the geometric product are used to represent random walks on hypercubes as sequences within the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k 0 by a formal power series expansion of elements in the algebra. Furthermore, by induc-

4 6 Operator Calculus on Graphs: Theory and Applications in Computer Science ing a walk on a larger Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. The work is then extended to homogeneous processes on Clifford algebras of arbitrary signature. Multiplicative random walks are induced by sequences of independent, uniformly distributed random variables taking values in the unit basis vectors and paravectors in the algebra. These walks can be viewed as random walks on directed hypercubes. Properties of such multiplicative walks are investigated. Sequences of multiplicative walks are then used to induce additive walks on the algebra. Finally, limit theorems for these walks are developed. In Chapter 9, the d-dimensional unit cube [0, 1] d is discretized to create a collection V of vertices used to define geometric graphs. Dynamic random walks are defined on the subsets of V, resulting in dynamic random walks on the collection of geometric graphs in the discretized cube. These walks naturally model addition-deletion networks and can be visualized as walks on hypercubes with loops. Adjacency operators are constructed using subalgebras of Clifford algebras and are used to recover information about the cycle structure and connected components of graphs in the sequence. In Chapter 10, a graph-theoretic approach to stochastic integrals is developed in which the m th iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued processes, Hermite and Poisson Charlier polynomials are recovered. In Chapter 11, a graph-theoretic perspective of partitions is investigated in which independent sets in graphs correspond to non-crossing partitions. By associating particular graphs with elements of zeon algebras, multiplicative functions can be summed over segments of lattices of partitions by employing methods of zeon operator calculus. In particular, properties of the algebra are used to sieve out the appropriate segments and sublattices. The chapter concludes with an application to joint moments of quantum random variables in free probability. Chapters 12 and 13 are devoted to pure operator calculus. In Chapter 12, motivated by evolution equations on Clifford algebras and illustrated with the n-particle fermion algebra, a theory of invertible left- and right- Appell systems is developed for Clifford algebras of an arbitrary quadratic form. A direct connection is also shown between blade factorization algo-

5 Introduction 7 rithms and the construction of Appell systems in these algebras. In Chapter 13, canonical raising and lowering operators defined on a Clifford algebra of arbitrary signature are used to define chains and cochains of vector spaces underlying the Clifford algebra, to compute the associated homology and cohomology groups, and to derive long exact sequences of underlying vector spaces. The vector spaces appearing in the chains and cochains correspond to the Appell system decomposition of the Clifford algebra. Using Mathematica, kernels of lowering operators and raising operators are explicitly computed. Connections with quantum probability and graphical interpretations of the lowering and raising operators are discussed. Chapters 14 and 15 turn to issues of computational complexity. Given a computing architecture based on Clifford algebras, an algorithm s time complexity can be expressed in terms of the number of geometric (Clifford) operations required. In Chapter 14 the existence of such a processor is assumed and the multivector-level complexity of a number of graphtheoretical and combinatorial problems is considered in detail. In Chapter 15, nilpotent adjacency matrix methods are employed to enumerate k-cycles in simple graphs on n vertices for any k n with attention given to the number of basis blade multiplications required. Discussed in detail are the worst-case and average-case blade-level time complexity of counting cycles with operator calculus methods. For reference, experimental results detailing computation times (in seconds) are included alongside similar computations performed with algorithms based on the approaches of Bax and Tarjan. Symbolic computations are the focus of Chapters 16 and 17. In Chapter 16, an innovative approach to minimal path algorithms based on operator calculus in graded semigroup algebras is described. Classical approaches to routing problems invariably require construction of trees and the use of heuristics to prevent combinatorial explosion. The operator calculus approach presented herein, however, allows such explicit tree constructions to be avoided. Moreover, the implicit tree structures underlying the problem are pruned automatically by the inherent properties of the semigroup algebras used in this approach. The operator calculus algorithm is applied to the problem of precomputed routing in a store-and-forward satellite constellation, which provides message communication services by relaying messages between satellites through gateways on the ground. Chapter 17 contains a user s guide for Mathematica packages devoted to Clifford algebra computations and operator calculus. These

6 8 Operator Calculus on Graphs: Theory and Applications in Computer Science packages are available online at the second-named author s web page, Notational Preliminaries The following notational conventions are used throughout the book. Uppercase Roman characters (e.g. X, V, I, etc.) are used to denote sets, vector spaces, and matrices. Given a set X, the power set of X is denoted 2 X. Set cardinality is typically denoted by X when X is a set, although the alternative notation { } is sometimes used in conjunction with set-builder notation. The identity operator and identity matrices are denoted I. Lowercase Roman characters in bold font (e.g. u, x) typically denote vectors. Given a positive integer n, the n-set is defined as [n] = {1,..., n}. The set of natural numbers (positive integers) is denoted by N, while the set of nonnegative integers N {0} is denoted by N 0. Uppercase Roman characters appearing as subscripts denote multiindices. That is, given a subset I [n] and an indexed collection {v 1,..., v n } on which some binary operation is defined, the notation v I denotes the ordered product v I = l I v l. When a lower case Roman character represents a nonnegative integer, its underlined counterpart represents the integer s binary subset representation. That is, i is defined by the relationship i N 0 i = l i 2 l. Consequently, i = 0 i =. The anticommutator is defined on pairs of elements in a ring or algebra by {a, b} + := ab + ba.

7 Introduction 9 The commutator is defined on pairs of elements in a ring or algebra by [a, b] := ab ba. The Kronecker delta function is defined for pairs of nonnegative integers by { 1 i = j, δ ij = (1.1) 0 otherwise. Recalling Dirac notation in a vector space V, an element u V is denoted u, while the corresponding element of the dual space V is denoted u. Let X and Y be algebraic structures of the same type (groups, semigroups, rings, fields, etc.). The notation X < Y is used to indicate X is a substructure (subgroup, subring, etc.) of Y. Let G and H be groups. If H is a normal subgroup of G, then H G.