Table Of ContentArmen H. Zemanian
Transfiniteness
For Graphs,
Electrical Networks,
and Random Walks
1996
Springer Science+Business Media, LLC
Armen H. Zemanian
Department of Electrical Engineering
State University of New York at Stony Brook
Stony Brook, NY 11794-2350
USA
Library of Congress Cataloging In-Publication Data
Zemanian, A. H. (Armen H.)
Transfiniteness for Graphs, Electrica! Networks, and Random Walks 1 Armen H.
Zemanian.
p. cm.
Includes bibliographical references and index.
1. Graph theory. 2. Transfinite Numbers. 1. Title.
QA166.Z46 1995 95-24601
511'.5--dc20 CIP
Printed on acid-free paper i5
Birkhăuser
© 1996 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 1996
Softcover reprint ofthe hardcover 1st edition 1996
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ISBN 978-1-4612-6894-9 ISBN 978-1-4612-0767-2 (eBook)
DOI 10.1007/978-1-4612-0767-2
9 8 7 6 5 4 3 2 1
CONTENTS
Preface vii
Acknowledgments
X
1 Introduction 1
1.1 Notations and Terminology . . . . . 1
1.2 Conventional Graphs - A Poor Start 5
1.3 0-Graphs ...... . 9
1.4 Why Transfiniteness? 12
2 Transfinite Graphs 19
2.1 1-Graphs . 20
2.2 f.L-Graphs . 29
2.3 w-Graphs . 36
2.4 w-Graphs . 42
2.5 Graphs of Higher Ranks 45
3 Connectedness 47
3.1 Transfinite Connectedness 48
3.2 A Transitivity Criterion for Transfinite Connectedness 53
3.3 About Nodes and Paths . . . . . . . . . . . . . . . . 57
3.4 v-Sequences . . . . . . . . . . . . . . . . . . . . . 60
3.5 Another Transitivity Criterion for Transfinite Connectedness 71
3.6 The Cardinality of the Branch Set . . . . . . . . . . . . . 74
4 Finitely Structured Transfinite Graphs 79
4.1 Subsection and Cores . . . . . . . 81
4.2 A Generalization of Konig's Lemma . 87
4.3 Isolating Sets and Cuts . . . 93
4.4 Contractions . . . . . . . . 100
4.5 Finitely Structured v-Graphs 104
4.6 Spanning trees . . . . . . . 109
5 Transfinite Electrical Networks 115
5.1 Kirchhoff's Laws, Tellegen's Equation, and Ohm's Law 116
5.2 The Voltage-Current Regime . . . . . . . . 126
5.3 Conditions Legitimizing Kirchhoff's Laws . . . . . . . 138
vi CONTENTS
5.4 The Regime Induced by a Pure Current Source. 143
5.5 Node Voltages . . . . . . . . . . . . . 149
6 Permissively Finitely Structured Networks 157
6.1 Permissively Finitely Structured v-Networks . 158
6.2 Excitations by Pure Sources . . . . . . 160
6.3 Maximum Principles for Node Voltages
and Branch Currents . . . . . . . . . . . . . . . . 170
7 Transfinite Random Walks 183
7.1 The Nash-Williams Rule 184
7.2 Transfinite Walks . . . . . . . . . . . . . . . . . 189
7.3 Transfinite Random Walks - The General Idea . 196
7.4 Reaching a Bordering Node ....... . . 202
7.5 Leaving a ,8-Node ............ . . 207
7.6 From a ,8-Node to a ,8-Adjacent (,8+ )-Node . 214
7.7 Wandering on a v-Network ........ . . 215
Appendix A: Ordinal and Cardinal Numbers . . . . . . . . 222
Appendix B: Summable Series ............... 229
Appendix C: Irreducible and Reversible Markov Chains . . 232
Bibliography . . . . 235
Index of Symbols . 239
Index . . . . . . . . 241
PREFACE
"What good is a newborn baby?"
Michael Faraday's reputed response when asked,
"What good is magnetic induction?"
But, it must be admitted that a newborn baby may die in infancy. What
about this one-the idea of transfiniteness for graphs, electrical networks,
and random walks? At least its bloodline is robust. Those subjects, along
with Cantor's transfinite numbers, comprise its ancestry.
There seems to be general agreement that the theory of graphs was born
when Leonhard Euler published his solution to the "Konigsberg bridge prob
lem" in 1736 [8]. Similarly, the year of birth for electrical network theory
might well be taken to be 1847 , when Gustav Kirchhoff published his volt
age and current laws [ 14]. Ever since those dates until just a few years ago,
all infinite undirected graphs and networks had an inviolate property: Two
branches either were connected through a finite path or were not connected
at all. The idea of two branches being connected only through transfinite
paths, that is, only through paths having infinitely many branches was never
invoked, or so it appears from a perusal of various surveys of infinite graphs
[17], [20], [29], [32]. Our objective herein is to explore this idea and some
of its ramifications. It should be noted however that directed graphs having
transfinite paths have appeared in set theory [6, Section 4.4 and Chapter
7], but, by virtue of that directedness and the sets the graphs represent, the
question of connections at infinite extremities of graphs is either avoided or
trivially resolved. This question is however a substantial issue in our theory
of undirected arbitrary transfinite graphs and is resolved by the invention
of a new kind of node.
Research on graphs, electrical networks, and random walks continues
to thrive, and new ideas for them are proliferating. It therefore is incum
bent upon us to argue why still another may be of interest. One reason is
offered in Section 1.4, where it is shown that a transfinite network is the
natural outcome of an effort to close a heretofore unaddressed lacuna in
the theory of infinite electrical ladders. A more compelling consideration
is that transfiniteness introduces radically new constructs and expands the
three aforementioned topics far beyond their conventional boundaries. For
instance, a transfinite graph is not a graph in any prior sense but is instead an
Vlll PREFACE
extension roughly analogous to Cantor's extension of the natural numbers
to the transfinite ordinals. It is undoubtedly way too much to expect that the
introduction of transfinite graphs will eventually be as important as Cantor's
invention of the transfinite numbers. However, even a small fraction of that
success would be ample justification for our endeavor.
Transfiniteness for undirected graphs and electrical networks was first
introduced through some embryonic ideas in [33] and [34] and then in all
its generality in [35, Chapters 3 and 5] and [37]. This book expands upon
those prior works with a variety of new results. It is in fact a sequel to
"Infinite Electrical Networks" [35] but can be read independently; there is
no need to read that book before embarking on this one. All definitions
and results within the mainstream of this present book are thoroughly de
veloped herein. Some peripheral ideas that broaden our discussions but are
not critical to them are borrowed from several sources, including "Infinite
Electrical Networks." Moreover, most of that book was restricted to con
ventional infinite networks, and the lesser part of it on transfinite networks
focussed upon the first rank of transfiniteness, that is, upon "!-networks."
Now, however, we exclusively examine transfinite graphs and networks with
due attention to higher ranks of transfiniteness. Moreover, we expand our
theory considerably with the introduction of several new topics related to
connectedness problems, discrete potential theory, uncountable networks,
and random walks.
A critical issue in the development of transfinite graphs and networks
was the choice of appropriate definitions. Much effort was expended in
pursuing likely possibilities, many of which failed because of inherent con
tradictions or unmanageable entanglements. An example of the latter is
described in Section 1.2. The definitions we have chosen appear to be
free of those obstacles and yet sufficiently general for our purposes. To be
sure, we could simplify a few definitions by weakening them, but this again
leads to complications because of the variety of situations that then arise.
We believe that we have arrived at a satisfactory compromise between the
complexity of definitions and the complexity of subsequent discussions.
On the other hand, we introduce many new concepts. This is due to
the unavoidable circumstance that transfinite undirected graphs are multi
faceted structures and entirely novel. We cannot borrow from established
theories but instead must construct everything we need. This is exacerbated
by the many ranks of trans finiteness, one for each countable ordinal. Were
we to restrict ourselves to the first rank, that is, to "1-graphs," the number
of definitions would be comparable to that for conventional graphs. But
the extensions to higher ranks through recursion expand that number sev
eralfold, especially since some of the definitions needed for a limit-ordinal
PREFACE ix
rank are significantly different from those for a successor-ordinal rank.
Almost all the ideas about transfinite graphs introduced in "Infinite
Electrical Networks" are employed herein without alteration, but there are
a few differences. One of them concerns the specification of a path, which
previously involved some ambiguity-with the result that a path could be
represented in different ways. The tighter definition we now use prevents
this. A related matter is that we now make a distinction between a path
"reaching" a node and "meeting" a node, the former concept being weaker
than the latter. Still another alteration is that we abandon the idea of a
"reduced graph" and use instead a "subgraph." A reduced graph is obtained
from a given transfinite graph by choosing just some of the branches and in
terconnecting them in accordance with the given graph. It is so defined that
a reduced graph is a transfinite graph too, but this requires the introduction
of other nodes. A subgraph uses the same interconnections of the chosen
branches but does not proliferate nodes; however, it is in general some
thing other than a transfinite graph. This replacement of reduced graphs by
subgraphs simplifies our theory.
Our book is arranged as follows. Chapter 1 is introductory. After ex
plicating notations and terminology in Section 1.1, we argue why transfinite
graphs and networks should be of interest and how they are needed to close
a gap in the theory of conventional infinite networks. We also indicate why
we adopt an unusual approach to conventional graphs in order to enable a
simpler transfinite generalization.
Transfinite graphs are treated in Chapters 2 through 4, electrical net
works in Chapters 5 and 6, and random walks in Chapter 7.
Chapter 2 is devoted almost entirely to the many definitions needed for
the transfiniteness of graphs. Here - and indeed throughout this book -
we illustrate new concepts with many examples. Connectedness problems
are taken up in Chapter 3. We show that transfinite connectedness need
not be transitive as a binary relation between branches. Conditions under
which transitivity does hold are then obtained. Chapter 3 closes with a
discussion of the cardinality of the branch set of a transfinite graph. In
Chapter 4 we develop the idea of "finite structuring" for transfinite graphs,
which mimics local-finiteness for conventional infinite graphs. This will be
needed when we discuss discrete potential theory and random walks. Here
we also present a transfinite generalization of Konig's classical lemma on
the existence of one-way infinite paths.
Chapter 5 presents some fundamental theorems asserting the existence
and uniqueness of voltage-current regimes in transfinite electrical networks
under very broad conditions. No restrictions on the transfinite graph of the
network are imposed. These theorems are based on a generalization of Tel-
PREFACE
X
legen's equation, since Kirchhoff's laws are no longer inviolate principles
in the transfinite case. Furthermore, node voltages need not be unique nor
even exist. The chapter ends by establishing additional conditions under
which node voltages are uniquely existent. Maximum principles for node
voltages and branch currents are established in Chapter 6. This last result
makes full use of the finite structuring developed in Chapter 4 and therefore
holds for a class of transfinite networks that is smaller than that for the
fundamental theorems.
Finally, a theory for random walks on a finitely structured transfinite
network is established in Chapter 7. The theory extends nearest-neighbor
random walks by allowing them to pass through transfinite nodes. This
is accomplished by exploiting the relationship between electrical network
theory and random-walk phenomena.
The appendices present brief surveys of three topics of importance to
our theory, namely ordinal and cardinal numbers, summable series, and
irreducible and reversible Markov chains. With this information our book
should be accessible to anyone familiar with the basic ideas about graphs,
Hilbert spaces, and resistive electrical networks.
ACKNOWLEDGMENTS. Much of the research on the subject of this
book was funded by the U.S. National Science Foundation; that support
is gratefully acknowledged. I am also obliged to the National Research
Council of Italy for sponsoring a series of lectures at the University of Milan
and the University of Rome "Tor Vergata." These presentations, along with
a special-topics course given at the University at Stony Brook, allowed me
to test early versions of the manuscript. In particular, my discussions with
Professors L. DeMichele, M. Picardello, P.M. Soardi, and W. Woess, and
also with the students of that course, namely A. Alexeyev, Y.R. Chan, V.A.
Chang, J.K. Fwu, H.T. Li, and H. Sundaram, led to many revisions and
improvements in this book.
Finally, to my wife Edna, for her unfailing patience, encouragement,
and support, once again and always -thanks.
