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Graph and Network Theory
Ernesto Estrada
Department of Mathematics and Statistics
University of Strathclyde, Glasgow
Introduction ................................................................................................................................ 2
1 The language of graphs and networks .................................................................................... 3
1.1 Graph operators ................................................................................................................ 3
1.2 General graph concepts .................................................................................................... 5
1.3 Types of graphs ................................................................................................................ 6
2 Graphs in condensed matter physics ....................................................................................... 7
2.1 Tight-binding models ....................................................................................................... 7
2.1.1 Nullity and zero-energy states ................................................................................... 9
2.2 Hubbard model ............................................................................................................... 10
3 Graphs in statistical physics .................................................................................................. 12
4 Feynman graphs .................................................................................................................... 16
4.1 Symanzik polynomials and spanning trees .................................................................... 17
4.2 Symanzik polynomials and the Laplacian matrix .......................................................... 20
4.3 Symanzik polynomials and edge deletion/contraction ................................................... 21
5 Graphs and electrical networks ............................................................................................. 21
6 Graphs and vibrations ........................................................................................................... 23
6.1 Graph vibrational Hamiltonians ..................................................................................... 24
6.2 Network of Classical Oscillators .................................................................................... 24
6.3 Network of Quantum Oscillators ................................................................................... 26
7 Random graphs ..................................................................................................................... 28
8 Introducing complex networks ............................................................................................. 30
9 Small-World networks .......................................................................................................... 32
10 Degree distributions ............................................................................................................ 34
10.1 ‘Scale-free’ networks................................................................................................... 36
11 Network motifs ................................................................................................................... 37
12 Centrality measures ............................................................................................................. 38
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13 Statistical mechanics of networks ....................................................................................... 41
13.1 Communicability in networks ...................................................................................... 42
14 Communities in networks ................................................................................................... 43
15 Dynamical processes on networks ...................................................................................... 45
15.1 Consensus ..................................................................................................................... 45
15.2 Synchronization in networks ........................................................................................ 47
15.3 Epidemics on networks ................................................................................................ 48
Glossary ................................................................................................................................... 50
List of works cited ................................................................................................................... 51
Further reading ......................................................................................................................... 53
Introduction
Graph Theory was born in 1736 when Leonhard Euler published “Solutio problematic
as geometriam situs pertinentis” (The solution of a problem relating to the theory of position)
(Euler, 1736). This history is well documented (Biggs et al., 1976) and widely available in
any textbook of graph or network theory. However, the word graph appeared for the first time
in the context of natural sciences in 1878, when the English mathematician James J. Sylvester
wrote a paper entitled “Chemistry and Algebra” which was published in Nature (Sylvester,
1877-78), where he wrote that “Every invariant and covariant thus becomes expressible by a
graph precisely identical with a Kekulean diagram or chemicograph”. The use of graph
theory in condensed matter physics, pioneered by many chemical and physical graph theorists
(Harary, 1968; Trinajstić, 1992), is today well established; it has become even more popular
after the recent discovery of graphene.
There are few, if any, areas of physics in the XXIst century in which graphs and
network are not involved directly or indirectly. Hence it is impossible to cover all of them in
this Chapter. Thus I owe the reader an apology for the incompleteness of this Chapter and a
promise to write a more complete treatise. For instance, quantum graphs are not considered in
this Chapter and the reader is referred to a recent introductory monograph on this topic for
details (Berkolaiko, Kuchment, 2013). In this chapter we will cover some of the most
important areas of applications of graph theory in physics. These include condensed matter
physics, statistical physics, quantum electrodynamics, electrical networks and vibrational
problems. In the second part we summarise some of the most important aspects of the study
of complex networks. This is an interdisciplinary area which has emerged with tremendous
impetus in the XXIst century which studies networks appearing in complex systems. These
systems range from molecular and biological to ecological, social and technological systems.
Thus graph theory and network theory have helped to broaden the horizons of physics to
embrace the study of new complex systems.
We hope this chapter motivates the reader to find more about the connections between
graph/network theory and physics, consolidating this discipline as an important part of the
curriculum for the physicists of the XXIst century.
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1 The language of graphs and networks
The first thing that needs to be clarified is that the terms graphs and networks are used
indistingtly in the literature. In this Chapter we will reserve the term graph for the abstract
mathematical concept, in general referred to small, artificial formations of nodes and edges.
The term network is then reserved for the graphs representing real-world objects in which the
nodes represent entities of the system and the edges represent the relationships among them.
Therefore, it is clear that we will refer to the system of individuals and their interactions as a
‘social network’ and not as a ‘social graph’. However, they should mean exactly the same.
For the basic concepts of graph theory the reader is recommended to consult the
introductory book by Harary (1967). We start by defining a graph formally. Let us consider a
finite set V v1 , v2 ,, vn of unspecified elements and let V V be the set of all ordered
pairs vi , v j of the elements of V . A relation on the set V is any subset E V V . The
relation E is symmetric if vi , v j E implies v j , vi E and it is reflexive if
v V , v, v E . The relation E is antireflexive if vi , v j E implies vi v j . Now we can
define a simple graph as the pair G V , E , where V is a finite set of nodes, vertices or
points and E is a symmetric and antireflexive relation on V , whose elements are known as
the edges or links of the graph. In a directed graph the relation E is non-symmetric. In
many physical applications the edges of the graphs are required to support weights, i.e., real
numbers indicating a specific property of the edge. In this case the following more general
definition is convenient. A weighted graph is the quadruple G V , E,W , f where V is a
finite set of nodes, E V V e1, e2 ,, em is a set of edges, W w1 , w2 ,
, wr is a set of
weights such that wi and f : E W is a surjective mapping that assigns a weight to
each edge. If the weights are natural numbers then the resulting graph is a multigraph in
which there could be multiple edges between pairs of vertices. That is, if the weight between
nodes p and q is k N it means that there are k links between the two nodes.
In an undirected graph we say that wo nodes p and q are adjacent if they are joined
by an edge e p, q . In this case we say that the nodes p and q are incident to the link e ,
and the link e is incident to the nodes p and q . The two nodes are called the end nodes of
the edge. Two edges e1 p, q and e2 r , s are adjacent if they are both incident to at
least one node. A simple but important characteristic of a node is its degree, which is defined
as the number of edges which are incident to it or similarly the number of nodes adjacent to
it. Slightly different definitions apply to directed graphs. The node p is adjacent to node q if
there is a directed link from p to q , e p, q . We also say that a link from p to q is
incident from p and incident to q ; p is incident to e and q is incident from e .
Consequently, we have two different kinds of degrees in directed graphs. The in-degree of a
node is the number of links incident to it and its out-degree is the number of links incident
from it.
1.1 Graph operators
The incidence and adjacency relations in graphs allow us to define the following graph
operators. We consider an undirected graph for which we construct its incidence matrix with
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an arbitrary orientation of its entries. This is necessary to consider that the incidence matrix is
a discrete analogous of the gradient. That is, for every edge p, q , p is the positive (head)
and q the negative (tail) end of the oriented link. Let the links of the graph be labeled as
e1 , e2 ,, em . Hence the oriented incidence matrix G :
1 node vi is the head of link e j
ij G 1 node vi is the tail of link e j
0 otherwise
We remark that the results obtained below are independent of the orientation of the
links but assume that once the links are oriented, this orientation is not changed. Let the
vertex LV and edge LE spaces be the vector spaces of all real-valued functions defined on V
and E , respectively. The incidence operator of the graph is then defined as
G : LV LE ,
(1.1)
such that for an arbitrary function f : V , G f : E is given by
(1.2)
G f e f p f q ,
where p are the starting (head) and q the ending (tail) points of the oriented link e . Here
we consider that f is a real or vector-valued function on the graph with f
being -
measurable for certain measure on the graph.
On the other hand, let H be a Hilbert space with scalar product , and norm . Let
G V , E be a simple graph. The adjacency operator is an operator acting on the Hilbert
space H : l 2 V defined as
(1.3)
Af p : f q , f H , i V .
u ,vE
The adjacency operator of an undirected network is a self-adjoint operator, which is
bounded on l 2 V . We recall that l 2 is the Hilbert space of square summable sequences with
inner product, and that an operator is self-adjoint if its matrix is equal to its own conjugate
transpose, i.e., it is Hermitian. It is worth pointing out here that the adjacency operator of a
directed network might not be self-adjoint. The matrix representation of this operator is the
adjacency matrix A , which for a simple graph is defined as
Aij 1 if i, j E
(1.4)
0 otherwise.
A third operator which is related to the previous two and which plays a fundamental
role in the applications of graph theory in physics is the Laplacian operator. This operator is
defined by
L G f f ,
(1.5)
and it is the graph version of the Laplacian operator
2 f 2 f
2 f
.
(1.6)
f
2
2
2
x1 x2
xn
The negative sign in (1.5) is used by convention. Then the Laplacian operator acting on
the function f previously defined is given by
LG f u f u f v,
(1.7)
u ,v E
which in matrix form is given by
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