Graph theory

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Institute for Design and Control of Mechatronical Systems

Introduction aux Systèmes Collaboratifs Multi-Agents

UPJV, Département EEA

Fabio MORBIDI

Laboratoire MIS

Équipe Perception Robotique E-mail: [email protected]

Année Universitaire 2016/2017

Jeudi 13h30-16h30, Salle 8

M1 EEAII - Découverte de la Recherche (ViRob)

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Graph theory: introduction

Graphs provide natural abstractions for how information is shared between agents in a network

The graph-based abstraction contains high-level descriptions of the network topology in terms of objects referred to as vertices and edges

In the next slides we will see:

Basic notions of:

Algebraic graph theory (adjacency, incidence, Laplacian matrices) Spectral graph theory (spectrum of the Laplacian)

i.e. = id est (latin) = à savoir, c’est à dire e.g. = exempli gratia (latin) = par exemple Remark:

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Institute for Design and Control of Mechatronical Systems

Harald Kirchsteiger 2011/04

Graph theory

A finite, undirected, simple graph, or graph for short, is built upon a finite set:

3

The graph is then formally defined as the pair :

the vertex (or node) set.

We also define the edge set:

It consists of elements of the form or such that

Fabio Morbidi

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Graph theory

When an edge exists between vertices and , they are called adjacent and denoted . In this case is called incident with vertices Example

The neighborhood of vertex is the set i.e., the set of all vertices adjacent to . In the example:

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Graph theory

A path of length

m

in is given by a sequence of distinct vertices

such that for , the vertices and are adjacent.

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When the vertices of the path are distinct except for its end vertices, the path is called a cycle.

A graph is called connected if for every pair of vertices in , there is a path that has them as end vertices. Otherwise the graph is called

disconnected (e.g. the graph below is connected).

Fabio Morbidi

Cycle

1

2

3 5 4

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Petite historique

The problem of the “7 bridges of Königsberg“ : first studied in graph theory Find a walk through the city that crosses each bridge once and only once

(i.e. find an Eulerian cycle in a graph)

Negative answer by Leonhard Euler in 1735

An Eulerian cycle exists if the graph is connected and has exactly zero or two nodes of odd degree (in the problem of Königsberg, 4 nodes have odd degree)

Leonhard Euler

Graph theory

(1707-1783)

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Harald Kirchsteiger 2011/04 7

: Complete or fully connected graph Each vertex is adjacent to every other vertex Example:

Fabio Morbidi

(7 vertices)

Standard graphs

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Standard graphs

: Path graph

where if and only if Example:

:

n

-cycle

where if and only if Example:

Note:

if is an integer multiple of n (i.e. and are congruent modulo n)

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Standard graphs

9

: Star graph

where if and only if

Examples:

Fabio Morbidi

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Generalizations of the notion of graph

When the edges in a graph are given directions, the resulting interconnection is no longer considered an undirected graph.

A directed graph (or digraph), denoted by can be obtained in two ways:

1. Drop the requirement that the edge set

E

contains unordered pairs of vertices

If the ordered pair then is said to be the tail (where the arrow starts) of the edge, while is its head (where the arrow ends) 2. Associate an orientation to the unordered edge set Directed graphs (digraphs)

Orientation +1 Orientation -1

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Generalizations of the notion of graph

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Directed graphs (digraphs) Example:

or

Fabio Morbidi

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Generalizations of the notion of graph

Directed graphs (digraphs)

The notions of adjacency, neighborhood, and connectness can be extended in the context of digraphs.

For example, a directed path of length m in is given by the sequence of distinct vertices:

such that for the vertices

A digraph is called:

a) Strongly connected: if for every pair of vertices there is a directed path between them

b) Weakly connected: if it is connected when viewed as a graph, i.e.

a “disoriented“ graph

For example, the graph in the previous slide is weakly connected but not strongly connected

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Graphs and matrices

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Graphs admit a representation in terms of matrices, beside a graphical representation in terms of vertices and edges.

For an undirected graph , the degree of a given vertex, , is the cardinality of the neighborhood set , i.e., the number of vertices that are adjacent to vertex in .

Example:

Fabio Morbidi

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Degree and adjacency matrix

The degree matrix of a graph is the

n × n

diagonal matrix containing the vertex-degree of on the diagonal:

The adjacency matrix is the symmetric

n × n

matrix encoding the adjacency relationships in the graph :

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Degree and adjacency matrix

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Example:

Symmetric matrix

Fabio Morbidi

,

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Exercise:

Compute and for

1

2 3 4

5

6

1

2 3 4

5

6

×

6 identity matrix

Degree and adjacency matrix

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Incidence matrix

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Remark:

captures not only the adjacency relationships, but also the orientation that the graph now enjoys

Under the assumptions that labels have been associated with edges in a graph whose edges have been arbitrarily oriented, the

n × m

incidence matrix is defined as (

n

= number of vertices;

m

= number of edges):

Fabio Morbidi

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Example:

Vertices

Edges

Incidence matrix

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Incidence matrix

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Remarks:

has a column sum equal to zero.

The incidence matrix of a digraph can be defined analogously by skipping the pre-orientation that is needed for graphs.

• The incidence matrix is denoted in this case by

Fabio Morbidi

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Graph Laplacian

The graph Laplacian associated with an undirected graph is:

From this definition, it follows that for all graphs, the rows of the Laplacian sum zero

Example:

Pierre-S. Laplace

where

(1749-1827)

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Graph Laplacian: alternative definition

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Given an arbitrary orientation to the edge set , the graph Laplacian of can be alternatively defined as:

This definition reveals that is a:

• Symmetric matrix (i.e. )

• Positive semidefinite matrix (i.e. )

The two definitions that we have given are equivalent and since no notion of orientation is needed in the first one, the graph Laplacian is orientation independent

Fabio Morbidi

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Weighted graph Laplacian

Given a weighted graph the weighted graph Laplacian is defined as:

where

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Weighted Laplacian for digraphs

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Let be a weighted digraph.

For the adjacency matrix, we let:

and for the diagonal degree matrix , we set:

where

is the weighted in-degree of vertex .

Fabio Morbidi

The corresponding (in-degree) weighted Laplacian is defined as:

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Weighted Laplacian for digraphs

Example:

5 2

7

Not symmetric !!

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Weighted Laplacian for digraphs

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Exercise:

3 1

4

Compute , and for the digraph reported below:

2

Fabio Morbidi

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Algebraic and spectral graph theory

Algebraic graph theory associates algebraic objects (e.g., the degree, adjacency, incidence and Laplacian matrices) to graphs Spectral graph theory studies the eigenvalues associated to the

adjacency and Laplacian matrices

Definition

A nonzero vector is a eigenvector of a matrix if and only if there exists a scalar such that:

where is called eigenvalue associated to .

We find the eigenvalues and eigenvectors of

by solving the equation:

Recall that ….

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Algebraic and spectral graph theory

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Consider the graph Laplacian . This matrix is symmetric and positive semidefinite, hence its

real eigenvalues can be ordered as:

Theorem:

The graph is connected if and only if

Fabio Morbidi

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Spectrum of the Laplacian matrix

• To find the Laplacian spectrum of an arbitrary graph is not trivial

• For some special graphs we can easily compute the eigenvalues (and the associated eigenvectors)

Since the spectrum of is that of shifted by

n

. Since the spectrum of the matrix is

the Laplacian spectrum of is:

Complete graph

n

^-cycle

The Laplacian spectrum of is:

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Algebraic and spectral graph theory

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is the second smallest eigenvalue of the Laplacian is called Fiedler value and the associated eigenvector is called Fiedler vector

The Fiedler value is important not only as a measure of robustness or level of connectedness of a graph, but also for the convergence properties of a collection of distributed coordination algorithms:

Remark:

Miroslav Fiedler

Fabio Morbidi

Consensus protocols

(1926 – 2015)