Introduction aux Systèmes Collaboratifs Multi-Agents

Transcription

1 M1 EEAII - Découverte de la Recherche (ViRob) Introduction aux Systèmes Collaboratifs Multi-Agents UPJV, Département EEA Fabio MORBIDI Laboratoire MIS Équipe Perception et Robotique fabio.morbidi@u-picardie.fr Jeudi 13h30-16h30, Salle 8 Année Universitaire 2015/2016

2 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) Remark: i.e. = id est (latin) = à savoir, c est à dire e.g. = exempli gratia (latin) = par exemple 2

3 Graph theory A finite, undirected, simple graph, or graph for short, is built upon a finite set: the vertex (or node) set. We also define the edge set: It consists of elements of the form or such that The graph is then formally defined as the pair : 3

4 Graph theory Example When an edge exists between vertices and, they are called adjacent and denoted. In this case is called incident with vertices The neighborhood of vertex is the set i.e., the set of all vertices adjacent to. In the example: 4

5 Graph theory A path of length m in is given by a sequence of distinct vertices such that for, the vertices and are adjacent. 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) Cycle 3 5

6 Graph theory Example 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 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 6

7 Standard graphs : Complete or fully connected graph Each vertex is adjacent to every other vertex Example: (7 vertices) 7

8 Standard graphs : Path graph where if and only if Example: : n-cycle where if and only if Example: Note: (i.e. and are congruent modulo n) if is an integer multiple of n 8

9 Standard graphs : Star graph where if and only if Examples: 9

10 Generalizations of the notion of graph Directed graphs (digraphs) 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. Orientation +1 Orientation -1 10

11 Generalizations of the notion of graph Directed graphs (digraphs) Example: or 11

12 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 12

13 Graphs and matrices 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: 13

14 Degree and adjacency matrix The degree matrix of a graph is the n x n diagonal matrix containing the vertex-degree of on the diagonal: The adjacency matrix is the symmetric n x n matrix encoding the adjacency relationships in the graph : 14

15 Degree and adjacency matrix Example: Symmetric matrix, 15

16 Degree and adjacency matrix Exercise: Compute and for G = S 6, G = C identity matrix

17 Incidence matrix Under the assumptions that labels have been associated with edges in a graph whose edges have been arbitrarily oriented, the n x m incidence matrix is defined as (n = number of vertices; m = number of edges): Remark: captures not only the adjacency relationships, but also the orientation that the graph now enjoys 17

18 Incidence matrix Example: Edges Vertices 18

19 Incidence matrix 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 19

20 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 Pierre S. Laplace Example: where 20

21 Graph Laplacian: alternative definition 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 21

22 Weighted graph Laplacian Given a weighted graph the weighted graph Laplacian is defined as: where 22

23 Weighted Laplacian for digraphs 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. The corresponding (in-degree) weighted Laplacian is defined as: 23

24 Weighted Laplacian for digraphs Example: Not symmetric!! 24

25 Weighted Laplacian for digraphs Exercise: Compute, and for the digraph reported below:

26 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 Recall that. 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 A by solving the equation: 26

27 Algebraic and spectral graph theory Consider the graph Laplacian. This matrix is symmetric and positive semidefinite, hence its n real eigenvalues can be ordered as: Theorem: The graph is connected if and only if 27

28 Spectrum of the Laplacian matrix To find the Laplacian spectrum {λ 1 (L), λ 2 (L),...,λ n (L)} of an arbitrary graph is not trivial For some special graphs we can easily compute the eigenvalues (and associated eigenvectors) Complete graph Since the spectrum of is that of shifted by n. Since the spectrum of the matrix is the Laplacian spectrum of is n-cycle The Laplacian spectrum of is 28

29 Algebraic and spectral graph theory is the second smallest eigenvalue of the Laplacian is called Fiedler value and the associated eigenvector is called Fiedler vector Remark: Miroslav Fiedler 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: the consensus protocols 29