Non-negative matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Connectivity Non-negative matrices and graphs MAA704 and the adjacency matrix Distance Christopher Engstr¨om matrix Shortest path and Dijkstra's algorithm Degree matrix November 20, 2018 of a graph Markov chains 1 / 50 Non-negative MAA704 Applied Matrix Analysis matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Connectivity and the Todays lecture: adjacency matrix I Non-negative matrices and introduction to graph theory. Distance matrix I Connectivity and Irreducibility. Shortest path and Dijkstra's I The Laplacian matrix of a graph algorithm Degree matrix of a graph Markov chains 2 / 50 Non-negative Non-negative matrices matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Connectivity A non-negative matrix is a matrix A where all elements and the I adjacency ai;j ≥ 0. matrix Distance I A positive matrix is a matrix A where all elements ai;j > 0. matrix Remark. Be careful to distinguish between Shortest path I and Dijkstra's positive/non-negative matrices and positive algorithm > Degree matrix definite/semi-definite matrices (recall x Ax). of a graph Markov chains 3 / 50 Non-negative Non-negative matrices in Graph-theory matrices and graphs MAA704 A graph is a collection of vertices (nodes) and edges (links) Christopher such as the one below. Engstr¨om lecture 2 Connectivity and the adjacency B matrix Distance matrix Shortest path and Dijkstra's algorithm A C Degree matrix of a graph Markov chains D 4 / 50 Non-negative Non-negative matrices in Graph-theory matrices and graphs MAA704 Christopher Engstr¨om There are many types of graphs those we will work with are: lecture 2 I Simple graphs, where we only allow a single edge from Connectivity and the one vertex to another. adjacency matrix Undirected graphs, where edges do not have a direction I Distance so we are only interested in if there is a edge between two matrix Shortest path vertices. and Dijkstra's algorithm I Directed graphs, where edges do have a direction, a edge Degree matrix A ! B does not necessary mean there is a edge B ! A. of a graph I Weighted graph, where we assign (positive) weights as Markov chains scalars to every edge. 5 / 50 Non-negative Non-negative matrices in Graph-theory matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Connectivity and the How can we represent a (simple) graph using a matrix? adjacency matrix Multiple ways such as the adjacency matrix, degree matrix, I Distance distance matrix, incidence matrix, Laplacian matrix, etc. matrix Shortest path I Which graph type and which matrix representation you and Dijkstra's use depend on your application. algorithm Degree matrix of a graph Markov chains 6 / 50 Non-negative Application, road network and connectivity matrices and graphs MAA704 Christopher We consider a road network between different cities where we Engstr¨om want to know if we can drive from one city to another, or if we lecture 2 need to go by for example air/boat. Connectivity and the I We represent the road network using a undirected graph adjacency by letting the vertices represent the cities, and a edge matrix Distance between two cities means there is a road between the two matrix cities. Shortest path and Dijkstra's I If there is a edge between, say, two cities A; B we know algorithm there is a road and we can take the car. However if there Degree matrix of a graph isn't we might still be able to take the car for example by Markov chains passing city C which is both connected to cities A and B. I To solve this problem we define connectivity for graphs, and irreducibility for matrices. 7 / 50 Non-negative Connectivity matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Connectivity I Two vertices are said to be connected if by traversing and the adjacency the edges there exists a path from both of the nodes to matrix the other one. Distance matrix A graph is said to be connected if it is connected for all I Shortest path pair of nodes in the undirected graph. and Dijkstra's algorithm I A directed graph is said to be strongly connected if it is Degree matrix connected for all pair of nodes in the directed graph. of a graph Markov chains 8 / 50 Non-negative Connectivity: connected components matrices and graphs MAA704 Christopher A connected component in an undirected graph is a Engstr¨om maximal part of the graph where all nodes are connected with lecture 2 each other. Connectivity and the adjacency matrix Distance matrix Shortest path and Dijkstra's algorithm Degree matrix of a graph Markov chains Figure: A undirected graph with 3 connected components 9 / 50 Non-negative Connectedness: strongly connected components matrices and graphs MAA704 Christopher A strongly connected component is a part of the graph Engstr¨om which is strongly connected. lecture 2 Connectivity and the adjacency matrix Distance matrix Shortest path and Dijkstra's algorithm Degree matrix of a graph Markov chains Figure: A directed graph with 3 strongly connected components 10 / 50 Non-negative Application, road network and adjacency matrix matrices and graphs MAA704 Christopher Engstr¨om lecture 2 So if our two cities belong to the same connected component, Connectivity and the there is a path between them and we can take the car. adjacency matrix I But how do we find the connected components? Distance Especially if we have a large graph it might even be hard matrix Shortest path to visualize the graph. and Dijkstra's algorithm To solve the problem we use a matrix representation called I Degree matrix the adjacency matrix. of a graph Markov chains 11 / 50 Non-negative Application, road network and adjacency matrix matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Definition Connectivity and the The adjacency matrix A of a graph G with n vertices is a adjacency matrix square n × n matrix with elements ai;j such that: Distance matrix 1, if there is a link from vertex i to vertex j Shortest path ai;j = and Dijkstra's 0, otherwise algorithm Degree matrix If the graph is undirected we consider every edge between two of a graph vertices as linking in both directions. Markov chains 12 / 50 Non-negative Non-negative matrices in graph theory matrices and graphs MAA704 Christopher Example of an undirected graph and corresponding adjacency Engstr¨om matrix (vertices ordered A,B,C,D). lecture 2 Connectivity and the adjacency B matrix Distance matrix 2 3 Shortest path 0 1 1 1 and Dijkstra's algorithm 61 1 1 07 A C 6 7 Degree matrix 41 1 0 05 of a graph 1 0 0 0 Markov chains D 13 / 50 Non-negative Application, road network and adjacency matrix matrices and graphs MAA704 Christopher Engstr¨om lecture 2 We note that the adjacency matrix is not unique in itself, we Connectivity and the need to choose in what order we put our vertices in the matrix! adjacency matrix I Sometimes changing the order of the vertices (essentially Distance re-labeling the graph) can make certain structures more matrix Shortest path obvious. and Dijkstra's algorithm For example we could group vertices in the same I Degree matrix connected component together. of a graph Markov chains 14 / 50 Non-negative Adjacency matrix, other applications matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Connectivity and the A undirected or directed graph and it's adjacency matrix could adjacency also be used to represent other things such as: matrix Distance I Links between homepages, and PageRank. matrix Shortest path I Electrical or water flow networks. and Dijkstra's algorithm Linguistic relations between words or phrases. I Degree matrix of a graph Markov chains 15 / 50 Non-negative Path, cycle and length matrices and graphs MAA704 Christopher Engstr¨om Definition lecture 2 Connectivity Given a weighted or unweighted graph: and the adjacency I A path in a graph is a sequence of vertices e1; e2;:::; en matrix such that for every vertex in the sequence there is an edge Distance matrix to the next vertex in the sequence. Shortest path and Dijkstra's I If the first vertex in a path is the same as the last vertex algorithm we call it a cycle. Degree matrix of a graph The length of a path is the number of edges in the path I Markov chains (counting multiple uses of the same edge). 16 / 50 Non-negative Application, road network and adjacency matrix matrices and graphs MAA704 Christopher Engstr¨om lecture 2 Returning to our road network example, while a 1 at element Connectivity a means there is a road between those two cities, we still and the ij adjacency don't know how to find the connected components using the matrix adjacency matrix. Distance matrix k I We take a look at the powers A of the adjacency matrix. Shortest path and Dijkstra's I A non-zero element of A1 means there is a path of algorithm Degree matrix "length" 1 between the two vertices. (length as in number of a graph of edges) Markov chains 17 / 50 Non-negative Application, road network and adjacency matrix matrices and graphs MAA704 Christopher Engstr¨om 2 2 We look at A and when a single element aij > 0. lecture 2 2 Connectivity I For ai;j > 0 we need that at least one of and the adjacency ai;k ak;j > 0; k = 1; 2;:::; n. matrix a = 1 if there is an edge between i; k and a = 1 if Distance I i;k k;j matrix there is an edge between k; j.