Presentation on theme: "Introduction to Network Theory: Modern Concepts, Algorithms"— Presentation transcript:
1 Introduction to Network Theory: Modern Concepts, Algorithms and ApplicationsErnesto EstradaDepartment of Mathematics, Department of PhysicsInstitute of Complex Systems at StrathclydeUniversity of Strathclyde
8 Directed Graph (digraph) Edges have directionsThe adjacency matrix is not symmetric
9 Simple GraphsSimple graphs are graphs without multiple edges or self-loops. They are weighted graphs with all edge weights equal to one.BEDCA
10 Local metricsLocal metrics provide a measurement of a structural property of a single nodeDesigned to characteriseFunctional role – what part does this node play in system dynamics?Structural importance – how important is this node to the structural characteristics of the system?
12 Betweenness centrality The number of shortest paths in the graph that pass through the node divided by the total number of shortest paths.
13 Betweenness centrality Shortest paths are:AB, AC, ABD, ABE, BC, BD, BE, CBD, CBE, DBEB has a BC of 5ABCED
14 Betweenness centrality Nodes with a high betweenness centrality are interesting because theycontrol information flow in a networkmay be required to carry more informationAnd therefore, such nodesmay be the subject of targeted attack
15 Closeness centralityThe normalised inverse of the sum of topological distances in the graph.
22 Extending the Concept of Degree Make xi proportional to the average of the centralitiesof its i’s network neighborswhere l is a constant. In matrix-vector notation wecan writeIn order to make the centralities non-negative we selectthe eigenvector corresponding to the principal eigenvalue(Perron-Frobenius theorem).
23 Eigenvalues and Eigenvectors The value λ is an eigenvalue of matrix A if there exists a non-zero vector x, such that Ax=λx. Vector x is an eigenvector of matrix AThe largest eigenvalue is called the principal eigenvalueThe corresponding eigenvector is the principal eigenvectorCorresponds to the direction of maximum change
24 Eigenvector Centrality The corresponding entry of the principal eigenvector of the adjacency matrix of the network.It assigns relative scores to all nodes in the network based on the principle that connections to high-scoring nodes contribute more.
26 Eigenvector Centrality: Difficulties In regular graphs all the nodes have exactly the same value of the eigenvector centrality, which is equal to
27 Subgraph CentralityA closed walk of length k in a graph is a succession of k (not necessarily different) edges starting and ending at the same node, e.g.1,2,8,1 (length 3)4,5,6,7,4 (length 4)2,8,7,6,3,2 (length 5)
28 Subgraph CentralityThe number of closed walk of length k starting at the same node i is given by the ii-entry of the kth power of the adjacency matrix
29 Subgraph CentralityWe are interested in giving weights in decreasing order of the length of the closed walks. Then, visiting the closest neighbors receive more weight that visiting very distant ones.The subgraph centrality is then defined as the following weighted sum
30 Subgraph Centrality By selecting cl=1/l! we obtain where eA is the exponential of the adjacency matrix.For simple graphs we have
34 CommunicabilityPath of length 6Walk of length 8Shortest path
35 Communicability Let be the number of shortest paths of length s between p and q.Let be the number of walks of length k>sbetween p and q.DEFINITION (Communicability):and must be selected such as the communicability converges.
36 Communicability By selecting bl=1/l! and cl=1/l! we obtain where eA is the exponential of the adjacency matrix.For simple graphs we have
40 Communicability & Communities A community is a group of nodes for wich the intra-cluster communicability is larger than the inter-cluster oneThese nodes communicates better among them than with the rest of extra-community nodes.
41 Communicability Graph LetThe communicability graph Q(G) is the graph whose adjacency matrix is given by Q(D(G)) results from the elementwise application of the function Q(G) to the matrix D(G).
47 ReferencesAldous & Wilson, Graphs and Applications. An Introductory Approach, Springer, 2000.Wasserman & Faust, Social Network Analysis, Cambridge University Press, 2008.Estrada & Rodríguez-Velázquez, Phys. Rev. E 2005, 71,Estrada & Hatano, Phys. Rev. E. 2008, 77,
48 Exercise 1 Identify the most central node according to the following criteria:(a) the largest chance of receiving information from closestneighbors;(b) spreading information to the rest of nodes in thenetwork;(c) passing information from some nodes to others.
49 Exercise 2 T.M.Y. Chan collaborates with 9 scientists in computational geometry. S.L. Abrams also collaborates withother 9 (different) scientists in the same network. However,Chan has a subgraph centrality of 109, while Abrams has 103.The eigenvector centrality also shows the same trend,EC(Chan) = 10-2; EC(Abrams) = 10-8.Which scientist has more chances of being informed aboutthe new trends in computational geometry?(b) What are the possible causes of the observed differencesin the subgraph centrality and eigenvector centrality?