Mathematics of Networks II 26.10.2016 1 / 30
Definition of a network Our definition (Newman): A network (graph) is a collection of vertices (nodes) joined by edges (links). More precise definition (Bollobàs): A graph G is an ordered pair of disjoint sets (V, E) such that E (the edges) is a subset of the set V (2) of unordered pairs of V (the vertices). 1 / 30
Graph theory The power of abstraction Königsberg bridge problem: Can one walk across all seven bridges and never cross the same one twice? Fig. from: Wikipedia, https://en.wikipedia.org/wiki/seven Bridges of Königsberg. 2 / 30
Graph theory Represent Königsberg as a graph: Euler proved: No such path is possible! 3 / 30
Graph theory Represent Königsberg as a graph: Euler proved: No such path is possible! Theorem Graph G has Eulerian path iff 1. G is connected 2. Zero or two vertices have odd degree 3 / 30
Graph theory Graph G = (V, E) with Set of vertices V Set of edges E V V 4 / 30
Graph theory Graph G = (V, E) with Set of vertices V Set of edges E V V Graph can be represented via Edgelist Adjacency matrix 4 / 30
Adjacency matrix A A ij = { 1 if there is an edge between vertices j and i 0 otherwise. 0 1 1 0 0 1 1 0 1 0 0 0 A = 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 2 1 3 6 4 5 5 / 30
Adjacency matrix A Graph property Adjacency matrix undirected symmetric simple diagonal(a) = 0, A ij {0, 1} multi-graph A ij N number of edges i j self-loop A ii = 2 if i i weighted A ij R directed asymmetric 6 / 30
Degree Degree k i of vertex i: Number of edges connected to i. Average degree of the network: k. In terms of the adjacency matrix A: n k i = A ij, k = 1 k i = 1 n n j=1 i n n A ij. i=1 j=1 5 k 5 = 1 k 2 = k 6 = 2 k 1 = k 3 = k 4 = 3 k = 2.33 2 3 6 4 1 7 / 30
Degree With n the number of vertices in the graph, and m the number of edges, it holds: n n n 2m = k i = A ij. i=1 i=1 j=1 For the average degree k of the graph this yields k = 1 n n k i = 1 n i=1 n n A ij = 2m n. i=1 j=1 8 / 30
Density / connectance Maximum possible number of edges in a simple graph with n vertices: 1 n(n 1). 2 Density or connectance of a graph: Fraction of maximum possible number of edges which are present in a given graph: ρ = 1 2 m 2m = n(n 1) n(n 1) = k n 1. 9 / 30
Degree distribution Number of vertices with degree k in a graph: n k 4 8 7 3 5 4 6 9 n k 2 3 1 10 1 2 11 12 13 0 0 2 4 6 k 10 / 30
Degree distribution Fraction of vertices in a graph that have degree k: p k = n k n. Degree distribution 0.3 8 7 0.2 5 4 6 9 p k 3 1 10 0.1 2 11 12 13 0.0 0 2 4 6 k 11 / 30
Degree distribution Hubs: well-connected vertices 0.0 0.2 0.4 0.6 0 10 20 k p k Degree distribution 12 / 30
Average degree from the degree distribution Degree distribution tells important information about a network, but doesn t contain the complete information. The average degree of a graph can be easily calculated from the degree distribution: k = 1 n n k i = 1 k max k max n k k = kp k. n i=1 k=0 k=0 13 / 30
Directed networks: in-degree, out-degree Number of vertices with k ingoing / outgoing edges. k in i = n j=1 A ij, k out i = n j=1 A ji 13 11 0.6 In degree distribution 0.4 Out degree distribution 12 10 9 0.3 2 1 6 8 0.4 3 7 p k p k 0.2 4 0.2 0.1 5 0.0 0.0 0 1 2 3 k 0 1 2 3 4 k 14 / 30
Paths Definition A path (of length r) in a graph G = (V, E) is a sequence v 0,..., v r V of vertices such that v i 1 v i E for all i = 1,..., r. Route through the network, from vertex to vertex along the edges Defined for both directed and undirected networks Special case: self-avoiding paths Length of a path: number of edges along the path ( hops ) Number of paths of length r between vertices i and j: N (r) ij = [A r ] ij 15 / 30
Geodesic / shortest paths A path between two vertices such that no shorter path exists Geodesic distance between vertices i and j is the smallest value of r such that [A r ] ij > 0. Self-avoiding In general not unique Diameter of a network: Length of the longest geodesic path between any pair of vertices 16 / 30
Shortest paths some examples Oracle of Bacon: https://oracleofbacon.org/ Network of movie actors (joint appearance in a movie, based on IMDB) Geodesic distance to Kevin Bacon 17 / 30
Shortest paths some examples Erd os number: Consult http://wwwp.oakland.edu/enp/ I Coauthorship network I Geodesic distance to Paul Erd os 18 / 30
Shortest paths Six degrees of separation Classic experiment by Stanley Milgram (also known for obedience to authority ) Average path lengths in social networks 19 / 30
Graph distance Consider a connected graph: The graph distance between vertices i and j is defined as the length of a shortest path from i to j In weighted graph it is often useful to consider the weighted distance Consider a path π from i to j, i.e. i = v 0 w1 v 1... wr v r = j Define the weighted distance of this path as d p ij = e w e where the sum runs over all edges e in path π 20 / 30
Graph distance vs geographic distance Network of global air traffic Nodes (airports) positioned according to geographic location Fig. from: D. Brockmann & D. Helbing, The Hidden Geometry of Complex, Network-Driven Contagion Phenomena, Science (342), 2013. 21 / 30
Graph distance vs geographic distance Weighted graph with flow matrix F mn quantifies (passengers per day) air traffic from node (airport) n to node m 4069 airports with 25,453 direct connections Total flow Φ = 8.91 10 6 passengers per day Define conditional probability of passenger starting at n to reach m P mn = F mn F n where F n = m F mn and effective distance of edge n m as d mn = (1 log P mn ) 1 Note: Probabilities are multiplicative, thus log-prob. summed up along path 22 / 30
Graph distance vs geographic distance Circular spread of epidemics when viewed in effective graph distance Fig. from: D. Brockmann & D. Helbing, The Hidden Geometry of Complex, Network-Driven Contagion Phenomena, Science (342), 2013. 23 / 30
Graph distance vs geographic distance Qualitative reconstruction of outbreak center Fig. from: D. Brockmann & D. Helbing, The Hidden Geometry of Complex, Network-Driven Contagion Phenomena, Science (342), 2013. 24 / 30
Components of networks Component: Largest connected subgroup of vertices There is a path between any two vertices in the same component There is no path between any two vertices in different components Adjacency matrix: Block-diagonal form 25 / 30
Components in directed networks Weakly connected components: connected in the sense of an undirected network Strongly connected components: directed path in both directions between every pair in the subset 26 / 30
Components in directed networks Out-component of a vertex i: set of vertices which are reachable via directed paths starting form i, including the vertex i itself In-component of a vertex i: set of vertices from which there is a directed path to i, including the vertex i itself One often considers the out- or in-component of a strongly connected component 27 / 30
Network of Global Corporate Control Ownership network of transnational corporations (TNCs) Vitali et al., PLOS One, 6 (2011) Ownership matrix W: W ij is the percentage of ownership that the owner (shareholder) i holds in firm j If W ij > 0 and W jl > 0, then vertex i has an indirect ownership of firm l Data: Orbis 2007 database Resulting network: 600508 vertices (economics actors), containing 43060 TNCs, 1006987 edges (ownership ties) 28 / 30
Network of Global Corporate Control Vitali et al., PLOS One, 6 (2011) 29 / 30