# Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis

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1 Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, Chapter 06: Network analysis Version: April 8, 04

2 / 3 Contents Chapter Description 0: Introduction History, background 0: Foundations Basic terminology and properties of graphs 03: Extensions Directed & weighted graphs, colorings 04: Network traversal Walking through graphs (cf. traveling) 05: Trees Graphs without cycles; routing algorithms 06: Network analysis Basic metrics for analyzing large graphs 07: Random networks Introduction modeling real-world networks 08: Computer networks The Internet & WWW seen as a huge graph 09: Social networks Communities seen as graphs

3 Network analysis Introduction Observation In real-world situations, graphs (or networks) may become very large, making it difficult to (visually) discover properties we need network analysis tools. Vertex degrees: Consider the distribution of degrees: how many vertices have high degrees versus the number of vertices with low degrees. Distance statistics: Focus on where vertices are positioned in the network: far away from each other, central in the network, etc. Clustering: To what extent are my neighbors also adjacent to each other? Centrality: Are there vertices that are more important than others? 3 / 3

4 Network analysis 6. Vertex degree Vertex degree Question Can you visually observe real (nonisomorphic) differences? 4 / 3

5 5 / 3 Network analysis 6. Vertex degree Vertex degree: Histogram (a): (b): n = 00, m = 300 n = 00, m = (a) (b)

6 6 / 3 Network analysis 6. Vertex degree Vertex degree: Ranked histogram

7 7 / 3 Network analysis 6. Distance statistics Distance statistics Definition G is connected, d(u,v) is distance between vertices u and v: the length of a shortest path between u and v. Eccentricity ε(u): Radius rad(g): Diameter diam(g): max{d(u, v) v V (G)} min{ε(u) u V (G)} max{d(u, v) u, v V (G)} Note Note that these definitions apply to directed as well as undirected graphs.

8 8 / 3 Network analysis 6. Distance statistics Path lengths Definition G is connected with vertex V ; d(u) is average length of shortest paths from u to any other vertex v: d(u) def = V d(u,v) v V,v u The average path length d(g): d(g) def = V u V d(u) = V V d(u,v) u,v V,u v

9 Network analysis 6. Distance statistics Path lengths Definition The characteristic path length is the median over all d(u). Note The median over n nondecreasing values x,x,...,x n : n odd x (n+)/ n even (x n/ + x n/+ )/ The median separates the higher values from the lower values into two equally-sized subsets. Example {3,4,4,6,0,6,} [0,,3,4,4,6,6] M = x (7+)/ = x 4 = 4 9 / 3

10 0 / 3 Network analysis 6. Distance statistics Example distance statistics Vertex ε(u) v u d(u,v) d(u)

11 Network analysis 6.3 Clustering coefficient Clustering coefficient Observation Many networks show a high degree of clustering: my neighbors are each other s neighbors. Note An extreme case is formed by having all my neighbors be adjacent to each other neighbors form a complete graph. Question What is the other extreme case? / 3

12 Network analysis 6.3 Clustering coefficient Clustering coefficient Definition G is simple, connected, undirected. Vertex v V (G) with neighborset N(v). Let n v = N(v). Note: max. number of edges between neighbors is ( n v ). Let m v is number of edges in subgraph induced by N(v): m v = E(G[N(v)]). Clustering coefficient cc(v): cc(v) def = { mv / ( n v ) = m v n v (n v ) if δ(v) > undefined otherwise / 3

13 Network analysis 6.3 Clustering coefficient Clustering coefficient Definition G is simple, connected and undirected. Let V def = {v V (G) δ(v) > }. Clustering coefficient CC(G) for G: CC(G) def = V v V cc(v) 3 / 3

14 Network analysis 6.3 Clustering coefficient Clustering coefficient: triangles Definition A triangle is a complete (sub)graph with exactly 3 vertices. A triple is a (sub)graph with exactly 3 vertices and edges. Definition G is simple and connected with n (G) distinct triangles and n Λ (G) distinct triples. The network transitivity τ(g) def = n (G)/n Λ (G). Notation A triple at v: v is incident to both edges ( in the middle ). n Λ (v) : number of triples at v. 4 / 3

15 5 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient: example Vertex: cc: /3 0 /3 undefined /3 n Λ : Vertex N() = {,5,7}; E(G[N()]) = 5,7 cc() = 3 Triples at : G[{,,5}],G[{,,7}],G[{5,,7}]

16 6 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient versus transitivity Observation Let n (v) be the number of triangles of which v is member cc(v) = n (v) n Λ (v) n Λ (v) = ( δ(v) ) n (G) = 3 v V n (v) (Note: V = {v V δ(v) > })

17 7 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient versus transitivity x x x v v v 3... v n v k v v v 3... v n y G k = G[{x,y,v,v,...,v k }] : cc(u) = y if u = v,...,v k ) = k k(k+) = k+ if u = x or u = y k ( k+ CC(G k ) = k + ( k + + k ) = k + k + 4 k + 3k + lim k CC(G k) = y

18 8 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient versus transitivity x v v v 3... v n G k = G[{x,y,v,v,...,v k }] { if u = v,...,v k n Λ (u) = ) ( = k+ ) if u = x,y ( δ(u) y τ(g k ) = n (G k ) n Λ (u) = k = k(k + ) + k k + lim τ(g k) = 0 k

19 Network analysis 6.4 Centrality Centrality Issue Are there any vertices that are more important than the others? Definition G is (strongly) connected. The center C(G) is the set of vertices with minimal eccentricity: Intuition C(G) def = {v V (G) ε(v) = rad(g)} At the center means at minimal distance to the farthest node. 9 / 3

20 0 / 3 Network analysis 6.4 Centrality Vertex centrality Definition G is (strongly) connected. The (eccentricity based) vertex centrality c E (u) of u: c E (u) def = ε(u) Intuition The higher the centrality, the closer to the center of a graph.

21 Network analysis 6.4 Centrality Closeness Definition G is (strongly) connected. The closeness c C (u) of u: c C (u) def = v V (G) d(u,v) Intuition How close is a vertex to all other nodes? / 3

22 / 3 Network analysis 6.4 Centrality Centrality: example Vertex: ε(u) d(u, ) c C (u):

23 Network analysis 6.4 Centrality Betweenness Intuition Important vertices are those whose removal significantly increases the distance between other vertices. Example: cut vertices. Definition G is simple and (strongly) connected. S(x,y) is set of shortest paths between x and y. S(x,u,y) S(x,y) paths that pass through u. Betweenness centrality c B (u) of u: c B (u) def = x y u S(x,u,y) S(x, y) 3 / 3

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