# Course on Social Network Analysis Graphs and Networks

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1 Course on Social Network Analysis Graphs and Networks Vladimir Batagelj University of Ljubljana Slovenia

2 V. Batagelj: Social Network Analysis / Graphs and Networks 1 Outline 1 Graph Graph / Sets Graph / Neighbors Graph / Matrix Subgraph Graph characteristics Walks Shortest paths Equivalence relations and Partitions Connectivity Reduction Biconnectivity

3 V. Batagelj: Social Network Analysis / Graphs and Networks 2 13 Special graphs Krebs Internet Industry Companies Cores core of Krebs Internet Industry Companies Networks Computational Geometry Valued Core Sources

4 V. Batagelj: Social Network Analysis / Graphs and Networks 1 Graph actor vertex, node relation line, edge, arc, link, tie arc = directed line, (a, d) a is the initial vertex, d is the terminal vertex. edge = undirected line, (c: d) c and d are end vertices.

5 V. Batagelj: Social Network Analysis / Graphs and Networks 2 Graph / Sets V = {a, b, c, d, e, f, g, h, i, j, k, l} A = {(a, b), (a, d), (a, f), (b, a), (b, f), (c, b), (c, c), (c, g), (c, g), (e, c), (e, f), (e, h), (f, k), (h, d), (h, l), (j, h), (l, e), (l, g), (l, h)} E = {(b: e), (c: d), (e: g), (f: h)} G = (V, A, E) L = A E A = undirected graph; E = directed graph. Pajek: GraphSet; TinaSet.

6 V. Batagelj: Social Network Analysis / Graphs and Networks 3 Graph / Neighbors N A (a) = {b, d, f} N A (b) = {a, f} N A (c) = {b, c, g, g} N A (e) = {c, f, h} N A (f) = {k} N A (h) = {d, l} N A (j) = {h} N A (l) = {e, g, h} N E (e) = {b, g} N E (c) = {d} N E (f) = {h} Pajek: GraphList; TinaList. N(v) = N A (v) N E (v), = N out (v), N in (v) Star in v, S(v) is the set of all lines with v as their initial vertex.

7 V. Batagelj: Social Network Analysis / Graphs and Networks 4 Graph / Matrix a b c d e f g h i j k l a b c d e f g h i j k l Pajek: GraphMat; TinaMat, picture picture. Graph G is simple if in the corresponding matrix all entries are 0 or 1.

8 V. Batagelj: Social Network Analysis / Graphs and Networks 5 Subgraph A subgraph H = (V, L ) of a given graph G = (V, L) is a graph which set of lines is a subset of set of lines of G, L L, its vertex set is a subset of set of vertices of G, V V, and it contains all end-vertices of L. A subgraph can be induced by a given subset of vertices or lines.

9 V. Batagelj: Social Network Analysis / Graphs and Networks 6 Graph characteristics number of vertices n = V number of lines m = L degree of vertex v, deg(v) = number of lines with v as end-vertex; indegree of vertex v, indeg(v) = number of lines with v as terminal vertex (end-vertex is both initial and terminal); outdegree of vertex v, outdeg(v) = number of lines with v as initial vertex. v V n = 12, m = 23, indeg(e) = 3, outdeg(e) = 5, deg(e) = 6 indeg(v) = outdeg(v) = A + 2 E, deg(v) = 2 L E 0 v V v V

10 V. Batagelj: Social Network Analysis / Graphs and Networks 7 Walks length s of the walk s is the number of lines it contains. s = (j, h, l, g, e, f, h, l, e, c, b, a) s = 11 A walk is closed iff its initial and terminal vertex coincide. If we don t consider the direction of the lines in the walk we get a semiwalk or chain. trail walk with all lines different path walk with all vertices different cycle closed walk with all internal vertices different A graph is acyclic if it doesn t contain any cycle.

11 V. Batagelj: Social Network Analysis / Graphs and Networks 8 Shortest paths A shortest path from u to v is also called a geodesic from u to v. Its length is denoted by d(u, v). If there is no walk from u to v then d(u, v) =. d(j, a) = (j, h, d, c, b, a) = 5 d(a, j) = ˆd(u, v) = max(d(u, v), d(v, u)) is a distance. The diameter of a graph equals to the distance between the most distant pair of vertices: D = max u,v V d(u, v).

12 V. Batagelj: Social Network Analysis / Graphs and Networks 9 Equivalence relations and Partitions Let C = {C i } be a set of subsets of V, C i V. C is a partition of V iff i C i = V and for i j, C i C j =. A relation R on V is an equivalence relation iff it is reflexive v V : vrv, symmetric u, v V : urv vru, and transitive u, v, z V : urz zrv urv. Each equivalence relation determines a partition into equivalence classes [v] = {u : vru}. Each partition C determines an equivalence relation urv C C : u C v C. k-neighbors of v is the set of vertices on distance k from v, N k (v) = {u v : d(v, u) = k}. The set of all k-neighbors, k = 0, 1,... of v is a partition of V. k-neighborhood of v, N (k) (v) = {u v : d(v, u) k}.

13 V. Batagelj: Social Network Analysis / Graphs and Networks 10 Connectivity Vertex u is reachable from vertex v iff there exists a walk with initial vertex v and terminal vertex u. Vertex v is weakly connected with vertex u iff there exists a semiwalk with v and u as its end-vertices. Vertex v is strongly connected with vertex u iff they are mutually reachable. Weak and strong connectivity are equivalence relations. Equivalence classes induce weak/strong components.

14 V. Batagelj: Social Network Analysis / Graphs and Networks 11 Reduction If we shrink every strong component of a given graph into a vertex, delete all loops and identify parallel arcs the obtained reduced graph is acyclic. For every acyclic graph an ordering / level function i : V IN exists s.t. (u, v) A i(u) < i(v).

15 V. Batagelj: Social Network Analysis / Graphs and Networks 12 Biconnectivity Vertices u and v are biconnected iff they are connected (in both directions) by two independent (no common internal vertex) paths. Biconnectivity determines a partition of the set of lines. A vertex is an articulation vertex iff its deletion increases the number of weak components in a graph.

16 V. Batagelj: Social Network Analysis / Graphs and Networks 13 Special graphs A weakly connected graph G is a tree iff it doesn t contain loops and semicycles of length at least 3. A graph G = (V, L) is bipartite iff its set of vertices V can be partitioned into two sets V 1 and V 2 such that every line from L has one end-vertex in V 1 and the other in V 2. A simple undirected graph is complete, K n, iff it contains all possible edges.

17 V. Batagelj: Social Network Analysis / Graphs and Networks 14 Krebs Internet Industry Companies This network shows a subset of the total internet industry during the period from 1998 to 2001, n = 219, m = 631. Two companies are connected with a line if they have announced a joint venture, strategic alliance or other partnership (red - content, blue - infrastructure, yellow - commerce). Network source:

18 V. Batagelj: Social Network Analysis / Graphs and Networks 15 Cores The notion of core was introduced by Seidman in A subgraph H = (W, L W ) induced by the set W in a graph G = (V, L) is a k-core or a core of order k iff v W : deg H (v) k, and H is a maximum subgraph with this property.

19 V. Batagelj: Social Network Analysis / Graphs and Networks Cores The core of maximum order is also called the main core. The core number of vertex v is the highest order of a core that contains this vertex. The degree deg(v) can be: in-degree, out-degree, in-degree + out-degree,..., determining different types of cores. The cores are nested: i < j = H j H i Cores are not necessarily connected subgraphs. The notion of cores can be generalized to valued cores.

20 V. Batagelj: Social Network Analysis / Graphs and Networks 17 6-core of Krebs Internet Industry Companies

21 V. Batagelj: Social Network Analysis / Graphs and Networks 18 Networks A graph with additional information on vertices and/or lines is called a network. In Pajek this information is represented using vectors (numerical properties of vertices) and partitions (categorical/nominal properties of vertices). Numerical values can be assigned also to lines line values. Example: Authors collaboration network based on the Computational Geometry Database. Two authors are linked with an edge, iff they wrote a common paper. The weight of the edge is the number of publications they wrote together. Problem of cleaning. Different names: Pankaj K. Agarwal, P. Agarwal, Pankaj Agarwal, and P.K. Agarwal. n = 9072, m = 13567/22577 n = 7343, m =

22 V. Batagelj: Social Network Analysis / Graphs and Networks 19 Computational Geometry Valued Core

23 V. Batagelj: Social Network Analysis / Graphs and Networks 20 Sources Vladimir Batagelj, Andrej Mrvar: Pajek. Vladimir Batagelj: Slides on network analysis. Vladimir Batagelj: Papers on network analysis. Andrej Mrvar: Social network analysis. B. Jones: Computational geometry database. jeffe/compgeom/biblios.html ftp://ftp.cs.usask.ca/pub/geometry/.

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