Network Analysis Antonio M. Chiesi Department of Social and Political Studies, Università degli Studi di Milano Antonio.chiesi@unimi.it
Essential references: Chiesi, A. M., Network Analysis, general, in International Encyclopedia of the Social and Behavioral Sciences, Pergamon, 2001 Chiesi A. M., Perspectives of Network Analysis applied to Social Sciences, Società Italiana di Statistica, Proceedings of the XLIII Scientific Meeting, Invited Sessions, Torino 2006. Chiesi, A. M., Measuring Social Capital and its Effectiveness, European Sociological Review, September 2007. Chiesi, A. M., L analisi dei reticoli, F. Angeli, Milano, 1999 Wasserman, S. and K. Faust, Social Network Analysis, Cambridge, Cambridge Univ. Press, 1994.
Getting a Job (Granovetter 1973)
Diffusion of Innovations (Coleman, Katz and Menzel 1957)
Careers as chains of opportunities (White 1970)
Social Density and Mental health (Marsden 1982)
Theoretical Origins Relational approach and formal sociology (Simmel 1908) Interlocking Directorates in Corporate America (Pujo Committee 1913) Sociograms in the 1930s (Moreno 1934) The Manchester School of Anthropology in the 1940s and 1950s (Barnes, Gluckman, Mitchell) Structural Analysis (White, Harary, Cartwright 1970s) Social capital theory (Coleman 1992, Lin 2001) Network dynamics (Siena models)
Analytical components of social relations The actor The tie The relation Groups and sub-groups Network structure
Content of ties T ip i di legam e B arnes (1 972) K noke- K uklinski (1 982) S cott (1991) W asserm an-f aust (1 994) scam b io eco nom ico X X X X dono X info rm azio ni X X co nversazio ne X interlo cking d irectorates X X relazio ni di tipo strum entale X atteggiam enti X relazio ni inform ali X X - am icizia X X - sim p atia X X - am m irazione X - deferenza, risp etto X X - ostilità X relazio ni fo rm ali X X - potere X X - auto rità X X parentela o d iscend enza X X affiliazio ne o appartenenza X X m obilità - sp aziale X - so ciale X co llegam ento sp aziale X
Types of ties
Data collection and matrix organization
Data files Matrix Matrix of ofties Numero della linea Origine Destinazione Intensità del legame Tipo di legame 1 Gianni Rosy 1 1 2 Daniela Rosy 1 1 3 Rosy Daniela 2 2 4 Rosy Davide 1 2 5 Davide Gigi 3 1 6 Vanda Davide 1 2 7 Piero Vanda 2 1 8 Piero Davide 3 2 9 Piero Gigi 1 2 10 Gigi Davide 3 1 11 Gigi Piero 2 1 Matrix Matrix of ofnodes Numero del punto Nome del punto Età Genere Titolo di studio Anzianità in mesi 1 Gianni 21 1 3 26 2 Rosy 19 2 2 24 3 Daniela 20 2 3 38 4 Davide 27 1 3 46 5 Piero 23 1 3 16 6 Vanda 18 2 2 21 7 Gigi 17 1 2 6 8 Nino 24 1 2 8
Limits of graphical representations (sociograms)
Isomorphic sociograms
Sub-graphs, components, cliques
Elementary structural statistics: 1. Size of the graph (n. of nodes): S = g 2. Graph density (probability that any couple of nodes be directly linked): 2 l = g (g-1)
Types of ties and corresponding matrices Type of tie Type of graph Type of matrix Presence/absence Simple graph Adjacency matrix: binary symmetric Directed Direct graph Adjacency matrix: binary asymmetric Positive/negative Signed graph Adjacency matrix: + signs in cells Weighted Valued graph Adjacency matrix: values in cells Multiplex Multigraph Several matrices according to level of multiplicity Membership Ipergraph Affiliation matrix: asymmetric
Matrix representation. 1 Affiliation Affiliation matrix matrix E 1 E 2 E 3 E 4 n 1 1 0 0 0 1 n 2 1 1 0 0 2 n 3 0 0 1 0 1 n 4 0 1 0 1 2 n 5 0 0 1 0 1 n 6 0 0 0 1 1 n 7 0 0 1 1 2 2 2 3 3
Matrix representation. 2 Adjacency Adjacency matrix matrix Gianni Daniela Nino Rosy Davide Vanda Gigi Piero Gianni - 0 0 1 0 0 0 0 1 Daniela 0-0 1 0 0 0 0 1 Nino 0 0-0 0 0 0 0 0 Rosy 0 1 0-1 0 0 0 2 Davide 0 0 0 0-0 1 0 1 Vanda 0 0 0 0 1-0 0 1 Gigi 0 0 0 0 1 0-1 2 Piero 0 0 0 0 1 1 1-3 0 1 0 2 4 1 2 1
Matrix representation. 3 Distance Distance matrix matrix in in a direct direct graph graph Gianni Daniela Nino Rosy Davide Vanda Gigi Piero Gianni - 0 0 0 0 0 0 0 Daniela 2-0 1 0 0 0 0 Nino 0 0-0 0 0 0 0 Rosy 1 1 0-0 0 0 0 Davide 2 2 0 1-1 1 1 Vanda 5 5 0 4 3-2 1 Gigi 3 3 0 2 1 2-1 Piero 4 4 0 3 2 3 1 -
Network Statistics
Centrality Measures Type of centrality Analytical concept Symbol: simple graphs directed graphs Short range centrality degree C D C ID Variable range centrality neighbourhood C N Global centrality geodesic distance C C C IR Information networks rush, betweenness C B Local prestige indegree P P Global prestige rank P R
Multiplicity of graphs
Operational definitions of sub-groups.1 Technical limits in the application of the concept of clique (maximal complete sub-graph): The definition is not flexible Its usefulness diminishes when applied to low density graphs Difficult interpretation of overlapping cliques in high density graphs The dimension of a clique (and the density of a graph) is function of the way sociometric questions are put (open choice or fixed number choice) By definition, within a clique it is not possible to identify different structural positions (structural equivalence by construction)
Operational definitions of sub-groups.2 Alternative concepts to clique: Component (maximal connected subgraph), useful only for low density graphs Clique-n, where n. is the maximal distance between pair of nodes (eg. distance 2) K-plex, where k is the number of possible missing ties if any node would be part of a clique.
Structural Equivalence A B C D E F A - 0 1 0 0 0 B 0-1 0 0 0 C 0 0-0 0 0 D 1 1 0-0 0 E 1 1 0 0-1 F 0 0 0 0 0 -
Alleances Exchanges Deals
A definition of Social Capital (SC) A set of steady interpersonal relations based on reputation through which Ego can mobilize the action of Alter in order to pursue his/her own goals.