Econ 430 Lecture 3: Significance and Structural Properties of N

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1 Econ 430 Lecture 3: Significance and Structural Properties of Networks Alper Duman Izmir University Economics, March 8, 2013

2 Prevalence of Networks Networks are everywhere! Even in this class. We can classify networks broadly into three: (1) technological/physical, (2) social/economic and (3) mixed. Transportation, power grids, internet, food web, chemical reaction and circuits are examples of the first class. Friendship, job, crime, insurance, risk sharing, sexual, supplier, customer, ownership and religious networks are some of the examples the second class of networks. Information, phone calls, communication, citation, patent and online-game players networks are from class three

3 Prevalence of Networks 1. Visit 2. Visit http : //www personal.umich.edu/ mejn/netdata/ 3. Visit http : //math.nist.gov/ RPozo/complex d atasets.html 4. Visit http : //wiki.gephi.org/index.php/datasets 5. Visit http : //vlado.fmf.uni lj.si/pub/networks/data/

4 Figure : Football Transfers Network from Countries in WC98 JAM CHE USA NLD BEL GBR IRN HRV ZAF DNK SCO CHL NGA ITA DEU AUT ROM NOR ARG TUR ESP FRA YUG GRE COL CMR BGR TUN BRA MAR KOR PRY JPN PRT MEX Pajek

5 Figure : Structural Equivalence -Hierarchical Clustering Pajek - Ward [0.00,6.79] ARG AUT BEL BGR BRA CHE DNK GRE IRN JAM KOR MAR MEX USA YUG ZAF HRV NLD NOR CHL CMR COL GBR NGA PRY TUN ITA ROM DEU JPN PRT SCO TUR ESP FRA

6 Random networks are benchmarks! Random network properties provide clues on what we can expect from other types of networks Believe or not sexual partnership network in US resemble a random network! Random networks on grids are important for games on networks.

7 Bipartite-Affiliation Networks We use adjacency matrices of bipartite (directed or undirected) networks in order to get one-mode projections; undirected networks. C = AA T gives cocitation matrix B = A T A gives bibliographic coupling matrix.

8 Acyclic Directed Networks Paper and patent citations are acyclic; the direction of edges can be represented all pointing downwards Acyclic directed networks also represent information geometry models in which there can be multiple causes and effects. The adjacency matrix of an acyclic directed network is a strictly triangular matrix.

9 Small Worlds Small world implies that large networks have small diameters, small average path lengths and greater clustering coefficients. Remember six degrees! Search for Erdös and Bacon numbers for mathematicians and movie stars respectively. How many steps does it take you to arrange a delivery of a mail to anyone in the world?

10 Figure : Ownership Network Giant Component by Gephi

11 Average degree: Diameter: 9 Average path length: Average Clustering coefficient: (Number of triangles: 4126) Short path lengths and high clustering suggest small world of corporations in Turkey.

12 Triad Closure and Clustering George Simmel -importance of mutually connected triplets. Norbert Elias -clustering in families, clans, tribes, communities It is easy to see why you would observe high clustering in the Facebook network But it is also true for WWW! Lara Adamic finds that WWW network has a clustering measure of ; the clustering coefficient of a random network with same number of links would be

13 Triad Closure and Clustering Triadic Closure occurs for many reasons One reason why B and C are more likely to become friends, when they have a common friend A, is simply based on the opportunity for B and C to meet Two friendship links suggest TRUST if A is friends with B and C, then it becomes a source of latent stress in these relationships if B and C are not friends with each other If a node A has edges to nodes B and C, then the B-C edge is especially likely to form if A s edges to B and C are both strong ties

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15 Degree Distributions Most networks are neither random nor regular. There are hubs in network with very high number of links and a lot of vertices with very few links. Fat-tails suggesting scale-free degree distributions are observed in many contexts. Word usage, plant classification, city size, article citations, internet servers! A simple preferential attachment model would give a scale-free (or power law) degree distribution.

16 Correlations and Assortativity When relatively high-degree nodes tend to be connected to other high-degree node, we call this relation positive assortativity. This relation is a common feature of social/economic networks in contrast to technological/physical networks. In the latter we often observe negative assortativity. In the world trade network, there is a hub-and-spoke system in which smaller countries (spokes) tend to have few partners and mainly trade with larger countries (hubs).

17 Homophily People are more likely to have links with other people similar to themselves. Give examples...ie football fans, ethnicity, employment status Why is the homophily effect important? Are you crime-prone and hence you become a member of a gang; or you know a gang member and she pulls you into crime business! Turkish migrants in Germany or in Holland; they are much more connected with each other

18 Weak and Strong Ties Mark Granovetter has come up with the idea while analysing job network in Massachusetts Strong ties relate to close/intense links, ie best friends, extended family members. Weak ties refer to links such as acquaintances. Weak ties are usually bridges! Weak ties are critical in terms of information flows

19 The embeddedness of an edge in a network to be the number of common neighbours the two endpoints have. Local bridges are precisely the edges that have an embeddedness of zero Social sanctions and reputation effects matter through embeddedness For isolates no such mechanism exist

20 Structural Holes and Social Capital A structural hole is a void between two connected subgraphs (components). This can be due to strategic decisions or due to physical/social constraints. An agent filling the void (a broker) can be very powerful/influential. Social capital related to the material/non-material asset that agents can have by having specific links or by being in a specific network. Community governance depends on the social capital; i.e. fishermen of Güzelbahçe.

21 Diffusion Diffusion of open source software, smart phones and new innovations Diffusion of diseases Diffusion of information Threshold models and rebellions (Chwe models in Collective Action)