Yves Zenou

# Course on Networks

There are two new books that need to be consulted for this course:

Sanjeev Goyal: *Connections: An Introduction to the Economics of Networks*, Princeton University Press, November 2007.

Matt O. Jackson: *Social and Economic Networks*, to be published by Princeton University Press in 2008.

Survey on "Social Networks" by Joan de Marti and Zenou (2009) [PDF] **NEW**

Slides overview of the course (based partly on Matt Jackson's slides) [PDF] **NEW**

**Lecture 1: Motivation, Definitions and Descriptive Evidence **

**Introduction 1:** Simple description of networks, based on Borgatti’s slides [PDF]

**Introduction 2:** Some definitions and notations [PDF]

**Introduction 3:** Some empirical evidences of co-authorships in economics from Goyal slides [PDF]

**Lecture 2: Theories of network formation**

**Random network models** based on Jackson (2007)'s survey [PDF]

Slides Growing Random Networks (based partly on Brian Rogers' Slides) [PDF] **NEW**

Erdös-Rényi (Bernoulli) Random Graphs;

Rewired Lattices and Clustering (Watts and Strogatz, 1998);

Preferential Attachment and Scale-Free Degree Distributions (Reka and Barabesi);

Hybrid Models (Jackson and Rogers AER 2007).

**Equilibrium Network Formation: Pairwise stability and Pairwise Nash Stability: **Slides [PDF]

We will define the equilibrium concept of pairwise stability (which is the most prominent equilibrium concept used in network formation games) introduced by Jackson and Wolinsky (1996) and illustrate it using two well-known examples:

The **Connections Model**: We will expose the results in both of equilibrium (pairwise stability) and efficiency.

Extensions of the “Connections Model”:

The *spatial *connection model (Johnson and Gilles, *Review of Eco Design* 2000)

The *small world *connection model (Jackson and Rogers, *JEEA* 2005)

The **Co-Author Model**. We will expose the results in both its equilibrium (pairwise stability) and efficiency.

This pairwise stability equilibrium concept is often viewed as a cooperative equilibrium concept. We will show that there is a tension between a pairwise stable network and an efficient network (from an overall society perspective). In particular, plenty of pairwise stable networks are not efficient (see, in particular, the surveys by Jackson, 2005, 2006). Another approach to network formation is the non-cooperative game introduced by Myerson and analyzed, for instance, by Bala and Goyal (2000) for the case of directed networks. We will expose and discuss this game for the case of un-directed networks, where link formation required mutual consent, using the paper by Calvó-Armengol and Ilkiliç (2006). This paper has also a very nice example for which the empty network, which happens to be a trembling-hand equilibrium network for the Myerson game, is not pairwise stable. This paper shows than one needs to resort to equilibrium notion of properness to reconcile the non-cooperative and cooperative approaches.

**References **

R. Albert and A. Barabási (2002), “Statistical mechanics of complex networks,” *Review of Modern Physics* 74, 47-97.

V. Bala and S. Goyal (2000), “A Non-Cooperative Model of Network Formation,” *Econometrica* 68, 1181-1230. [PDF]

A. Calvó-Armengol and R. Ilkiliç (2006), “Pairwise Stability and Nash Equilibria in Network Formation,” mimeo, Universitat Autonoma de Barcelona. [PDF]

M.O. Jackson and A. Wolinsky (1996), “A Strategic Model of Social and Economic Networks,” *Journal of Economic Theory* 71, 44-74. [PDF]

M.O. Jackson (2005), “A Survey of Models of Network Formation: Stability and Efficiency,” in *Group Formation in Economics: Networks, Clubs and Coalitions*, G. Demange and M. Wooders (Eds.), Cambridge: Cambridge University Press, pp. 11-88. [PDF]

M.O. Jackson (2007), “The Economics of Social Networks,” in *Proceedings of the Ninth World Congress of the Econometric Society*, R. Blundell, W. Newey and T. Persson (Eds.), Cambridge: Cambridge University Press, pp. 1-56. [PDF]

M.O. Jackson and B.W. Rogers (2005), "The economics of small worlds," *Journal of the European Economic Association* 3, 617-627.

M.O. Jackson and B.W. Rogers (2007), "Meeting strangers and friends of friends: How random are social networks? *American Economic Review* 97, 890-915.

C. Johnson and R.P. Gilles (2000), "Spatial social networks," *Review of Economic Design* 5, 273-299.

F. Vega-Redondo (2007), *Complex Social Networks*, Econometric Society Monograph Series, Cambridge: Cambridge University Press.

**Lecture 3: Games on Networks **

Slides [PDF] **NEW**

In this lecture, we take networks as given (thus we leave aside the issue of network formation) and analyze the consequence of network structures on economic outcomes. The starting point of this analysis will be the paper by Ballester, Calvó-Armengol and Zenou (2006). They use a linear-quadratic utility function that exhibits both strategic substituabilities and complementarities between agents and each agent chooses the optimal amount of an action by maximizing this utility function. They show that, if the largest eigenvalue of the adjency matrix (the matrix that represents the graph of the network) is bounded above, then there is a unique Nash equilibrium and each action is proportional to the Bonacich centrality (in the network) of each agent. We develop further this approach by using the paper of Ballester and Calvó-Armengol (2006), who show the robustness of the first paper by reformulating the model as a linear-complementarity problem, a well-known issue in applied mathematics. One interesting result is to show that, using a suitable linear transformation of the interaction matrix (the one that gives the cross-derivatives), a linear-quadratic utility function that does not initially exhibit strategic complementarities can have complementarities in the induced game. This is referred to as hidden complementarities. One example of a model that exhibits hidden complementarities is that of Bramoullé and Kranton (2005), which a model of public goods in networks. We will expose the model, and analyze the case with hidden complementarities as well as that with pure substituabilities. A recent work on games on networks with incomplete information about the network structure is Galeotti et al. (2006).

Bramoulle Kranton JET 2007 [PDF]

Slides Bramoulle Kranton JET 2007 [PDF]

**References**

C. Ballester, A. Calvó-Armengol and Y. Zenou (2006), “Who’s Who in Networks. Wanted: the Key Player,” *Econometrica*. [PDF]

C. Ballester and A. Calvó-Armengol (2006), “Interaction Patterns with Hidden Complementarities,” mimeo, Universitat Autonoma de Barcelona. [PDF]

Y. Bramoullé and R. Kranton (2007), “Public goods in Networks,” *Journal of Economic Theory*. [PDF]

A. Galeotti, S. Goyal, M.O. Jackson, F. Vega-Redondo and L. Yariv (2006), “Network Games,” mimeo, Caltech University. [PDF]

**Mixing network formation and games on networks**: Slides [PDF] **NEW**

**Lecture 4: Applications to Labor Economics **

Slides Calvo JET 2004 [PDF] Slides Calvo-Jackson AER 2004 [PDF]

Slides Calvo-Jackson AER Tryads [PDF] Slides Calvo-Jackson AER Dyads [PDF]

We will use the tools learned in Lectures 1 and 2 to deal with labor issues. Direct applications are made in the papers by Calvó-Armengol (2004) and Calvó-Armengol and Zenou (2005). We will expose these papers and show how they explain the role of networks in the labor market, in particular how people transmit job information to their friends. Another approach, which is dynamic and has a more explicit structure (though there is no network formation), is that of Calvó-Armengol and Jackson (2004). This is a beautiful model that has very strong implications. In particular, it explains unemployment duration dependence. A non-technical introduction to this literature can be found in Calvó-Armengol (2006).

**References**

A. Calvó-Armengol (2004), “Job Contact Networks,” *Journal of Economic Theory*, 115, 191-206. [PDF]

A. Calvó-Armengol (2006), “Social Networks and Labour Market Outcomes,” Els Opuscles del CREI. [PDF]

A. Calvó-Armengol and M.O. Jackson (2004), “The Effects of Social Networks on Employment and Inequality,” *American Economic Review*, 94, 426-454. [PDF]

A. Calvó-Armengol and Y. Zenou (2005), “Job matching, social network and word-of-mouth communication”, *Journal of Urban Economics*, 57, 500-522. [PDF]

**Lecture 5: Applications to Crime **

Slides [PDF]** NEW**

Crime is a “social” activity. Peers and friends have an important impact on crime decisions and crime activities. However, few theoretical models have investigated this issue. The first paper on this issue is the one by Glaeser, Sacerdote and Scheinkman (1996), where agents are located on the circumference of a circle and decide criminal activities by looking at their neighbors. They show that there are multiplier effects in the sense that the variance of crime is much higher when there are social interactions. Calvó-Armengol and Zenou (2004) develop another explicit network model where any network structure (and not only the circle) is studied. They also analyze the network formation of criminals (using the pairwise-stability equilibrium concept) and show how social interactions affect crime decisions. Another interesting approach is to differentiate between weak and strong ties in crime and to see how they affect both crime and labor activities. For that, we will expose the paper by Calvó-Armengol, Verdier and Zenou (2005). We finally study the policy implications of crime networks using the paper by Ballester, Calvó-Armengol and Zenou (2004). Instead of punishing randomly criminals, they study the key-player policy, which consists in getting rid of the criminal whose removal results in the maximal decrease in overall activity.

**References**

C. Ballester, A. Calvó-Armengol and Y. Zenou (2004), « Who’s Who in Crime Networks. Wanted: the Key Player », CEPR Discussion Paper No. 4421. [PDF]

C. Ballester, A. Calvó-Armengol and Y. Zenou (2009), « Delinquent Networks », *Journal of the European Economic Association*, forthcoming.

A. Calvó-Armengol, T. Verdier and Y. Zenou (2007), “Strong Ties and Weak Ties in Employment and Crime,” *Journal of Public Economics*. [PDF]

A. Calvó-Armengol and Y. Zenou (2004), “Social Networks and Crime Decisions: The Role of Social Structure in Facilitating Delinquent Behavior,” *International Economic Review*, 45, 935-954. [PDF]

E.L. Glaeser, B. Sacerdote and J. Scheinkman (1996), “Crime and Social Interactions,” *Quarterly Journal of Economics*, 111, 508-548. [PDF]

**Lecture 6: Empirical Aspects of Social Networks **

Slides [PDF] **NEW**

In this last lecture, we will explore the empirical studies of some of the theoretical papers mentioned above. We will study how peer effects and social networks affect labor and crime activities. In particular, we will expose the papers by Calvó-Armengol, Patacchini and Zenou (2005) and Wahba and Zenou (2005). Other important papers that have test social network effects are, among others, Topa (2001) for unemployment rates in the US, Conley and Udry (2005), who investigate the role of social learning in the diffusion of a new agricultural technology in Ghana and Fafchamps and Lund (2003) who study risk-sharing networks in rural Philippines. There is a recent paper by Laschever (2005), who has an extraordinary natural experiment (the World World I draft), that allows him to test network effects in the labor market without having the standard problem of endogenous network formation. He also able to decompose the outcome effect into the endogeneous and the contextual effects, avoiding the standard reflection problem highlighted by Manski (1993).

**References**

A. Calvó-Armengol, E. Patacchini and Y. Zenou (2009), “Peer Effects and Social Networks in Education », *Review of Economics Studies*. [PDF]

T.G. Conley and C.R. Udry (2005), “Learning About a New Technology: Pineapple in Ghana,” mimeo, University of Chicago. [PDF]

M. Fafchamps and S. Lund (2003), “Risk Sharing Networks in Rural Philippines,” *Journal of Development Economics* 71, 261-87. [PDF]

R. Laschever (2005), "The doughboys network: Social interactions and the labor market outcomes of World War I veterans," mimeo, Northwestern University. [PDF]

C.F. Manski (1993), "Identification of endogenous social effects: The reflection problem," *Review of Economic Studies* 60, 531-542.

G. Topa (2001), “Social interactions, local spillovers and unemployment,” *Review of Economic Studie*s 68, 261-295. [PDF]

J. Wahba and Y. Zenou (2005), “Density, Social Networks and Job Search Methods: Theory and Applications to Egypt,” *Journal of Development Economics* 78, 443-473. [PDF]

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