Skip to main content

The structure of risk-sharing networks

Abstract

We examine the structure of risk-sharing networks in developing countries using data from the Tanzanian village of Nyakatoke. We first show that the Nyakatoke network exhibits: (1) the “small-world” phenomenon, where two households who are not themselves risk-sharing partners are separated only by a short chain of intermediaries; (2) preferential attachment, which is a network formation process where the probability of a household receiving a partner is proportional to that household’s existing number of partners; and (3) assortative mixing, as similarly connected households tend to link to each other. We then examine the implications of these features for network performance by comparing the Nyakatoke network to simulated networks with alternative structural traits. Our simulations show that the Nyakatoke network displays optimal or near-optimal performance along multiple dimensions. In particular, the Nyakatoke network has a notable ability to withstand perturbations of multiple types, a property that none of the counterfactual networks possess.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Availability of data and material

The data are publicly available and can be found at www.uantwerpen.be/en/staff/joachim-deweerdt.

Notes

  1. 1.

    Section 5 provides a discussion of the implications of our results for the theoretical network formation literature. Here we only comment on those studies with direct relevance to our research question. See Wahhaj (2010), Ambrus et al. (2014, 2015), Grandjean (2014), Milán (2016), or Ambrus et al. (2017) for other theoretical work in this area.

  2. 2.

    In a similar model, Bramoullé and Kranton (2007b) studied risk-sharing networks across communities. While they found that the formation of cross-community links reduces risk sharing within a village, the net welfare effect can be positive.

  3. 3.

    Caudell et al. (2015) also considered some network formation-oriented questions. In particular, the authors examined the relationship between individual attributes and the number of times a person was named as a lender. They found that males are more likely to be cited as a lender and the probability of them being so increases with their herd size.

  4. 4.

    A couple of other studies do explicitly recognize the role of structure in risk-sharing networks, but this is typically in the context of examining network formation. For example, Comola (2010) examined whether agents choose risk-sharing partners based on their location in the network. Using data from Tanzania, the author found that location matters, as agents prefer wealthy contacts with few additional risk-sharing partners. For an example of a similar finding in the context of labor-sharing networks in Ethiopia, see Krishnan and Sciubba (2009).

  5. 5.

    See Jackson and Rogers (2007) for a discussion of structural features commonly observed in social networks.

  6. 6.

    Using only internal links is common with these data. See, for example, De Weerdt (2004) or Comola and Fafchamps (2014).

  7. 7.

    The authors report that of all \(119 \times 118 = 14\),042 dyads, 700 are discordant. See their Table 2 for more information.

  8. 8.

    Comola and Fafchamps (2014) assume that discordant responses are the result of underreporting and test whether the data are best interpreted as representing existing links or desired links. As mentioned above, they further distinguish between two interpretations of existing links: bilateral and unilateral. Bilateral links are existing links formed in mutual self-interest, whereas unilateral links are formed at the request of either party. In the Nyakatoke data, the desire-to-link model outperforms both the bilateral and unilateral models in a likelihood ratio-based framework for testing non-nested models.

  9. 9.

    See Bloch et al. (2008) for an example of how the flow of information is central to the behavior of risk-sharing networks.

  10. 10.

    Reciprocated links are counted twice under the desire-to-link assumption because the network is directed. The underreporting and overreporting scenarios result in undirected networks, so each link is only counted once. Multiplying the number of links for the underreporting and overreporting scenarios by two allows one to see how many links are added or deleted (relative to the desire-to-link scenario) when imposing these assumptions. For example, the desire-to-link scenario has 630 links, 350 of which are discordant. When imposing the underreporting assumption, we add the “missing” links until all relationships are reciprocated. This yields a total of \(630 + 350 = 980\) links, which is twice that reported for the number of links associated with underreporting.

  11. 11.

    As a result, the average degree cannot vary across simulations for the random network, which is why we do not report a standard deviation.

  12. 12.

    Note that there is a natural ordering of these statistics: as we go from underreporting, to desire-to-link, and then to overreporting, we are progressively deleting links, which serves to increase average distance.

  13. 13.

    The CCDF is simply one minus the cumulative distribution function.

  14. 14.

    See Table 1 for information on the average degree for each case.

  15. 15.

    Preferential attachment is embodied in this case as nodes with higher degrees are more likely to be found through the network-based meeting process.

  16. 16.

    The Jackson–Rogers model is a growing random network, so the degree distribution for this case differs from the static Erdös–Rényi network.

  17. 17.

    The authors put forth the following two-step procedure for estimating r: first, set \(\eta _0=0\) and m to the average degree observed in the Nyakatoke data. For undirected cases, m should be set to half of the average degree. Second, estimate r through an iterative least squares procedure. Starting with some initial value \(r_0\), regress \(\ln [1 - F(\eta )]\) on \(\ln (\eta + r_0 m)\) to estimate \(-(1 + r)\) and get an estimate \(r_1\). Repeat this process until a fixed point \(r^*\) is located. Note that we estimated each of our models with different starting values, and the results do not change.

  18. 18.

    Recall that assortative mixing occurs when high-degree nodes tend to link to other high-degree nodes, and disassortative mixing occurs when high-degree nodes link to low-degree nodes. See Appendix for further discussion and a definition of the degree correlation coefficient.

  19. 19.

    Perfect assortativity results when all nodes of a particular degree only connect to other nodes with that same degree. That is, perfect assortativity is associated with a group-like network structure.

  20. 20.

    Recall that a node’s excess degree is the node’s degree minus one.

  21. 21.

    This statistical significance also occurs for the in-degree correlation coefficient for that network (p-value = 0.02), for which we find \(\rho = 0.09\).

  22. 22.

    Note that a full consideration of diffusion processes (i.e., how shocks propagate through the network) is beyond the scope of this paper. See Newman (2010) for detailed discussion.

  23. 23.

    For the process to be well defined, the network must be initialized properly. The reader is referred to Jackson and Rogers (2007) for details regarding initialization.

  24. 24.

    These results are averages across 10,000 simulations of the corresponding network. By corresponding network, we mean a network with the same number of nodes and links. For example, to simulate the counterfactual processes for the desire-to-link network, we generate growing random networks with 119 nodes and 630 links. Further, as the desire-to-link network is directed, the counterfactual networks are also generated as directed.

  25. 25.

    Note that our results are affected by the fact that our network is of a finite size, while theoretical results are based on networks of an arbitrary (infinite) size. See Newman (2010) or Boguñá et al. (2004) for a detailed discussion.

  26. 26.

    For the counterfactual processes, we generate a new network for each iteration. That is, for each iteration, we generate a network and then progressively remove nodes from that network at random.

  27. 27.

    The simple explanation for this result is that the variance of the overreporting network’s degree distribution is sufficiently close to that of the corresponding PA network.

  28. 28.

    The PA network lacks robustness to targeted attack because its structural integrity relies heavily on high-degree hubs, and so is quickly decimated by targeted attack. Purely random networks lack hubs, which implies that the effect of targeted attack is similar to random node removal (Barabási 2016).

  29. 29.

    We consider as valid any pixel that is even partially inside that region.

  30. 30.

    The rewiring performed once inside the target pixel is to ensure more uniform sampling. It is important that this rewiring does not lead the network to leave the pixel and, as such, any rewiring that does so is discarded.

  31. 31.

    Note that the white space in each panel represents invalid pixels. As such, we see that the shape of the valid region is consistent with the notion of a positive relationship between assortativity and clustering coefficients in social networks (Newman and Park 2003).

  32. 32.

    That component size tends to be smaller for assortative networks is consistent with Newman (2002, 2003).

  33. 33.

    It is also possible that assortativity is simply an unintentional by-product of a dynamic network formation process. In the Jackson and Rogers (2007) model, for example, assortativity results from the fact that older nodes are more likely to have higher degrees and be linked to each other.

  34. 34.

    For example, consider a star network, which is a tree that is perfectly disassortative.

  35. 35.

    Regular networks have undefined assortativity coefficients because the denominator in Eq. (A.3) is zero. While it is possible to have a well-defined assortativity coefficient for almost 2-regular networks, the coefficient will generally be close to one because nearly all nodes are connected to nodes with the same degree.

  36. 36.

    The data are publicly available at Banerjee et al. (2013b).

  37. 37.

    Specifically, the survey gathered network information along multiple dimensions, including friendship, family, religious affiliation, credit relationships, etc. We approximate the risk-sharing network in each village by taking the union of two different types of networks: borrowing/lending small amounts of cash and borrowing/lending kerosene or rice. We believe that these networks capture risk sharing, as the related survey questions specifically reference times of need. For example, the question related to borrowing small amounts of cash asks “If you suddenly needed to borrow Rs. 50 for a day, who would you ask?” See Sect. 2 for the comparable question from the Nyakatoke survey.

  38. 38.

    That is, for each village we simulated 10,000 random networks with the same number of nodes and links. We then calculated the actual clustering coefficient for each village and subtract from that number the average clustering coefficient across all associated random networks.

  39. 39.

    See Figure 1 in Huisman (2009). Note that Huisman did not find similar levels of bias for clustering coefficients. See Lee et al. (2006) for additional analysis of the bias associated with sampled networks.

  40. 40.

    For example, when randomly removing 54% of nodes in our overreporting network, we find an average clustering coefficient of 0.07 across 10,000 replications. This is comparable to the true clustering coefficient of 0.08.

  41. 41.

    Other studies examining the interaction between formal and informal insurance include Cox and Jakubson (1995), Cox and Jimenez (1995), Maitra and Ray (2003), Fan (2010), Janssens and Kramer (2016), and Strupat and Klohn (2018). Also see Chih (2016) for a related paper regarding the role of social networks in influencing the extent to which the government provision of public goods crowds out voluntary contributions.

  42. 42.

    For example, consider a situation where household 1 reports a link with household 2, but household 2 does not report household 1. In this case, we impute \(g^u_{12}=g^u_{21}=1\), \(g^o_{12}=g^o_{21}=0\), but let \(g^d_{12}=1 \ne 0 = g^d_{21}\).

  43. 43.

    See Jackson and Rogers (2007) for details and a discussion of alternative clustering coefficients. Here we have adopted what is known as a “global” clustering coefficient. An alternative “local” clustering coefficient calculates the proportion of transitive triples on a node-by-node basis and then averages across nodes.

  44. 44.

    A node’s excess degree is the node’s degree minus one.

  45. 45.

    Where \(p_j\) is the probability that a randomly chosen node has a degree of j, we have \(q_j = (j + 1)p_{j+1}/\sum _k k p_k\). It must also be the case that \(\sum _i e_{ij} = q_j\).

References

  1. Albert R, Jeong H, Barabási A (1999) Diameter of the world-wide web. Nature 401(6749):130–131

    Article  Google Scholar 

  2. Albert R, Jeong H, Barabási A (2000) Error and attack tolerance of complex networks. Nature 406(6794):378–382

    Article  Google Scholar 

  3. Ambrus A, Mobius M, Szeidl A (2014) Consumption risk-sharing in social networks. Am Econ Rev 104(1):149–182

    Article  Google Scholar 

  4. Ambrus A, Chandrasekhar A, Elliott M (2015) Social investments, informal risk sharing, and inequality. eRID Working Paper No. 179

  5. Ambrus A, Gao W, Milán P (2017) Informal risk sharing with local information. Retrieved from http://public.econ.duke.edu/~aa231/AGM170226.pdf. Accessed 19 Jan 2018

  6. Attanasio O, Barr A, Cardenas J, Genicot G, Meghir C (2012) Risk pooling, risk preferences, and social networks. Am Econ J Appl Econ 4(2):134–167

    Article  Google Scholar 

  7. Bakshi RK, Mallick D, Ulubaşoğlu MA (2019) Social capital as a coping mechanism for seasonal deprivation: the case of the Monga in Bangladesh. Empir Econ 57(1):239–262

    Article  Google Scholar 

  8. Baltagi BH, Deng Y, Ma X (2018) Network effects on labor contracts of internal migrants in China: a spatial autoregressive model. Empir Econ 55(1):265–296

    Article  Google Scholar 

  9. Banerjee A, Chandrasekhar AG, Duflo E, Jackson MO (2013a) The diffusion of microfinance. Science 341(6144):1236498

    Article  Google Scholar 

  10. Banerjee A, Chandrasekhar AG, Duflo E, Jackson MO (2013b) Replication data for: the diffusion of microfinance. Harv Dataverse. https://doi.org/10.7910/DVN/U3BIHX

    Article  Google Scholar 

  11. Barabási A (2016) Network science. Cambridge University Press, Cambridge

    Google Scholar 

  12. Barabási A, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512

    Article  Google Scholar 

  13. Barabási A, Albert R, Jeong H (1999) Mean-field theory for scale-free random networks. Physica A Stat Mech Appl 272(1):173–187

    Article  Google Scholar 

  14. Barr A, Genicot G (2008) Risk sharing, commitment, and information: an experimental analysis. J Eur Econ Assoc 6(6):1151–1185

    Article  Google Scholar 

  15. Barr A, Dekker M, Fafchamps M (2012) Who shares risk with whom under different enforcement mechanisms? Econ Dev Cult Change 60(4):677–706

    Article  Google Scholar 

  16. Bloch F, Genicot G, Ray D (2008) Informal insurance in social networks. J Econ Theory 143(1):36–58

    Article  Google Scholar 

  17. Boguñá M, Pastor-Satorras R, Vespignani A (2004) Cut-offs and finite size effects in scale-free networks. Eur Phys J B 38(2):205–209

    Article  Google Scholar 

  18. Boorman S (1975) A combinatorial optimization model for transmission of job information through contact networks. Bell J Econ 6(1):216–249

    Article  Google Scholar 

  19. Bramoullé Y, Kranton R (2007a) Risk-sharing networks. J Econ Behav Organ 64(3):275–294

    Article  Google Scholar 

  20. Bramoullé Y, Kranton R (2007b) Risk sharing across communities. Am Econ Rev 97(2):70–74

    Article  Google Scholar 

  21. Callaway D, Newman M, Strogatz S, Watts D (2000) Network robustness and fragility: percolation on random graphs. Phys Rev Lett 85(25):5468–5471

    Article  Google Scholar 

  22. Caudell M, Rotolo T, Grima M (2015) Informal lending networks in rural Ethiopia. Soc Netw 40:34–42

    Article  Google Scholar 

  23. Chih Y (2016) Social network structure and government provision crowding-out on voluntary contributions. J Behav Exp Econ 63:83–90

    Article  Google Scholar 

  24. Cohen R, Erez K, Ben-Avraham D, Havlin S (2000) Resilience of the internet to random breakdowns. Phys Rev Lett 85(21):4626–4628

    Article  Google Scholar 

  25. Comola M (2010) The network structure of mutual support links: evidence from rural Tanzania. pSE Working papers no. 2008-74

  26. Comola M, Fafchamps M (2014) Testing unilateral and bilateral link formation. Econ J 124(579):954–976

    Article  Google Scholar 

  27. Comola M, Fafchamps M (2017) The missing transfers: estimating misreporting in dyadic data. Econ Dev Cult Change 65(3):549–582

    Article  Google Scholar 

  28. Cox D, Jakubson G (1995) The connection between public transfers and private interfamily transfers. J Publ Econ 57(1):129–167

    Article  Google Scholar 

  29. Cox D, Jimenez E (1995) Private transfers and the effectiveness of public income redistribution in the Philippines. In: van de Walle D, Nead K (eds) Public spending and the poor: theory and evidence. Johns Hopkins University Press, Baltimore

    Google Scholar 

  30. De Weerdt J (2004) Risk-sharing and endogenous network formation. In: Dercon S (ed) Insurance against poverty. Oxford University Press, Oxford, pp 197–216

    Chapter  Google Scholar 

  31. De Weerdt J, Dercon S (2006) Risk-sharing networks and insurance against illness. J Dev Econ 81(2):337–356

    Article  Google Scholar 

  32. De Weerdt J, Fafchamps M (2011) Social identity and the formation of health insurance networks. J Dev Stud 47(8):1152–1177

    Article  Google Scholar 

  33. Dercon S, De Weerdt J, Bold T, Pankhurst A (2006) Group-based funeral insurance in Ethiopia and Tanzania. World Dev 34(4):685–703

    Article  Google Scholar 

  34. Erdös P, Rényi A (1959) On random graphs. Publ Math Debr 6:290–297

    Google Scholar 

  35. Erdös P, Rényi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Sci 5:17–61

    Google Scholar 

  36. Fafchamps M, Gubert F (2007) The formation of risk sharing networks. J Dev Econ 83(2):326–350

    Article  Google Scholar 

  37. Fafchamps M, Lund S (2003) Risk-sharing networks in rural Philippines. J Dev Econ 71(2):261–287

    Article  Google Scholar 

  38. Fan E (2010) Who benefits from public old age pensions? Evidence from a targeted program. Econ Dev Cult Change 58(2):297–322

    Article  Google Scholar 

  39. Furusawa T, Konishi H (2007) Free trade networks. J Int Econ 72(2):310–335

    Article  Google Scholar 

  40. Gao W, Moon E (2016) Informal insurance networks. BE J Theor Econ 16(2):455–484

    Google Scholar 

  41. Goyal S, Van Der Leij M, Moraga-González J (2006) Economics: an emerging small world. J Polit Econ 114(2):403–412

    Article  Google Scholar 

  42. Grandjean G (2014) Risk-sharing networks and farsighted stability. Rev Econ Des 18(3):191–218

    Google Scholar 

  43. Granovetter M (2005) The impact of social structure on economic outcomes. J Econ Perspect 19(1):33–50

    Article  Google Scholar 

  44. Holme P, Zhao J (2007) Exploring the assortativity-clustering space of a network’s degree sequence. Phys Rev E 75(4):046111

    Article  Google Scholar 

  45. Huisman M (2009) Imputation of missing network data: some simple procedures. J Soc Struct 10(1):1–29

    Article  Google Scholar 

  46. Inekwe JN, Jin Y, Valenzuela MR (2019) Income inequality, financial flows and political institution: Sub-Saharan African financial network. Empir Econ. https://doi.org/10.1007/s00181-019-01634-3

    Article  Google Scholar 

  47. Jackson M (2005) The economics of social networks. California Institute of Technology Social Science working paper 1237

  48. Jackson M (2008) Social and economic networks. Princeton University Press, Princeton

    Book  Google Scholar 

  49. Jackson M (2011) An overview of social networks and economic applications. In: Benhabib J, Bisin A, Jackson M (eds) Handbook of social economics, vol 1. North-Holland, Amsterdam, pp 511–585

    Google Scholar 

  50. Jackson M (2014) Networks in the understanding of economic behaviors. J Econ Perspect 28(4):3–22

    Article  Google Scholar 

  51. Jackson M, Rogers B (2007) Meeting strangers and friends of friends: How random are social networks? Am Econ Rev 97(3):890–915

    Article  Google Scholar 

  52. Jackson M, Rodriguez-Barraquer T, Tan X (2012) Social capital and social quilts: network patterns of favor exchange. Am Econ Rev 102(5):1857–1897

    Article  Google Scholar 

  53. Jackson M, Rogers B, Zenou Y (2017) The economic consequences of social-network structure. J Econ Lit 55(1):49–95

    Article  Google Scholar 

  54. Jacoby H, Skoufias E (1997) Risk, financial markets, and human capital in a developing country. Rev Econ Stud 64(3):311–335

    Article  Google Scholar 

  55. Janssens W, Kramer B (2016) The social dilemma of microinsurance: free-riding in a framed field experiment. J Econ Behav Organ 131:47–61

    Article  Google Scholar 

  56. Jensen R (2004) Do private transfers ‘displace’ the benefits of public transfers? Evidence from South Africa. J Public Econ 88(1):89–112

    Article  Google Scholar 

  57. Jeong H, Tombor B, Albert R, Oltvai Z, Barabási A (2000) The large-scale organization of metabolic networks. Nature 407(6804):651–654

    Article  Google Scholar 

  58. Kranton R, Minehart D (2001) A theory of buyer–seller networks. Am Econ Rev 91(3):485–508

    Article  Google Scholar 

  59. Krishnan P, Sciubba E (2009) Links and architecture in village networks. Econ J 119(537):917–949

    Article  Google Scholar 

  60. Kurosaki T, Fafchamps M (2002) Insurance market efficiency and crop choices in Pakistan. J Dev Econ 67(2):419–453

    Article  Google Scholar 

  61. Lee SH, Kim PJ, Jeong H (2006) Statistical properties of sampled networks. Phys Rev E 73(1):016102

    Article  Google Scholar 

  62. Maitra P, Ray R (2003) The effect of transfers on household expenditure patterns and poverty in South Africa. J Dev Econ 71(1):23–49

    Article  Google Scholar 

  63. Milán P (2016) Network-constrained risk sharing in village economies. Barcelona GSE working paper no. 912

  64. Morduch J (1999) Between the state and the market: Can informal insurance patch the safety net? World Bank Res Obs 14(2):187–207

    Article  Google Scholar 

  65. Murgai R, Winters P, Sadoulet E, De Janvry A (2002) Localized and incomplete mutual insurance. J Dev Econ 67(2):245–274

    Article  Google Scholar 

  66. Newman M (2002) Assortative mixing in networks. Phys Rev Lett 89(20):1–5

    Article  Google Scholar 

  67. Newman M (2003) Mixing patterns in networks. Phys Rev E 67(2):1–14

    Article  Google Scholar 

  68. Newman M (2010) Networks: an introduction. Oxford University Press, New York

    Book  Google Scholar 

  69. Newman M, Park J (2003) Why social networks are different from other types of networks. Phys Rev E 68(3):036122

    Article  Google Scholar 

  70. Noldus R, Van Mieghem P (2015) Assortativity in complex networks. J Complex Netw 3(4):507–542

    Article  Google Scholar 

  71. Paxson C (1992) Using weather variability to estimate the response of savings to transitory income in Thailand. Am Econ Rev 82(1):15–33

    Google Scholar 

  72. Pennock D, Flake G, Lawrence S, Glover E, Giles C (2002) Winners don’t take all: characterizing the competition for links on the web. Proc Natl Acad Sci 99(8):5207–5211

    Article  Google Scholar 

  73. Price D (1965) Networks of scientific papers. Science 149(3683):510–515

    Article  Google Scholar 

  74. Ryan B, Gross N (1943) The diffusion of hybrid seed corn in two Iowa communities. Rural Sociol 8(1):15–24

    Google Scholar 

  75. Strupat C, Klohn F (2018) Crowding out of solidarity? Public health insurance versus informal transfer networks in Ghana. World Dev 104:212–221

    Article  Google Scholar 

  76. Townsend R (1994) Risk and insurance in village India. Econometrica 62(3):539–91

    Article  Google Scholar 

  77. Van den Broeck K, Dercon S (2011) Information flows and social externalities in a Tanzanian banana growing village. J Dev Stud 47(2):231–252

    Article  Google Scholar 

  78. Wahhaj Z (2010) Social norms and individual savings in the context of informal insurance. J Econ Behav Organ 76(3):511–530

    Article  Google Scholar 

  79. Watts D (1999) Networks, dynamics, and the small-world phenomenon. Am J Sociol 105(2):493–527

    Article  Google Scholar 

  80. Watts D, Strogatz S (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440–442

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank Margherita Comola and Marcel Fafchamps for providing the data for this project, and Brais Alvarez Pereira for detailed comments. We would also like to thank attendees of the 43rd and 44th Annual Conference of the Eastern Economic Association, as well as participants of Drake University’s First Friday Brown Bag Seminar.

Funding

This work was supported by Drake University’s College of Business and Public Administration. Computing resources used for this project were provided by the American University High Performance Computing System, which is funded in part by the National Science Foundation (BCS-1039497). See www.american.edu/cas/hpc for information on the system.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Heath Henderson.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflicts of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

Code availability

Computer code used to generate the results in this paper is available at https://github.com/arnoblalam/risksharing.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

In this appendix, we first discuss some notational considerations and then provide definitions of the statistics used in our analysis. Let \(N = \{1, 2, \ldots , n \}\) denote the set of nodes or households in the Nyakatoke risk-sharing network. The network can be represented by an \(n \times n\) adjacency matrix \(g^m\) where \(m \in \{u, o, d\}\) signifies the network that results from employing the underreporting, overreporting, or desire-to-link assumption, respectively (see Sect. 2). For a given m, let \(g^m_{ij} \in \{0,1\}\) denote the risk-sharing relationship between any two households \(i,j \in N\) where \(g^m_{ij}=1\) indicates that household i cited household j, and otherwise \(g^m_{ij}=0\). When the survey responses of any two households i and j agree, note that we have \(g^u_{ij} = g^o_{ij} = g^d_{ij}\). When the survey responses of i and j are discordant, we ensure \(g^u_{ij}=g^u_{ji}=1\) and \(g^o_{ij}=g^o_{ji}=0\), but permit \(g^d_{ij} \ne g^d_{ji}\).Footnote 42

In our analysis, we treat the underreporting and overreporting networks as undirected networks in which two nodes are either connected or not. That is, it cannot be the case that node i links to node j without j linking to i, and thus relationships can be represented by a single, undirected link. The desire-to-link network is a directed network, which means that node i may link to node j without j linking to i. Directionality provides important information in such networks so we represent unreciprocated relationships by one directed link and reciprocated relationships by two directed links. The distinction between undirected and directed networks is important primarily because it affects how network statistics are calculated. In what follows, we provide definitions of the statistics used in our analysis, including definitions for average degree and distance, the clustering coefficient, and the degree correlation coefficient.

The out-degree of a node \(\eta ^+_i (g^m) = \sum _{j \in N} g^m_{ij}\) is the number nodes that household i links to in network \(g^m\), whereas the in-degree \(\eta ^-_i (g^m) = \sum _{j \in N} g^m_{ji}\) is the number of nodes that link to i in \(g^m\). For both undirected and directed networks, the average degree of the network is \(\eta (g^m) = \sum _{i \in N} \eta ^+_i (g^m)/n =\sum _{i \in N} \eta ^-_i (g^m)/n\). That is, the average degree is the same whether it is taken over the out- or in-degree of the network. A path between households i and j in network \(g^m\) is a sequence of households such that \(g^m_{i_k i_{k+1}}=1\) for each \(k \in \{1, 2, \ldots , K-1 \}\) where \(i_1 = i\) and \(i_K = j\). If there exists a path in each direction between two households, we say that those two households belong to the same component. (Note that path direction is only relevant under the desire-to-link assumption.) A network has a giant component if one component contains a relatively large fraction of the network’s nodes.

The distance or length of the shortest path from household i to j in the network \(g^m\) is denoted by \(d(i,j; g^m)\), and set to be infinite if no path exists. When the network \(g^m\) is connected (i.e., there exists a path between every pair of households), we can calculate the average distance as

$$\begin{aligned} d(g^m) = \frac{\sum _{i,j} d(i,j; g^m)}{n(n-1)}. \end{aligned}$$
(A.1)

The clustering coefficient measures the degree to which a household’s links are linked with each other. That is, we consider situations where node i links to node j and j links to node k, and calculate the percentage of times i links to k. For the undirected or directed network \(g^m\), the clustering coefficient is given by

$$\begin{aligned} C(g^m) = \frac{\sum _{i; j \ne i; k \ne j,i} g^m_{ij}g^m_{jk}g^m_{ik}}{\sum _{i; j \ne i; k \ne j,i} g^m_{ij}g^m_{jk}}, \end{aligned}$$
(A.2)

which gives the proportion of such “transitive triples”.Footnote 43

Finally, a network is said to exhibit (1) “assortative mixing” if high-degree nodes tend to link to other high-degree nodes, or (2) “disassortative mixing” if high-degree nodes tend to link to low-degree nodes. The assortativity of a network can be examined with the degree correlation coefficient. Following Newman (2002, 2003), let \(e_{ij}\) denote the fraction of links connecting nodes with excess degrees i and j.Footnote 44 Further, let \(q_j\) denote the probability that a node at the end of a randomly chosen link has an excess degree of j.Footnote 45 The degree correlation coefficient for the undirected network is then

$$\begin{aligned} \rho = \frac{\sum _{i,j} ij(e_{ij} - q_i q_j)}{\sigma _q^2} \end{aligned}$$
(A.3)

where \(\sigma _q\) is the standard deviation of the distribution \(q_j\). The coefficient lies in the interval \([-1, 1]\) with positive values associated with assortative mixing and negative values with disassortative mixing. Equation (A.3) is equivalent to the Pearson correlation coefficient of the degrees at either ends of the network’s links, and can be easily generalized for directed networks by letting i and j denote in- or out-degrees.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Henderson, H., Alam, A. The structure of risk-sharing networks. Empir Econ (2021). https://doi.org/10.1007/s00181-021-02037-z

Download citation

Keywords

  • Risk sharing
  • Informal insurance
  • Social networks
  • Network structure