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Empirical Economics

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14.03.2021

The structure of risk-sharing networks

verfasst von: Heath Henderson, Arnob Alam

Erschienen in: Empirical Economics | Ausgabe 2/2022

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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.

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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.

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.

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.

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).

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

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

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

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.

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

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.

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

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.

The CCDF is simply one minus the cumulative distribution function.

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

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

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

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.

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.

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.

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

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

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.

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.

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.

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.

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.

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.

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).

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

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.

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).

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

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.

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

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.

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

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.

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.

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.

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.

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.

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 \(g^d_{12}=1 \ne 0 = g^d_{21}\).

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.

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

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 \(\sum _i e_{ij} = q_j\).

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Titel: The structure of risk-sharing networks
verfasst von: Heath Henderson
Arnob Alam
Publikationsdatum: 14.03.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 2/2022
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-021-02037-z

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