## Abstract

This paper reports results of a simulation designed to evaluate the precision of Neumann approximations of outcomes of networked economic systems. We simulate systems with Erdös-Renyi, Watts-Strogatz, and Barabási-Albert networks. We discuss the conditions under which second-order approximations of economic outcomes generate small errors. We find that in certain economic systems data requirement is significantly reduced if the research goal is to predict economic outcomes of targeted agents (as opposed to outcomes of the entire system). Despite the systems’ complex network structure, our simulations indicate that sampling the targeted group, its connections, and the connections of its connections is sufficient to predict outcomes. As a result, economic policy that targets this specific group should focus on agents in the two-order of distance sample as opposed to the entire network. This sample of agents can be thought of as the group’s agents of change or opinion leaders.

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## Notes

- 1.
Neumann approximations are formally defined in Section 3.

- 2.
The peer-elasticity

*b*indicates the weight agents place on their social networks, hence, it can be thought of as the strength of the system’s network effect. The social multiplier is computed as follows:*η*= 1/(1-*b*). Refer to Section 2 for a discussion. - 3.
- 4.
We impose

*w*_{ i i }=0 indicating that an agent is not connected to herself. Notice that**W**is not assumed to be symmetric. - 5.
Notice that the row sums of

*b***A**are equal to 0 <*b*< 1, and the average column sum is also*b*. - 6.
The effect of the connections of all

*i*’s connections is therefore captured by the*i*-th row of (*b***A**)^{2}. - 7.
Network density is defined as the number of existing connections divided by the number of total possible connections.

- 8.
Intransitivity is the the probability that adjacent agents of an agent are not connected. Thus, high intransitivity implies low clustering.

- 9.
Note that

*k*_{ i }correspond to the number of adjacent links of agent*i*which were not initiated by*i*itself. - 10.
The power law generates degree distributions with positive skew.

- 11.
225 models correspond to: 9 (number of

*b*s) × 9 (number of Erdös-Renyi networks) + 9 (number of*b*s) × 8 (number of Watts-Strogatz networks) + 9 (number of*b*s) × 8 (number of Barabási-Albert networks). - 12.
Notice that the only random component of each model is the matrix

**A**. For each model, the simulation has the potential to generate 500 different outcome profiles based on 500 different matrices**A**. The data generated are 112,500 vectors**Y**(225 models × 500 replications) and 3,375,000 (112,500 × 30) approximations \(\hat {\mathbf {Y}}^{(\mathbf {k})}\). - 13.
Since approximation errors are always non-negative, there is no reason to compute squared errors.

- 14.
See rows of Table 1.

- 15.
See rows of Table 2.

- 16.
Approximated using the middle order of 1.5, i.e. −2.36=−2.45+0.06*(1.5).

- 17.
Approximated using the middle order of 29.5, i.e. -0.68 = -2.45 + 0.06*(29.5).

- 18.
**W**^{(S)}has (*n*−*n*^{S}) rows of zeros. The data of the non-zero rows of**W**^{(S)}correspond to the second-order snowball sample of the targeted group. Refer to Wasserman and Faust (1994, Chapter 2) for a detailed discussion of network data collection. - 19.
The outcome variance among

*n*^{G}agents may be very different from the variance among*n*agents, especially when elements of**X**are not uniformly distributed. Distributions with a bell shape (mass point in the middle range of**X**) and a U-shape (mass point on low and high ranges of**X**) are studied in Section 6. - 20.
All realizations

*x*_{ i }were positive. - 21.
Data was obtained from the UCINET project’s website. Available at: https://sites.google.com/site/ucinetsoftware/datasets. Four inmates reported no connections. To comply with the assumptions of the model we randomly assigned one connection to each one of these inmates.

- 22.
Data was obtained from Mathew Jackson’s webpage. Available at: http://www.stanford.edu/jacksonm/Data.html. An adjacent matrix was constructed from the matrix with number of citations assuming that a connection exists if there are positive citations. The diagonal, i.e. self-citation, was converted to zero.

- 23.
For example, (Duflo and Saez 2002) investigate the participation decisions in a retirement plan and estimate a peer effect parameter of approximately 0.2, i.e. social multiplier of 1.25. Lin (2010) estimates peer effects in student academic achievement and finds a social multiplier of approximately 1.38. Anselin et al. (2010) study spatial effects with a spatial hedonic model to value the access to potable water and estimate a spatial multiplier of 1.32. Baller et al. (2001) estimate spatial effects for the homicide rates of southern US counties and find spatial multipliers of 1.22 and 1.30 for 1980 and 1990, respectively.

- 24.
See Neilson and Wichmann (2014) for a discussion about network centrality and social welfare.

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Wichmann, B. Agents of Change and the Approximation of Network Outcomes: a Simulation Study.
*Netw Spat Econ* **15, **17–41 (2015). https://doi.org/10.1007/s11067-014-9266-2

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### Keywords

- Neumann series
- Neumann approximations
- Networks
- Snowball sampling
- Networked systems