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Online Appendix to “Asset Pricing in Large Information Networks”
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ozsoylev, Han N. Waldén, Johan |
| Copyright Year | 2011 |
| Abstract | Proof of Proposition 1: We construct a growing sequence of “caveman” networks that converge to a given degree distribution. A caveman network is one which partitions the set of agents in the sense that if agent i is connected with j and j is connected with k, then i is connected with k (see Watts [2]). We proceed as follows: First we observe that for d(1) = 1, the result is trivial, so we assume that d(1) = 1. For a given d ∈ S∞, define k = mini{i = 1 : i ∈ supp[d]}. For m > k, we define d̂ ∈ S by d̂(i) = d(i)/∑mj=1 d(j). Clearly, limm→∞ ∑mi=1 |d̂m(i)−d(i)| = 0. For an arbitrary n ≥ k, choose m = n . For 1 < ≤ m, = k, choose z = d̂( )× n/ , and z k = (n− ∑ =k z m )/k . Now, define Gn, with degree distribution d, as a network in which there are z clusters of tightly connected sets of agents, with members, 1 < ≤ m and n−∑m =2 z singletons. With this construction, |zn /n − d̂m(i)| ≤ /n for > 2 and = k. Moreover, |zn 1 /n − d̂m(1)| ≤ (k + 1)/n, and |zn kk/n − d̂m(k)| ≤ (k + 1)/n + m/n, so ∑m =1 |zn − d̂( )| ≤ 2(k + 1)/n+ 2m/n = O(n−1/3). Thus, ∑ n1/3 i=1 |dn(i)− d̂ n 1/3 (i)| → 0, when n → ∞ and since ∑ n1/3 i=1 |d̂ n1/3 (i)−d(i)| → 0, when n → ∞, this sequence of caveman networks indeed provides a constructive example for which the degree distribution converges to d. Moreover, it is straightforward to check that if d(i) = O(i−α), α > 1, then (10) is satisfied in the previously constructed sequence of caveman networks, and that if α > 2, then (11) is satisfied. If d(i) ∼ i−α, α ≤ 2, on the other hand, then clearly ∑i d(i)i = ∞, so (11) will fail. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://faculty.haas.berkeley.edu/walden/HaasWebpage/apinformationnetworksappendix2011jet.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
