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Unformatted text preview: rationale for this terminology
is as follows. Consider a Markov chain on D with transition matrix AD . For each i ∈ D, ui is the expected
19 number of periods before exiting D, starting from node i.11 Expression (46) shows that the node depths
measure the ampliﬁcation of losses due to interconnections among nodes in the default set.
We remark that the concept of node depth is dual to the notion of eigenvector centrality in the
networks literature (see for example Newman 2010). To see the connection let us restart the Markov
chain uniformly in D whenever it exits D. This modiﬁed chain has an ergodic distribution proportional
to 1D · [ID + AD + A2 + ...], and its ergodic distribution measures the centrality of the nodes in D. It
D
follows that node depth with respect to AD corresponds to centrality with respect to the transpose of
AD .
Although they are related algebraically, the two concepts are quite diﬀerent. To see why let us return
to the example of Figure 2(b). Suppose that node 1 (the central node) suﬀers a shock x1 > 80. This
causes all nodes to default, that is, the default set is D = {1, 2, 3, 4, 5}. Consider any node j > 1. In
the Markov chain described above the expected waiting time to exit the set D , starting from node j , is
given by the recursion uj = 1 + βj uj , which implies
uj = 1/(1 − βj ) = 1 + y/55. (48) From node 1 the expected waiting time satisﬁes the recursion
u1 = 1(100/140) + (1 + uj )(40/140). (49) u1 = 9/7 + 2y/385. (50) Hence Comparing (48) and (50) we ﬁnd that node 1 is deeper than the other nodes (u1 > uj ) for 0 ≤ y < 22
and shallower than the other nodes for y > 22. In contrast, node 1 has lower eigenvector centrality than
the other nodes for all y ≥ 0 because it cannot be reached directly from any other node.
The magnitude of the node depths in a default set can be bounded as follows. In the social networks
literature a set D is said to be αcohesive if every node in D has at least α of its obligations to other
nodes in D, that is, j ∈D aij ≥ α for every i ∈ D (Morris, 2000). The cohesiveness of D is the maximum such α, which we shall denote by αD . From (46) it follows that
∀i ∈ D, ui ≥ 1/(1 − αD ). (51) Thus the more cohesive the default set, the greater the depth of the nodes in the default set and the
greater the ampliﬁcation of the associated shock.
Similarly we can bound the node depths from above. Recall that βi is the proportion of i’s obligations
to other nodes in the ﬁnancial system. Letting βD = max{βi : i ∈ D} we obtain the upper bound,
11 Liu and Staum (2012) show that the node depths can be used to characterize the gradient of the clearing vector p(x)
with respect to the asset values. 20 assuming βD < 1,
∀i ∈ D, ui ≤ 1/(1 − βD ) . (52) The bounds in (51) and (52) depend on the default set D, which depends on the shock x. A uniform
upper bound is given by
∀i, ui ≤ 1/(1 − β + ) where β + = max βi , (53) assuming β + < 1.
We are now in a position to compare the...
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This document was uploaded on 02/20/2014 for the course ECON 101 at Pontificia Universidad Católica de Chile.
 Spring '11
 Traferri

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