Demoralization and Obesity in Upper Manhattan

Abstract

Here, following Wallace et al. (2003b), we apply the analysis of the previous chapters to a neighborhood geographic level that is intermediate between individual and simple aggregate scales, and where multiple factors may synergistically interact in a kind of ecosystem “mesoscale resonance” analogous to that described in Wallace and Wallace (2008b) and Holling (1992).

Appendix

The essential driving factor in the analysis is the relative strength of “weak” and “strong” social ties within the community. Weak ties, in Granovetter’s (1973) terminology, are those which do not disjointly partition a community. Meeting and talking with someone regularly in a park might well be a weak tie, while age cohort, ethnicity, and family relationship would be strong ties that indeed disjointly partition a community.

We can envision the culture of a geographically focused community in terms of a language, in the largest sense, combining behavioral and spoken, and other means of communication between individuals, and between an individual and his or her embedding social structures (Wallace and Wallace 1998c,1999a; Wallace and Fullilove 1999). Languages are characterized by an information-theoretic “source uncertainty” which quantifies their efficiency in delivering a message. Languages with high source uncertainty can say much with little, according to theory (e.g., Ash 1990).

As discussed in previous chapters, expressing source uncertainty in this form allows identification of a homology with the free energy density of a physical system that permits importation of techniques from thermodynamics and statistical mechanics into information theory.

The essential point is that we can parameterize the source uncertainty characteristic of a geographically focused community in terms of an inverse index of its strength of weak ties, that we will call K. If P is, for example, the ensemble average probability of weak ties across the community, then we take K ≡ 1∕P, and write H = H(K) = lim _{n → ∞} log[N(n, K)]∕n.

Thus we assume that K is monotonic in our index of chronic community stress, that is, K increases uniformly with increase of the index.

Note that, for simplicity, we assume the “strong” ties to be of a single fixed average probability across the community.

Imposing a thermodynamic formalism, we can write an “equation of state” for an information system dominated by the relative magnitude of strong and weak social ties as the relation

$$\displaystyle{ S(K) \equiv H(K) - KdH/dK }$$

(11.2)

S is defined as the disorder of the system, and the quantity I = S − H is defined as its instability.

Wallace and Fullilove (1999) propose that patterns of risk behavior are proportional to the instability I as the average probability of weak ties (or its inverse K​​ = ​​ 1∕P) changes across the community. Here we postulate that, for our study population, which is well embedded in community, not using drugs, not alienated from authority and, in general, strongly connected with family and friends, individual demoralization is proportional to the community instability index I.

Let the function I(K) follow the functional form of the SNR of a stochastic resonance:

$$\displaystyle{ I(K) = \frac{1} {K^{2}}[1 +\alpha \exp (1/(2K))]^{-1} }$$

(11.3)

Then, from above, I(K) = −KdH∕dK and we can write

$$\displaystyle\begin{array}{rcl} H(K)& =& \int -\frac{I(K)} {K} dK = \frac{1} {2K^{2}} - \frac{2} {K}\log [1 +\alpha \exp (1/2K)] \\ & & -4\mathtt{Polylog}[2,-\alpha \exp (1/2K)] {}\end{array}$$

(11.4)

where we have set the constant of integration to zero. Polylog[x,y] is a standard tabulated function.

An explicit plot shows H(K) is a reverse S-shaped curve, compared to I(K)’s inverted “U”. The richness of trans-community “language” declines monotonically with increasing K, the inverse of the average probability of weak ties within the community. This is also a way of saying that the capacity of the community as a communication channel declines with increasing K, since elementary information theory arguments show H(K) ≤ C where C is the channel capacity.

Note that the decrease of calculated H(K) is most rapid across the “hump” of the I(K) plot. If the inverse average probability of weak ties across the community, K = 1∕P, is monotonically determined by the community chronic stress index, this suggests that the functionality of the Upper Manhattan study zone as a social communication channel decreases very sharply indeed with increasing community stress.

We have a paradoxical result applicable to any reverse S-shaped curve, in that instability will be greatest at intermediate points of stress: the “falling off a cliff” effect.