A STRUCTURAL APPROACH TO THE REPRESENTATION OF LIFE HISTORY DATA¹

*Densities undefined where class size = 1.

Abstract

We propose a structural approach to the representation of life history data, based on an interval graph formalism. Various “life history graphs” are defined using this approach, and properties of such graphs are related to life course concepts. The analysis of life history graphs using standard network methods is illustrated, and suggestions are made regarding the use of network analysis for life course research. A duality is demonstrated between individual life history graphs and social networks, and connections between life history graph analysis and conventional life course methods are discussed.

Keywords:

• Life history data
• Interval graphs
• Life course
• Data representation
• Network analysis

Notes

¹The Cornell Community Study was conducted by the Cornell Employment and Family Careers Institute (with acknowledgments to Phyllis Moen, Stephen Sweet, and the Cornell Computer Assisted Survey Team) and sponsored by the Alfred P. Sloan Foundation (Sloan FDN #96-6-9, #99-6-23, Phyllis Moen, Principal Investigator).

²This work was supported in part by the Center for the Computational Analysis of Social and Organizational Systems and the Institute for Complex Engineered Systems at Carnegie Mellon University.

³This work was supported in part by the Cornell Employment and Family Careers Institute.

¹For instance, one could form an abstract ordering of life events based on current income; this might be of value in studies of career trajectories.

²We use the term “career” in the more general sense, applicable to any life domain, of a pattern of states over time (Han and Moen, 1999).

³This follows from the reflexive property of ∼.

⁴Identical onsets should in principle be very rare, but may in practice be fairly common due to data collection procedures which round dates to large time intervals (e.g., months or years). Even when spell transitions take place on the same day, such as leaving a relationship and moving, they rarely coincide instantaneously.

⁵Or other quantity, depending on the definition of 𝒯.

⁶For instance, we would expect more similarity across representations in analyses involving whole life history structures (e.g., comparison using structural distances (Butts and Carley, 2001) than in analyses which consider only small subgraphs (e.g., analysis of vertex degree scores).

⁷These structures, drawn in the plane with vertices placed in a circular arrangement, appear like stars inscribed within a polygon. The chordal property of the SLHG also implies that the gap graph is a comparability graph, i.e., that it has a transitive orientation.

⁸For a discussion of the interpretation of structural equivalence within life history structures, see below.

⁹Recall that if a set of spells are mutually coterminous, there exists some t ∈ T such that t belongs to all of the spells within the set.

¹⁰This problem is equivalent to constructing a node-minimal loopless digraph having 2|ℐ| edges under the constraints that two vertices have outdegree 1 and all other vertices have outdegrees of two or less. A minimal structure is given by a path (with bidirectional dyads) with |ℐ| + 1 vertices.

¹¹A slightly modified form of the construction in Theorem 4 must be used for both necessity and sufficiency to hold (not shown here), but the intuition is similar.

¹²The life course concept of life role (Elder, 1992) is distinct from that of structural role (Nadel, 1957; Lorrain and White, 1971) as used within network analysis; we will qualify our use of the term “role” accordingly, save when there is little danger of ambiguity.

¹³Or, possibly, narrative cycles, if demand is invoked for accounts which begin and end with reflections on the same spell.

¹⁴It is a standard result (Erdös and Rényi, 1960; Bollobás, 1984) that large graphs rapidly connect above a minimal threshold density; somewhat similar results hold for interval graphs (Maehara, 1991; Godehardt and Jaworski, 1996), and informal examination of both constructed and empirical life histories (the latter taken from the Cornell Community Study (Moen et.al., 2001)) suggests relatively few epochs per person on average.

¹⁵Though see the discussion below for suggestions regarding interpersonal extensions.

¹⁶Or, in the terms of social network analysis, the degree centrality.

¹⁷In the directed case, substitute the ordered pair (v _i , v _j ) for the unordered pair {v _i , v _j }.

¹⁸Such indices should be of particular interest to scholars applying narrative approaches to studying life histories, such as those outlined by Clausen (1998) and McAdams (1993).

¹⁹Induced life history subgraphs were constructed for each individual using those spells which were active within each of the four decades in question (i.e., 20s, 30s, etc.), and structural distances were calculated for all pairs of such graphs (Butts and Carley, 2001). For the 468 individuals with nondegenerate life history structures in each decade, these resulting distances were summed across decades to produce an aggregate measure (see Pixley and Butts, 2001 for additional details).

²⁰For Subject 1, closeness is poorly-defined due to the presence of multiple epochs. Here, we follow the common (but not universal) convention of replacing undefined geodesic distances by one greater than the maximum path length in analyses which require such values; scores are otherwise computed normally.

²¹Alternatives to this notion of centralization include odd moments of the distribution (particularly skew) and even upper semi-moments (e.g., upper semi-variance, upper semikurtosis). In practice, however, these measures are rarely if ever deployed within SNA.

²³These patterns follow from the types of spells the researcher uses to characterize the life course. For instance, the elderly would be more likely to exhibit high multiple domain involvement if spells of physical disability, hospitalization, or co-residence with adult children are included.

²²These results do not hold for some eigenvector centrality measures, for which non-star configurations (e.g., K ₂ ∪ N _n ) can be maximally centralized.

²⁴While social network density also tends to fall off in the number of vertices (since individual degree generally cannot increase without bound), the potential for long-running spells complicates this somewhat for life history graphs; nevertheless, since many spells cannot overlap within substantive classes, life history graphs with many more spells than classes will typically have low density.

²⁵As before, we substitute |V| for undefined path lengths.

²⁶Under a somewhat stricter definition, the condition is required over all V(G); note that where loops are not permitted, this implies that only non-adjacent vertices may be structurally equivalent.

²⁷Although this is far less true of spells in the SLHG. Consider, e.g., multiple jobs covered by a marriage and a child care spell, with no other spells present.

²⁸Recall that the similarity here is in terms of structural equivalence. This can also be generalized to other equivalences, e.g., automorphic or regular.

²⁹Actually, it could be argued that they are problematic even today. However, certain classifications (e.g., full-time employment) derive much of their significance from specific institutional (here legal) regimes which can at least be said to have some meaning for persons within a narrow socio-geographic context.

³⁰Note that this does not imply that the predictions in question will not hinge on sociohistorical variables, only that the underlying models themselves will be constant across place and time. By analogy, we do not expect a ball on an inclined plane to behave like a ball on a flat surface (under uniform gravitation), but the laws by which we make these predictions are the same in both circumstances.

³¹Recall that, as per the above discussion, these substantive classes may themselves be seen as emerging from structural equivalence classes on a set of universal spells.

*Densities undefined where class size = 1.

³²In point of fact, the relation described here exists with minor variations in a wide range of cultures; for the moment, however, we will restrict ourselves to the modern western version, as typified by norms followed in Europe, Canada, and the United States in the late 20th and early 21st centuries.

³³Presuming spell types for each relational type, we can in principle recover all networks at any point in time (for a given population) using the duality; where life histories are specified solely in terms of relational spells, the reverse process would allow us to recover all individual life histories from the set of networks across time.