Representing State Changes With Directed Graphs

Why Direction Matters in a State Graph

A state model describes an individual’s condition at a point in time and the changes that may take that individual from one condition to another. A directed graph gives a compact picture of those changes: each state is a node, and each permitted direct change is an arrow. The direction matters because a change from state \(A\) to state \(B\) does not, by itself, say that a change from \(B\) to \(A\) is possible.

For actuarial work, a graph is a useful first check on how a model has been described. It makes the states and the stated transitions visible before numerical calculations are introduced. The graph records which direct changes are represented; it does not, on its own, provide probabilities, rates, or the time spent in a state.

Definition: A directed state graph consists of a set of states \(\mathcal{S}\), represented by vertices, and a set of directed edges \(\mathcal{E}\), where each edge is an ordered pair \((j,k)\) of states. The edge \((j,k)\), drawn \(j\to k\), represents a direct transition with source state \(j\) and destination state \(k\).

The source is the state the individual is leaving; the destination is the state the individual enters. An arrow must therefore be read in order. If both \(A\to B\) and \(B\to A\) are represented, they are two distinct edges, not one reversible edge.

Edges, Paths, and Continued Occupancy

A direct edge represents one state change. A path is a sequence of one or more directed edges that can be followed in order: the destination of each edge must be the source of the next. For example, \(A\to B\to C\) is a path with two edges and one intermediate state, \(B\). It includes the direct transition \(A\to B\) and then the direct transition \(B\to C\); it does not imply a direct edge \(A\to C\).

Remaining in a state is different. If an individual is in state \(B\) at one time and is still in \(B\) later, no change of state has occurred between those observations. A graph focused on state changes therefore does not need an arrow from \(B\) back to \(B\) to represent continued occupancy. Such a self-loop would be a distinct representation, not a necessary way to show that time passes while the individual stays in \(B\).

Key distinction: An edge \(j\to k\) records a direct change from \(j\) to \(k\). Continued occupancy is not an edge between different states. A path with several edges records successive direct changes; it does not turn those changes into a single direct edge between the path’s first and last states.

Worked Examples: Constructing and Reading Graphs

Worked Example: A Hypothetical Disability-Income Model

A hypothetical income-protection model uses three states: \(A\) for active and not receiving disability benefits, \(D\) for disabled and receiving benefits, and \(X\) for dead. For this exercise, the specified direct transitions are \(A\to D\), \(D\to A\), \(A\to X\), and \(D\to X\). Represent the graph, list its edges, and determine whether \(X\to D\) is represented.

Method. Write each state as a vertex and translate every stated direct transition into an ordered pair. Do not infer an edge by reversing another arrow.

The vertex set is \(\mathcal{S}=\{A,D,X\}\). Translating the four specified changes gives \(\mathcal{E}=\{(A,D),(D,A),(A,X),(D,X)\}\). A labelled representation is:

A \xrightarrow{(A,D)} D \qquad D \xrightarrow{(D,A)} A \qquad A \xrightarrow{(A,X)} X \qquad D \xrightarrow{(D,X)} X

The edge \((A,D)\) has source \(A\) and destination \(D\); the edge \((D,A)\) is a separate transition in the reverse direction. Likewise, \((D,X)\) is explicitly present, but it does not create an edge in the reverse direction. Since \((X,D)\) is not in the specified edge set, \(X\to D\) is not represented.

The sequence \(A\to D\to X\) is a path with two edges: \((A,D)\) followed by \((D,X)\). The graph contains no direct edge \((A,X)\) by implication from that path—but in this particular model \((A,X)\) is also independently specified, so it is present as its own direct transition.

Common mistake avoided: Treating an arrow as reversible would incorrectly add \(X\to D\) or \(D\to A\) without checking the stated edges. Here \(D\to A\) is present because it was specified, not because another arrow exists.

Worked Example: Long-Term-Care Pathways

A hypothetical long-term-care model has states \(I\) (living independently), \(H\) (receiving care at home), \(F\) (receiving facility care), and \(X\) (dead). The stated direct transitions are \(I\to H\), \(H\to I\), \(H\to F\), \(F\to X\), and \(I\to X\). Draw the graph, identify a route from \(I\) to \(X\) through both care settings, count its edges, and assess whether \(F\to H\) is shown.

Method. Use one vertex for each of the four states, then make one arrow for each ordered pair in the supplied list. Finally, check a proposed route by matching each edge’s destination to the next edge’s source.

The edge set is \(\mathcal{E}=\{(I,H),(H,I),(H,F),(F,X),(I,X)\}\). A labelled representation is:

I \xrightarrow{(I,H)} H \qquad H \xrightarrow{(H,I)} I \qquad H \xrightarrow{(H,F)} F \qquad F \xrightarrow{(F,X)} X \qquad I \xrightarrow{(I,X)} X

The route through both care settings is \(I\to H\to F\to X\). Its successive edges are \((I,H)\), \((H,F)\), and \((F,X)\), so it has three edges and two intermediate states, \(H\) and \(F\). It is a three-step route, not a two-step route.

The graph does not show \(F\to H\), because \((F,H)\) is absent from the edge set. The presence of \(H\to F\) does not imply the reverse direction. The direct edge \(I\to X\) is also present, so this graph represents both a direct route from \(I\) to \(X\) and a separate route through the two care states.

Common mistake avoided: Counting the intermediate states instead of the edges gives the wrong number of steps. Count the arrows actually traversed: this route uses three arrows.

Worked Example: Pension Statuses and Direct Versus Indirect Changes

A hypothetical pension model uses \(A\) for active employment, \(D\) for deferred membership, \(R\) for retired, and \(X\) for dead. Its specified edges are \(A\to D\), \(A\to R\), \(A\to X\), \(D\to R\), \(D\to X\), and \(R\to X\). List the outgoing edges from each state, decide whether \(A\to R\) is direct, and identify the edges in the route \(A\to D\to R\to X\).

Method. Group the ordered pairs by their first entry to find outgoing edges. To test for a direct transition, look for the exact ordered pair in the edge set; a route through another state is not a substitute.

The edge set is \(\mathcal{E}=\{(A,D),(A,R),(A,X),(D,R),(D,X),(R,X)\}\). Grouping by source gives:

From \(A\): \((A,D)\), \((A,R)\), and \((A,X)\).
From \(D\): \((D,R)\) and \((D,X)\).
From \(R\): \((R,X)\).
From \(X\): no outgoing edge is specified.

The transition \(A\to R\) is direct because the ordered pair \((A,R)\) is explicitly in the edge set. Separately, \(A\to D\to R\) is a two-edge path through \(D\). The longer route \(A\to D\to R\to X\) uses the three successive edges \((A,D)\), \((D,R)\), and \((R,X)\), so it has three transitions.

Common mistake avoided: A reader might reason that \(A\to R\) is only indirect because the graph also contains the route through \(D\). The exact edge list resolves this: both the direct edge \((A,R)\) and the indirect route are represented.

Why the Ordered-Pair Representation Works

The graph is not merely a picture. Its ordered pairs provide a precise test for whether a proposed description is a direct transition or a path. Suppose the edge set is \(\mathcal{E}\). A proposed change from \(j\) to \(k\) is represented directly exactly when \((j,k)\in\mathcal{E}\). For a proposed sequence \(j_0\to j_1\to\cdots\to j_n\), every consecutive ordered pair must be an edge.

This follows directly from the definition of an edge: each arrow records one source and one destination. A sequence is possible in the graph only when the destination reached at one step is the source of the next. Consequently, having a path from \(j\) to \(k\) does not establish that \((j,k)\) itself belongs to \(\mathcal{E}\).

j_0\to j_1\to\cdots\to j_n \text{ is a path if and only if } (j_{r-1},j_r)\in\mathcal{E} \text{ for every }r=1,\ldots,n

This rule also explains why direction must be checked at every step. A sequence that uses an edge backward is not a path in the directed graph unless the reverse ordered pair is separately included.

Common Checks Before Using a State Graph

A graph helps prevent ambiguity when describing a model, but only if it is read carefully. In particular, it should not be used to silently add assumptions to the stated transitions.

Check the arrow’s direction. Read \(j\to k\) as source \(j\), destination \(k\); do not assume \(k\to j\) is also present.
Separate a route from a direct edge. A multi-edge path can connect two states without there being a direct arrow between them.
Count edges to count transitions. Intermediate states are not transitions; each arrow traversed is one edge.
Do not draw occupancy as a change. Remaining in a state over time is not an arrow to a different state.
Respect the stated edge set. A graph represents the specified direct changes; an absent ordered pair should not be inserted merely because it seems plausible.

These checks are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In later calculations, it is important to know which direct state changes a model represents. At this stage, the essential task is to read and encode those changes accurately, without confusing a direct transition, a route through intermediate states, and continued occupancy.

Check Your Understanding

For each problem, use only the states and edges stated in the question. Treat each arrow as directed.

A hypothetical model has states \(A\), \(B\), and \(X\), with edges \(A\to B\) and \(B\to X\). Write the ordered-pair edge set. Is \(A\to X\) a direct edge, and how many edges are in the route from \(A\) through \(B\) to \(X\)? (Answer: \(\{(A,B),(B,X)\}\); no direct \(A\to X\) edge is specified; the route has 2 edges.)
A care model has states \(U\) (independent), \(C\) (receiving care), and \(X\) (dead), with edges \(U\to C\), \(C\to X\), and \(U\to X\). Name the direct route from \(U\) to \(X\) and the route through \(C\), and count the edges in each. (Answer: direct route \(U\to X\), 1 edge; route \(U\to C\to X\), 2 edges.)
A hypothetical membership graph has states \(R\), \(S\), and \(X\), with edges \(R\to S\), \(S\to R\), and \(S\to X\). Is \(X\to S\) represented? Is continued occupancy in \(S\) represented by one of these three edges? (Answer: No, \((X,S)\) is absent; no, continued occupancy in \(S\) is not represented by an edge between different states, and none of the listed edges is a self-loop.)
A pension graph has states \(A,D,R,X\) and edges \(A\to D\), \(A\to R\), \(D\to R\), \(D\to X\), and \(R\to X\). List the outgoing edges from \(D\), and state whether \(A\to D\to R\to X\) is a valid path and how many transitions it contains. (Answer: Outgoing edges from \(D\) are \((D,R)\) and \((D,X)\); the route is valid and contains 3 transitions.)
A graph has states \(Q,R,S,T\) and edges \(Q\to R\), \(R\to S\), \(Q\to S\), and \(S\to T\). Compare \(Q\to R\to S\to T\) with \(Q\to S\to T\), and assess \(Q\to R\to T\). (Answer: The first route is valid with 3 edges; the second is valid with 2 edges; \(Q\to R\to T\) is not a valid path because \((R,T)\) is absent.)

What You'll Learn