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Mathematics LibreTexts

3.3: Kinds of Graphs

Now we need to introduce some terminology to describe different kinds of graphs. Figure 3.2 is an example of a binary (as opposed to a signed or ordinal or valued) and directed (as opposed to a co-occurrence or co-presence or bonded-tie) graph. Figure 3.3 is an example of a "co-occurrence" or "co-presence" or "bonded-tie" graph that is binary and undirected (or simple). The social relations being described here are also simplex (in Figures 3.2 and 3.3). Figure 3.4 is an example of one method of representing multiplex relational data with a single graph.

Let's take a moment to review some of this terminology in a little more detail.

Levels of Measurement: Binary, Signed, and Valued Graphs

In describing the pattern of who describes whom as a close friend, we could have asked our question in several different ways. If we asked each respondent "is this person a close friend or not," we are asking for a binary choice: each person is or is not chosen by each interviewee. Many social relationships can be described this way: the only thing that matters is whether a tie exists or not. When our data are collected this way, we can graph them simply: an arrow represents a choice that was made, no arrow represents the absence of a choice. But, we could have asked the question a second way: "for each person on this list, indicate whether you like, dislike, or don't care." We might assign a + to indicate "liking," zero to indicate "don't care" and - to indicate dislike. This kind of data is called "signed" data. The graph with signed data uses a + on the arrow to indicate a positive choice, a - to indicate a negative choice, and no arrow to indicate neutral or indifferent. Yet another approach would have been to ask: "rank the three people on this list in order of who you like most, next most, and least." This would give us "rank order" or "ordinal" data describing the strength of each friendship choice. Lastly, we could have asked: "on a scale from minus one hundred to plus one hundred - where minus 100 means you hate this person, zero means you feel neutral, and plus 100 means you love this person - how do you feel about...". This would give us information about the value of the strength of each choice on a (supposedly, at least) ratio level of measurement. With either an ordinal or valued graph, we would put the measure of the strength of the relationship on the arrow in the diagram.

Directed or "bonded" ties in the graph

In our example, we asked each member of the group to choose which others in the group they regarded as close friends. Each person (ego) then is being asked about ties or relations that they themselves direct toward others (alters). Each alter does not necessarily feel the same way about each tie as ego does: Bob may regard himself as a good friend to Alice, but Alice does not necessarily regard Bob as a good friend. It is very useful to describe many social structures as being composed of "directed" ties (which can be binary, signed, ordered, or valued). Indeed, most social processes involve sequences of directed actions. For example, suppose that person A directs a comment to B, then B directs a comment back to A, and so on. We may not know the order in which actions occurred (i.e. who started the conversation), or we may not care. In this example, we might just want to know that "A and B are having a conversation." In this case, the tie or relation "in conversation with" necessarily involves both actors A and B. Both A and B are "co-present" or "co-occurring" in the relation of "having a conversation." Or, we might also describe the situation as being one of an the social institution of a "conversation" that by definition involves two (or more) actors "bonded" in an interaction (Berkowitz).

"Directed" graphs use the convention of connecting nodes or actors with arrows that have arrow heads, indicating who is directing the tie toward whom. This is what we used in the graphs above, where individuals (egos) were directing choices toward others (alters). "Simple" or "Co-occurrence" or "co-presence" or "bonded-tie" graphs use the convention of connecting the pair of actors involved in the relation with a simple line segment (no arrow head). Be careful here, though. In a directed graph, Bob could choose Ted, and Ted choose Bob. This would be represented by headed arrows going from Bob to Ted, and from Ted to Bob, or by a double-headed arrow. But, this represents a different meaning from a graph that shows Bob and Ted connected by a single line segment without arrow heads. Such a graph would say "there is a relationship called close friend which ties Bob and Ted together." The distinction can be subtle, but it is important in some analyses.

Simplex or multiplex relations in the graph

Social actors are often connected by more than one kind of relationship. In our simple example, we showed two graphs of simple (sometimes referred to as "simplex" to differentiate from "multiplex") relations. The friendship graph (figure 3.2) showed a single relation (that happened to be binary and directed). The spouse graph (figure 3.3) showed a single relation (that happened to be binary and un-directed). Figure 3.4 combines information from two relations into a "multiplex" graph.

There are, potentially, different kinds of multiplex graphs. We graphed a tie if there was either a friendship or spousal relation. But, we could have graphed a tie only if there were both a friendship and spousal tie (what would such a graph look like?).

We also combined the information about multiple ties into a single line. Alternatively, one might use different symbols, colors, line widths, or other devices to keep all of the information about multiple relations visible in a multiplex graph -- but the result can often be too complicated to be useful.