This link has been bookmarked by 4 people . It was first bookmarked on 22 Nov 2006, by Ole C Brudvik.
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01 Jul 14
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15 Oct 13
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Role algebras for multiplex data
Let's suppose that we were looking at a single matrix on who was friends with whom. An obvious way of characterizing what we see is to classify the each pair as "friends" or "not friends." But now, let's extend our analysis one step further (or look at paths of length 2). Now each pair of actors could be characterized as friend, not friend, friend of friend, friend of not-friend, not-friend of friend, or not-friend of not friend. If we wanted to consider paths of length three...well, you get the idea.
The notion of a "role algebra" is to understand the relations between actors as realizations of the logically possible "compounds" of relations of selected path lengths. Most often in network analysis, we focus on path of length one (two actors are connected or not). But, sometimes it is useful to characterize a graph as containing more complex kinds of relations (friend of friend, not-friend of friend, etc.). Lists of these kinds of relations can be obtained by taking Boolean products of matrices (i.e. 0*0 = 0, 0*1 = 0, 1*0 = 0, and 1*1 = 1). When applied to single matrix, we raise a matrix to a power (multiply it by itself) and take the Boolean product; the result generates a matrix that tells us if there is a relation between each pair of nodes that is of a path length equal to the power. That is, to find whether each pair of actors is connected by the relation "friend of a friend" we take the Boolean product of the friendship matrix squared.
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- When we multiply the information matrix by its transpose and take Boolean products, we are identifying linkages like "sends information to a node that sends information to..."
- When we multiply the money matrix by its transpose and take Boolean products, we are identifying the linkage: "sends money to a node that sends money to ..."
- When we multiply the information matrix times the money matrix, we are identifying the relationship: "sends information to a node that sends money to..."
- When we multiply the money matrix times the information matrix, we are identifying the relationship: "sends money to a node that sends information to..."
This (elegant, but rather mysterious) method of finding "compound relations" can be applied to multi-plex data as a way of identifying the kinds of relations that exist in a multi-plex graph. The Transform>Semigroup algorithm can be used to identify these more complex qualitative kinds of relations among nodes.
It is easier for most people to understand this with an example, than in the abstract. So let's do a somewhat extended examination of the Knoke data for both information and money ties.
If we consider just direct relations, there are two: organizations can be tied by information; organizations can be tied by money. What if consider relations at two steps (what are called "word lengths" in role algebra)? In addition to the original two relations, there are now four more:
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These four new (two-step) relations among nodes are "words" of length two, or "compounds."
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It is possible, of course, to continue to compound to still greater lengths. In most sociological analyses with only two types of ties, longer lengths are rarely substantively meaningful. With more kinds of ties, however, the number of types of compound relationships can become quite large quite quickly.
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04 Apr 09
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that actors behavior is strongly shaped by the complex interaction of many simultaneous constraints and opportunities arising from how the individual is embedded in multiple kinds of relationships
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multi-valued, data combine information about multiplex relations into a single matrix. The values, however, don't represent strength, cost, or probability of a tie, but rather distinguish the qualitative type of tie that exists between each pair of actors
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22 Nov 06
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Introduction: Multiple relations among actors
Most of tools of social network analysis deal with structures defined by patterns in a single kind of relationship among actors: friendship, kinship, economic exchange, warfare, etc. Social relations among actors, however, are usually more complex, in that actors are connected in multiple ways simultaneously.
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In this chapter we will look at some of the tools that social network analysts have used grapple with the complexity of analyzing simultaneous multiple relations among actors. We'll begin by examining some basic data structures for multi-plex data, and how they can be visualized. To be useful in analysis, however, the information about multiple relations among a set of actors must somehow be represented in summary form.
There are two general approaches: reduction and combination. The "reduction" approach seeks to combine information about multiple relations among the same set of actors into a single relation that indexes the quantity of ties. All of these issues are dealt with in the section on multiplex data basics.
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The "combination" approach also seeks to create a single index of the multi-plex relations, but attempts to represent the quality of ties. Summarizing the information about multiple kinds of ties among actors as a single qualitative typology is discussed in the section on "role algebra." We won't actually explore the complexities of role algebra analysis, but we will provide a brief introduction to this way of approaching multi-relational complexity.
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