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OPTIMAL CONNECTIONS: STRENGTH AND DISTANCE IN VALUED GRAPHS

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- Yang, Song and David Knoke
- RESEARCH QUESTION:
- How to identify optimal connections, that is, direct or indirect paths between dyads that permit the highest exchange volumes while taking into account the actors’ costs of interaction?

- CONNECTIONS IN BINARY GRAPHS
- Graph is depicted as a two dimensions by a set of nodes representing actors and a set of lines representing the direct ties between a pair (dyad) respectively.
- We are concentrating on undirected, symmetric graphs that reflect mutual interactions. Marriages between persons, and contracts between corporations are two good cases in point. If A is married to B, B must be married to A as well.

- In binary graphs, the presence of connection between a pair of nodes is indicated by a constant value of 1. In contrast, the absence of connection is indicated by a value of 0.
- In a graph, a path is a set of distinct nodes and lines that connect a specific pair of nodes. A length of a path refers to the number of lines in it. The path distance between two nodes is defined as the length of the shortest path.

- In binary graphs, path distance is normally used to indicate the optimal connections between a pair of nodes. This solution assumes that intermediaries are costly. If more intermediaries are necessary to connect a pair of actors, they may extract higher commissions for their services, distort the information content exchanged, and increase the time required to complete a transaction.

- An Illustration

- EXAMPLE FOR THE DYAD AB
- PATHLENGTHOPTIMAL CONNECTION
- A-B11
- A-E-B2N/A
- A-E-D-C-B4N/A

- Valued graph is defined as a graph whose lines carry numerical values indicating the intensities of the relationships between all dyads. These numbers typically represent frequencies or durations of interactions among social actors; for example, volumes of communications, levels of friendship and trust, or dollar amounts of economic transactions. For organizations engaging in strategic alliances, a valued graph might indicate the numbers of distinct partnerships formed between each pair.

- Illustration

- In valued graphs, using path length to indicate optimal connection is not applicable because the shortest path is less identifiable.
- Previous researchers propose two solutions to measure optimal connections in valued graphs. Peay (1980) states that path value, defined as the smallest value attached to any line in a path, indicates the optimal path between a pair of nodes.

- EXAMPLE FOR THE DYAD AB
- PATHOPTIMAL CONNECTION
- A-B1
- A-E-B3
- A-E-D-C-B2
- This solution assumes that lower path values represent bottlenecks that impede the interactions between two nodes.

How to determine the path value/optimal connection when multiple paths/path values present between two nodes.

How to account for the transaction costs of exchanges involving many go-betweens.

- Flament (1963) uses path length, defined as the sum of the values of the lines included in a path, to represent the optimal connection between a pair of nodes.
- EXAMPLE FOR THE DYAD AB
- PATHOPTIMAL CONNECTION
- A-B1
- A-E-B6
- A-E-D-C-B15

- · No standard for which stands for optimal connection among results from Flament’s path length. whether larger or smaller path lengths are viewed as optimal for connecting dyads.
- ·If larger values indicate optimal connection. Then a high number can result when either (1) the lines in a path have high values, or (2) a path has many lines with low values that sum to a large total. And the solution fails when the second situation occurs.
- ·Else if lower values represent optimal connection. Then a low number can result when either (1) the lines in a path have low values, or (2) a path has few lines that add up to a small value.

- Bring binary distance back to the equations. We argue that including binary distance is especially crucial for measuring path strength in a valued graph because it takes into account the costs (in time, energy, or decay of information) required for indirectly connected dyads to reach one another through varying numbers of intermediaries.
- We now formally define two measures of path strength applicable to valued graphs. A valued graph G consists of three sets of information

- · A set of nodes N = {n1, n2, … ng}
- A set of lines between pairs of nodes L = {l1, l2, … lg}
- A set of values attached to the lines V = {v1, v2, … vg}.
- A path between nodes ni and nj consists of a sequence of distinct lines connecting the pair through one or more intermediaries, expressed as:

- A set of values attached to the lines V = {v1, v2, … vg}.
- · {li,i+1, li+1,i+2, … lj-2,j-1, lj-1,j},

- The dual subscripts indicate the origin and terminus nodes of each line.
- The minimum value Mij of a path between nodes ni and nj is the smallest value of any line in that path, indicated as
- ·Mij = min (vi,i+1, vi+1,i+2, … vj-2,j-1, vj-1,j).
- Notice that Mij is actually Peay’s path value.

- The distance of that path Dij is the total number of lines where each line has a value of one, which is indicated as
- ·Dij = (li,i+1 + li+1,i+2 … + lj-2,j-1 + lj-1,j ).
- Note that this sum is identical to distance in a corresponding binary graph, obtained by counting the number of lines in a path connecting nodes ni and nj.

- Illustrate

- Solution

- How UCINET chooses a different result to represent the optimal connections? The algorithm works like this,
- Find the highest path value among the multiple paths Between a pair of nodes, thinking this is the optimal path.
- In our example, UCINET picks 3 for the path
- BEDC, thinking it is the optimal path connecting the dyad BC.
- Calculating the binary distance associated with the
- optimal path it just picked up between the pair of
- nodes. In our example, it was 3 for the path BEDC.
- Dividing the highest path value by its binary distance, saying that I get the APV. In our example, it was 3/3=1.

- We want,
- Finding the path values for all the paths between a dyad.
- Calculating the binary distances for all the paths.
- Dividing each path values by its binary distance,
- producing multiple APVs for a dyad.
- Picking up the highest APV to represent the optimal connection between the dyad, which is 2/1=2 in our example.

- Such a difference in computing optimal connection between UCINET and our solution produces only one discrepancy in our example with five nodes and 10 dyads.
C52 = 5!/3!*2!=10, which is the maximum number of dyad relationships for 10 actors.

- However, social scientists rarely deal with 5 by 5 matrix. Instead, many of the matrices contain 10s, 100s, or even 1000s of actors, forming symmetrical matrices with many dimensions.
- Suppose we have a matrix with 100 actors. It can have a maximum C1002 = 100!/2!*98!=4,950 dyads. If UCINET and our solution have 10% disagreement, we are expecting 495 discrepancies between UCINET output and our expected output, which is less tolerable.

- Choose a right algorithm such as Floyd-Walshall algorithm, used in computing shortest path in valued graphs.
- Its implementation appears in web search of a shortest path between two locations in “mapblast” or “yahoo map.”
- Implement the algorithm using any languages such as C, C++, JAVA, or FORTRAN.
- Keeping track of the binary distances for each and every Paths between a pair of nodes turns out to be a difficult task. Thus,
- We are waiting for a successful implementation of a right algorithm to solve our research problem.