OPTIMAL CONNECTIONS: STRENGTH AND DISTANCE IN VALUED GRAPHS

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OPTIMAL CONNECTIONS: STRENGTH AND DISTANCE IN VALUED GRAPHS. 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?.

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OPTIMAL CONNECTIONS: STRENGTH AND DISTANCE IN VALUED GRAPHS
• 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?
Binary
• 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.
Binary 2
• 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.
Binary 3
• 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.
A binary graph
• An Illustration
• PATHLENGTHOPTIMAL CONNECTION
• A-B 1 1
• A-E-B 2 N/A
• A-E-D-C-B 4 N/A
CONNECTIONS IN VALUED GRAPHS
• 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.
Valued Graph
• Illustration
Problems in Measuring OP in Valued Graphs
• 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.
Path Valued
• EXAMPLE FOR THE DYAD AB
• PATHOPTIMAL CONNECTION
• A-B 1
• A-E-B 3
• A-E-D-C-B 2
• This solution assumes that lower path values represent bottlenecks that impede the interactions between two nodes.
The problems of Peay’s path value solutions

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’s Solution
• 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-B 1
• A-E-B 6
• A-E-D-C-B 15
The Problems with Flament’s path length solution.
• · 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.
OUR SOLUTION
• 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
Definitions
• ·   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:
• ·       {li,i+1, li+1,i+2, … lj-2,j-1, lj-1,j},
Definitions
• 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.
Solution
• 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.
Solution
• Illustrate
Why differs
• 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.
What We Want
• 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.
How big of a difference
• 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.

It can be worse
• 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.
Real Solutions
• 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.