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Island Network AnalysisMSTs and Dominating sets

By Aaron Desrochers

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Every network system created by man or by nature, has a corresponding graph model of that system. For example, one can take a group of islands and create a plethora of different types of graphs, as shown by anthropologist Per Hage and mathematician Frank Harary in their co-written books Island Networks and Exchange in Oceania.

I will highlight two graph theoretic concepts; the use of minimum spanning trees and dominating sets, and their applications to anthropology and archeology.

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Oceania is located to the northeast of the continent of Australia.

Around 1500 B.C, travel between these islands was a necessary part of survival for the people of Oceania. In the study of these early cultures, archeologists andanthropologists can use graph theory to verify their research on the spread of different cultures and also better understand interactions between islands inan island network.

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Some definitions

A spanning tree of a graph G is a subgraph of G that is a tree containing all thevertices of G.

In a weighted graph, a minimum spanning tree is a spanning tree whose sum of edge weights is as small as possible. It is the most economical tree of a graph with weighted edges.

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An Adjacency Matrix of an undirected graph is a (0,1)-matrix with a one for vertexes that are adjacent and a zero for those vertexes not adjacent.

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A B C D E F GA * 1 1 1 0 0 0 B 1 * 1 0 1 1 0C 1 1 * 1 1 0 1D 1 0 1 * 1 0 1E 0 1 1 1 * 1 1F 0 1 0 0 1 * 1G 0 0 1 1 1 1 *

When a graph is weighted one can use a similar adjacency matrix, where the value of the edge between two vertexes is recorded instead of a one to find a minimum spanning tree.

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Long ago, travel between the islands of Oceania was hazardous, mainly because the boats used were nothing more than wooden canoes. So traveling islanders would island hop as much as possible during their long sea voyages, traveling the shortest distances possible in their necessary interaction between other islands.

Thus, a finding a minimum spanning tree of an island networks gives us the paths that ancient islanders paddling wooden canoes would have taken thousands of years ago.

MSTs have been used to verify archeological data that show the spread and influence of cultures in different island clusters in Oceania.

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Trees in Oceania

Even before the study of graph theory began, the early natives of Polynesia used a tree to illustrate the hierarchy in their society.

([2] page 6)

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Minimal Spanning TreesOne can turn an island cluster into a complete graph with paths from each island to all of its neighbors. The distance between these islands can be measured very easily and applied as weights to the edges of the graph. From this we can construct an adjacency matrix, and from this matrix we can find with a minimum spanning tree that hits all the islands.

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• Prim’s algorithm states: Setting n= to the number of vertices, repeat the following step until the tree T has n-1 edges: Add to T the shortest edge between a vertex in T and a vertex not in T (initially pick any edge of the shortest length).

A B C D E F GA * 3 1 4 6 9 10 B 3 * 2 6 4 10 12C 1 2 * 2 2 5 7D 4 6 2 * 3 3 3E 6 4 2 3 * 2 6F 9 10 5 3 2 * 1G 10 12 7 3 6 1 *

Here n = 7; T will have 6 edges

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This minimum spanning tree is simple to find, but it has a very interesting application in anthropology. The paths between the islands in the MST were the safest and thus the most traveled by the ancient mariners of Oceania.

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According to Hage and Harary, “solutions to many network problems can be found by decomposing a graph into sub graphs…. In certain types of communication problems, a graph may be decomposed into its dominating sets.”

DOMINATING SETS

A dominating set S of a graph G has every node (vertex) of G either in it or adjacent to it. Consider this graph.

[2] page 205

{1,6,8,9} is a dominating set

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The Five Queens problem

A good example of a dominating set is the Five Queens problem in chess. How can one position five Queens on a chess board so that they command or occupy every square on the board without threatening each other?

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Another important feature of dominating sets, the minimum dominating set, illustrated by the 5 Queens problem is:

“ There may also be six, seven, or eight Queens in a minimal dominating set (so that the removal of one Queen leaves a non-dominating set). Hence the five Queens constitute a minimum dominating set. ”  

[2] page 205

The set {1,8,9} is a minimum dominating set.

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[2] page 205

Harary defines independent dominating sets as a set in which no nodes are adjacent.

{1,4,7,10} is an independent dominating set.

One can look at the number of nodes in each set and determine a domination number. The domination number is “the smallest number of nodes in any dominating set, hence it is the cardinality of a minimum dominating set.

In the graph on the previous slide, =3 Lastly, “the independent domination number (G) is the smallest number of nodes in any independent dominating set.

''( ) 4G

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( )G

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How does Domination apply in Oceania?

Hage and Harary apply the concept of dominating sets in the western Caroline Islands in order “to analyze distributional aspects of power relations.” They assume that, “given the traditions of warfare, conquest and hierarchy in the Carolines…all the islands in the network were either dominated or dominating.” According to Harary and , ,so there must be 4 dominating islands in the network.

( )G'( ) 4G

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“Elato and Satawal paid tribute to Lamotrek in return for the right to exploit neighboring uninhabited islands for marine recourses.”

“Puluwat dominated, through conquest and colonization, the neighboring islands of Pulusuk, Pulap, and Namonuito”

“Ulithi dominated Fais and Sorol in an informal but fundamental way: These two islands, alone amoung all others in the western Carolines, had, at an early date, lost their navigational skills and had to rely on Ulithi for communication with all other islands.”

Since dominating islands do not threaten each other, either Woleai or Ifaluk must make up the last member of the independent dominating set.

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“Woleai did not dominate its neighbors in the recent past but instead turned in on itself and developed an intra-atoll structure…We suppose that either Woleai dominated Eaurpik, Faraulep, and Ifaluk or that Ifaluk dominated Eaurpik, Faraulep or perhaps most likely, that Woleai and Ifaluk dominated at different times.”

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Class Exercise

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Find an independent dominating set and minimal dominating set in this make believe island network.

{B,E,I} and {A,F,H} are independent dominating sets with they are also a minimal dominating set. =3

'( ) 3G ( )G