Encapsulation theory: the transformation equations of absolute information hiding
📝 Abstract
This paper describes how the maximum potential number of edges of an encapsulated graph varies as the graph is transformed, that is, as nodes are created and modified. The equations governing these changes of maximum potential number of edges caused by the transformations are derived and briefly analysed.
💡 Analysis
This paper describes how the maximum potential number of edges of an encapsulated graph varies as the graph is transformed, that is, as nodes are created and modified. The equations governing these changes of maximum potential number of edges caused by the transformations are derived and briefly analysed.
📄 Content
1 Encapsulation theory: the transformation equations of absolute information hiding. Edmund Kirwan* www.EdmundKirwan.com Abstract This paper describes how the maximum potential number of edges of an encapsulated graph varies as the graph is transformed, that is, as nodes are created and modified. The equations governing these changes of maximum potential number of edges caused by the transformations are derived and briefly analysed. Keywords Encapsulation theory, encapsulation, maximum potential number of edges, transformation equation. 1. Introduction The maximum potential number of edges (M.P.E.) of an encapsulated graph was introduced in [1], which derived the equations for the M.P.E. of any given encapsulated graph of absolute information hiding. These equations, however, were static, offering no insight into the evolution of a graph over time. This paper addresses this evolutionary aspect by deriving the equations which describe not the overall M.P.E. of a graph but the changes in M.P.E. as a graph undergoes an arbitrary series of transformations. This paper considers encapsulated graphs of absolute information hiding only. 2. Standard deviation Before examining the transformation equations themselves, let us peform some experiments whose results we shall compare with those we might intuitively expect. Theorem 1.11 in [1] showed that, given two otherwise equivalent encapsulated graphs, the graph whose information hidden nodes are unevenly distributed over encapsulated regions can never have an M.P.E. of less than that of the graph with evenly distributed information hidden nodes. This may be understood qualitatively by considering the internal M.P.E. of an encapsulated region which, as also shown in [1], was shown to be proportional to the square of the number of nodes in that region. Thus consider an encapsulated graph of evenly distributed nodes where each encapsulated region has 10 nodes; each region will have an internal M.P.E. of 90 (=102 – 10). A node moved from one encapsulated region to another will (in a sense we shall later define precisely) increase the, ”Unevenness,” of the distribution: now one region will have 11 nodes and an M.P.E. of 110 (=112 – 11), whereas the donor region will have an M.P.E. of 72 (=92 – 9): moving this node has caused an overall net M.P.E. increase of 2. * © Edmund Kirwan 2009. Revision 1.1, Jan 14th 2009. arXiv.org is granted a nonexclusive and irrevocable license to distribute this article; all other entities may republish, but not for profit, all or part of this material provided reference is made to the author and title of this paper. The latest revision of this paper is available at [2]. 2 Loosely speaking, being proportional to the square of the number of nodes in a region, the internal M.P.E. tends to amplify deviations from even distribution, so the more unevenly distributed a graph is, the greater its M.P.E. Can we establish a more formal basis for investigation this relationship? Can we rigorously measure this, ”Unevenness?” Indeed we can, by using a tool of the statistician: the standard deviation. The standard deviation measures how widely spread the values in a dataset are. We shall use it first to measure how widely spread the number of information hidden nodes per encapsulated region is, that is, to measure the hidden node distribution. If we take a graph of r encapsulated regions where xi is the number of hidden nodes per region and where x is the average number of hidden nodes per region, then the standard deviation is defined by the equation: = 1 r ∑ i=1 r xi−∣x∣ 2 The standard deviation of the hidden node distribution for an evenly distributed encapsulated graph is 0; this figure then rises as the graph becomes increasingly unevenly distributed. Instead of examing how the M.P.E. behaves as the standard deviation of the hidden node distribution increases, however, it is useful to instead examine how the isoledensal configuration efficiency (also defined in [1]) behaves, as the configuration efficiency, being defined between 0 and 1, helps to normalise the trend for graphs of different cardinalities. Thus, whereas we expect that the M.P.E. of a graph will rise as the standard deviation of its hidden node distribution increases, we expect the configuration efficiency of that graph to fall as its standard deviation increases. Finally, we need only state the actual means of increasing the unevenness of a graph distribution. We shall begin, not with a perfectly evenly distributed graph, but with an graph of, say, 100 encapsulated regions, each region having one information hiding violational node, and a random number – between 0 and 30 – of information hidden nodes. Being thus unevenly distributed, the the graph will have a standard deviation of hidden node distribution of some nonzero number. We shall then take one hidden node from a region and move it to an arbitrarily designated target region. We shall th
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