Encapsulation theory: the transformation equations of absolute information hiding

1 Encapsulation theor y: the transf ormation equations of absolute infor mation hiding. Edmund Kirwan * www .EdmundKir w an.com Abstract This paper describes how t he maximum potential number of edg es of an encapsulated g r aph varies as the g raph i s tr ansformed, that is, as nodes ar e creat ed and modified. The equations gov e rning these c hang e s of maximum po tential number of edg es caused by the tr ansf or mations ar e derived and brief ly analys ed. Ke y w ords Encapsulation theor y , encapsulation, maximum potential number of edg es, t ransf or mation equation. 1 . Introduction The maximum potential number of edg es (M.P .E.) of an encapsulated graph was int roduced in [1], which deriv ed the equations f or the M.P .E. of an y giv en encapsulated graph of absolute inf or mation hiding. These equations, ho we ver , w ere static, off eri ng no insight into the e volution of a graph o ver time. This paper addresses this e v olutionar y aspect b y der iving the equations which describe not the o v er all M.P .E. of a graph but the chang es in M.P .E. as a graph undergoes an arbitrary ser ies of transf ormations. This paper considers encapsulated graphs of absolute inf orm ation hiding onl y . 2. Standard deviation Bef ore e xamining the transf or mation equations themsel ves, le t us pef or m some e xper iments whose results w e shall compare with those w e might intuitiv ely e xpect. Theorem 1 . 1 1 in [1] show ed t hat, giv en tw o other wise equiv a lent encapsulated graphs, the graph whose inf ormation hidden nodes are unev enly dis tr ibuted o v er encapsulated regions can nev er ha v e an M.P .E. of less than t hat of the graph wit h e venl y distri buted inf or mation hidden nodes. This ma y be underst ood qualitativ ely b y consider ing the inter nal M.P .E. of an encapsulated region which, as also sho wn in [1], was sho wn to be propor tional to the squa re of the number of nodes in t hat region. Thus consider an encapsulated graph of ev enly dis tributed node s where eac h encapsulated region has 1 0 nodes; each region will ha ve an internal M.P .E. of 90 (=1 0 2 – 1 0). A node mo v ed from one encapsulated region to another will (in a sense w e shall later define precisely) incr ease the, ”Une venness, ” of t he distribution: no w one region will ha v e 1 1 nodes and an M.P .E. of 1 1 0 (=1 1 2 – 1 1), whereas the donor region will ha v e an M.P .E. of 7 2 (=9 2 – 9): mo ving this node has caused an o v erall net M.P .E. increase of 2. * © Edmund Kirwan 2009. R evision 1 . 1 , Jan 1 4 th 2009. arXiv .org is granted a non-e x c lusiv e and i rrev ocable license to distribute this ar ticle; all other entities may r epublish, but not f or profit, all or par t of this mater ial pro vided refere nce is made to the author and title of this pa per . The latest re vision of this paper is av ailable at [2]. 2 Loosely s peaking, being proport ional to the squar e of the number of nodes in a region, t he internal M.P .E. tends to ampl ify deviations fr om ev en distribution, so t he more une venl y distri buted a graph is, the greater its M.P .E. Can w e establish a more f or mal basis f or in ves tigation this relationship? Can w e r igorousl y measure this, ”Une venness ?” Indeed w e can, b y using a tool of the statistician: the standard de viation. The standard de viation measures ho w widel y spread the values in a dataset are. W e shall use it first to measure ho w widely spread the number of inf or mation hidden nodes per encapsulated region is, that is, to measure the hidden node distribution. If w e take a graph of r encapsulated regions where x i is the number of hidden nodes per region and where  x is the av erage number of hidden nodes per re gion, t hen the standard de viation is def ined b y the equation: =  1 r ∑ i = 1 r  x i − ∣  x ∣  2 The standard de vi ation of the hidden node dis tr ibution f or an ev enly distributed encapsulated graph is 0; this f igure then rises as t he graph becomes increasingl y une v enl y distributed. Instead of e xaming ho w the M.P .E. beha v es as the standard de viation of the hidden node distribution increases, ho w e v er , it is useful to instead e xamine ho w t he isoledensal configuration eff iciency (also defined in [1]) beha v es, as the conf iguration efficiency , being def ined betw een 0 and 1 , he lps to normalise the trend f or graphs of different cardinalities. Thus, whereas w e e xpe ct that the M.P .E. of a graph will r ise as the standard de viation of its hidden node distribution increases, w e e xpec t the configuration eff iciency of that graph to fall as its s tandard de viation increases. Finall y , we need onl y state the actual means of increasing the unev enness of a graph distri bution. W e shall begin, not with a perfectl y e v enly dis tributed g raph, but with an graph of, say , 1 00 encapsulated regions, each region ha ving one information hiding violational node, and a random number – betw een 0 and 30 – of inf or mation hidden nodes. Being thus une venl y distri buted, the the g raph will ha ve a s tandard de viation of hidden node distribution of some non-zero number . W e shall then t ak e one hidden node from a region and mo v e it to an arbitrarily designated targ et region. W e shall then record the chang e in t he hidden node distribution standard de viation and its resulting chang e in conf iguration efficiency . W e shall then mo v e a second hidden node from a region into the targ et region and perf orm t he measurement ag ain. This shall be repeated until all the hidden nodes of the g raph are in the targ et region, thus maximising the hidden node distribution standard de viation. Figure 1 sho ws the resulting configuration eff iciency plotted as a function of the changi ng hidden node distribution standard de viation. 3 F igur e 1: Isoledensal configuration eff iciency as a function of incr easing standar d deviation of the hidden node distribution. This figure sho ws the e xpected result: the isoledensal conf iguration efficiency falls as the hidden node distribution standard de viation increases, i.e., as the encapsulated graph becomes increasingly une v enly distributed in hidden nodes. T o sho w a slightl y broader e xample of this trend, Figure 2 sho ws a furt her ten graphs randomly - ge nerated but with t he same cons traints as t hat in figure 1 . F igur e 2: Isoledensal configuration eff iciency as a function of incr easi ng s tandar d deviation, multiple gr aphs. W e shall no w e xamine the f iv e transf or mation equations and att empt to confir m the abo v e result in ter ms of the appropriate transf or mation. 4 3. The transforma tion equations From the point of vie w of in v esigat ing chang es in M.P .E. there are onl y tw o fundamental transf ormations we can mak e to an y e ncapsulated graph: w e can add a giv en number of inf ormation hiding violational nodes to an encapsulated region of the graph and w e can add a giv en number of information hidden nodes to an encapsulated region of the graph. Let us denote this, ”Giv en number , ” ref erenced abov e, m . It ma y seem that w e could also transform a graph by a dding an encapsulated region itself, but in encapsulation theor y edg es can only be f or med betw een nodes, not regions, and so adding an y number of empty regions to a gr aph cannot chang e the M.P .E. of that g raph. Of course, b y de finition, no node can e xist outside an encapsulated region, these transf or mations theref ore presume the exis t ence of an emp ty encapsulated region into whic h new nodes ma y be introduced, where necessary . Although only tw o transformations are fundamental, it is possible to derive a fur ther three transf ormations from t hese tw o fundamental transf or mation. These three derived transf or mations both co ver common chang e s to encapsulated graphs and yield important insight into the nature of the changing M.P .E. in their ow n r ight. The three derived transf or mations are: mo ving m inf or mation hiding violational nodes from one encapsulated region to ano ther, m o ving m inf orma tion hidden nodes from one encapsulated region to another , and conv er ting m inf ormation hidden nodes in an encapsulated region into m inf ormation hiding violational nodes. W e no te that m may be neg ativ e and w e establish the con v ention that t he addition of a neg ativ e number of nodes to an encapsulated region ma y be inter preted as the remo v al of ∣ m ∣ nodes from that region. Where m is negat iv e, it ma y not e x ce ed the number of nodes that actually reside wi thi n an encapsulated region: no region ma y cont ain a negativ e number of nodes at an y time. Bef ore proce eding, we r ecall the def initions of the ter ms from [1]: G : An encapsulated graph s(G) : The M.P .E. of encapsulated graph G .  s  G  : The chang e of M.P .E. of encapsulated g raph G . ∣ h  G  ∣ : The number of inf orma tion hiding violational nodes in encapsulated graph G . K x , K s and K t : Encapsulated regions in encapsulated graph G . K x is used when onl y one encapsulated region is in v olv ed. T ranslation transf orma tions in v olv e two e ncapsulated regions: K s is the source region from which nodes are mo v ed, K t is the targe t region to whic h nodes are mo v ed. ∣ K x ∣ : The total number of nodes in encapsulated region K x . ∣ h  K x  ∣ : The number of inf or mation hiding violational nodes in encapsulated region K x . n : the total number of nodes in an encapsulated graph. T able 1 lists the fi v e transformation equations. 5 Equation Description  s  G  = m  n  ∣ K x ∣  ∣ h  G  ∣ − ∣ h  K x  ∣  m − 1  The inf ormation hi ding violation transf ormation equation (see theorem 3. 1 1), which giv es the chang e of M.P .E. of an encapsulated graph G when m inf or mation hiding violational nodes are added to encapsulated region K x .  s  G  = m  ∣ h  G  ∣ − ∣ h  K x  ∣  2 ∣ K x ∣  m − 1  The inf ormation hi dden transf or mation equation (see theorem 3. 1 7), which giv es the chang e of M.P .E. of an encapsulated graph G when m information hidden nodes are added to encapsulated region K x .  s cumulative  G = m  ∣ K t ∣ − ∣ h  K t  ∣ − ∣ K s ∣ − ∣ h  K s  ∣  The inf ormation hi ding violation translation transf ormation equation (see theorem 3. 1 8), which giv es the change of M.P .E. of an encapsulated graph G when m inf ormation hidi ng violational nodes are mo v ed from sour ce encapsulated region K s to targ et encapsulated region K t .  s cumulative  G = m  2 ∣ K t ∣ − 2 ∣ K s ∣  ∣ h  K s  ∣ − ∣ h  K t  ∣  2m  The inf ormation hi dden translation transf ormation equation (see theorem 3. 1 9), which giv es the change of M.P .E. of an encapsulated graph G when m inf or mation hidden nodes are mo v ed from source encapsulated region K s to targ et encapsulated region K t .  s cumulative  G = m  n − ∣ K x ∣  The con v ersion transf ormation equation (see theorem 3.20), which giv es t he chang e of M.P .E. when m inf or mation hidden nodes are con v erte d int o inf or mation hiding violational nodes. T able 1: The f iv e tr ansformation equations. 4. Reflections on the equations 4. 1 The non-conservativ e transformation equations Consider the f irs t tw o transf orma tion eq uations. These are the fundament al eq uations and they are non-conservativ e in that they c hange the to tal number of nodes in t he graph; the other t hree eq uations are conser v ativ e in t hat the y do not chang e the total number of nodes in the graph. Pe rhaps the most important aspect of the tw o non-conservativ e transformation equations is that it is tr ivial to sho w (b y subtracting the second from the f irst) that adding a violational node to a graph causes a larg er increase in M.P .E. than adding a hidden node, as w e intuitiv ely e xpect . 6 4.2 The translation transf ormation equations The third and f ourt h equations are de r iv ed from the first tw o. These equations are translation equations in that they sho w ho w M.P .E. chang es as nodes are translated or mo v ed from one encapsulated region to another . W e shall e xamine them in rev erse order . 4.2. 1 The f our th equation Consider the f our th transf ormation equation, descr ibing the chang e of M.P .E. as inf or mation hidden nodes are mo ved bet ween encapsul ated regions. This is the equation go v e rning the c hanges that w e f ound in section 2, where all the hidden nodes of a graph were incr ementally translated from their original encapsulated regions to a specific targ et encapsulated region, thereb y maximising the standard de viation of the hidden node distribution. If w e look at the ter ms of the f our th equation, w e see t hat there are three components of the M.P .E. chang e (ignoring t he common scaling m f actor): (i) 2 ∣ K t ∣ − 2 ∣ K s ∣ (ii) ∣ h  K s  ∣ − ∣ h  K t  ∣ (iii) 2m The 2m component is clear ly independent of t he encapsulated regions aff ected by the transf or mation. Component (i) is the diff erence in the total number of nodes (multiplied b y tw o) betw een the source and targe t encapsulated regions. Component (ii) is the diff erence in the number of inf or mation hiding violational nodes betw een the source and targ et encapsulated regions, though in the opposite sense of component (i) in that component (i) is targ et minus source but component (ii) is sour ce minus targe t. The interaction betw ee n these tw o components is complicated, but in our e xperi ment in section 2, w e mo v ed more and more hidden nodes into a single, targ et encapsulated region, causing the targ et region to become increasingl y larg e while its violational nodes remained unc hang ed: thus component (i) grew to dominate component (ii) and repeated translations increasingl y added to the M.P .E. of the graph. Increasing the M.P .E. of a fixed number of nodes b y de finition decreases the graph's conf iguration efficiency and t his is precisel y what the g raph in figures 1 and 2 sho w . The re v er se is also true: mo ving inf or mation hidden nodes fr om a larg er to a smaller encapsulated region must necessaril y decrease the M.P .E. of a graph and t hus increase the configuration eff iciency . This e xplains wh y a graph of unif or ml y distributed hidden nodes cannot ha v e an M.P .E. greater than one of une v enly dis tributed hidden nodes (information hiding violation distribution being equal). 4.2.2. The third equation Whereas the f ourt h transf or mation mo v e d inf or mation hidden nodes betw een encapsulate d regions , the t hir d transf or mation mo ves i nf or mation hiding violational nodes betw een encapsulated regions. W e might e xpect this e q uation to yield results quite similar to the f our th equation giv en t hat the y are both translation transf ormations, but this is not the case. T o appreciate this cur ious diff erence w e shall per f or m another e xperiment. In section 2 w e plotted the f alling confi guration efficiency of a g raph as its hidden nodes w ere increasinl y piled into just one encapsulated r egion. Let us perform a similar e xperim ent but this time w e shall incrementally mo v e only the inf or mation hiding violational nodes into one encapsulated region. (A minor diff erence in procedure must be obser v ed: ev er y encapsulated region must contain at leas t one violational node as otherwis e it is uncontactable by node s in other regions, so instead of mo ving all violation nodes from 7 the source regions, w e shall mo v e all but one violation node. This diff erence in itself should not materially chang e the outcome.) Let us ag ain take an encapsulated graph of 1 00 encapsulated regions. In section 2 we put one inf or mation hiding violational node in each region and put a random number – betw een 0 and 30 – of inf or mation hidden nodes in each region. F or our ne w e xper iment w e shal l do the opposite, putting one inf or mation hidden node in each region and puting a random number – betw een 0 and 30 – of information hiding violational nodes in each region. In section 2, it w as the stanard de viation of the hidden node distr ibution that w e measured; this time w e shall measure the standard de viation of the violational node distribution: t he number of inf or mation hiding violational nodes per region. Incrementall y movi ng one violational node into one targ et region will continuously incr ease the standard de viation of the violational node distri bution of the entire graph, e xactl y analogous to the fir s t e xper iment. The que stion is: ho w will t he configuration eff iciency chang e as t he standard de viation of the violational node distribution increases? The result is sho w n in figure 3. F igur e 3: Isoledensal configuration eff iciency as a function of incr easi ng s tandar d deviation of the violational node distribution. Figure 3 sho ws tw o t hings but onl y one is rele vant here. F irstl y , the lo w the conf iguration efficiency of this gr aph ma y be s tr iking but this is simpl y due to the w a y w e created the gr aph. The graph w as created with just one hidden node in each r egion and up to 30 violational nodes in each region; the graph is theref ore composed o verwhelming of violational nodes and as such e xploits little encapsulation: hence the low configuration eff iciency . More rele vant to our e xper iment and perhaps more surpr ising is the result that in our sample graph the conf iguration efficiency is independent of violational distribution standard deviation; i n other wor ds, the M.P .E. of the graph is unchanged b y mo ving all the possible violational nodes into one encapsulated region. Wh y is this result so differ ent from the fi rs t e xper iment with inf ormation hidden nodes? The answ er lies in the thi r d equation. Let us ag ain break the equation into its component parts; we s ee it has just tw o (i gnoring the common scaling m fac tor) : (i) ∣ K t ∣ − ∣ h  K t  ∣ (ii) ∣ K s ∣ − ∣ h  K s  ∣ 8 Looking at the fi rs t component, w e ma y ask ourselv es: ho w do w e inter pret this quantity? This component is the total number of nodes in the targe t encapsulated region minus the number of violational nodes in the targe t region. W e are already f amiliar wit h this quantity , how ev er: it all the violational nodes are remo v ed from a region, all this is lef t are the inf or mation hidden nodes. Thus the f irst component is simpl y the number of hidden nodes in t he targ et region. The second component is just the number of hidden nodes in the source region. This e xplains figure 3. Mo ving violational nodes betw e en regions depends onl y on t he diff erence betw een the number of hidden nodes of the source and targe t region, and in our e xperiment all t he regions ha v e the same number of hidden nodes, t hus the diff erence betw een t he number of hidden nodes in an y tw o regions is zero. So it doesn't matter ho w many violational nodes are mo v ed betw een regions: these translations cannot chang e t he M.P .E. and canno t chang e the conf iguration efficiency of t he graph. 4.2.3. Equal unev enness In both e xperiments per f ormed so far , one set of nodes w ere alwa ys e v enly dis tr ibuted and minimised: in the f irst experiment, each re gion had just one violational node whereas the hidden nodes w ere une v enly dis tributed; in t he second e xper iment, each region had one hidden node wherea s the viol ational nodes w ere une v enl y distributed. T o model more, ”R e al w orld, ” problems, w e mus t e xamine graphs whose hidden nodes and violational nodes are both une v enl y distributed. Let us re-visit the first e xperiment and look at the tr anslation of hidden nodes in a graph again of 1 00 encapsulated regions with each region ha ving a random number – betw een 0 and 30 – of hidden nodes and a random number – betw ee n 1 and 30 – of violational nodes. Bef ore we do so, ho w ev er , w e shall attempt to predict the results b y e xamining the translation transf or mation equation f or hidden nodes, the f ourt h equation in table 1 . As w e noted bef ore, the dominant component of translation transf or mation equation f or hidden nodes is simpl y the total number of nodes in the t ar get minus the total number of nodes in the source. As we are chosing at random the targ et region into whic h all the hidden nodes will be mo v ed, then thi s region wil l initially ha v e betw een 0 and 60 nodes in total (there will be at most 30 hidden and at mos t 30 violational nodes). It is theref ore conceiv able that the f irst source r egion chosen f or a translation will hav e more nodes than our ta r ge t region, and so the total number of nodes in the targe t region minus the total number of nodes in the source region will be negativ e; this negativ e chang e in M.P .E. im plies that t he configuration eff iciency of the g raph w ould initially rise. This mo v ement of hidden nodes from a larg er to a small region, ho w ev er , also implies that the standard de viation of the hidden node distribution falls, as the graph is being more, ”Smoothed-out, ” by s uch a translation. Thus a negativ e chang e of M.P .E. corresponds to a negativ e chang e of standard de viation. As more translations are perf orme d, ho we ver , w e should quic kly r each a situation, in a randoml y distributed graph, where t he number of hidden nodes in the targ et region becomes greater than the number of nodes in an y other single region; this will cer tainl y be the case when the targ et region contains 6 1 nodes and will probabl y be the case much sooner . Af ter this point, all the hidden node translations will increase the M.P .E. and increase the standard de viation of the hidden node distribution. This will yield a picture v ery similar to that already sho wn in figure 1 , where t he configuration eff iciency will f all as the hidden node distribution standard de viation increases. The results are sho wn in f igure 4. 9 F igur e 4: Isoledensal configuration eff iciency as a function of incr easing standar d deviation of the hidden node distribution f or a gr aph with unev enly dist ri but ed hidden and violational nodes. There are three aspects of figure 4 that require e xplanation but only tw o are rele vant to our e xperi ment. Compared with figure 1 , t he initial configuration eff iciency of the graph in f igure 4 is rather lo w , but this is due to the gr aph's no w cont aining multiple hidden and multiple violational nodes randoml y distributed: an y such graph with a non-tr ivial number of encapsulated regions will usuall y ha v e a configuration eff iciency of around 0.5. (R ecall that t he graph in f igure 1 had onl y one violational node per region: it w as e xtremel y we ll encapsulated and hence its conf iguration efficiency was muc h higher than our latest graph.) The second and most intere sting aspect of figure 4 is that, as suspected, t he configuration eff iciency of a graph of unev enly dis tr ibuted hidden and violational nodes f alls with increasing hidden node distribution standard de viation, just as w as t he case with the unev enly dis tributed hidden node s and e venl y distr ibuted violational nodes of the f irs t expe r iment. There onl y remains to be e xplained why the terminal conf iguration efficiency of t he graph in figure 4 is not as lo w as t hat in figure 1 . The expl anation comes again from the f our th equation. The larg est c hang e of M.P .E. occurs when the differ ence in total number of nodes betw een t he source and targ et re gions is maximised. In our first e xper iment, all the regions contained only one violational node but in thi s lates t e xperi ment there w ere alwa ys a random number of violational nodes left behind when the hidden nodes were e xtracted, thus the dif f erences in to tal number of nodes betw een targ et and source regions w ere not as great as those in t he first e xper iments causing the M.P .E. to rise by a lesser amount than in the first e xper iment. This directl y translates to the conf iguration efficiency's not f alling as far in our lates t e xperim ent. As bef ore, merely t o demonstrate the trend, figure 5 show s ten randomly g enerated graphs, each constrained as w as the graph in f igure 4, subjected to repea ted hidden translation transf orm ations. 1 0 F igur e 5: Isoledensal configuration eff iciency as a function of incr easing standar d deviation of the hidden node distribution f or a gr aph with unev enly dist ri but ed hidden and violational nodes, multiple gr aphs. Let us also re-visit the second e xper iment and look at the translation of violational nodes in a gr aph again of 1 00 encapsulated regions with each region ha ving a random number – betw een 0 and 30 – of hidden nodes and a random number – betw e en 1 and 30 – of violational nodes. Bef ore w e do so, ho w ev er , w e shall attempt to pre dict the results by e xamining the translation transf orm ation equat ion f or violational nodes, the third equation in table 1 . As w e sa w , the equation sho w ed us that the chang e in M.P .E. g enerated by the translation depends onl y on the diff erence betwe en the numbers of hidden nodes in t he targ et and source regions. In our pre vious violation translation e xperi ment, that diff erence w as zero, and so mo ving all the violational nodes to one targ et region had no eff ect on the g raph's M.P .E . or conf iguration efficiency . In our randoml y generated graph, ho we v er, the diff erence will usuall y be non-zero and so there will usuall y be a chang e of M.P .E. It will not, ho we ver , resemble the M.P .E. caused b y hidden node translations. In hidden node translations, the chang e in M.P .E. w as proporti onal t o the dif f e re nce of the number of nodes in the ta r ge t and source nodes, and this chang e grew incr easingl y larg e as the t ar ge t node grew increa singly larg e. The v ery act of translating a hidden node to the t ar ge t region increased the chang e in M.P .E. caused b y translating all subsequent hidden nodes to the targ et region. The third equation e xhibits very diff erent beha viour . The change in M.P .E. is proportional to the diff erence in the number of hidden nodes only and mo ving violational nodes does not chang e the number of hidden nodes in source or targ et region. Thus repeat ed violational node translations will not generate M.P .E. chang es proportional to the increasing size of the t ar get region: the change in M.P .E . is fixe d b y the choice of source and targ et regions. Also, suppose the targ et region has 1 5 hidden node s; then translating violational nodes from a source region with f e wer than 1 5 hidden nodes will cause an increase of M.P .E . and translating violational nodes from a source region with more than 1 5 nodes wi ll cause a decrease of M.P .E. So unlike the hidden node translations, which after an initial time w ere guaranteed to onl y increase M.P .E., violational node translations can lead to small increases of M.P .E. which can then be offse t by s ubsequent small decreases of M.P .E. 1 In other w ords, although the increasing violational distribution standard de viation will usuall y chang e the conf iguration efficiency , we do no t e xpect configuration eff iciency to chang e by an yt hing lik e as much as was caused b y the hidden node t ranslations: the configuration eff iciency should be quite insenitiv e to 1 This justifies the notion of the inter fa ce repository in computer programming, a subsystem holding onl y public interfaces that act as facades to v ar ious other subsys tems of hidden implementations. 1 1 increasing violational distribution standard de vi ation. The results of repeated violational node translations of a graph une v enly dis tributed in both hidden and violational nodes is sho w n in figure 6. F igur e 6: Isoledensal configuration eff iciency as a function of incr easing standar d deviation of the violation node distribution f or a gr aph with unev enly dist ri but ed hidden and violational nodes, multiple gr aphs. As figure 6 indeed sho ws, configuration effi ciency does c hange w it h increa sing violational distribution standard de viation, but onl y negligibl y . Figure 7 sho ws the results f or multiple g raphs. F igur e 7 : Isoledensal configuration ef ficiency as a function of incr easing standar d deviation of the violation node distribution f or a gr aph with unev enly dist ri but ed hidden and violational nodes, multiple gr aphs. 4.3.The fifth equation The fif th equation in table 1 is the conv ersion transf ormation equation, which giv es t he chang e of M.P .E. when m information hidden nodes are con v erte d into inf orm ation hiding violational nodes. The equation simpl y states t hat when a hidden node in a region is con verted into a violational node in that same region, then the M.P .E. must rise. This just confir ms our e xpectation that increasing the a ccess to a node 1 2 within a region must raise the M.P .E . of an encapsulated graph. The re v er se is also true: the change of M .P .E. in con v er ting a violational node to a hidden node can be calculated b y changing the sign of the m , w hic h then sho w s that such a con v ersion must reduce the M.P .E. of a g raph. 5. Conclusions Sy stems e v olv e. T o c ontrol this e v olution means be able to deter minis tically pre dict the af f ects of chang es bef ore thos e cha nges occur . F or sys tems that can be modelled by e ncapsulated gra phs, this primar il y means predicting the M.P .E . of the graph bef ore the chang es occur . This paper proposed that t he e v olution of an encapsulated graph may be modelled as an arbitraril y comple x ser ies of transf or mations that ma y be applied to that graph. The tw o fundamental transf or mations w ere t hen es tablished that describe all chang es to a graph and the two M.P .E. equations corresponding to those transf orma tions w ere proposed. Three furt her equations w ere derived from t hese fundamental equations to de scr ibe the more common chang es t hat graphs undergo. 6. Related work - 7 . Appendix A 7 . 1 . Definitions [D3. 1] Giv en encapsulated region K in encapsulated graph G and the transformation T , t he chang e of interna l maximum potential number of edg es  s in  K  eff ected b y applying T to K is giv en b y equation:  s in  K = s in  T  K  − s in  K  The chang e of inter nal maximum potential number of edg es  s in  G  eff ected by appl ying T to G is giv en b y equation:  s in  G  = s in  T  G  − s in  G  [D3.2] Giv en encapsulated region K in encapsulated graph G and the transformation T , the chang e of e xter nal maximum potential numbe r of edges  s ex  K  eff ected by appl ying T to K is giv en b y equation:  s ex  K = s ex  T  K   − s ex  K  The chang e of external maximum potential number of edg es  s ex  G  eff ected by appl ying T to G is giv en b y equation:  s ex  G  = s ex  T  G  − s ex  G  [D3.3] Giv en encapsulated region K in encapsulated graph G and the transformation T , the chang e of maximum potential number of edg es  s  K  eff ected by appl ying T to K is giv en b y equation:  s  K = s  T  K  − s  K  The chang e of maximum po tential number of edges  s  G  eff ected by appl ying T to G is giv en b y equation:  s  G = s  T  G  − G  [D3.4] Giv en the encapsulated graph G wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xter nal inf or mation hiding violation of h(K i ) , let T h be the violational transf or mation whic h maps K i onto K i * 1 3 where K i * diff ers from K i b y m inf or mation hiding violational nodes where m ≥ − ∣ K i ∣ or: T h  K i , m = { K i ∈ G : ∣ h  K i  ∣  ∣ h  K i  ∣  m } Where T h is applied to jus t the x th encapsulated region K x of G , the transformation becomes: T h  x , G , m ={ G  T h  G  : ∣ h  K i  ∣  ∣ h  K i  ∣  m ∀ i = x ; ∣ h  K i  ∣  ∣ h  K i  ∣ ∀ i ≠ x } No te that as m may be positiv e or negativ e, K i * ma y ha v e more or f ew er inf or mation hiding violational nodes than K i . [D3.5] Giv en the encapsulated graph G wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xter nal inf or mation hiding violation of h(K i ) , let T z be the hidden transf or mation which maps K i onto K i * where K i * diff ers from K i b y m inf or mation hidden nodes where m ≥ − ∣ K i ∣ and where the inf orm ation hiding violational nodes remain unchang ed, t hat is: T z  K i , m = { K i ∈ G : ∣ K i ∣  ∣ K i ∣  m ; ∣ h  K i  ∣  ∣ h  K i  ∣ } Where T z is applied to just the x th encapsulated region K x of G , the transf or mation becomes: T z  x , G , m = { G  T z  G  : ∣ K i ∣  ∣ K i ∣  m ∀ i = x ; ∣ K i ∣  ∣ K i ∣ ∀ i ≠ x ; ∣ h  K i  ∣  ∣ h  K i  ∣ ∀ i } No te that as m may be positiv e or negativ e, K i * ma y ha v e more or f ew er inf or mation hidden nodes than K i . 7 .2. Theorems The theorems are org anised as f ol lo ws. Theorems 3. 1 – 3.5 e s tablish some general results concerning the sum of changes of maximum potential number of edg es and the application of transf or mations to graphs. Theorems 3.6 – 3. 1 1 establish the fundamental transf or mation equation f or the appli cation of the violational transf ormation to a g raph. Theorems 3. 12 – 3. 1 7 establish the fundamenta l transf or mation equation f or the application of t he hidden transf ormation to a gra ph. Theorems 3. 1 8 – 3.20 establish t he three deriv ed transf or mation eq uations f or the translation and con v ersion transf ormations. All theorems relate to encapsulated graphs of absolute inf ormation hiding only . Theorem 3. 1 . Giv en encapsulated region G and the transf ormation T , t he c hange of maximum po tential number of edg es s(G) effected b y applying T to G is giv en by :  s  G  =  s in  G   s ex  G  Pr oof: By definition: s(G) = s in (G) + s ex (G) (i) Let K * = T(K) . Theref ore: s(G * ) = s in (G * ) + s ex (G * ) (ii) Also be definintion:  s  G = s  T  G  − s  G  (iii) 1 4 Substituting (i) and (ii) into (i ii) giv es:  s  G = s  G *  − s  G  = s in  G *  s ex  G * − s in  G − s ex  G  = s in  G * − s in  G  s ex  G * − s ex  G  =  s in  G   s ex  G  QED Theorem 3.2. Giv en the encapsulated region K and the violational transf orm ation T h defined in [D3.4], t hen the number of nodes in T h (K) diff ers from the number of nodes in K by m, or: ∣ T h  K , m  ∣ = ∣ K ∣  m Pr oof: Let K contain a inf ormation hidden nodes and ∣ h  K  ∣ inf or mation hiding violational nodes. Thus, b y def inition: ∣ K ∣ = a  ∣ h  K  ∣ (i) Let K * =T h (K,m) . By definintion T h lea v es the number of inf or mation hidden nodes unchang ed, theref ore: ∣ K * ∣ = a  ∣ h  K *  ∣ (ii) Also b y definition of T h : ∣ h  K *  ∣ = ∣ h  K  ∣  m (iii) Substituting (iii) int o (ii) give s: ∣ K * ∣ = a  ∣ h  K  ∣  m (iv) Substituting (i) int o (iv) giv es: ∣ T h  K , m  ∣ = ∣ K * ∣ = ∣ K ∣  m QED Theorem 3.3. Giv en the encapsulated graph G wit h an i th encapsulated region K i and giv en that a par ticular x th encapsulated region K x onl y is subject to the violational transf or mation T h defined in [D3.4], t he number of inf or mation hiding violation nodes in T h ( G ) is giv en by : ∣ h  T h  x , G , m  ∣ = ∣ h  G  ∣  m Pr oof: By definintion: ∣ h  G  ∣ = ∑ i = 1 r ∣ h  K i  ∣ 1 5 = ∑ i = 1 ≠ x r ∣ h  K i  ∣  ∣ h  K x  ∣ (i) Let G * =T h (x,G,m) and let K i * =T h (K i ,m). By definition: ∣ h  G *  ∣ = ∑ i = 1 r ∣ h  K i *  ∣ = ∑ i = 1 ≠ x r ∣ h  K i *  ∣  ∣ h  K x *  ∣ (ii) By the def inintion of T h : ∀ i ≠ x : ∣ h  K i  ∣  ∣ h  K i  ∣ (iii) Substituting (iii) int o (ii) give s: ∣ h  G *  ∣ = ∑ i = 1 ≠ x r ∣ h  K i  ∣  ∣ h  K x *  ∣ (iv) Also b y the defi nintion of T h : ∀ i = x : ∣ h  K i  ∣  ∣ h  K i  ∣  m (v) Substituting (v) into (i v) giv es: ∣ h  G *  ∣ = ∑ i = 1 ≠ x r ∣ h  K i  ∣  ∣ h  K x  ∣  m (vi) Substituting (i) int o (vi) giv es: ∣ h  T h  x , G , m  ∣ = ∣ h  G *  ∣ = ∣ h  G  ∣  m QED Theorem 3.4. Giv en the encapsulated region K and the hidden node transf or mation T z defined in [D3.5], t hen the number of nodes in T z (K) diff ers from the number of nodes in K b y m, or: ∣ T z  K , m  ∣ = ∣ K ∣  m Pr oof: Let K contain a inf ormation hidden nodes and ∣ h  K  ∣ inf or mation hiding violational nodes. Thus, b y def inition: ∣ K ∣ = a  ∣ h  K  ∣ (i) Let K * =T z (K,m) . By definintion T z lea v es the number of inf or mation hiding violational nodes unchang ed, theref ore: ∣ K * ∣ = a *  ∣ h  K  ∣ (ii) Also b y definition of T z : a * = a  m (iii) Substituting (iii) int o (ii) give s: ∣ K * ∣ = a  m  ∣ h  K  ∣ (iv) 1 6 Substituting (i) int o (iv) giv es: ∣ T z  K , m  ∣ = ∣ K * ∣ = ∣ K ∣  m QED Theorem 3.5. Giv en the encapsulated graph G wit h an i th encapsulated region K i and giv en that a par ticular x th encapsulated region K x onl y is subject to the hidden node transformation T z defined in [D3.5], t he number of inf or mation hiding violation nodes in T z ( G ) is giv en b y : ∣ h  T z  x , G , m   ∣ = ∣ h  G  ∣ Pr oof: Let G * =T z (x,G,m) and let K i * =T z (K i ,m). By definition: ∣ h  G *  ∣ = ∑ i = 1 r ∣ h  K i *  ∣ (i) By the def inintion of T z : ∀ i : ∣ h  K i *  ∣ = ∣ h  K i  ∣ (ii) Substituting (ii) into ( i) giv es: ∣ h  T z  x , G , m   ∣ = ∑ i = 1 r ∣ h  K i  ∣ = ∣ h  G  ∣ QED Theorem 3.6. Giv en the encapsulated graph G wit h an i th encapsulated region K i of ∣ K i ∣ nodes and inter nal maximum potential number of edg es s in (K i ) , the chang e of s in (K i ) when the number of inf or mation hiding violational nodes in K i chang e s b y m where m ≥ − ∣ K i ∣ is giv en by :  s in  K i = 2 m ∣ K i ∣  m 2 − m Pr oof: Let T h be the violational transf or mation defined in [D3.4] and let K i * =T h (K i ,m). By definition [D3. 1], the chang e of inter nal maximum potential number of edg es ef f ected b y appl ying T h to K i is:  s in  K i = s in  K i * − s in  K i  (i) By theorem 1 .2: s in  K i * = ∣ K i * ∣  ∣ K i * ∣ − 1  (ii) By theorem 3.2: ∣ K i * ∣ = ∣ K i ∣  m (iii) Substituting (iii) int o (ii) give s: s in  K i * =  ∣ K i ∣  m   ∣ K i ∣  m − 1  = ∣ K i 2 ∣  m ∣ K i ∣ − ∣ K i ∣  m ∣ K i ∣  m 2 − m 1 7 = ∣ K i ∣  ∣ K i ∣ − 1  2m ∣ K i ∣  m 2 − m (iv) But: s in  K i = ∣ K i ∣  ∣ K i ∣ − 1  (v) Substituting (v) into (i v) giv es: s in  K i * = s in  K i  2m ∣ K i ∣  m 2 − m s in  K i * − s in  K i = 2 m ∣ K i ∣  m 2 − m And theref ore b y (i):  s in  K i = 2 m ∣ K i ∣  m 2 − m QED Theorem 3.7 . Giv en the encapsulated graph G wit h an i th encapsulated region K i of inter nal maximum potential number of edg es s in (K i ) , the chang e of inter nal maximum potential number of edg e s of the entire graph when the number of inf or mation hiding violational nodes in a particular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is equal to the cha nge of internal maximum potential number of edg es of K x , or:  s in  G  = s in  K x  Pr oof: By definintion, t he internal maximum potential number of edg es of G is the sum of the inter nal maximum potential number of edg es of all its encapsulated regions: s in (G) = ∑ i = 1 r s in  K i  = ∑ i = 1 ≠ x r s in  K i  s in  K x  (i) Let T h be the violational transf or mation defined in [D3.4] and let K i * =T h (K i ,m). Futhermore, let T h apply t o the x th encapsulated region onl y such that K x * =T h (K x ,m) and G * =T h (x,G,m). By definintion: s in (G * ) = ∑ i = 1 r s in  K i *  = ∑ i = 1 ≠ x r s in  K i *  s in  K x *  (ii) By definition [D3.3], t he c hange of internal maximum potential number of edg es of G eff ected by applying tr ansformation T h to G is giv en by :  s in  G = s in  G * − s in  G  (iii) Substituting (i) and (ii) into (i ii) giv es:  s in  G  = ∑ i = 1 ≠ x r s in  K i *  s in  K x * − ∑ i = 1 ≠ x r s in  K i − s in  K x  (iv) But as T h is only appl ied to K x then all encapsulated regions e x cept K x are unchang ed, or: 1 8 K i * = K i ∀ i ≠ x And theref ore: s in  K i * = s in  K i  ∀ i ≠ x (v) Substituting (v) into (i v) giv es:  s in  G  = ∑ i = 1 ≠ x r s in  K i  s in  K x * − ∑ i = 1 ≠ x r s in  K i  − s in  K x  = s in  K x *  − s in  K x  (vi) By definition [D3. 1], t he c hange of internal maximum potential number of edg es effe cted by applying T h to K x is then:  s in  K x = s in  K x * − s in  K x  (vii) Substituting (vii) into ( vi) giv es:  s in  G  = s in  K x  QED Theorem 3.8. Giv en the encapsulated graph G wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xternal maximum potential number of edg es s ex (K i ) , the chang e of s ex (K i ) when the number of inf or mation hiding violational nodes in K i chang e s b y m where m ≥ − ∣ K i ∣ is giv en by :  s ex  K i = m ∣ h  G  ∣ − m ∣ h  K i  ∣ Pr oof: Let T h be the violational transf or mation defined in [D3.4]; let K i * =T h (K i ,m) and let G * =T h (G,m). By definition [D3.2], t he c hange of e xter nal maximum potential number of edg es eff ected b y applying T h to K i is then:  s ex  K i = s ex  K i * − s ex  K i  (i) By theorem 1 .4: s ex  K i * = ∣ K i * ∣  ∣ h  G *  ∣ − ∣ h  K i *  ∣  (ii) By theorem 3.3: ∣ h  G *  ∣ = ∣ h  G  ∣  m (iii) Substituting (iii) int o (ii) give s: s ex  K i * = ∣ K i * ∣  ∣ h  G  ∣  m − ∣ h  K i *  ∣  (iv) By theorem 3.2: ∣ K i * ∣ = ∣ K i ∣  m (v) Substituting (v) into (i v) giv es: s ex  K i * = ∣ K i  m ∣  ∣ h  G  ∣  m −  ∣ h  K i  ∣  m   = ∣ K i  m ∣  ∣ h  G  ∣  m − ∣ h  K i  ∣ − m  1 9 = ∣ K i  m ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  = ∣ K i ∣ ∣ h  G  ∣ − ∣ K i ∣ ∣ h  K i  ∣  m ∣ h  G  ∣ − m ∣ h  K i  ∣ = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  m ∣ h  G  ∣ − m ∣ h  K i  ∣ (vi) But: s ex  K i  = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  (vii) Substituting (vii) into ( vi) giv es: s ex  K i * = s ex  K i  m ∣ h  G  ∣ − m ∣ h  K i  ∣ s ex  K i * − s ex  K i = m ∣ h  G  ∣ − m ∣ h  K i  ∣ And theref ore b y (i):  s ex  K i = m ∣ h  G  ∣ − m ∣ h  K i  ∣ QED Theorem 3.9. Giv en the encapsulated graph G of n nodes wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xter nal maximum potential numbe r of edges s ex (K i ) , the chang e of e xter nal maximum po tential number of edg es of the e ntire graph when the number of inf ormation hiding violational nodes in a par ticular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is giv en by:  s ex  G = mn − m ∣ K x ∣   s ex  K x  Pr oof: By definintion, t he e xter nal maximum potential number of edg es of G is the sum of t he e xter nal maximum potential number of edg es of all its encapsulated regions: s ex (G) = ∑ i = 1 r s ex  K i  = ∑ i = 1 ≠ x r s ex  K i  s ex  K x  (i) Let T h be the violational transf or mation defined in [D3.4] and let K i * =T h (K i ,m). Futhermore, let T h apply t o the x th encapsulated region onl y such that K x * =T h (K x ,m) and G * =T h (x,G,m). By definintion: s ex (G * ) = ∑ i = 1 r s ex  K i *  = ∑ i = 1 ≠ x r s ex  K i *  s ex  K x *  (ii) By definintion [D3.2]:  s ex  G = s ex  G *  − s ex  G  (iii) Substituting (i) and (ii) into (i ii) giv es:  s ex  G  = ∑ i = 1 ≠ x r s ex  K i *  s ex  K x * − ∑ i = 1 ≠ x r s ex  K i − s ex  K x  (iv) 20 By definintion [D3.2]:  s ex  K x = s ex  K x * − s ex  K x  (v) Substituting (v) into (i v) giv es: s ex  G = ∑ i = 1 ≠ x r s ex  K i * − ∑ i = 1 ≠ x r s ex  K i   s ex  K x  (vi) By theorem 1 .4: s ex  K i * = ∣ K i * ∣  ∣ h  G *  ∣ − ∣ h  K i *  ∣  (vii) If w e consider K i where i ≠ x then as T h is applied only t o K x then: ∣ K i * ∣ = ∣ K i ∣ ∀ i ≠ x (viii) Substituting (viii) int o (vii) give s: s ex  K i * = ∣ K i ∣  ∣ h  G *  ∣ − ∣ h  K i  ∣  (ix) By theorem 3.3: ∣ h  G *  ∣ = ∣ h  G  ∣  m (x) Substituting (x) into (i x) giv es: s ex  K i * = ∣ K i ∣  ∣ h  G  ∣  m − ∣ h  K i  ∣  = ∣ K i ∣ ∣ h  G  ∣ − ∣ K i ∣ ∣ h  K i  ∣  m ∣ K i ∣ = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  m ∣ K i ∣ (xi) But: s ex  K i  = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  (xii) Substituting (xi) int o (xii) giv es: s ex  K i * = s ex  K i  m ∣ K i ∣ As this holds ∀ i ≠ x w e can take the sum o v er all encapsulated regions e x cept x : ∑ i = 1 ≠ x r s ex  K i * = ∑ i = 1 ≠ x r s ex  K i  ∑ i = 1 ≠ x r m ∣ K i ∣ = ∑ i = 1 ≠ x r s ex  K i  m ∑ i = 1 ≠ x r ∣ K i ∣ (xiii) By definintion, n = ∑ i = 1 r ∣ K i ∣ = ∑ i = 1 ≠ x r ∣ K i ∣  ∣ K x ∣ So: n − ∣ K x ∣ = ∑ i = 1 ≠ x r ∣ K i ∣ (xiv) Substituting (xiv) into (vii i) give s: 2 1 ∑ i = 1 ≠ x r s ex  K i * = ∑ i = 1 ≠ x r s ex  K i  m  n − ∣ K x ∣  ∑ i = 1 ≠ x r s ex  K i * = ∑ i = 1 ≠ x r s ex  K i  mn − m ∣ K x ∣ (xv) Substituting (xv) into (vi ) giv es:  s ex  G = ∑ i = 1 ≠ x r s ex  K i  mn − m ∣ K x ∣ − ∑ i = 1 ≠ x r s ex  K i   s ex  K x  = mn − m ∣ K x ∣   s ex  K x  QED Theorem 3. 1 0. Giv en the encapsulated graph G of n nodes wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xter nal maximum potential numbe r of edges s ex (K i ) , the chang e of e xter nal maximum po tential number of edg es of the e ntire graph s ex (G) when the number of inf or mation hiding violational nodes in a par ticular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is giv en by :  s ex  G  = mn − m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣ Pr oof: Let T h be the violational transf or mation defined in [D3.4] and let it appl y to the x th encapsulated region onl y . By theorem 3.8, t he c hange of external maximum potential number of edg es of K x b y the application of T h to K x is giv en b y:  s ex  K x = m ∣ h  G  ∣ − m ∣ h  K x  ∣ (i) By theorem 3.9, the chang e of e xter nal maximum potential number of edg es of t he entire graph G b y the application of T h to K x is giv en b y :  s ex  G  = mn − m ∣ K x ∣  s ex  K x  (ii) Substituting (ii) into ( i) giv es:  s ex  G  = mn − m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣ QED Theorem 3. 1 1 . Giv en the encapsulated graph G of n nodes wit h an i th encapsulated region K i of ∣ K i ∣ nodes, the chang e of maximum potential number of edg e s of the entire graph s(G) when the number of information hiding violational nodes in a par ticular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is giv en b y:  s  G = mn  m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣  m 2 − m Pr oof: Let T h be the violational transf or mation defined in [D3.4] and let K i * =T h (K i ,m). Futhermore, let T h apply t o the x th encapsulated region onl y such that K x * =T h (K x ,m) and G * =T h (x,G,m). From theorem 3.7 , when the number of inf or mation hiding violational nodes in K x chang e s b y m , t he cha nge of internal maximum 22 potential number of edg es of the entire graph is given b y:  s in  G =  s in  K x  (i) By theorem 3.6, when the number of inf or mation hiding violational nodes in K x chang e s b y m , t he chang e of internal maximum potential number of edges of K x is giv en b y  s in  K x = 2m ∣ K x ∣  m 2 − m (ii) Substituting (ii) into ( i) giv es:  s in  G  = 2m ∣ K x ∣  m 2 − m (iii) From theorem 3. 1 0, when the number of inf or mation hiding violational nodes in K x chang e s b y m , t he chang e of e xt eral maximum pote ntial number of edges is giv en by :  s ex  G  = mn − m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣ (iv) By theorem 3. 1 , the c hange of maximum po tential number of edges of G is giv en by :  s  G =  s in  G   s ex  G  (v) Substituting (iii) and (iv) int o (v) giv es:  s  G = mn − m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣  2m ∣ K x ∣  m 2 − m = mn  m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣  m 2 − m QED Theorem 3. 1 2. Giv en the encapsulated graph G wit h an i th encapsulated region K i of ∣ K i ∣ nodes and inter nal maximum potential number of edg es s in (K i ) , the chang e of s in (K i ) when the number of inf or mation hidden nodes in K i chang e s b y m where m ≥ − ∣ K i ∣ is giv en b y :  s in  K i = 2 m ∣ K i ∣  m 2 − m Pr oof: Let T z be the hidden transf or mation defined in [D3.5] and let K i * =T z (K i ,m). By definition [D3. 1], the chang e of internal maximum potential number of edges eff ected b y appl ying T z to K i is:  s in  K i = s in  K i * − s in  K i  (i) By theorem 1 .2: s in  K i * = ∣ K i * ∣  ∣ K i * ∣ − 1  (ii) By theorem 3.4: ∣ K i * ∣ = ∣ K i ∣  m (iii) Substituting (iii) int o (ii) give s: s in  K i * =  ∣ K i ∣  m   ∣ K i ∣  m − 1  = ∣ K i 2 ∣  m ∣ K i ∣ − ∣ K i ∣  m ∣ K i ∣  m 2 − m = ∣ K i ∣  ∣ K i ∣ − 1  2m ∣ K i ∣  m 2 − m (iv) But: 23 s in  K i = ∣ K i ∣  ∣ K i ∣ − 1  (v) Substituting (v) into (i v) giv es: s in  K i * = s in  K i  2m ∣ K i ∣  m 2 − m s in  K i * − s in  K i = 2 m ∣ K i ∣  m 2 − m And theref ore b y (i):  s in  K i = 2 m ∣ K i ∣  m 2 − m QED Theorem 3. 1 3. Giv en the encapsulated graph G wit h an i th encapsulated region K i of inter nal maximum potential number of edg es s in (K i ) , the chang e of inter nal maximum potential number of edg e s of the entire graph when the number of inf or mation hidden nodes in a particular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is equal to the cha nge of internal maximum potential number of edg es of K x , or:  s in  G  = s in  K x  Pr oof: By definintion, t he internal maximum potential number of edg es of G is the sum of the inter nal maximum potential number of edg es of all its encapsulated regions: s in (G) = ∑ i = 1 r s in  K i  = ∑ i = 1 ≠ x r s in  K i  s in  K x  (i) Let T z be the hidden transf or mation defined in [D3.5] and let K i * =T h (K i ,m). Futherm ore, le t T z appl y to the x th encapsulated region onl y such that K x * =T h (K x ,m) and G * =T z (x,G,m). By definintion: s in (G * ) = ∑ i = 1 r s in  K i *  = ∑ i = 1 ≠ x r s in  K i *  s in  K x *  (ii) By definition [D3. 1], t he c hange of internal maximum potential number of edg es of G effected b y applying tr ansformation T z to G is:  s in  G = s in  G * − s in  G  (iii) Substituting (i) and (ii) into (i ii) giv es:  s in  G  = ∑ i = 1 ≠ x r s in  K i *  s in  K x * − ∑ i = 1 ≠ x r s in  K i − s in  K x  (iv) But as T z is onl y applied to K x then all encapsulated regions e x cept K x are unchang ed, or: K i * = K i ∀ i ≠ x And theref ore: 24 s in  K i * = s in  K i  ∀ i ≠ x (v) Substituting (v) into (i v) giv es:  s in  G  = ∑ i = 1 ≠ x r s in  K i  s in  K x * − ∑ i = 1 ≠ x r s in  K i  − s in  K x  = s in  K x *  − s in  K x  (vi) By definition [D3. 1], t he c hange of internal maximum potential number of edg es effe cted by applying T z to K x is then:  s in  K x = s in  K x * − s in  K x  (vii) Substituting (vii) into ( vi) giv es:  s in  G  = s in  K x  QED Theorem 3. 1 4. Giv en the encapsulated graph G wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xternal maximum potential number of edg es s ex (K i ) , the chang e of s ex (K i ) when the number of inf or mation hidden nodes in K i chang e s b y m where m ≥ − ∣ K i ∣ is giv en b y :  s ex  K i = m ∣ h  G  ∣ − m ∣ h  K i  ∣ Pr oof: Let T z be the hidden transf or mation defined in [D3.5]; let K i * =T z (K i ,m) and let G * =T z (G,m). By definition [D3.2], t he c hange of e xter nal maximum potential number of edg es eff ected b y applying T z to K i is then:  s ex  K i = s ex  K i * − s ex  K i  (i) By theorem 1 .4: s ex  K i * = ∣ K i * ∣  ∣ h  G *  ∣ − ∣ h  K i *  ∣  (ii) By theorem 3.5: ∣ h  G *  ∣ = ∣ h  G  ∣ (iii) Substituting (iii) int o (ii) give s: s ex  K i * = ∣ K i * ∣  ∣ h  G  ∣ − ∣ h  K i *  ∣  (iv) By definintion of T z : ∣ h  K i *  ∣ = ∣ h  K i  ∣ (v) Substituting (v) into (i v) giv es: s ex  K i * = ∣ K i * ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  (vi) By theorem 3.4: ∣ K i * ∣ = ∣ K i ∣  m (vii) Substituting (vii) into ( vi) giv es: 25 s ex  K i * = ∣ K i  m ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  = ∣ K i ∣ ∣ h  G  ∣ − ∣ K i ∣ ∣ h  K i  ∣  m ∣ h  G  ∣ − m ∣ h  K i  ∣ = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  m ∣ h  G  ∣ − m ∣ h  K i  ∣ (viii) But: s ex  K i  = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  (ix) Substituting (ix) int o (viii) giv es: s ex  K i * = s ex  K i  m ∣ h  G  ∣ − m ∣ h  K i  ∣ s ex  K i * − s ex  K i = m ∣ h  G  ∣ − m ∣ h  K i  ∣ And theref ore b y (i):  s ex  K i = m ∣ h  G  ∣ − m ∣ h  K i  ∣ QED Theorem 3. 1 5. Giv en the encapsulated graph G of n nodes wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xter nal maximum potential numbe r of edges s ex (K i ) , the chang e of e xter nal maximum po tential number of edg es of the e ntire graph when the number of inf ormation hidden nodes in a par ticular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is giv en b y:  s ex  G  = s ex  K x  Pr oof: By definintion, t he e xter nal maximum potential number of edg es of G is the sum of t he e xter nal maximum potential number of edg es of all its encapsulated regions: s ex (G) = ∑ i = 1 r s ex  K i  = ∑ i = 1 ≠ x r s ex  K i  s ex  K x  (i) Let T z be the hidden transf or mation defined in [D3.5] and let K i * =T z (K i ,m). Futhermore, let T z apply to the x th encapsulated region onl y such that K x * =T z (K x ,m) and G * =T z (x,G,m). By definintion: s ex (G * ) = ∑ i = 1 r s ex  K i *  = ∑ i = 1 ≠ x r s ex  K i *  s ex  K x *  (ii) By definintion [D3.2]:  s ex  G = s ex  G *  − s ex  G  (iii) Substituting (i) and (ii) into (i ii) giv es:  s ex  G  = ∑ i = 1 ≠ x r s ex  K i *  s ex  K x * − ∑ i = 1 ≠ x r s ex  K i − s ex  K x  (iv) 26 By definintion [D3.2]:  s ex  K x = s ex  K x * − s ex  K x  (v) Substituting (v) into (i v) giv es: s ex  G = ∑ i = 1 ≠ x r s ex  K i * − ∑ i = 1 ≠ x r s ex  K i   s ex  K x  (vi) By theorem 1 .4: s ex  K i * = ∣ K i * ∣  ∣ h  G *  ∣ − ∣ h  K i *  ∣  (vii) If w e consider K i where i ≠ x then as T z is applied onl y to K x then: ∣ K i * ∣ = ∣ K i ∣ ∀ i ≠ x (viii) Substituting (viii) int o (vii) give s: s ex  K i * = ∣ K i ∣  ∣ h  G *  ∣ − ∣ h  K i  ∣  (ix) By theorem 3.5: ∣ h  G *  ∣ = ∣ h  G  ∣ (x) Substituting (x) into (i x) giv es: s ex  K i * = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  (xi) But b y theorem 1 .4: s ex  K i  = ∣ K i ∣  ∣ h  G  ∣ − ∣ h  K i  ∣  (xii) Substituting (xii) into ( xi) giv es: s ex  K i * = s ex  K i  As this holds ∀ i ≠ x w e can take the sum o v er all encapsulated regions e x cept x : ∑ i = 1 ≠ x r s ex  K i * = ∑ i = 1 ≠ x r s ex  K i  (xiii) Substituting (xiii) int o (vi) giv es: s ex  G =  s ex  K x  QED Theorem 3. 1 6. Giv en the encapsulated graph G of n nodes wit h an i th encapsulated region K i of ∣ K i ∣ nodes and e xter nal maximum potential numbe r of edges s ex (K i ) , the chang e of e xter nal maximum po tential number of edg es of the e ntire graph s ex (G) when the number of inf or mation hidden nodes in a particular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is giv en b y :  s ex  G  = m ∣ h  G  ∣ − m ∣ h  K x  ∣ Pr oof: Let T z be the hidden transf or mation defined in [D3.5] and let K i * =T z (K i ,m). Futhermore, let T z apply to the x th encapsulated region onl y such that K x * =T z (K x ,m) and G * =T z (x,G,m). By theorem 3. 1 4, the chang e of 27 e xter nal maximum potential numbe r of edges of K x b y the application of T z to K x is giv en b y :  s ex  K x = m ∣ h  G  ∣ − m ∣ h  K x  ∣ (i) By theorem 3. 1 5, the chang e of e xternal maximum potential number of edg es of the entire graph G b y the application of T z to K x is the same as the change of e xter nal maximum potential number of edg es of K x or: s ex  G  = s ex  K x  (ii) Substituting (i) int o (ii) giv es:  s ex  G  = m ∣ h  G  ∣ − m ∣ h  K x  ∣ QED Theorem 3. 1 7 . Giv en the encapsulated graph G of n nodes wit h an i th encapsulated region K i of ∣ K i ∣ nodes, the chang e of maximum potential number of edg e s of the entire graph s(G) when the number of information hidden nodes in a par ticular x th encapsulated region K x chang e s b y m where m ≥ − ∣ K x ∣ is giv en b y :  s  G = m ∣ h  G  ∣ − m ∣ h  K x  ∣  2m ∣ K x ∣  m 2 − m Pr oof: Let T z be the hidden transf or mation defined in [D3.5] and let K i * =T z (K i ,m). Futhermore, let T z apply to the x th encapsulated region onl y such that K x * =T z (K x ,m) and G * =T z (x,G,m). From theorem 3. 13, when the number of inf ormation hidde n nodes in K x chang e s b y m , t he cha nge of internal maximum potential number of edge s of t he entire graph is giv en by :  s in  G =  s in  K x  (i) By theorem 3. 1 2, when the number of inf or mation hidden nodes in K x chang e s b y m , t he chang e of interna l maximum potential number of edg es of K x is giv en b y  s in  K x = 2m ∣ K x ∣  m 2 − m (ii) Substituting (ii) into ( i) giv es:  s in  G  = 2m ∣ K x ∣  m 2 − m (iii) From theorem 3. 1 6, when the number of inf or mation hidden nodes in K x chang e s b y m , t he chang e of e xteral maximum potential number of edg es is giv en b y:  s ex  G  = m ∣ h  G  ∣ − m ∣ h  K x  ∣ (iv) By theorem 3. 1 , the c hange of maximum po tential number of edges of G is giv en by :  s  G =  s in  G   s ex  G  (v) Substituting (iii) and (iv) int o (v) giv es:  s  G = m ∣ h  G  ∣ − m ∣ h  K x  ∣  2m ∣ K x ∣  m 2 − m QED Theorem 3. 1 8. Giv en the encapsulated graph G of n nodes, t he cumulativ e chang e of maximum potential number of 28 edg es of the e ntire graph s(G) when m inf or mation hiding violational nodes are mo ved from a particular source encapsulated region K s to a particular ta r ge t encapsulated region K t is giv en b y :  s cumulative  G = m  ∣ K t ∣ − ∣ h  K t  ∣ −  ∣ K s ∣ − ∣ h  K s  ∣  Pr oof: Let T h1 be the violational transformation defined in [D3.4] and let T h1 appl y to the s th encapsulated region K s only suc h t hat K s * =T h1 (K s ,m) and G * =T h1 (s,G,m). T h1 will remo ve m inf or mation hiding violational nodes from K s . Let T h2 be the violational transformation defined in [D3.4] and let T h2 appl y to the t th encapsulated region K t only suc h t hat K t * =T h2 (K t ,m) and G ** =T h2 (t,G * ,m). T h2 will add m inf or mation hiding violational nodes from K t . Let us define the violational translation transformation T Th as the combination of the two translations T h1 and T h2 such that: T Th (G)= T h2 (T h1 (G)) As these transf orm ations are linear , the chang e of the maximum potential number of edges of T Th is equal to the sum of the chang es of the maximum potential number of edg es of the transformations T h2 and T h1 , or:  s cumulative  G = s  G  s  G *  (i) Considering nodes remov ed from an encapsulated region as neg ativ e, then T h1 will add -m nodes to G . By theorem 3. 1 1 , the change of maximum potential number of edges in G caused b y the remov al of t hese m inf ormation hidi ng violational nodes from K s is giv en b y :  s  G = mn  m ∣ K s ∣  m ∣ h  G  ∣ − m ∣ h  K s  ∣  m 2 − m (ii) Substituting -m f or m in (ii) giv es:  s  G  = − mn − m ∣ K s ∣ − m ∣ h  G  ∣  m ∣ h  K s  ∣  m 2  m (iii) By theorem 3. 1 1 again, the chang e of maximum potential number of edge s in G * caused b y the addition of m inf or mation hiding violational nodes t o K t is giv en b y :  s  G * = mn *  m ∣ K t ∣  m ∣ h  G *  ∣ − m ∣ h  K t  ∣  m 2 − m (iv) G * has m f ew er nodes than G and t he y a re all inf or mation hiding violational, and thus both h(G * ) and n * are chang ed in compar ison with G , such that: ∣ h  G *  ∣ = ∣ h  G  ∣ − m (v) n * = n − m (vi) Substituting (v) and (vi) int o (iv) giv es:  s  G * = m  n − m   m ∣ K t ∣  m  ∣ h  G  ∣ − m − m ∣ h  K t  ∣  m 2 − m = mn − m 2  m ∣ K t ∣  m ∣ h  G  ∣ − m 2 − m ∣ h  K t  ∣  m 2 − m = mn  m ∣ K t ∣  m ∣ h  G  ∣ − m ∣ h  K t  ∣ − m 2 − m (vii) Substituting (iii) and (vii) into (i) giv es:  s cumulative  G = − mn − m ∣ K s ∣ − m ∣ h  G  ∣  m ∣ h  K s  ∣  m 2  m  mn  m ∣ K t ∣  m ∣ h  G  ∣ − m ∣ h  K t  ∣ − m 2 − m 29 = m  ∣ K t ∣ − ∣ h  K t  ∣ −  ∣ K s ∣ − ∣ h  K s  ∣   QED Theorem 3. 1 9. Giv en the encapsulated graph G of n nodes, t he cumulativ e chang e of maximum potential number of edg es of the e ntire graph s(G) when m inf or mation hidden nodes are mo ved from a pa r ticular source encapsulated region K s to a particular ta r ge t encapsulated region K t is giv en b y :  s cumulative  G = m  2 ∣ K t ∣ − 2 ∣ K s ∣  ∣ h  K s  ∣ − ∣ h  K t  ∣  2m  Pr oof: Let T z1 be the hidden transf or mation defined in [D3.5] and let T z1 appl y to the s th encapsulated region K s only suc h t hat K s * =T z1 (K s ,m) and G * =T z1 (s,G,m). T z1 will remo ve m inf or mation hidden nodes from K s . Let T z2 be the hidden transf or mation defined in [D3.5] and let T z2 appl y to the t th encapsulated region K t only suc h t hat K t * =T z2 (K t ,m) and G ** =T z2 (t,G * ,m). T z2 will add m inf or mation hidden nodes from K t . Let us define the hidden translation transformation T T as the combination of t he tw o translations T z1 and T z2 such that: T T (G)= T z2 (T z1 (G)) As these transf orm ations are linear , the chang e of the maximum potential number of edges of T T is equal to the sum of the chang es of the maximum potential number of edg es of the transformations T z2 and T z1 , or:  s cumulative  G = s  G  s  G *  (i) Considering nodes remov ed from an encapsulated region as neg ativ e, then T z1 will add -m nodes to G . By theorem 3. 1 7 , the change of maximum po tential number of edges in G caused b y the remov al of these m inf ormation hidden nodes from K s is giv en b y :  s  G = m ∣ h  G  ∣ − m ∣ h  K s  ∣  2m ∣ K s ∣  m 2 − m (ii) Substituting -m f or m in (ii) giv es:  s  G = − m ∣ h  G  ∣  m ∣ h  K s  ∣ − 2m ∣ K s ∣  m 2  m (iii) By theorem 3. 1 7 again, the chang e of maximum potential number of edge s in G * caused b y the addition of m inf or mation hidden nodes to K t is giv en b y:  s  G * = m ∣ h  G *  ∣ − m ∣ h  K t  ∣  2m ∣ K t ∣  m 2 − m (iv) G * has m f e wer nodes than G but t he y ar e all inf or mation hidden, and thus h(G * ) is unchang ed, or: ∣ h  G *  ∣ = ∣ h  G  ∣ (v) Substituting (v) into (i v) giv es:  s  G * = m ∣ h  G  ∣ − m ∣ h  K t  ∣  2m ∣ K t ∣  m 2 − m (vi) Substituting (iii) and (vi) int o (i) giv es:  s cumulative  G = − m ∣ h  G  ∣  m ∣ h  K s  ∣ − 2m ∣ K s ∣  m 2  m  m ∣ h  G  ∣ − m ∣ h  K t  ∣  2m ∣ K t ∣  m 2 − m = m  2 ∣ K t ∣ − 2 ∣ K s ∣  ∣ h  K s  ∣ − ∣ h  K t  ∣  2m  30 QED Theorem 3.20. Giv en the encapsulated graph G of n nodes, t he cumulativ e chang e of maximum potential number of edg es of the e ntire graph s(G) when m inf or mation hidden nodes in a par ticular encapsulated region K x are con v er ted to inf or mation hiding violational nodes within the same encapsulated region is giv en b y:  s cumulative  G = m  n − ∣ K x ∣  Pr oof: Let T z be the hidden transf or mation defined in [D3.5] and let T z appl y to the x th encapsulated region K x only s uch that K x * =T z (K x ,m) and G * =T z (x,G,m). T z will remo ve m inf or mation hidden nodes from K x . Let T h be the violational transf or mation defined in [D3.4] and let T h appl y to the x th encapsulated region K x only s uch that K x * =T h (K t ,m) and G ** =T h (x,G * ,m). T h will add m inf or mation hiding violational nodes from K x . Let us define the conv ersion transf or mation T C as the combination of the two translations T z and T h such that: T C (G)= T h (T z (G)) As these transf orm ations are linear , the chang e of the maximum potential number of edges of T C is equal to the sum of the chang es of the maximum potential number of edg es of the transformations T h and T z , or:  s cumulative  G = s  G  s  G *  (i) Considering nodes remov ed from an encapsulated region as neg ativ e, then T z will add -m nodes to G . By theorem 3. 1 7 , t he c hange of ma ximum potential number of edges in G caused b y the remo val of these m inf or mation hidden nodes from K x is giv en b y:  s  G = m ∣ h  G  ∣ − m ∣ h  K x  ∣  2m ∣ K x ∣  m 2 − m (ii) Substituting -m f or m in (ii) giv es:  s  G  = − m ∣ h  G  ∣  m ∣ h  K x  ∣ − 2m ∣ K x ∣  m 2  m (iii) By theorem 3. 1 1 , the change of m aximum potential number of edg es in G * caused b y the addition of m inf ormation hidi ng violational nodes t o K x is giv en b y :  s  G * = mn *  m ∣ K x * ∣  m ∣ h  G *  ∣ − m ∣ h  K x *  ∣  m 2 − m (iv) K x * has m f ew er nodes than K and t he y a re all inf or mation hidden, and thus both K x * and n * are chang ed in comparison with G , but h(K x * ) and h(G * ) are unchang ed such that: ∣ h  G *  ∣ = ∣ h  G  ∣ (v) ∣ h  K x *  ∣ = ∣ h  K x  ∣ (vi) n * = n − m (vii) ∣ K x * ∣ = ∣ K x ∣ − m (viii) Substituting (v), (vi), (vii) and (viii) into ( iv) give s:  s  G * = m  n − m  m  ∣ K x ∣ − m  m ∣ h  G  ∣ − m ∣ h  K x  ∣  m 2 − m 3 1 = mn − m 2  m ∣ K x ∣ − m 2  m ∣ h  G  ∣ − m ∣ h  K x  ∣  m 2 − m = mn  m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣ − m 2 − m (ix) Substituting (iii) and (ix) int o (i) giv es:  s cumulative  G = − m ∣ h  G  ∣  m ∣ h  K x  ∣ − 2m ∣ K x ∣  m 2  m  mn  m ∣ K x ∣  m ∣ h  G  ∣ − m ∣ h  K x  ∣ − m 2 − m = m  n − ∣ K x ∣  QED 8. References [1] "Encapsulation theor y fundamentals," Ed Kir w an, www .EdmundKirwan.com/pub/paper1 .pdf [2] "Encapsulation theor y: the transf or mation equations of absolute inf or mation hiding," Ed Kirwan, www .EdmundKirwan.com/pub/paper3.pdf