COMPLETE CALCULATION OF DISCONNECTION PROBABILITY IN PLANAR GRAPHS. G. Tsitsiashvili. IAM, FEB RAS, Vladivostok, Russia s:

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1 G. Tstsashvl COMPLETE CALCULATION OF ISCONNECTION PROBABILITY IN PLANAR GRAPHS RT&A # 0 (24) (Vol.) 202, March COMPLETE CALCULATION OF ISCONNECTION PROBABILITY IN PLANAR GRAPHS G. Tstsashvl IAM, FEB RAS, Vladvostok, Russa e-mals: guram@am.dvo.ru ABSTRACT In ths paper complete asymptotc formulas for an dsconnecton probablty n random planar graphs wth hgh relable arcs are obtaned. A defnton of coeffcents n these formulas have geometrc complexty by a number of arcs. But a consderaton of planar graphs and dual graphs allow to solve ths problem wth no more than cubc complexty by a number of graph faces.. INTROUCTION A problem of a calculaton of a connectvty probablty n random graphs wth unrelable arcs s consdered n manfold artcles and monographs devoted to the relablty theory [] - [4] etc. It occurs n an analyss of electro techncal devces, computer networks and has manfold applcatons to a research of honeycomb structures [5], [6], and nanosystems [7] [9]. In [0] [2] upper and low estmates of the connectvty probablty are constructed for general type networks on a base of maxmal systems of dsjont frames. For small numbers of arcs n [3] accelerated algorthms of a calculaton of relablty polynomal coeffcents are constructed. These algorthms showed good results n a comparson wth drect calculatons. In [4] ths problem s solved usng the Monte-Carlo method wth some combnatory formulas. To calculate the connectvty probablty n rectangle lattces the transfer matrx method s used [5]. But an ncreasng of arcs number leads to large complexty and so t s worthy to develop asymptotc methods. In ths paper an analog of the Burtn-Pttel asymptotc formula [6] for dsconnecton probablty of random graph wth hgh relable arcs s constructed. Its parameters are the mnmal number of arcs n cross sectons and the number C of cross sectons wth volume. A defnton of for a random port demands to fnd a maxmal flow and has cubc complexty [7]. But a defnton of C has geometrc complexty. So we consder wdely used planar graphs for whch we prove that a defnton of coeffcents, C has no more than cubc complexty by a number of faces. And there s a lot of graphs [8, Ch. IV] for whch ths complexty s lnear and smaller. These results are based on a consderaton of dual graphs [9, [20], n whch cross sectons generate cycles [2], [22]. Numercal experment confrms an accuracy and a performance of suggested method. 2. ASYMPTOTIC FORMULAS Consder non orented connected graph G wth fnte sets of nodes U and of arcs W. Suppose that each par of nodes n G may be connected wth no more than sngle arc and there are not loops. enote L u, v the set of all cross sectons n G whch dvde nodes u, v U, u v, and defne the set L L u, v of all cross sectons n G. Graph cross secton uv 54

2 G. Tstsashvl COMPLETE CALCULATION OF ISCONNECTION PROBABILITY IN PLANAR GRAPHS RT&A # 0 (24) (Vol.) 202, March a number of arcs n the cross secton L and u, v mn d L : L L u, v, mn u, v uv, L L L : d L, C - s a number of cross sectons n the set L. Suppose that graph arcs work ndependently wth probabltes p( w), w W. s such set of arcs whch deleton makes the graph non connected. Put dl Theorem. If Theorem 2. If p( w) p w h, w W, then graph dsconnecton probablty P ~ Ch, h 0. () p( w) ~ c h, h 0, w W, then w P ~ h c, ww, h 0. LL Theorems, 2 are generalzatons of the Burtn-Pttel asymptotc formula [6]. wl w 3. CALCULATION OF CONSTANTS C, Theorem 3. The set of arcs whch do not belong to any cycle concdes wth the set of cross sectons L and. Assume that the graph G s planar and ts each arc belongs to some cycle. Arcs of planar graph dvde a plane nto faces [9,Сh. ]. }. Confront the graph G ts dual graph G. Each face z n G accords the node z n G, each arc w n G belongng faces z, z2 accords an arc w connectng nodes z, z n G. 2 A set of arcs w n G accords some subgraph R n G. For ts defnton each arc w, d, accords a par of faces whch contan ths arc. Then ths par of faces accords a par of nodes n R connected by the arc w. Say that the graph R s generated by the set of arcs w. Theorem 4. The set of cross sectons L conssts of all sets of arcs w whch generate cycles wth mnmal length n the dual graph G and 5. Ths statement s a corollary of the Whthney theorem and the Euler formula [9, Theorem.5, Corollary of Theorem.6], [20]. In fg. there are examples of planar graphs arranged on a sphere wth 4, 5. Fg.. Suppose that elements a j of the matrx A defne a number of arcs whch belong to z z j, j, a 0, n the planar graph G wth n faces and m arcs and wthout loops and multple arcs. 55

3 G. Tstsashvl COMPLETE CALCULATION OF ISCONNECTION PROBABILITY IN PLANAR GRAPHS RT&A # 0 (24) (Vol.) 202, March Corollary. If j, then, jn 2, C a j aj 4 (2), jn and a complexty of constants, C calculaton by the formula (2) s squared by n. If for j a only for j n then ths complexty s lnear. u j efne u 2... u k u c the number of cycles wth length, 3, 4,5, n, consst of same set of nodes u,..., and by a drecton of a bypass concde. Elements of a power Corollary 2. If max a, then, jn j G. Assume that all cycles u k and dffer by an ntal node u l A, l, of a matrx A denote by mn : c 0, 3,4,5, C c, (3) tra, c 3 3, 4 2 c4 tra 2m 2 aj 6 8 jn n n c 5 tra 5trA 5 a j 2 aj. 8 j Complexty of the constants, C calculaton usng the formula (3) s cubc by n. The formulas of с, с с calculaton are obtaned n [2], see also [22, Formulas (6), (7)]. 3 4, 5 Consder a connected graph G whch conssts of plane faces n three dmensonal space. Suppose that each par of faces has not jont ponts or has jont node or has jont arc and each arc belongs at least to two faces. Take a set of arcsw, d from G and confront each arc w a par of faces accords some (non unque) graph d whch connect these nodes. Theorem 5. If the graph whch generates t s not cross secton n G. Corollary 3. Suppose that the set L of arcs sets whch generate cycles wth mnmal length and whch are cross sectons n G s not empty. Then, L L. 4. EXAMPLES z, z, whch contan ths arc. Then the set of arcs w wth the nodes z, z, d, and arcs w d s acyclc then the set of arcs w l a. j Results of the number defnton and an enumeraton of cross sectons wth mnmal volume are based on lsted theorems and smple geometrc constructons. Example. On fg. 2 there are examples of planar graphs wth representatves of cross sectons from the set L : ) an nteger rectangle wth the length M an nteger rectangle wth the length N ( L conssts of arcs pars connected wth angle nodes), 2 a honeycomb structure ( L conssts of all possble pars of arcs whch belong to nternal and external faces smultaneously), 56

4 G. Tstsashvl COMPLETE CALCULATION OF ISCONNECTION PROBABILITY IN PLANAR GRAPHS RT&A # 0 (24) (Vol.) 202, March 3) a tube whch s constructed by a glung of opposte sdes (wth a length M ) of nteger rectangle ( L conssts of arcs trplets whch have common butt node), f N>3. Fg. 2. Planar graphs wth cross sectons dedcated by bold type. Example 2. On fg. 3 there are graphs wth examples of ther cross sectons from the set : ) a graph constructed from nteger rectangle by a glung of pars of ts opposte sdes ( L conssts of arcs quads whch have common node), 2) a graph constructed from unt cubes wth nteger coordnates of ther nodes ( L conssts of arcs trplets whch contan a cube node, n ths node the cube does not ntersect or has only common node wth another cube). L Fg. 3. Graph G wth dedcated cross sectons. 5. NUMERICAL EXPERIMENT Calculate the dsconnecton probablty of honeycomb structure (fg., n center) usng Theorem and Corollary and by the Monte-Carlo method wth 0 6 realzatons. Falure probablty of each arc s Results of calculatons are represented n the table. Tme of calculatons by asymptotc method s few seconds and by the Monte-Carlo method s some hours. Sze structure Asymptotc method Monte-Carlo method Relatve error % 4.3 % 3.5 % 4.5 % 4.8 % 2.4 % 57

5 G. Tstsashvl COMPLETE CALCULATION OF ISCONNECTION PROBABILITY IN PLANAR GRAPHS RT&A # 0 (24) (Vol.) 202, March The author thanks A.S. Losev for numercal experment realzaton. 6. PROOFS OF MAIN STSTEMENTS Proof of Theorem. Suppose that V L s a random event that all arcs n cross secton L fal. Then P P VL VL ~ P VL, h 0. LL LL\ L LL As P VL o h, L L \ L, h 0, so P VL ~ Ch, h 0. LL Proof of Theorem 5. Suppose that arcs set w from the graph G generates acyclc graph R. Prove that each arc w, d, may by bypassed n G by a way whch does not contan arcs of ths set. The subgraph R conssts of trees S,..., Sm whch do not connect wth each other. Arrange each tree S, m, on a plane so that n each node z arcs connected wth ths node follow each other as ther pre mages on the face z f we bypass ths face n some drecton. Confront each tree S closed way whch bypasses once all ts arcs from both sdes, m (fg. 4). Fg. 4. Bypass of tree arcs. Accord the way bypassng tree S arcs a closed way whch passes n the graph G through all nodes of arcs w, whch generate the tree S (fg. 5). The way has not arcs from the set w. Consequently each arc from w may be bypassed n G by a way whch does not contan arcs from ths set. So the set w from the graph G, d, whch does not generate a cycle n G, does not belong to the set of cross sectons L. Fg. 5. Bypassng of arcs n a tree and n G. 58

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