Random walks on networks: Cumulative distribution of cover time

Transcription

1 PHYSICAL REVIEW E 8, 9 Radom walks o eworks: Cumulaive disribuio of cover ime Nikola Zlaaov Macedoia Academy for Scieces ad Ars, Bul. Krse Misirkov,, P.O. Box 8, Skopje, Republic of Macedoia Ljupco Kocarev Macedoia Academy for Scieces ad Ars, Bul. Krse Misirkov,, P.O. Box 8, Skopje, Republic of Macedoia ad Isiue for Noliear Sciece, Uiversiy of Califoria Sa Diego, 95 Gilma Drive, La Jolla, Califoria 99-, USA Received April 9; revised mauscrip received 8 July 9; published Ocober 9 We derive a exac closed-form aalyical expressio for he disribuio of he cover ime for a radom walk over a arbirary graph. I special case, we derive simplified exac expressios for he disribuios of cover ime for a complee graph, a cycle graph, ad a pah graph. A accurae approximaio for he cover ime disribuio, wih compuaioal complexiy of O, is also preseed. The approximaio is umerically esed oly for graphs wih odes. DOI:./PhysRevE.8. PACS umbers: 5..Fb,.5.Ga,.5.Cw I. INTRODUCTION The radom walk is a fudameal dyamic process which ca be used o model radom processes ihere o may impora applicaios, such as raspor i disordered media, euro firig dyamics, spreadig of diseases, or raspor ad search processes 8. I his paper, we ivesigae radom walks o graphs 9 ad derive exac expressios for he cumulaive disribuio fucios for hree quaiies of a radom walk ha play he mos impora role i he heory of radom walks: hiig ime h ij or firs-passage ime, which is he umber of seps before ode j is visied sarig from ode i; commue ime ij =h ij +h ji ; ad cover ime, ha is he umber of seps o reach every ode. Average hiig ime, average commue ime, ad average cover ime have bee recely sudied i several papers. I he auhors ivesigaed radom walks o complex eworks ad derived a exac expressio for he mea firspassage ime bewee wo odes. For each ode he radom walk ceraliy is iroduced, which deermies he relaive speed by which a ode ca receive ad spread iformaio over he ework i a radom process. Usig boh umerical simulaios ad scalig argumes, he behavior of a radom walker o a oe-dimesioal small-world ework is sudied i. The average umber of disic sies visied by he radom walker, he mea-square displaceme of he walker, ad he disribuio of firs-reur imes obey a characerisic scalig form. The expeced ime for a radom walk o raverse bewee wo arbirary sies of he Erdos-Reyi radom graph is sudied i. The properies of radom walks o complex rees are sudied i. Boh he verex discovery rae ad he mea opological displaceme from he origi prese a cosiderable slowig dow i he ree case. Moreover, he mea firs-passage ime displays a logarihmic degree depedece, i coras o he iverse degree shape exhibied i looped eworks. The radom walk o eworks has also much relevace o algorihmic applicaios. The expeced ime ake o visi every verex of coeced graphs has recely bee exesively sudied. I a series of papers, Cooper ad Frieze sudied he average cover ime of various models of a radom graph, see for example. This is a oulie of he paper. I Sec. II we derive closed formulas of he cumulaive disribuio fucio for hiig ime, commue ime, ad cover ime; we also prese a simple example of a graph wih four odes, ad derive closed formulas of he cumulaive disribuio fucio for cover ime of complee graphs, cycle, ad pah graphs. A approximaio of he cumulaive disribuio fucio for cover ime is proposed i he Sec. III; we also prese some umerical resuls of he cumulaive disribuio fucio for cover ime of differe graphs Sec. IV. We fiish he paper wih coclusios. II. EXACT RANDOM WALK DISTRIBUTIONS FOR HITTING TIME, COMMUTE TIME, AND COVER TIME Le G=V,E be a coeced graph wih odes ad m edges. Cosider a radom walk o G: we sar a a ode v ; if a he h sep we are a a ode v, we move o eighbor of v wih probabiliy /dv, if a edge exiss bewee ode v ad is eighbor, where dv is he degree of he ode v. Clearly, he sequece of radom odes v :=,,... is a Markov chai. We deoe by M =m ij i,jv he marix of rasiio probabiliies of his Markov chai, = m /di, if ij E ij, oherwise, where di is he degree of he ode i. Recall ha he probabiliy m ij of he eve ha sarig a i, he radom walk will be a ode j afer seps, is a ery of he marix M.Iis well kow ha m ij dj/m as. We ow iroduce hree quaiies of a radom walk ha play he mos impora role i he heory of radom walks: hiig ime h ij is he umber of seps before ode j is firs visied sarig from ode i; commue ime ij =h ij +h ji is he umber of seps i a radom walk sarig a i before ode j is visied ad he ode i is reached agai; ad cover ime is he umber of seps o reach every ode. A. Hiig ime We firs calculae he probabiliy mass fucio PMF for he hiig ime. To calculae he hiig ime from i o j, we /9/8/8-9 The America Physical Sociey

2 NIKOLA ZLATANOV AND LJUPCO KOCAREV replace he ode j wih a absorbig ode. Le D j be a marix such ha d ik =m ik for all k j, ad d ij = for all i j ad d jj =. This meas ha he marix D j is obaied from M by replacig he origial row j wih he basis row-vecor e j for which he jh eleme is ad all oher elemes are. Le d ij be he ij ery of he marix D j, deoig he probabiliy ha sarig from i he walker is i he ode j by ime. Sice j is a absorbig sae, d ij is he probabiliy of reachig j, origiaig from i, i o more he seps, i.e., d ij is he cumulaive disribuio fucio CDF of hiig ime. Noe ha he jh colum of he marix D j approaches he all- vecor, as. The probabiliy mass fucio of he hiig ime h ij o reach j sarig from i is, herefore, give by p hij = d ij d ij,. Le E xj be he eve of reachig he ode x j sarig from he ode ix j by ime. Cosider a sequece of eves E x,e x,...,e xk. Wha is he probabiliy of he eve sarig from ode i, ha he walker visis oe of he odes x,x,...,x k by ime? Obviously, i is he probabiliy of k he uio j= E xj. To calculae his probabiliy, we replace he odes x,x,...,x k wih absorbig odes. Le D x be a marix obaied from M by replacig he rows x,x,...,x k wih he basis row-vecors e x,e x,...,e xk, respecively. Le d ixj be he ix j ery of he marix D k x. j= d ixj is he probabiliy ha sarig from i we reach for he firs ime oe of he x,x,...,x k odes i seps. Therefore, k F x,...,x k = j= d ixj is he CDF of he hiig ime h ix = of he uio of eves. The probabiliy of reachig oe of he odes x,x,...,x, sarig from i, iheh sep is give by p hix = F x,...,x k F x,...,x k,, which acually gives he PMF of hiig ime h ix of he uio k j= E xj. B. Commue ime Probabiliy mass fucio of he commue ime ij =h ij +h ji is obaied as he covoluio of PMFs of he wo radom variables h ij ad h ji, p ij = p hij p hji = p hij p hji. The cumulaive disribuio fucio of he commue ime ca also be derived as follows: we copy our Markov chai ad we modify he origial Markov chai by deleig all ougoig edges of he ode j, we modify he origial Markov chai by deleig all ougoig edges of he ode j, ad we modify he copied Markov chai by replacig all ougoig edges of he ode i which is a copy of he ode i of he origial Markov chai wih a self-loop. We he coec he wo chais by addig oe direced edge from ode j o is copy j of he copied chai. Le O be marix of all s, = O j =o kl be he marix for which all elemes are excep o jj =, ad D j be he marix obaied from M by replacig he jh row wih all. Defie he marix C as C = D j O D i. The marix C is a rasiio marix of he modified Markov chai wih elemes origial Markov chai ad is copy. Le c i,i+ be he i,i+ eleme of he marix C. This eleme is he cumulaive disribuio fucio for he commue ime ij. C. Cover ime Cover ime is defied as he umber of seps o reach all odes i he graph. I order o deermie he CDF of he cover ime, we cosider he eve j=,jz E xj, ad use he well-kow equaio for he iclusio-exclusio of muliple eves, P E xk = PE xi PE xi E xj k=,kz,iz,iz j=i+,jz O j PE x E x... E x, From he las equaio ad Eq., we deermie he cumulaive disribuio fucio of he cover ime as F cover =,iz F xi,iz j=i+,jz F xi,x j F x,x,...,x, where z is he sarig ode of he walk. Equaio is he mai resul of his paper. The probabiliy mass fucio of he cover ime ca be easily compued from he Eq.. We oe ha Eq. is pracically applicable oly for small values of ; i fac he compuaioal complexiy of Eq. a a sigle ime sep is j= PHYSICAL REVIEW E 8, 9! j! j! =. D. Example We ow prese a simple example o illusrae our resuls. Cosider a radom walk o a ework wih four odes, see Fig., such ha he marix of rasiio probabiliies of he correspodig Markov chai is give by = / / / / / M / / / / /. Le m ij be he i, j-h eleme of he marix M. Sice, i his example, M isa marix, oe ca compue aalyically, usig for example he sofware package MATHEMATICA, he -

3 RANDOM WALKS ON NETWORKS: CUMULATIVE PHYSICAL REVIEW E 8, 9 FIG.. Radom walk o a ework wih four odes. elemes of he marix M. Thus, i ca be foud, for example, m = 5, which is he probabiliy ha he walker sarig from a he ime is i j=. Noe ha lim m =/5. To compue he hiig ime o reach he ode sarig from a arbirary ode, we modify he exisig radom walk o he radom walk show i Fig.. The rasiio marix of he modified walk is = D / / / / / / / /. Le d ij be he elemes of he marix D. Agai he elemes of he marix D ca be compued aalyically. For example, he probabiliy of reachig he ode sarig form i ime seps is equal o FIG.. Modified radom walk for compuig he hiig ime o reach he ode sarig from a arbirary ode. = C / / / / / / / / / / / / / / /. The eleme c 5 of he marix C is he cumulaive disribuio fucio of he commue ime ad i is give by c 5 = / + + / / 9 6. Noice ha agai lim c 5 =. As he las example, we cosider he probabiliy of reachig he ode or he ode from he ode i ime seps. ' d = ' ' Clearly, as for ay cumulaive disribuio, lim d =. Le us ow compue he probabiliy sarig from ode o reach he ode ad he o reach sarig from i ime. For his, we cosider he modified radom walk show o Fig., wih he rasiio marix give by ' FIG.. Modified radom walk for compuig he commue ime sarig from ode o reach for he firs ime he ode ad he o reach for he firs ime sarig from. -

4 NIKOLA ZLATANOV AND LJUPCO KOCAREV PHYSICAL REVIEW E 8, FIG.. Modified radom walk for compuig he probabiliy of reachig he ode or he ode from arbirary ode. The modified radom walk is show o Fig. ad he rasiio probabiliy marix of he modified walk is give by he marix D ;, which has he form = / / / D ; / / /. The elemes d, ad d, d of he marix D ; = d =. The cumulaive disribuio fucio of he eve: he ode or he ode is reached from he ode for he firs ime by he h sep, is give by d +d. are E. Cover ime for complee, cycle, ad pah graph I his subsecio we derive exac expressios for he CDF of cover ime for hree paricular graphs: complee, cycle, ad pah graph.. Complee graph A complee graph is a simple graph i which every pair of disic verices is coeced by a edge. The complee graph o verices has verices ad / edges, ad is deoed by K. We ca ow easily derive aalyical resuls for he PMF of a complee graph. I is easy o see ha for he complee graph we have PE xi =, PE xi E xj =, Therefore, PE xi P E xj k E xk =. k E xi =. Thus, he cumulaive disribuio fucio of he cover ime for complee graph wih odes ca be expressed as F cover = 8 FIG. 5. Cycle graph wih odes. =!!! P =! = = E xi. Therefore, he probabiliy mass fucio is f c =.. Cycle graph A cycle graph is a graph ha cosiss of a sigle cycle, or i oher words, some umber of verices coeced i a closed chai. Le us deoe he cycle graph wih verices as C. The umber of verices i a C equals he umber of edges, ad every verex has degree ; ha is, every verex has exacly wo edges icide wih i. A example of a cycle graph wih odes is give i Fig. 5. Le us assume ha he firs ode of he cycle graph is he sarig ode of he walk. We eed o fid he iersecio of he eves of reachig odes,, o. These eves form a pah. A pah i a graph is a sequece of verices such ha, from each of is verices, here is a edge o he ex verex i he sequece. A cycle is a pah such ha he sar verex ad ed verex are he same. Noe ha he choice of he sar 7 6 -

5 RANDOM WALKS ON NETWORKS: CUMULATIVE verex i a cycle is arbirary. By exploiig he remark of corollary..6 give i 5, ad proved i 6 for eves ha form a pah, we fid ha he cumulaive disribuio fucio of he cover ime for a cycle graph is F cover = i= PE i i=. Pah graph PE i E i+. A pah graph is a paricularly simple example of a ree, amely, oe which is o brached a all, ha is, i coais oly odes of degree ad. I paricular, wo of is verices have degree ad all ohers if ay have degree. A example of a pah graph wih seve odes is give i Fig. 6. To fid he cumulaive disribuio fucio of he cover ime for a pah graph we oe ha all he odes will be covered if he firs ad he las odes are reached by he radom walker. Therefore, he cumulaive disribuio fucio of he cover ime for a pah graph is F cover = PE E = PE + PE PE E. We oe ha if he firs ode is he sarig ode he PE E = PE ad if he las ode is he sarig ode he PE E = PE. III. APPROXIMATION OF THE CDF OF COVER TIME The cumulaive disribuio fucios for hiig ad commue ime ca be compued for reasoable large graphs. The complexiy of marix muliplicaio, if carried ou aively, is O, bu more efficie algorihms do exis, which are compuaioally ieresig for marices wih dimesios 7. The iclusio-exclusio formula has lile pracical value i graphs wih large umber of odes sice i he requires exesive compuaioal imes. I he followig, we prese a accurae ad useful approximaio of Eq. ha ca be evaluaed i a reasoable ime. The firs iequaliy for he iclusio-exclusio was discovered by Ch. Jorda 8 ad from he uil ow a lo of work has bee doe i sharpeig he bouds or he approximaio. A excelle survey of he various resuls for he iclusio-exclusio is give i 5. We propose he followig approximaio for iclusioexclusio formula: P k= FIG. 6. Pah graph wih 7 odes. E xk PE xi E xi+ PE xi PE xi PE xi+, where PE xi E xi+ = PE xi + PE xi+ PE xi E xi+. The ode idexes mus be arraged i such way ha here exiss a edge bewee odes x i ad x i+. This codiio is o ime sep sric, ad here ca exis a small umber of odes ha do o saisfy his codiio. The Appedix preses he heurisic derivaio of Eq. 5 by usig he mehod proposed i 9. As ca be see, he sigle-sep compuaioal complexiy of Eq. 5 is O. The proposed approximaio is very accurae for srogly coeced graphs like he complee graph ad is less accurae for poorly coeced graphs like he pah graph, as ca be see from he figures below. We oe ha he error of he approximaio for he pah graph is he upper boud of he error whe he middle ode is he sarig ode. This is due o he fac ha he proposed approximaio equaio reduces o he exac equaio for idepede eves, while dimiishig as he eves become more ad more depede. Whe almos all he odes are subse of he res of he odes he he approximaio formula is he leas accurae. Thus he formula is he leas accurae whe is applied o a pah graph wih odes ad he walk sars from he middle, /-h ode assumig ha is a eve umber. I his case he eve of reachig odes o/ is he eve of reachig ode ad he eve of reachig odes /+ o is he eve of reachig ode. Ieresigly, if we sar chagig he sarig ode ad evaluae he error of he approximaio for a pah graph, he whe he sarig ode approaches he firs or he h ode he formula becomes more ad more accurae ad whe he sarig ode is he firs or he las ode, he he approximaio formula reduces o he exac formula. To prove his we le he firs ode be he sarig ode, he he eve of reachig ode i ad ode j, ifji is PE i E j = PE j. Thus he approximaio formula is give by P k= PHYSICAL REVIEW E 8, 9 E k = PE i Exac Approximaio odes, sarig ode is he h ode odes, sarig ode is he h ode FIG. 7. Exac ad he approximae formula of he CDF for a pah graph wih wo differe sarig odes. PE i = PE. A similar proof is whe he h ode is he sarig ode. A example is give i Fig. 7 where he CDF of a pah graph is give by he exac ad he approximae formula for wo pah graphs wih ad odes whe sarig ode is he fourh ad he h ode, respecively. The secod wors case error is whe he approximaio formula is applied o a cycle graph ad i his case he error is idepede of he sarig ode. 6-5

6 NIKOLA ZLATANOV AND LJUPCO KOCAREV PHYSICAL REVIEW E 8, Simulaio Approximaio Aalyical (a) ime sep...8 Simulaio Approximaio Aalyical.6.. Exac Approximaio 5 5 (a) ime sep.8 pmf (b) ime sep FIG. 8. Aalyical, approximaed ad simulaed a cumulaive disribuio fucio ad b probabiliy mass fucio of cover ime for a radom graph wih odes. IV. NUMERICAL EXAMPLES I his secio, several umerical examples are preseed. Firs, we validae he cover ime formula ad he approximaio by Moe Carlo simulaios, Fig. 8, for a Erdos-Reyi radom graph wih odes. Figure 8a illusraes he CDF, while Fig. 8b he PDF of cover ime. We illusrae he accuracy of he approximaio for a pah graph ad a complee graph, Figs. 9 ad, respecively, where he sarig ode of he walk for he pah graph is he middle ode for boh Figs. 9a ad 9b. We have performed various umerical simulaios of he cumulaive disribuio fucios usig exac ad approximae expressios for complee, pah, ad cycle graphs wih up o odes. For small we foud ha icreasig o up o, he accuracy of he approximaio is maiaied. We believe ha Eq. 5 is a good approximaio for cumulaive disribuio of cover ime eve for larger graphs, bu sice a he mome we do o have esimaes for accuracy of our approximaio, we leave his as a subjec of our ex research. More deailed aalysis o how he CDF of cover ime depeds o graph opology will be discussed i a forhcomig paper. V. CONCLUSIONS I his paper we have derived he exac closed-form expressios for he PMF ad CDF of hree radom walk parameers ha play pivoal role i he heory of radom walks:. Exac Approximaio (b) ime sep x 5 FIG. 9. Exac vs approximaed CDF for a a pah graph wih 5 odes ad b a pah graph wih odes. hiig ime, commue ime, ad cover ime. We also have derived simpler closed formulas for he cumulaive disribuio fucio of cover ime for complee, cycle, ad pah graphs. A approximaio of he cumulaive disribuio fucio for cover ime is proposed, ad several umerical resuls for he CDF of cover ime for differe graphs are preseed. ACKNOWLEDGMENT W wish o graefully ackowledge he suppor of EU projec MANMADE Gra No. 6. APPENDIX If A is he uio of he eves A,A,...,A he, wriig p i for he probabiliy of A i, p ij for he probabiliy of A i A j, p ijk for he probabiliy of A i A j A k, ec., he probabiliy of A is give by pa = i p i p ij + p ijk + p.... ij ijk The iclusio-exclusio priciple ells us ha if we kow he p i, p ij, p ijk... he we ca fid PA. However, i pracice we are ulikely o have full iformaio o he p i, p ij, p ijk... Therefore, we are faced wih he ask of approximaig PA, akig io accou whaever parial iformaio we are -6

7 RANDOM WALKS ON NETWORKS: CUMULATIVE PHYSICAL REVIEW E 8, Exac Approximaio PB = P Ā i = PA i. The approximaed form A of he eve B is PB PB i j=i+ A PB i B j. A5 PB ipb j give. I cerai cases where he eves A i are i some sese close o beig idepede, he here are a umber of kow resuls approximaig PA. I his paper we use he followig resul 9, Eq. A: where PA q i q i = PĀ i, j=i+ q ij, A A q ij = PĀ i Ā j. A PĀ i PĀ j Le he eve B i be defied as B i =Ā i. The B= Ā i ad he probabiliy of his eve is = (a) pmf (b). 6 8 ime sep x Exac.5 Approximaio ime sep FIG.. Exac vs approximaed CDF a ad PMF b for complee graph wih 5 odes. B i Replacig Eq. A ad B i=a i io Eq. A5, wege P A i PA i where PA i A j ca be expressed as PA i A j j=i+ PA i PA j, A6 PA i A j = PA i + PA j PA i A j. Whe he eves A i ad A j are o close o beig idepede bu o he corary, oe of he eves is a subse of he oher, as he case for he eves i he cover ime formula, he approximaio formula A6 is o accurae. The iaccuracy ca be see from he followig example: Le he eves A j for ji are all subses of he eve A i. The he probabiliy of he eve A i A j is PA i A j = PA j. If we ow replace his expressio i Eq. A6 we ge P A i PA i i. PA i The if is large ad he probabiliies PA i are a very small umbers, his probabiliy expressio ca be a umber much bigger he oe. Oe way o solve he accuracy problem is o o ake he secod produc over all ode pairs, bu jus over differe eighborig pairs. We sugges he followig approximaio for he cumulaive disribuio of cover ime: P A i PAi A PA i i+ PA i PA i+. We oe ha his approximae probabiliy expressio reduces o he exac probabiliy expressio i he wo limiig cases: firs, whe all eves are muually idepede, ad secod, whe all eves are subse of jus oe eve. The firs claim ca be proved jus by oig ha PA i A i+ = PA i PA i+ ad he secod claim was previously proved, see Eq. 6 whe he eves E i for,..., are all subses of he eve E. D. Be-Avraham ad S. Havli, Diffusio ad Reacios i Fracals ad Disordered Sysems Cambridge Uiversiy Press, Cambridge,. H. C. Tuckwell, Iroducio o Theoreical Neurobiology Cambridge Uiversiy Press, Cambridge, 988. A. L. Lloyd ad R. M. May, Sciece 9, 6. O. Beichou, M. Coppey, M. Moreau, P. H. Sue, ad R. Voiuriez, Phys. Rev. Le. 9, M. F. Shlesiger, Naure Lodo, I. Eliazar, T. Kore, ad J. Klafer, J. Phys.: Codes. Maer -7

8 NIKOLA ZLATANOV AND LJUPCO KOCAREV 9, L. A. Adamic, R. M. Lukose, A. R. Puiyai, ad B. A. Huberma, Phys. Rev. E 6, R. Guimera, A. Diaz-Guilera, F. Vega-Redodo, A. Cabrales, ad A. Areas, Phys. Rev. Le. 89, L. Lovasz, i Combiaorics; Paul Erdos is Eighy Jaos Bolyai Mahemaical Sociey, Budapes, 996, Vol., pp J. D. Noh ad H. Rieger, Phys. Rev. Le. 9, 87. E. Almaas, R. V. Kulkari, ad D. Sroud, Phys. Rev. E 68, 565. V. Sood, S. Reder, ad D. Be-Avraham, J. Phys. A 8, 9 5. A. Barochelli, M. Caazaro, ad R. Pasor-Saorras, Phys. PHYSICAL REVIEW E 8, 9 Rev. E 78, 8. C. Cooper ad A. Frieze, Radom Sruc. Algorihms, 7; C. Cooper ad A. Frieze, Proceedigs of SODA SIAM, New York, 9 p K. Dohme, Habiliaiosschrif, Mah.-Na. Fak. II, Humbold-Uiversia zu Berli, available olie a hp:// 6 D. Q. Naima ad H. P. Wy, A. Sa., W. H. Press, B. P. Flaery, S. A. Teukolsky, ad W. T. Veerlig, Numerical Recipes: The Ar of Scieific Compuig, rd ed. Cambridge Uiversiy Press, Cambridge, 7. 8 Ch. Jorda, Ma. Phys. Lapok, D. Kessler ad J. Schiff, Elecro. Commu. Probab. 7,