Spreading Processes and Large Components in Ordered, Directed Random Graphs

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1 Sreadig Processes ad Large Comoets i Ordered, Directed Radom Grahs Paul Hor Malik Magdo-Ismail Setember 2, 202 Abstract Order the vertices of a directed radom grah v,..., v ; edge v i, v j for i < j exists ideedetly with robability. This radom grah model is related to certai sreadig rocesses o etworks. We cosider the comoet reachable from v ad rove existece of a shar threshold = log / at which this reachable comoet trasitios from o to Ω. Itroductio I this ote we study a radom grah model that catures the dyamics of a articular tye of sreadig rocess. Cosider a set of ordered vertices {v,..., v } with vertex v iitially ifiltrated at time ste. At time stes 2, 3,...,, vertex v attemts to ideedetly ifiltrate, with robability, each of v 2, v 3..., v i tur oe er ste. Either v i gets ifiltrated or immuized. If v i is ifected, it attemts to ifect v i+,..., v, also each with robability ; v i does ot attemt to ifect v,..., v i, however, as rior vertices are already either ifiltrated or immuized. At time ste i, all ifiltrated vertices v j with j < i are attemtig to ifiltrate v i, ad v i gets ifiltrated if ay oe of these attemts succeeds. Ituitively, v i is more likely to get ifiltrated if more vertices are already ifiltrated at the time that v i becomes succetible. Oe examle of such a cotagio rocess is give i?]. This sreadig rocess is equivalet to the followig radom model of a ordered, directed grah G: order the vertices v,..., v, ad for i < j, the directed edge v i, v j exists i G with robability ideedetly. Vertex v i is ifected if there is a directed ath from v to v i. The questio we address is, What is the size of the set of vertices reachable from v? the size of the ifectio. We rove the followig shar result. Theorem. Let R be the set of vertices reachable from v, ad suose = c log ξ = o log ad c > 0 is fixed. The:. If c <, the R = c+o, a.a.s. Deartmet of Mathematics, Harvard Uiversity, hor@math.harvard.edu Deartmet of Comuter Sciece, Resselaer Polytechic Istitute, magdo@cs.ri.edu + ξ, where

2 2. If c =, the R = o, a.a.s. 3. If c >, the R = c + o, a.a.s. Recall that a evet holds a.a.s.asymtotically almost surely, if it holds with robability o; that is it holds with robability tedig to oe as teds to ifiity. Note that we do ot exlicitly care whether ξ is ositive or egative i the results above. Similar hase trasitios are well kow for various grah roerties i other radom grah models. As show by Erdős ad Réyi i 2], i the G, M model of radom grahs, where a grah is chose ideedetly from all grahs with M edges, there is a similar emergece of a comoet of size Θ aroud M = 2 edges. Likewise, a threshold for coectivity was show for M = log 2 edges. For the more familiar G, model, where edges are reset ideedety with robability, this traslates ito a threshold at = log for a giat comoet, ad at = for coectivity. A much more comrehesive accout of results o roerties of radom grahs ca be foud i ]. Luczak i 4] ad more recetly Luczak ad Seierstad i 5], studied the emergece of the giat comoet i a radom directed grahs, i both the directed model where M radom edges are reset ad i the model where edges are reset with robability. Thresholds for strog coectivity were established for radom directed grahs by Palásti 6] for radom directed grahs with M edges ad Graham ad Pike 3] for radom directed grahs with edge robability. We are ot aware of ay results for ordered directed radom grahs where edges coect vertices of lower idex to higher idex. 2 A Proof of Theorem Uer bouds: For i >, let R i deote the evet that v i is reachable, ad let X i deote the umber of aths to vertex v i i G. If P i deotes the set of all otetial aths from v to v i, the X i = x P i Ix where Ix is a {0, } idicator radom variable idicatig whether the ath x exists i G; Ix = if ad oly if all edges i the ath x are reset i G. The, PR i = PX i EX i ] = x P i EIx] = i 2 l=0 x P i x =l+ i 2 i 2 = l l=0 EIx] l+ = + i 2 e i. Let X deote the umber of reachable vertices other tha v. EX] = PR i e i = e+ e. i=2 For = c log + ξ with c <, e + = ex c log + olog = c+o, 2

3 ad Thus, e = k k! EX] c+o. = + O. Alyig Markov s iequality yields that PX > logex] = o, so X logex] = c+o, a.a.s. Now cosider c >. Let ξ := 3 { } c max log log, 2 ξ c log t := c ξ Note that by our choice of ξ, ad the fact that ξ = o log, that ξ = o. The, PR t e t = ex = ex c log log + ξ c + oc ex + ξ c ξ clog ξ ξξ log log + ξ clog ξ c = o Here, the last iequality comes from the fact that, by our choice of ξ, clogξ log log ξ c 3 ξ. Sice e i is icreasig i i, the exected umber of reachable vertices v i with i t is at most tpr t = o. Alyig Markov s iequality, R {v,..., v t } = o a.a.s. Thus, R t + R {v,..., v t } = c + o a.a.s. For = log + ξ with ξ = o log, we will write ξ = ω log, where ω 0. Let t = ω log log. The, ] PR t e t = ex + ω ω log log log + log + ω log ] = ex ω 2 log log + ω + log log + log + ω log log = o, Thus the exected value of R {v,..., v t } is o ad by Markov s iequality, this is also true a.a.s. Now, sice t is also o, we have that R = o a.a.s. 3

4 To rove the lower bouds, we require a simle lemma similar to Dirichlet s theorem. Let di deote the umber of divisors of i ad let d t i deote the umber of divisors of i that are at most t. Dirichlet s Theorem states that k di = k log k + 2γ k + O k, where γ is Euler s costat. For our uroses, we eed a refiemet of this result, summig d t i. k Lemma. d t i = k log mit, k + Ok. Proof. For t > k the result follows from Dirichlet s theorem as we may relace d t i with di i the summatio. For t k, k k k k d t i = k kh t, 2 3 t where H t is the t-th harmoic umber. Lower bouds: For exositio, assume that we costruct our grah o coutably may vertices ad that we the restrict our attetio to the first vertices. Let X i deote the idex of the i-th reachable vertex that is ot v. If X i > the R i. Set X 0 =, ad for i, X i X i is geometrically distributed with arameter i. Fix t, ad cosider EX t ]: EX t ] = t EX k X k ] = j=0 t Each term is a ifiite geometric series, ad so t EX t ] = kj. k. As this series is absolutely summable as EX t ] is clearly fiite, Fubii s theorem allows us to rearrage terms i the summatio to get t EX t ] = t + kj = t + d t i i. j= because the term i aears i the origial summatio where i = kj oce for every divisor i has that is at most t. We ow use summatio by arts to maiulate the secod term: i d t i i = i d t l l= = i i logmi{t, i} + Oi log t + O i i. 4

5 Sice i i = / 2, we have that EX t ] = t + log t + O Furthermore, sice X k+ X k ad X k X k are ideedet, 2 k 3 VarX t = t t k 2 t 2 k 3. k t EX t]. 2 Here, the first iequality follows from a alicatio of Cauchy-Schwarz, ad the secod from 2 = 3. Now, suose that = c log + ξ for c <, ad set t = c ex ξ log log. The, from, EX t ] c c log 2 ξ log log log ex ξ + + O 3 c log + ξ c ex ξ + 2 ξ log log log + O 4 c log + ξ ξ + log log ξ + log log = + o. 5 c log + ξ log For sufficietly large, EX t ]. Meawhile, from 2, c VarX t + o log ξ +log log 2c log 2 +c + o c log EX t] = log because EX t ] = O ad c <. Chebyshev s iequality asserts that ] P X t EX t ] 2 ξ + log log 2c log ] log log Thus, P X t EX t ] + 2c log = o. Usig 5, P X t log log 2c log + o EXt ] = O c 3/2 log 4c 2 log 2 VarX t 2 ξ + log log 2 = o. ] log log = o, c log i.e., X t < a.a.s. Sice X t < imlies R t, we have that R > c ex ξ log log = c+o a.a.s. For c >, take t = log log log. The, usig, EX t ] c + o., 5

6 Agai, by 2 ad because EX t ] = O, VarX t = O 3/2 log log / log = o 3/2. Chebyschev s iequality asserts that P X t EX t ] 3/4] o3/2 3/2 = o. Hece, P X t EX t ] + 3/4] = o. 6 So, X t c + o a.a.s. We ow cosider the vertices idexed higher tha X t ad show that essetially all of them are reachable. Let Y be the vertices with idex higher tha X t which are ot adjacet to oe of the first t reachable vertices i v,..., v Xt. The E Y ] = j=x t+ t = X t t e t = log c+o = o. Alyig Markov s iequality, Y = o with robability o. Sice the set of vertices idexed above X t that is ot reachable is a subset of Y, R t + X t Y. Sice Y, t are o ad X t = c + o, we have that R c + o with robability o, as desired. Ackowledgemet. Magdo-Ismail ackowledges that this research was sosored by the Army Research Laboratory ad was accomlished uder Cooerative Agreemet Number W9NF The views ad coclusios cotaied i this documet are those of the authors ad should ot be iterreted as reresetig the official olicies, either exressed or imlied, of the Army Research Laboratory or the U.S. Govermet. The U.S. Govermet is authorized to reroduce ad distribute rerits for Govermet uroses otwithstadig ay coyright otatio here o. Refereces ] B. Bollobás. Radom grahs, volume 73 of Cambridge Studies i Advaced Mathematics. Cambridge Uiversity Press, Cambridge, secod editio, ] P. Erdős ad A. Réyi. O the evolutio of radom grahs. Magyar Tud. Akad. Mat. Kutató It. Közl., 5:7 6, ] A. J. Graham ad D. A. Pike. A ote o thresholds ad coectivity i radom directed grahs. Atl. Electro. J. Math., 3: 5, ] T. Luczak. The hase trasitio i the evolutio of radom digrahs. J. Grah Theory, 42:27 223, ] T. Luczak ad T. G. Seierstad. The critical behavior of radom digrahs. Radom Structures Algorithms, 353:27 293, ] I. Palásti. O the strog coectedess of directed radom grahs. Studia Sci. Math. Hugar, :205 24,