A New Graph Model with Random Edge Values: Connectivity and Diameter

Transcription

1 A New Graph Model wth Random Edge Values: Connectvty and Dameter Rchard J. La and Maya Kabkab Abstract We ntroduce a new random graph model. In our model, n, n, vertces choose a subset of potental edges by consderng the (estmated) benefts or utltes of the edges. More precsely, each vertex selects k, k, ncdent edges t wshes to set up, and an edge between two vertces s present n the graph f and only f both of the end vertces choose the edge. Frst, we examne the scalng law of the smallest k needed for graph connectvty wth ncreasng n and prove that t s Θ(log(n)). Second, we study the dameter of the random graph when t s connected and demonstrate that, under certan condtons on k, ts dameter s close to log(n)/ log(log(n)) wth hgh probablty. In addton, as a byproduct of our fndngs, we show that, for all suffcently large n, f k > β log(n), where β.466, there exsts a connected Erdös-Rény random graph that s embedded n our random graph wth hgh probablty. I. INTRODUCTION In recent years, there has been a renewed nterest n varous types of random graph models. Some of these random graph models nclude famous Erdös-Rény random graphs 6, random ntersecton graphs 4, random graphs wth hdden varables 5, 7, random threshold graphs 4, and random geometrc graphs, 6, just to name a few. A short descrpton of these models and some of ther applcatons are provded n Secton III. These random graphs are constructed n a very dfferent manner. Each model attempts to capture unque characterstcs or mechansms by whch edges are selected n the graph. Such edge selecton mechansms often make use of attrbutes or varables assgned to each vertex and lead to dfferent correlatons n edges and statstcal behavor. In the new random graph model we propose, unlke n (most of) the exstng random graph models, we assgn attrbutes to (potental) edges between vertces (as opposed to the vertces themselves). Ths allows us to capture the strategc or selfoptmzng nature of vertces we expect n many applcatons. More precsely, we nterpret the attrbutes of the edges as ther benefts or utltes, and each vertex attempts to select a subset of ncdent edges that are most benefcal to t. Such stuatons may arse naturally, for nstance, n dstrbuted systems: Suppose that each of dstrbuted agents needs to dentfy a set of edges that t wshes to set up, where the edges may represent ether physcal/wreless lnks or (logcal) relatons. However, rather than choosng the set of edges at random, t seeks nstead to maxmze ts (expected) beneft or Ths work was supported n part by the Natonal Scence Foundaton under Grant CCF Authors are wth the Department of Electrcal & Computer Engneerng (ECE) and the Insttute for Systems Research (ISR) at the Unversty of Maryland, College ark. E-mal: hyongla@umd.edu utlty from ts choce of edges. To ths end, the agent should frst estmate the beneft of each potental edge, whch may be computed based on prevous nteractons wth other agents or on other pror nformaton, and pck the edges wth the hghest benefts. Moreover, n some cases, the number of edges agents can select may be lmted due to, for nstance, a budget constrant (when settng up or mantanng edges ncurs costs), physcal constrants, or prohbtve communcaton overhead resultng from exchanges of control nformaton wth a large number of neghbors. We only consder smple cases n ths paper, n whch edge selectons are carred out only once by the vertces. However, n some applcatons, edge selectons may be repeated over tme and edges can be updated dynamcally based on more up-to-date nformaton. Ths may happen, for example, when the benefts or utltes of the edges are tme-varyng or whle the vertces are n the process of dscoverng or learnng the values of the edges. A lne of research related to ths topc can also be found n the lterature on network formaton (e.g., 6, 0 and references theren). Most of these studes model the problem of network formaton as a game among the vertces, where each vertex s a ratonal player and s nterested n maxmzng ts payoff, e.g., the number of reachable vertces dscounted accordng to ther hop dstances. These game theoretc models typcally assume that the vertces have the necessary knowledge to compute ther payoffs as a functon of strategy profle,.e., the set of actons chosen by the vertces. Ths payoff computaton often requres global knowledge, whch may not be avalable at the vertces n many applcatons. In contrast, our model allows us to capture the strategc nature of the vertces wthout assumng ther full ratonalty or avalablty of global nformaton at them. Ths s because the vertces base ther selectons only on the estmated benefts of potental edges, whch do not depend on the strategy profle, and choose the edges that they beleve to be most benefcal. In our model, settng up an edge requres mutual consent between ts two end vertces; when only one of the end vertces chooses the edge, the edge s not set up. The questons we are nterested n explorng are the followng: Suppose that there are n, n, vertces n the graph, and each vertex s allowed to choose k, k n, potental edges that provde the largest estmated benefts. Then, Q: How large should k be as a functon of n n order for the graph to be connected (wth hgh probablty)? Q: When the graph s connected, what s ts dameter? To offer a partal answer to these questons, we prove the

2 followng man results: For large n,. f k > β log(n), where β.466, the graph s connected wth hgh probablty;. f k < 0.5 log(n), the graph contans at least one solated node,.e., a node wth no neghbor, and s not connected wth hgh probablty; and. when k s of the order log(n) and s larger than β log(n) (so that the graph s connected wth hgh probablty), the dameter of a connected graph s close to log(n)/ log(log(n)). In the process of provng the frst result, we also brng to lght the followng nterestng fact: Under the condton k > β log(n), n spte of ther seemngly dsparate constructons, we can fnd an Erdös-Rény random graph that s embedded n our random graph and s also connected wth hgh probablty. The rest of the paper s organzed as follows: Secton II descrbes the random graph model we propose. We brefly dscuss some of exstng random graph models n Secton III. Secton IV defnes the graph connectvty and the dameter of a graph, and formally states the questons of nterest to us. We present the man results n Secton V, followed by numercal results n Secton VI. Throughout the paper, we assume that all random varables (rvs) are defned on a common probablty space ( Ω, F, ). II. A NEW RANDOM GRAH MODEL For each n IN + := {, 3,...}, let V (n) = {,,..., n} be the set of n nodes or vertces. We assume that all edges between nodes are undrected n our model and denote the undrected edge between nodes and j ( j) by (, j) = (j, ). For fxed n IN +, a subset of n(n )/ possble undrected edges s dentfed through the followng edge selecton process. a) Edge values For every par of dstnct nodes, say, j V (n) ( j), there s a value V,j assocated wth the edge between them. Edge value V,j s used to model an (expected) beneft node may enjoy f t has an edge wth node j. We assume that edge values are symmetrc,.e., V,j = V j,. Whle ths may not be true n some cases, we beleve that ths a reasonable assumpton n many applcatons, ncludng cooperatve dstrbuted systems; n such systems, the value of an edge should sgnfy to both of ts end vertces ts utlty to the overall system performance. The values of n(n )/ possble edges are modeled as n(n )/ ndependent and dentcally dstrbuted (..d.) rvs wth a common contnuous dstrbuton F. { b) Edge selecton Based on realzed edge values V,j, j V (n), j }, each node V (n) selects k ncdent edges wth the k largest values. Denote the set of edges chosen by node by E (n) { (, j) j V (n) \ {} }. Throughout the rest of the paper, we use the words nodes and vertces nterchangeably. In more general cases, however, edges may be drected, and edge values V,j and V j, may be correlated (as opposed to beng dentcal as assumed n ths paper). The edge set E (n) of the graph s then gven by { } E (n) := (, j) (, j) E (n) E (n) j V (n) V (n). It s clear from the defnton n () that an edge s n E (n) f and only f both of ts end nodes pck the edge. In other words, we only consder scenaros where mutual consent of the end nodes s requred for an edge to be present n the graph. When (, j) E (n), we say that nodes and j are neghbors and denote ths relaton by j. Thus, when node chooses an edge (, j) E (n), t expresses ts desre to be node j s neghbor, and we say that node pcks node j (as a potental neghbor). The par ( V (n), E (n)) gves rse to a random graph, whch we denote by G(n, k), because the edge set E (n) s a random set. We frst pont out a few observatons related to the random graph: O-. Snce each node s allowed to choose only k potental neghbors, a node degree s upper bounded by k. O-. As the selecton of potental neghbors s carred out usng symmetrc edge values, there are correlatons n the selecton of E (n), V (n). For nstance, when k < n, gven { (, j) E (n) that (, j) E (n) j (, j) E (n) j () }, the condtonal probablty s larger than the pror probablty,.e., (n) (, j) E > (, j) E (n) j. O-3. The choce of dstrbuton F (of edge values) s not mportant n that our results do not depend on the dstrbuton F. Ths s because only the orderng of the edge values matters n edge selectons. In order to see ths, note that, gven two contnuous dstrbutons F and F, we can construct a set of..d. rvs wth dstrbuton F usng another set of..d. rvs wth dstrbuton F n such a way that the orderng of the rvs s preserved wth probablty one (w.p.). Ths can be done, for nstance, usng the probablty ntegral transform and quantle functon theorems. As a result, the dstrbuton of random graph G(n, k) s not dependent on edge value dstrbuton F as long as t s contnuous. We wll make use of ths observaton n the proofs of our man results. III. EXISTING RANDOM GRAH MODELS In ths secton we brefly summarze some of well-known random graph models, and hghlght the dfferences between these models and our model delneated n Secton II. ) Erdös-Rény random graphs: One of earler random graph models s the Erdös-Rény random graph model. In the socalled G(n, p) model, there are n vertces n the graph, and each undrected edge between two vertces and j s present n the edge set E wth probablty p, ndependently of other edges 6. Snce each undrected edge s added wth probablty p ndependently, the degree of a node has a bnomal(n, p) dstrbuton. Thus, not only the degree dstrbuton n the Erdös-Rény random graphs s qute dfferent from that of our

3 3 random graphs, but more mportantly the Erdös-Rény random graphs do not model any correlaton among the edges. Another related Erdös-Rény random graph model s known as G(n, m) random graph model. In ths model, there are n vertces as before. However, nstead of addng each undrected edge wth probablty p, a set of m edges are selected at random. More precsely, each subset of m edges s equally lkely to be selected from the set contanng ( ) n(n )/ m subsets of m edges out of n(n )/ possble edges. However, t s shown that when m n(n ) p/, G(n, p) and G(n, m) behave smlarly. ) Random ntersecton graphs: In the smplest random ntersecton graph model, denoted by G(n, m, k), there are n vertces and each vertex s assgned k dstnct colors at random from a set of m colors 4. The set of k colors assgned to each vertex s selected ndependently of those of other vertces. There exsts an undrected edge between two vertces and j f the two vertces share at least one common color. The constructon of a random ntersecton graph ntroduces correlatons n the selecton of edges as does our model. However, the degree of a node s stll bnomal(n, p) dstrbuted, where p = k ( ) l=0 k/(m l). Random ntersecton graphs have been used to model the Eschenauer-Glgor key dstrbuton n (wreless) networks 4, e.g., 8; each node n a network s assgned k keys out of a pool of m keys, and two nodes can communcate wth each other only f they share a common key. Hence, the connectvty of the network can be studed by examnng the graph connectvty of G(n, m, k) random ntersecton graph. ) Random graphs wth hdden varables: In a random graph wth hdden varables studed n 7 (and later generalzed n 5) wth n vertces, each vertex, =,,..., n, s assgned a ftness (or vertex-mportance ), denoted by ς. These ftness values are modeled usng n..d. rvs wth some common dstrbuton. For gven ftness values of the vertces, there s an undrected edge between two vertces and j wth probablty f(ς, ς j ) for some functon f. It s clear that ths constructon of a random graph ntroduces correlatons n the edges va the ftness values of vertces. Ths type of random graphs s used to generate graphs wth wdely varyng degree dstrbutons, ncludng power law dstrbuton (so-called scale-free networks) 7, wthout havng to resort to preferental attachment 3, 5. In fact, somewhat surprsngly, t s shown 7 that scale-free networks can be generated even by non-scale-free dstrbutons for ftness values. v) Random threshold graphs: A random threshold graph s a specal case of random graphs wth hdden varables 4; a random threshold model s a random graph wth hdden varables where { f ς + ς f(ς, ς j ) = j >, 0 f ς + ς j, where s some threshold. v) Random geometrc graphs: In a random geometrc graph wth n vertces on a spatal doman D, whch s often a subset of IR k for some k IN := {,,...}, the locatons of the n vertces are gven by n..d. rvs X (n), =,,..., n, that are dstrbuted over D accordng to a common spatal dstrbuton. 3 Two vertces and j then have an undrected edge between them f X (n) X (n) j γ, where denotes some norm (e.g., L norm) used to measure the dstance between two vertces, and γ s some threshold. Random geometrc graphs are often used to model the onehop connectvty n wreless networks, where the threshold γ can be vewed as a proxy to the communcaton range employed by wreless nodes (e.g., 8,, ). The communcaton range s n turn determned by the transmt power of wreless devces and path losses, whch are governed by the operatng envronment. One thng to note s that the last four random graph models ntroduce correlatons n edge selecton through attrbutes assocated wth vertces. In contrast, n our random graph model, correlatons are ntroduced through attrbutes assocated wth edges. IV. GRAH CONNECTIVITY AND DIAMETER Before we state the questons of nterest, we frst ntroduce the defnton of graph connectvty we adopt throughout the paper and defne the dameter of an undrected graph. Defnton : We say that an undrected graph G = (V, E) s connected f t s possble to reach any node from any other node through a sequence of neghbors. In other words, for every par of dstnct nodes and j, we can fnd K IN and a sequence of nodes,,..., K such that. = and K = j, and. ( k, k+ ) E for all k =,,..., K. The subgraph ( V p, E p ) =: n Defnton, where Vp = {,,..., K } and E p = {(, ), (, 3 ),..., ( K, K )}, s called a path between nodes and j. The length of path s defned to be K. Suppose G = ( V, E ) s an undrected graph. The dstance between two dstnct nodes, j V s the mnmum among the lengths of all paths between the two nodes,.e., the length of the shortest path between nodes and j. We denote ths dstance by d(, j). When there s no path between two nodes and j, we set d(, j) =. Defnton : The dameter of an undrected graph G s defned to be D(G) = max { d(, j), j V, j }. In other words, the dameter of G s smply the largest dstance among all pars of nodes. We are nterested n the followng questons: Let G(n, k) be a random graph descrbed n Secton II. 3 The ndependence between nodes or homogenety of node locatons fals to hold n many practcal scenaros. In, La studed the heterogeneous moblty cases, stll under the ndependence assumpton. In another work, La and Seo relaxed the ndependence assumpton n smple one-dmensonal stuatons and ntroduced correlatons va group moblty.

4 4 Queston #: How large should k be n order to ensure that the random graph G(n, k) s connected wth hgh probablty for large values of n? Queston #: When the random graph G(n, k) s connected, what s ts dameter? Unfortunately, computng the exact probablty that the random graph G(n, k) s connected for fxed values of n and k s dffcult. Smlarly, fndng the dameter of G(n, k), whle n prncple computatonally feasble once the edge set s revealed, can be tme-consumng for large n. For these reasons, we turn to the asymptotc analyss wth an ncreasng number of nodes n. We frst examne how k should scale as a functon of n (denoted by ) n order for the random graph G(n, ) to be connected wth hgh probablty. In partcular, we are nterested n fndng the smallest that ensures connectvty of G(n, ) wth hgh probablty. Second, we nvestgate the dameter of G(n, ) when s of the same order as the smallest value that yelds a connected graph. V. MAIN RESULTS In ths secton, we provde a partal answer to the questons we posed n the prevous secton. More specfcally, we show that, wth hgh probablty, a) the smallest requred to ensure graph connectvty s of the order log(n), and b) the dameter of the random graph s approxmately log(n)/ log(log(n)) when s of the same order as the smallest value that produces graph connectvty. A. Connectvty of random graph G(n, ) Frst, we derve a suffcent condton on so that G(n, ) s connected (wth hgh probablty) for large values of n. Let β be the unque soluton to β = exp ( /β ). Ths soluton s approxmately.466. Theorem : Suppose that β > β and β log(n) for all suffcently large n. Then, G(n, ) s connected as n. roof: A proof of the theorem s provded n Appendx A. The proof of Theorem reveals the followng very nterestng fact. Under the condton n Theorem, wth probablty approachng one as n, we can fnd an Erdös-Rény random graph G(n, p(n)) wth p(n) = ζ log(n)/n, ζ >, whose edge set s contaned n the edge set of G(n, k). Snce the Erdös-Rény random graph G(n, p(n)) s connected wth probablty gong to one when ζ > 6, ths mples that our random graph s connected wth probablty tendng to one as n ncreases. Theorem tells us that the smallest necessary for graph connectvty only needs to scale as O ( log(n) ). The followng theorem states that the smallest needed for graph connectvty s also Ω ( log(n) ). Theorem : Suppose that α < 0.5 and α log(n) for all suffcently large n. Then, G(n, ) contans an solated node as n. roof: A proof of the theorem s gven n Appendx C. Snce the probablty that the graph s not connected s lower bounded by the probablty that there s at least one solated node n the graph, Theorem mples that, under the stated condton, G(n, ) s not connected wth probablty gong to one as n. Our fndngs n Theorems and ndcate that, for large n, the smallest we need so that G(n, ) s connected les n 0.5 log(n), β log(n) wth hgh probablty. Hence, for graph connectvty, the parameter should scale as Θ(log(n)). B. Dameter of random graph G(n, ) In ths subsecton, we study the dameter of G(n, ) when the condton n Theorem holds. Theorem 3: Suppose that = Θ ( log(n) ) such that lm nf n / log(n) > β. Then, the dameters of G(n, ), n IN +, satsfy the followng: For any postve ɛ > 0, lm D ( G(n, ) ) n log(n)/ log(log(n)) > ɛ = 0. In other words, D ( G(n, ) ) / ( log(n)/ log(log(n)) ) converges to one n probablty. roof: See Appendx D for a proof. Theorem 3 mples that f κ log(n) for some κ > β for all suffcently large n, the dameter of G(n, ) s close to log(n)/ log(log(n)) wth hgh probablty. For example, when n s 0 6 and 0 9, we have log(n)/ log(log(n)) 5.65 and , respectvely. Hence, the dameter of G(n, ) ncreases rather slowly wth the number of nodes n the graph under the condton n Theorem 3. VI. NUMERICAL RESULTS Our fndngs n Secton V-A provde us wth an nterval to whch the smallest that yelds graph connectvty belongs (wth hgh probablty). However, t does not establsh the exstence of a sharp phase transton that we observe, for nstance, wth the Erdös-Rény random graphs 6. In ths secton, we provde some numercal results to look for an answer to ths queston. In our numercal examples, for n = 50, 00, 800, and 3,000, we vary the value of parameter k to see how the probablty of graph connectvty changes wth k. The emprcal probablty s obtaned as the fracton of tmes the graph s connected out of 00 realzatons. Ths s plotted n Fg.. The dotted vertcal red lnes n the fgure ndcate k = log(n) for dfferent values of n. There are two observatons we can make from Fg.. Frst, t reveals that the probablty of graph connectvty ncreases sharply as k ncreases over a short nterval around log(n) (as opposed to over the nterval 0.5 log(n), β log(n)).

5 5 robablty of graph connectvty robablty of no node solaton robablty of connectvty n=50 n=00 n=800 n=3000 robablty of no node solaton n=50 n=00 n=800 n= k Fg.. robablty of graph connectvty as a functon of k for n = 50, 00, 800, and 3, k Fg.. robablty of no solated node as a functon of k for n = 50, 00, 800, and 3,000. Second, the transton wdth,.e., the length of the nterval over whch the probablty ncreases from (close to) zero to (close to) one, does not appear to depend on n. Together, these two observatons suggest that, for large n the dstrbuton of the smallest needed for graph connectvty s lkely to be concentrated around log(n). Ths leads to the followng conjecture. Conjecture : Suppose that β > and β log(n) for all suffcently large n. Then, G(n, ) s connected as n. Smlarly, f β < and β log(n) for all suffcently large n. Then, G(n, ) s connected 0 as n. a) Node solaton: It s well known that for the Erdös- Rény random graphs, when the graph does not have an solated node, the graph s lkely connected as well. Ths s captured by the followng sharp result: () If the edge selecton probablty p(n) α log(n)/n wth α <, G(n, p(n)) contans an solated node as n ; and () If the edge selecton probablty p(n) β log(n)/n wth β >, G(n, p(n)) s connected as n. A smlar observaton holds for random geometrc graphs and many other graphs. Hence, t s of nterest to see f the same observaton holds wth our random graphs. Fg. plots the probablty that there s no solated node n the graph G(n, ) for the same set of values of n and k used n Fg.. By comparng Fgs. and, t s clear that the probablty that the graph contans no solated node s very close to the probablty that the graph s connected. Hence, the fgures suggest that, for large n, wth hgh probablty the mnmum k necessary for graph connectvty s close, f not equal, to the smallest k requred to elmnate solated nodes n the graph. VII. CONCLUSION We proposed a new random graph model that allows us to capture the strategc nature of nodes whch arses naturally n some cases. The proposed random graphs are constructed n a very dfferent fashon than exstng random graph models and lkely exhbt dfferent statstcal behavor. We beleve that the new random graph model wll prove to be useful for modelng the network formaton n dstrbuted systems n whch the agents are free to choose ther neghbors based on the estmated benefts and/or local nformaton. We are currently workng to generalze the proposed random graph model. Frst, we are studyng the case where the edge values are not symmetrc, but nstead are correlated. Second, we are also examnng the scenaros where the number of potental edges chosen by each node vares and s modeled as a random varable. Acknowledgment The authors would lke to thank rofessor Armand M. Makowsk and Dr. Vjay G. Subramanan for many helpful dscussons. REFERENCES J.E. Angus, The probablty ntegral transform and related results, SIAM Revew, 36(4):65-654, Dec M.J.B. Appel and R.. Russo, The connectvty of a graph on unform ponts on 0, d, Statstcs & robablty Letters, 60:35-357, R. Albert and A.-L. Barabás, Statstcal mechancs of complex networks, Revew of Modern hyscs, 74():47-97, Jan-Mar S.R. Blackburn and S. Gerke, Connectvty of the unform random ntersecton graph, Dscrete Mathematcs, 309(6): , Aug M. Bogu ná and R. astor-satorras, Class of correlated random networks wth hdden varables, hyscal Revew E, 68, artcle 036, B. Bollobás, Random Graphs, Cambrdge Studes n Advanced Mathematcs, nd ed., Cambrdge Unversty ress, G. Caldarell, A. Capocc,. De Los Ros and M.A. Muñoz, Scale-free networks from varyng vertex ntrnsc ftness, hyscal Revew Letters, 89(5), artcle 5870, Dec. 00.

6 6 8 H.A. Davd and H.N. Nagaraja, Order Statstcs, 3rd ed., Wley, L. de Haan and A. Ferrera, Extreme Value Theory: An Introducton, Sprnger Seres n Operatons Research and Fnancal Engneerng, Sprnger, Dacons and D. Freedman, Fnte exchangeable sequences, The Annals of robablty, 8(4): , 980. R. Durrett, Random Graph Dynamcs, Cambrdge Seres n Statstcal and robablstc Mathematcs, Cambrdge Unversty ress, Erdös and A. Rény, On random graphs. I, ublcatones Mathematcae, 6: 90-97, Erdös and A. Rény, The evoluton of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl., 5:7-6, L. Eschenauer and V.D. Glgor, A key-management scheme for dstrbuted sensor network, roc. of the ACM Conference on Computer and Commmuncatons Securty (CSS), pp. 4-47, Washngton (DC), Nov M. Falk, A note on unform asymptotc normalty of ntermedate order statstcs, Annals of the Insttute of Statstcal Mathematcs, 4():9-9, March S. Goyal, Connectons: An Introducton to the Economcs of Networks, rnceton Unversty ress, G. Grmmett and D. Strzaker, robablty and Random rocesses, thrd ed., Oxford Unversty ress, Gupta and.r. Kumar, Crtcal power for asymptotc connectvty n wreless networks, Stochastc Analyss, Control, and Optmzaton and Applcatons, pp , Y. Isukapall and B.D. Rao, An analytcally tractable approxmaton for the Gaussan Q-functon, IEEE Communcatons Letters, (9):668-67, M.O. Jackson, Socal and Economc Networks, rnceton Unversty ress, 00. R.J. La, Network connectvty wth heterogeneous moblty, roc. of IEEE Internatonal Conference on Communcatons (ICC), Jun. 0. R.J. La and E. Seo, Network connectvty wth a famly of group moblty models, IEEE Transactons on Moble Computng, (3):504-57, Mar R.J. La and M. Kabkab, A new graph model wth random edge values: connectvty and dameter, reprnt. Avalable at hyongla/aers/allerton3 TR.pdf 4 A.M. Makowsk and O. Yağan, Scalng laws for connectvty n random threshold graph models wth non-negatve ftness varables, preprnt. Avalable at oyagan/journals/threshold.pdf 5 M. Newman, Networks: An Introducton, Oxford Unversty ress, M.D. enrose, Random Geometrc Graphs, Oxford Studes n robablty, Oxford Unversty ress, M. Shaked and J.G. Shanthkumar, Stochastc Orders, Sprnger Seres n Statstcs, Sprnger, O. Yağan and A.M. Makowsk, Zero-one laws for connectvty n random key graphs, IEEE Transactons on Informaton Theory, 58(5): , May 0. AENDIX A ROOF OF THEOREM In order to prove the theorem, we show that when = β log(n), n IN +, for any β > β, 4 the probablty that the random graph G(n, ) s connected goes to one as n ncreases. For each n IN +, we frst order the edge values seen by each node by decreasng value: For each V (n), let U (n),,..., U (n),n denote the order statstcs of V,j, j V (n) \ {}. 5 Then, t s clear that U (n),,..., U (n), are the values of the edges selected by node. Recall from observaton O-3 n Secton II that we can assume any contnuous dstrbuton for the edge value dstrbuton F wthout affectng our result. Takng advantage of 4 When β log(n) s not an nteger, we can assume that = β log(n) wthout any problem because we can always fnd β satsfyng β < β < β and fnte n (β) such that, for all n n (β), we have β log(n) β log(n). 5 Throughout the proofs, we assume that the order statstcs are ordered by decreasng value. ths fact, we assume F exponental() n the proof. In other words, edge values V,j, j, are assumed to be..d. exponental rvs wth mean of one. We frst ntroduce a lemma that wll be used to complete the proof of the theorem. Its proof s gven n Appendx B. Lemma : Suppose that the edge values are gven by..d. exponental() rvs. For each n IN + and V (n), let {( ( )) ( )} A (n) = U (n) n, log β, β and A (n) := V (n) A(n). Then, A (n) as n. Lemma tells us that, as n ncreases, wth hgh probablty ( ( ) ( ) ) n n log β, log + β =: B (n) U (n), smultaneously for all V (n). For each n IN +, let p(n) = V, > log ( ) n + = e log(n ) log()+ /β = n e /β. Substtutng β log(n) for, β p(n) = β log(n) n e /β = ζ log(n) n, () where ζ := β exp ( /β ). From the assumpton β > β = exp ( /β ) > exp ( /β ), we have ζ >. To complete the proof, we make use of followng Erdös- Rény random graphs constructed from the same vertex sets and edge values: For each n IN +, let G(n, p(n)), where p(n) s gven n (), be an Erdös-Rény random graph wth vertex set V (n) and edge set comprsng all edges (, j), j, satsfyng ( ) n V,j > log + β =: v n. Ths s ndeed an Erdös-Rény random graph because an edge between two nodes s present wth probablty p(n) n (), ndependently of each other. Snce ζ >, 3, G(n, p(n)) s connected as n. (3) Usng the law of total probablty, we can fnd an upper bound on the probablty that G(n, ) s not connected: G(n, ) s not connected = ( A(n) ) c {G(n, ) s not connected} + A(n) {G(n, ) s not connected} ( A(n) ) c + A(n) {G(n, ) s not connected} (4)

7 7 Frst, Lemma states that ( A(n) ) c goes to zero as n ncreases. Second, n the event A(n), we have U (n), B (n), hence, U (n), < v n for every V (n). Thus, all edges whose value s greater than v n wll be n the edge set E (n) of G(n, ). Snce the edges n the Erdös-Rény random graph G(n, p(n)) are those wth values larger than v n, they also belong to E (n). Ths tells us that when the Erdös-Rény random graph G(n, p(n)) s connected, so s G(n, ), yeldng the followng upper bound for the second term n (4). A(n) {G(n, ) s not connected} A(n) {G(n, p(n)) s not connected} G(n, p(n)) s not connected (5) From (3), as n ncreases, the Erdös-Rény random graph G(n, p(n)) s connected wth probablty approachng one, mplyng that (5) goes to zero as n. Hence, the second term n (4) goes to zero as n ncreases, and G(n, ) s not connected 0 as n. Defne B(n) := AENDIX B ROOF OF LEMMA ( log(n), ) log(n), n IN +. From 5, (.), p. 4 and 8, Theorem 0.8., p. 3, for any δ > 0, there exsts fnte n(δ) such that, for all n n(δ), ( U (n), log ( (n )/ )) B(n) ( ) c = A (n) ( ) ( + δ) Q log(n), where Q s the complementary cumulatve dstrbuton functon of a standard normal rv. Usng a well known upper bound for Q(x), namely exp( x /)/( πx) 9, ( A (n) ( ) ) c exp log(n) ( + δ) π log(n) exp ( log(n)) = ( + δ) π log(n) + δ = n π log(n). (6) Snce A (n) = V (n) A(n), usng a unon bound and (6), (A (n)) c ( ) c V (n) + δ π log(n). A (n) Therefore, (A (n) ) c 0 or, equvalently, A (n) as n. AENDIX C ROOF OF THEOREM We show that when α < 0.5 and = α log(n), 6 the probablty that G(n, ) contans at least one solated node converges to one as n. To ths end, we use the (second) moment method: Defne Z + := {0,,,...} to be the set of nonnegatve ntegers. Suppose {Z n ; n =,,...} s a sequence of Z + -valued rvs wth fnte second moment,.e., E Zn < for all n IN. Then, lm Z n = 0 = 0 f n (E Z n ) lm n E Zn =. (7) Equaton (7) can be easly proved usng Cauchy-Schwarz nequalty 7, p. 65. For each n IN + and V (n), defne and = {node s solated n G(n, )} C (n) = n =. Clearly, C (n) denotes the total number of solated nodes n G(n, ). We wll prove that C (n) = 0 0 as n by means of (7). Frst, because, V (n), are dentcally dstrbuted, E C (n) = n E. (8) Second, snce, V (n), are also exchangeable, ( E C (n)) ( n ) = E = n E Usng (8) and (9), we get ( ) E C (n) (C ) = E (n) = ( n E = + n(n )E n E ) + n n. (9) ( ) n E + n(n )E E ( ) E. (0) The proof of the theorem can be completed wth help of Lemmas and 3, whose proofs can be found n 3 and are omtted here due to a space constrant. Lemma : Under the condton n Theorem, the expected values of C (n), n IN +, satsfy lm E C (n) = lm n E n n =. Lemma 3: Suppose ( that the condton ) n Theorem holds.. Then, E E 6 For a smlar reason stated n the proof of Theorem n Appendx A, when α log(n) s not an nteger, we can assume that = α log(n).

8 8 Lemmas and 3, together wth (0), mply that ( E C (n) ) /E (C (n) ) as n. Therefore, (7) tells us that C (n) = 0 0 as n, and the probablty that G(n, ) has an solated node converges to one. AENDIX D ROOF OF THEOREM 3 Theorem 3 can be proved wth the help of the followng result for Erdös-Rény random graphs. Theorem.8.6, p. 67: Suppose that the edge selecton probabltes p(n), n IN +, of Erdös-Rény random graphs G(n, p(n)) ( satsfy () lm nf n ) np(n)/ log(n) > and () lm n log(np(n))/ log(n) = 0. Then, the dameters of the Erdös-Rény random graphs G(n, p(n)) satsfy D ( G(n, p(n)) ) log(n)/ log(n p(n)) > ɛ 0 as n for all ɛ > 0. Let κ = lm nf n log(n) and κ = lm sup n log(n). Then, for any ɛ > 0, there exsts fnte ñ(ɛ) such that, for all n ñ(ɛ), we have (κ ɛ) log(n), (κ + ɛ) log(n). ck ɛ satsfyng 0 < ɛ < κ β, and let ñ(ɛ) be the smallest n IN + that satsfes the above condton. In the rest of the proof, utlzng observaton O-3 n Secton II, we assume F exponental(). For n ñ(ɛ), let κ(n) = / log(n) κ ɛ, κ + ɛ. Defne p (n) = V, > log ( n ) + κ(n) = ζ (n) log(n) n, () where ζ (n) := κ(n)e /κ(n) > because κ(n) κ ɛ > β, and ( ) n p (n) = V, > log κ(n) = ζ (n) log(n) n, () where ζ (n) := κ(n)e /κ(n) > ζ (n). For n IN +, denote by E (n) (n) ER, (resp. E ER, ) the set of edges (, j),, j V (n) ( j), satsfyng V,j > log ( (n )/ ) + /κ(n) (resp. V,j > log ( (n )/ ) /κ(n)). Let G(n, pl (n)) := ( V (n), E ER,l) (n), l =,. As explaned n Appendx A, G(n, p l (n)), l =,, are Erdös- Rény random graphs wth edge selecton probablty p l (n) gven n () and (). Clearly, p l (n), l =,, meet the condtons n Theorem.8.6, p. 67, whch was stated earler. Hence, the dameters of Erdös-Rény random graphs G(n, p l (n)), n IN +, satsfy D ( G(n, p l (n)) ) log(n)/ log ( n p l (n) ), l =,, (3) where denotes convergence n probablty. Note that, for all n ñ(ɛ), both ζ (n) and ζ (n) le n the fnte nterval (κ ɛ)e /(κ ɛ), (κ + ɛ)e /(κ+ɛ). As a result, we have log(n) log ( n p l (n) ) log(n) log ( ), l =,. (4) log(n) Equatons (3) and (4) tell us D ( G(n, p l (n)) ) log(n)/ log ( log(n) ), l =,. (5) To complete the proof, we wll prove that, wth probablty tendng to one as n, we have D ( G(n, p (n)) ) D ( G(n, ) ) D ( G(n, p (n)) ). (6) To ths end, we wll show lm E (n) n ER, E (n) E (n) ER, =, (7) whch mples (6). Defne {( (  (n) = U (n) n ) ) }, log ˆΛ(n) V (n), where ( ) ˆΛ(n) := κ(n),. κ(n) Followng smlar steps as n the proof Â(n) of Lemma (n Appendx B), we can show lm n =. It s clear that, n the event  (n), the edge set E (n) of G(n, ) satsfes (7); condtonal on the event  (n), a) all edges (, j) wth V,j > log ( (n )/ ) + /κ(n) belong to E (n), and b) no edge (, j) such that V,j log ( (n )/ ) /κ(n) s n E (n). The theorem s then a corollary of the followng facts: Â(n). as n ;. In the event  (n), nequaltes n (6) hold; and. The dameters D ( G(n, p l (n)) ), l =,, satsfy (5).