k -connectivity of inhomogeneous random key graphs with unreliable links
aa r X i v : . [ c s . CR ] O c t k -connectivity of inhomogeneous random keygraphs with unreliable links Rashad Eletreby,
Student Member, IEEE, and Osman Ya ˘gan,
Member, IEEE,
Abstract —We consider secure and reliable connectivity inwireless sensor networks that utilize a heterogeneous random keypredistribution scheme. We model the unreliability of wirelesslinks by an on-off channel model that induces an Erd˝os-R´enyigraph, while the heterogeneous scheme induces an inhomoge-neous random key graph. The overall network can thus bemodeled by the intersection of both graphs. We present conditions(in the form of zero-one laws) on how to scale the parameters ofthe intersection model so that with high probability i) all of itsnodes are connected to at least k other nodes; i.e., the minimumnode degree of the graph is no less than k and ii) the graph is k -connected, i.e., the graph remains connected even if any k − nodes leave the network. We also present numerical results tosupport these conditions in the finite-node regime. Our resultsare shown to complement and generalize several previous workin the literature. Index Terms —General Random Intersection Graphs, WirelessSensor Networks, Security, Inhomogeneous Random Key Graphs, k -connectivity, Mobility.
1. I
NTRODUCTION
A. Motivation and Background
Wireless sensor networks (WSNs) enable a broad rangeof applications including military, health, and environmentalmonitoring, among others [1]. A typical WSN consists ofhundreds, thousands, or hundreds of thousands of nodes thatare often deployed randomly in hostile environments. The easeof deployment, low cost, low power consumption, and smallsize have paved the way for the proliferation of WSNs, butalso rendered them vulnerable to various types of attacks. Infact, security of WSNs is a key challenge given their uniquefeatures [2]; e.g., limited computational capabilities, limitedtransmission power, and vulnerability to node capture attacks.Random key predistribution schemes were proposed to tacklethose limitations, and they are currently regarded as the mostfeasible solutions for securing WSNs; e.g., see [3, Chapter 13]and [4], and references therein.Random key predistribution schemes were first introducedin the pioneering work of Eschenauer and Gligor [5]. Theirscheme, hereafter referred to as the EG scheme, operates asfollows: prior to deployment, each sensor node is assigneda random set of K cryptographic keys, selected from a keypool of size P (without replacement). After deployment, twonodes can communicate securely over an existing channel if they share at least one key. The EG scheme led the way toseveral other variants, including the q -composite scheme [6],and the random pairwise scheme [6] among others. R. Eletreby and O. Ya˘gan are with the Department of Electrical andComputer Engineering and CyLab, Carnegie Mellon University, Pittsburgh,PA, 15213 USA. E-mail: [email protected], [email protected].
Recently, a new variation of the EG scheme, referred toas the heterogeneous key predistribution scheme, was intro-duced [7]. The heterogeneous scheme considers the case whenthe network includes sensor nodes with varying levels ofresources, features, or connectivity requirements (e.g., regularnodes vs. cluster heads); it is in fact envisioned [8] thatmany WSN applications will be heterogeneous. The schemeis described as follows. Given r classes, each sensor isindependently classified as a class- i node with probability µ i > for each i = 1 , . . . , r . Then, sensors in class- i areeach assigned K i keys selected uniformly at random (withoutreplacement) from a key pool of size P . Similar to the EGscheme, nodes that share key(s) can communicate securelyover an available channel after the deployment; see Section 2for details.In [9], the authors considered the reliability of secure WSNsunder the heterogeneous key predistribution scheme; namely,when each wireless link fails with probability − α inde-pendently from other links. From a wireless communicationperspective, this is similar with investigating the secure con-nectivity of a WSN under an on/off channel model, whereineach wireless channel is on with probability α independentlyfrom other links. There, we established critical conditions onthe probability distribution µµµ = { µ , µ , . . . , µ r } , and scalingof the key ring sizes KKK = { K , K , . . . , K r } , the key poolsize P , and the channel parameter α as a function of networksize n , so that the resulting WSN is securely connected withhigh probability. Although these results form a crucial startingpoint towards the analysis of the heterogeneous key predistri-bution scheme, there remains to establish several importantproperties of the scheme to obtain a full understanding of itsperformance in securing WSNs. In particular, the connectivityresults given in [9] do not guarantee that the network wouldremain connected when sensors fail due to battery depletionor get captured by an adversary. Moreover, the results arenot applicable for mobile WSNs; wherein, the mobility ofsensor nodes may render the network disconnected. In essence,sharper results that guarantee network connectivity in theaforementioned scenarios are needed.
B. Contributions
The objective of our paper is to address the limitationsof the results in [9]. We consider the heterogeneous keypredistribtuion scheme under an on/off communication modelconsisting of independent wireless channels each of which iseither on (with probability α ), or off (with probability − α ).We focus on the k -connectivity property which implies that the network connectivity is preserved despite the failure ofany ( k − nodes or links [10]. Accordingly, k -connectivityprovides a guarantee of network reliability against the potentialfailures of sensors or links. Moreover, for a k -connected mobile WSN, any ( k − nodes are free to move anywherewhile the rest of the network remains at least -connected.Our approach is based on modeling the WSN by an appro-priate random graph and then establishing scaling conditionson the model parameters such that certain desired propertieshold with high probability (whp) as the number of nodes n getslarge. The heterogeneous key predistribution scheme inducesan inhomogeneous random key graphs [7], denoted hereafterby K ( n, µµµ, KKK, P ) , while the on-off communication modelleads to a standard Erd˝os-R´enyi (ER) graph [11], denoted by G ( n, α ) . Hence, the appropriate overall random graph modelis the intersection of an inhomogeneous random key graphwith an ER graph, denoted K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) .We establish two main results for the intersection model K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) ; namely, i) a zero-one law for theminimum node degree of K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) to be no lessthan k for any non-negative integer k and ii) a zero-one law forthe k -connectivity property of K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) for anynon-negative integer k . More precisely, we present conditionson how to scale the parameters of K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) sothat i) its minimum node degree is no less than k and ii) it is k -connected, both with high probability when the number ofnodes n gets large. Furthermore, we show by simulations thatminimum node degree being no less than k and k -connectivityproperties exhibit almost equal (empirical) probabilities. Notonly do our results complement and generalize several previ-ous work in the literature, but they also have broad range ofapplications to other interesting problems (See Section 3 fordetails). C. Notation and Conventions
All limiting statements, including asymptotic equivalenceare considered with the number of sensor nodes n going toinfinity. The random variables (rvs) under consideration are alldefined on the same probability triple (Ω , F , P ) . Probabilisticstatements are made with respect to this probability measure P , and we denote the corresponding expectation by E . Theindicator function of an event E is denoted by E ] . We saythat an event holds with high probability (whp) if it holds withprobability as n → ∞ . For any event E , we let E denotethe complement of E . For any discrete set S , we write | S | forits cardinality. For sets S a and S b , the relative compliment of S a in S b is given by S a \ S b . In comparing the asymptoticbehaviors of the sequences { a n } , { b n } , we use a n = o ( b n ) , a n = w ( b n ) , a n = O ( b n ) , a n = Ω( b n ) , and a n = Θ( b n ) , withtheir meaning in the standard Landau notation. Namely, wewrite a n = o ( b n ) as a shorthand for the relation lim n →∞ a n b n =0 , whereas a n = O ( b n ) means that there exists c > suchthat a n ≤ cb n for all n sufficiently large. Also, we have a n =Ω( b n ) if b n = O ( a n ) , or equivalently, if there exists c > such that a n ≥ cb n for all n sufficiently large. Finally, wewrite a n = Θ( b n ) if we have a n = O ( b n ) and a n = Ω( b n ) atthe same time. We also use a n ∼ b n to denote the asymptoticequivalence lim n →∞ a n /b n = 1 . 2. T HE M ODEL
We consider a network consisting of n sensor nodes labeledas v , v , . . . , v n . Each sensor is assigned to one of the r possible classes (e.g., priority levels) according to a probabilitydistribution µµµ = { µ , µ , . . . , µ r } with µ i > for each i = 1 , . . . , r ; clearly it is also needed that P ri =1 µ i = 1 . Priorto deployment, each class- i node is given K i cryptographickeys selected uniformly at random from a pool of size P .Hence, the key ring Σ x of node v x is a P K tx -valued randomvariable (rv) where P A denotes the collection of all subsets of { , . . . , P } with exactly A elements and t x denotes the classof node v x . The rvs Σ , Σ , . . . , Σ n are then i.i.d. with P [Σ x = S | t x = i ] = (cid:18) PK i (cid:19) − , S ∈ P K i . After the deployment, two sensors can communicate securelyover an existing communication channel if they have at leastone key in common.Throughout, we let
KKK = { K , K , . . . , K r } , and assumewithout loss of generality that K ≤ K ≤ . . . ≤ K r .Consider a random graph K induced on the vertex set V = { v , . . . , v n } such that distinct nodes v x and v y are adjacentin K , denoted by the event K xy , if they have at least onecryptographic key in common, i.e., K xy := [Σ x ∩ Σ y = ∅ ] . (1)The adjacency condition (1) characterizes the inhomogeneousrandom key graph K ( n ; µµµ, KKK, P ) that has been introducedrecently in [7]. This model is also known in the literatureas the general random intersection graph ; e.g., see [12]–[14].The inhomogeneous random key graph models the cryp-tographic connectivity of the underlying WSN. In particular,the probability p ij that a class- i node and a class- j have acommon key, and thus are adjacent in K ( n ; µµµ, KKK, P ) , is givenby p ij = P [ K xy ] = 1 − (cid:18) P − K i K j (cid:19),(cid:18) PK j (cid:19) (2)as long as K i + K j ≤ P ; otherwise if K i + K j > P , we clearlyhave p ij = 1 . We also find it useful define the mean probability λ i of edge occurrence for a class- i node in K ( n ; µµµ, KKK, P ) .With arbitrary nodes v x and v y , we have λ i = P [ K xy | t x = i ] = r X j =1 p ij µ j , i = 1 , . . . , r, (3)as we condition on the class t y of node v y .In this work, we consider the communication topologyof the WSN as consisting of independent channels that areeither on (with probability α ) or off (with probability − α ).More precisely, let { B ij ( α ) , ≤ i < j ≤ n } denotei.i.d Bernoulli rvs, each with success probability α . Thecommunication channel between two distinct nodes v x and v y is on (respectively, off) if B xy ( α ) = 1 (respectively if B xy ( α ) = 0 ). This simple on-off channel model captures theunreliability of wireless links and enables a comprehensiveanalysis of the properties of interest of the resulting WSN,e.g., its connectivity. It was also shown that on-off channel model provides a good approximation of the more realisticdisk model [15] in many similar settings and for similarproperties of interest; e.g., see [16], [17]. The on/off channelmodel induces a standard Erd˝os-R´enyi (ER) graph G ( n ; α ) [18], defined on the vertices V = { v , . . . , v n } such that v x and v y are adjacent, denoted C xy , if B xy ( α ) = 1 .We model the overall topology of a WSN by the intersectionof an inhomogeneous random key graph K ( n ; µµµ, KKK, P ) and anER graph G ( n ; α ) . Namely, nodes v x and v y are adjacent in K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) , denoted E xy , if and only if theyare adjacent in both K and G . In other words, the edgesin the intersection graph K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) representpairs of sensors that can securely communicate as they havei) a communication link in between that is on , and ii) ashared cryptographic key. Therefore, studying the connectivityproperties of K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) amounts to studying thesecure connectivity of heterogenous WSNs under the on/offchannel model.Hereafter, we denote the intersection graph K ( n ; µµµ, KKK, P ) ∩ G ( n ; α ) by the graph H ( n ; µµµ, KKK, P, α ) . To simplify the nota-tion, we let θθθ = ( KKK, P ) , and ΘΘΘ = ( θθθ, α ) . The probability ofedge existence between a class- i node v x and a class- j node v y in H ( n ; ΘΘΘ) is given by P [ E xy (cid:12)(cid:12)(cid:12) t x = i, t y = j ] = P [ K xy ∩ C xy | t x = i, t y = j ] = αp ij by independence. Similar to (3), the mean edge probability fora class- i node in H ( n ; µµµ, ΘΘΘ) as Λ i is given by Λ i = r X j =1 µ j αp ij = αλ i , i = 1 , . . . , r. (4)Throughout, we assume that the number of classes r isfixed and does not scale with n , and so are the probabilities µ , . . . , µ r . All of the remaining parameters are assumed tobe scaled with n .We close this section with some additional notation that willbe useful in the rest of the paper. For any three distinct nodes v x , v y and v j , we define E xj ∩ yj := E xj ∩ E yj , E xj ∩ yj := E xj ∩ E yj , E xj ∩ yj := E xj ∩ E yj , and E xj ∩ yj := E xj ∩ E yj .3. M AIN R ESULTS AND D ISCUSSION
A. Results
We refer to a mapping K , . . . , K r , P : N → N r +10 as a scaling (for the inhomogeneous random key graph) as long asthe conditions ≤ K ,n ≤ K ,n ≤ . . . ≤ K r,n ≤ P n / (5)are satisfied for all n = 2 , , . . . . Similarly any mapping α : N → (0 , defines a scaling for the ER graphs. As a result,a mapping ΘΘΘ : N → N r +10 × (0 , defines a scaling for theintersection graph H ( n ; µµµ, ΘΘΘ n ) given that condition (5) holds.We remark that under (5), the edge probabilities p ij will begiven by (2).We first present a zero-one law for the minimum nodedegree being no less than k in the inhomogeneous randomkey graph intersecting ER graph. Theorem 3.1.
Consider a probability distribution µµµ = { µ , . . . , µ r } with µ i > for i = 1 , . . . , r and a scaling ΘΘΘ : N → N r +10 × (0 , . Let the sequence γ : N → R bedefinedthrough Λ ( n ) = α n λ ( n ) = log n + ( k −
1) log log n + γ n n , (6)foreach n = 1 , , . . . .(a)If λ ( n ) = o (1) ,wehave lim n →∞ P " Minimumnodedegreeof H ( n ; µµµ, ΘΘΘ n ) ≥ k = 0 if lim n →∞ γ n = −∞ (b)Wehave lim n →∞ P " Minimumnodedegreeof H ( n ; µµµ, ΘΘΘ n ) ≥ k = 1 if lim n →∞ γ n = ∞ . Next, we present a zero-one law for the k -connectivity of H ( n ; µµµ, ΘΘΘ) . Theorem 3.2.
Consider a probability distribution µµµ = { µ , . . . , µ r } with µ i > for i = 1 , . . . , r and a scaling ΘΘΘ : N → N r +10 × (0 , . Let the sequence γ : N → R bedefinedthrough(6)foreach n = 1 , , . . . .(a)If λ ( n ) = o (1) ,wehave lim n →∞ P [ H ( n ; µµµ, ΘΘΘ n ) is k -connected ] = 0 if lim n →∞ γ n = −∞ (b)If P n = Ω( n ) , (7) K r,n P n = o (1) , (8) K r,n K ,n = o (log n ) , (9)wehave lim n →∞ P [ H ( n ; µµµ, ΘΘΘ n ) is k -connected ] = 1 if lim n →∞ γ n = ∞ . (10)In words, Theorem 3.1 (respectively Theorem 3.2) statesthat the minimum node degree in H ( n ; µµµ, ΘΘΘ n ) is greater thanor equal to k (respectively H ( n ; µµµ, ΘΘΘ n ) is k -connected) whpif the mean degree of class- nodes, i.e., n Λ ( n ) , is scaled as (log n + ( k −
1) log log n + γ n ) for some sequence γ n satis-fying lim n →∞ γ n = ∞ . On the other hand, if the sequence γ n satisfies lim n →∞ γ n = −∞ , then whp H ( n ; µµµ, ΘΘΘ n ) has atleast one node with degree strictly less than k , and hence is not k -connected. This shows that the critical scaling for theminimum node degree of H ( n ; µµµ, ΘΘΘ n ) being greater than orequal to k (respectively for H ( n ; µµµ, ΘΘΘ n ) to be k -connected)is given by Λ ( n ) = log n +( k −
1) log log nn , with the sequence γ n : N → R measuring the deviation of Λ ( n ) from thecritical scaling.The scaling condition (6) can be given a more explicit formunder some additional constraints. In particular, it was shownin [7, Lemma 4.2] that if λ ( n ) = o (1) then λ ( n ) ∼ K ,n K avg ,n P n (11) where K avg ,n = P rj =1 µ j K j,n denotes the mean key ring sizein the network. This shows that the minimum key ring size K ,n is of paramount importance in controlling the connec-tivity and reliability of the WSN; as explained previously, itthen also controls the number of mobile sensors that can beaccommodated in the network. For example, with the meannumber K avg ,n of keys per sensor is fixed, we see that reducing K ,n by half means that the smallest α n (that gives the largestlink failure probability − α n ) for which the network remains k -connected whp is increased by two-fold for any given k ;e.g., see Figure 3 for a numerical example demonstrating this. B. Comments on the additional technical conditions
We first comment on the additional technical condition λ ( n ) = o (1) . This is enforced here mainly for technicalreasons for the proof of the zero-law of Theorem 3.1 (and thusof Theorem 3.2) to work. A similar condition was also requiredin [19, Thm 1] for establishing the zero-law for the minimumnode degree being no less than k in the homogeneous randomkey graph intersecting ER graph. In view of (11), this conditionis equivalent to K ,n K avg ,n = o ( P n ) . (12)In real-world WSN applications the key pool size P n isenvisioned to be orders of magnitude larger than any keyring size in the network [5], [20]. As discussed below inmore details, this is needed to ensure the resilience of thenetwork against adversarial attacks. Concluding, (12) (and thus λ ( n ) = o (1) ) is indeed likely to hold in most applications.Conditions (7) and (8) are also likely to be needed inpractical WSN implementations in order to ensure the re-silience of the network against node capture attacks; e.g., see[5], [20]. To see this, assume that an adversary captures anumber of sensors, compromising all the keys that belong tothe captured nodes. If P n = O ( K r,n ) contrary to (8), then itwould be possible for the adversary to compromise a positivefraction of the key pool (i.e., Ω( P n ) keys) by capturing onlya constant number of sensors that are of type r . Similarly, if P n = o ( n ) , contrary to (7), then again it would be possible forthe adversary to compromise Ω( P n ) keys by capturing only o ( n ) sensors (whose type does not matter in this case). Inboth cases, the WSN would fail to exhibit the unassailability property [21], [22] and would be deemed as vulnerable againstadversarial attacks. We remark that both (7) and (8) wererequired in [7], [19] for obtaining the one-law for connectivityand k -connectivity, respectively, in similar settings to ours.Finally, the condition (9) is enforced mainly for technicalreasons and takes away from the flexibility of assigning verysmall key rings to a certain fraction of sensors when k -connectivity is considered; we remark that (9) is not neededfor the minimum node degree result given at Theorem 3.1. Anequivalent condition was also needed in [7] for establishingthe one-law for connectivity in inhomogeneous random keygraphs. We refer the reader to [7, Section 3.2] for an extendeddiscussion on the feasibility of (9) for real-world WSN imple-mentations, as well as possible ways to replace it with milderconditions. We close by providing a concrete example that demonstrateshow all the conditions required by Theorem 3.2 can be met ina real-world implementation. Consider any number r of sensortypes, and pick any probability distribution µµµ = { µ , . . . , µ r } with µ i > for all i = 1 , . . . , r . For any channel probability α n = Ω( log nn ) , set P n = n log n and use K ,n = (log n ) / ε √ α n and K r,n = (1 + ε )(log n ) / − ε µ r √ α n with any ε > . Other key ring sizes K ,n ≤ K ,n , . . . , K r − ,n ≤ K r,n can be picked arbitrarily. In viewof Theorem 3.2 and the fact [7, Lemma 4.2] that λ ( n ) ∼ K ,n K avg ,n P n , the resulting network will be k -connected whp forany k = 1 , , . . . . Of course, there are many other parameterscalings that one can choose. C. Comparison with related work
In comparison with the existing literature on similar models,our result can be seen to extend the work by Zhao et al.[19] on the homogeneous random key graph intersecting ERgraph to the heterogeneous setting. There, zero-one laws forthe property that the minimum node degree is no less than k and the property that the graph is k -connected were establishedfor H ( n, K, P, α n ) . With r = 1 , i.e., when all nodes belong tothe same class and thus receive the same number K of keys,Theorem 3.1 and Theorem 3.2 recover the result of Zhao etal. (See [19, Theorems 1-2]).Our paper also extends the work by Ya˘gan [7] who con-sidered the inhomogeneous random key graph K ( n, µµµ, KKK, P ) under full visibility; i.e., when all pairs of nodes have acommunication channel in between. There, Ya˘gan establishedzero-one laws for the absence of isolated nodes (i.e., absenceof nodes with degree zero) and -connectivity. Our workgeneralizes Ya˘gan’s results on two fronts. Firstly, we con-sider more practical WSN scenarios where the unreliabilityof wireless communication channels are taken into accountthrough the on/off channel model. Secondly, in addition tothe properties that the graph has no isolated nodes (i.e., theminimum node degree is no less than ) and is -connected,we consider general minimum node degree and connectivityvalues, k = 0 , , . . . .Finally, our work (with α n = 1 for each n = 2 , , . . . )improves upon the results by Zhao et al. [12]; therein, thismodel was referred to as the general random intersectiongraph. Our main argument is that the additional conditionsrequired by their main result renders them inapplicable inpractical WSN implementations. This issue is discussed atlength in [7, Section 3.3], but we give a summary herefor completeness. With X n denoting the random variablerepresenting the number of keys assigned to an arbitrary nodein the network, the main result in [12] requiresvar [ X n ] = o ( E [ X n ]) n (log n ) ! (13)that puts a prohibitively stringent limit on the variance of thekey ring sizes. For instance, it precludes using K ,n = cK ,n for some c > , and forces key ring sizes to be asymptotically equivalent; i.e., K r,n ∼ K ,n . In fact, under (13), even thesimplest case where key ring sizes vary by a constant ispossible only when E [ X n ] = ω ( √ n log n ) . Put differently,the results in [12] are useful only if the mean number of keysassigned to a sensor node is much larger than √ n log n ; andeven then only small variations among key ring sizes would bepossible. However, in most WSN applications, sensor nodeswill have very limited memory and computational capabilities[1] and such large key ring sizes are not likely to be feasible;typically key rings on the order of log n are envisioned inapplications [5], [20]. These arguments show that conditionsenforced in [12] are not likely to hold in practice. In contrast,our results allow for much larger variation in key ring sizes andrequire parameter conditions that are likely to hold in practice;e.g., we only need E [ X n ] = o ( P n ) .4. N UMERICAL R ESULTS
We now present numerical results to support Theorems 3.1and 3.2 in the finite node regime. In all experiments, we fixthe number of nodes at n = 500 and the size of the key poolat P = 10 . To help better visualize the results, we use thecurve fitting tool of MATLAB.In Figure 1, we consider the channel parameters α = 0 . , α = 0 . , α = 0 . , and α = 0 . , while varying the parameter K , i.e., the smallest key ring size, from to . The numberof classes is fixed to , with µµµ = { . , . } . For each value of K , we set K = K + 10 . For each parameter pair ( KKK, α ) ,we generate independent samples of the graph H ( n ; µµµ, ΘΘΘ) and count the number of times (out of a possible 200) that theobtained graphs i) have minimum node degree no less than and ii) are -connected. Dividing the counts by , weobtain the (empirical) probabilities for the events of interest.In all cases considered here, we observe that H ( n ; µµµ, ΘΘΘ) is -connected whenever it has minimum node degree no lessthan yielding the same empirical probability for both events.This supports the fact that the properties of k -connectivity andminimum node degree being larger than k are asymptoticallyequivalent in H ( n ; µµµ, ΘΘΘ n ) .In Figure 1 as well as the ones that follow we show thecritical threshold of connectivity “predicted” by Theorem 3.2by a vertical dashed line. More specifically, the vertical dashedlines stand for the minimum integer value of K that satisfies λ ( n )= X j =1 µ j − (cid:0) P − K j K (cid:1)(cid:0) PK (cid:1) ! > α log n + ( k −
1) log log nn (14)with any given k and α . We see from Figure 1 that theprobability of k -connectivity transitions from zero to onewithin relatively small variations in K . Moreover, the criticalvalues of K obtained by (14) lie within the transition interval.In Figure 2, we consider four different values for k , namelywe set k = 4 , k = 6 , k = 8 , and k = 10 while varying K from to and fixing α to . . The number of classesis fixed to with µµµ = { . , . } and we set K = K + 10 for each value of K . Using the same procedure that producedFigure 1, we obtain the empirical probability that H ( n ; µµµ, θθθ, α ) is k -connected versus K . The critical threshold of connectiv-ity asserted by Theorem 3.2 is shown by a vertical dashed K P r o b a b ili t y o f - c o nn ec t i v i t y α = 0 . α = 0 . α = 0 . α = 0 . Fig. 1. Empirical probability that H ( n ; µµµ, θθθ, α ) is -connected as a functionof KKK for α = 0 . , α = 0 . , α = 0 . , α = 0 . with n = 500 and P =10 ; in each case, the empirical probability value is obtained by averagingover experiments. Vertical dashed lines stand for the critical threshold ofconnectivity asserted by Theorem 3.2. K
15 20 25 30 35 40 P r o b a b ili t y o f k - c o nn ec t i v i t y α = 0 . Fig. 2. Empirical probability that H ( n ; µµµ, θθθ, α ) is k -connected as a functionof K for k = 4 , k = 6 , k = 8 , and k = 10 , with n = 500 and P =10 ; in each case, the empirical probability value is obtained by averagingover experiments. Vertical dashed lines stand for the critical threshold ofconnectivity asserted by Theorem 3.2. line in each curve. Again, we see that numerical results are inparallel with Theorem 3.2.Figure 3 is generated in a similar manner with Figure 1,this time with an eye towards understanding the impact of theminimum key ring size K on network connectivity. To thatend, we fix the number of classes at with µµµ = { . , . } andconsider four different key ring sizes KKK each with mean ;we consider KKK = { , } , KKK = { , } , KKK = { , } , and KKK = { , } . We compare the probability of -connectivityin the resulting networks while varying α from zero to one.We see that although the average number of keys per sensor iskept constant in all four cases, network connectivity improvesdramatically as the minimum key ring size K increases; e.g.,with α = 0 . , the probability of connectivity is one when K = K = 40 while it drops to zero if we set K = 10 while increasing K to so that the mean key ring size isstill 40.Finally, we examine the reliability of H ( n ; µµµ, θθθ, α ) by α P r o b a b ili t y o f - C o nn ec t i v i t y K = 10 , K = 70 K = 20 , K = 60 K = 30 , K = 50 K = 40 , K = 40 Fig. 3. Empirical probability that H ( n ; µµµ, θθθ, α ) is -connected with n =500 , µµµ = (1 / , / , and P = 10 ; we consider four choices of KKK =( K , K ) each with the same mean. looking at the probability of 1-connectivity as the numberof deleted (i.e., failed) nodes increases. From a mobilityperspective, this is equivalent to investigating the probabilityof a WSN remaining connected as the number of mobile sensors leaving the network increases. In Figure 4, we set n = 500 , µµµ = { / , / } , α = 0 . , P = 10 , and select K and K = K + 10 from (14) for k = 8 , k = 10 , k = 12 , and k = 14 . With these settings, we would expect (for very large n ) the network to remain connected whp after the deletion ofup to 7, 9, 11, and 13 nodes, respectively. Using the sameprocedure that produced Figure 1, we obtain the empiricalprobability that H ( n ; µµµ, θθθ, α ) is connected as a function of thenumber of deleted nodes in each case. We see that even with n = 500 nodes, the resulting reliability is close to the levelsexpected to be attained asymptotically as n goes to infinity. Inparticular, we see that the probability of remaining connectedwhen ( k − nodes leave the network is around . for thefirst two cases and around . for the other two cases.5. P RELIMINARIES
A number of technical results are collected here for easyreferencing.
Proposition 5.1 ([7, Proposition 4.1]) . For any scaling K , K , . . . , K r , P : N → N r +10 , we have λ ( n ) ≤ λ ( n ) ≤ . . . ≤ λ r ( n ) , n = 2 , , . . . . (15)In view of (4), Proposition 5.1 implies that Λ ( n ) ≤ Λ ( n ) ≤ . . . ≤ Λ r ( n ) , n = 2 , , . . . . (16) Proposition 5.2.
Consider a scaling K , K , . . . , K r , P : N → N r +10 and a scaling α : N → (0 , . Let the sequence We choose the nodes to be deleted from the minimum vertex cut of H ,defined as the minimum cardinality set whose removal renders it disconnected.This captures the worst-case nature of the k -connectivity property in acomputationally efficient manner (as compared to searching over all k -sizedsubsets and deleting the one that gives maximum damage). Number of Deleted Nodes P r o b a b ili t y o f C o nn ec t i v i t y K = 30 K = 33 K = 36 K = 38 Fig. 4. Empirical probability that H ( n ; µµµ, θθθ, α ) remains connected afterdeleting nodes from the minimum vertex cut set. We fix n = 500 , µµµ =(1 / , / , α = 0 . , P = 10 , and choose K and K = K + 10 from(14) for each k = 8 , k = 10 , k = 12 , and k = 14 ; i.e., we use K =30 , , , , respectively. γ : N → R be defined through (6) for each n = 1 , , . . . .Under (7) and (9), we have K ,n = ω (1) (17) when lim n →∞ γ n = + ∞ .Proof: From (6), we clearly have λ ( n ) > log nnα n (18)for all n sufficiently large when lim n →∞ γ n = + ∞ . We alsoknow from [23, Lemmas 7.1-7.2] that p j ( n ) ≤ K ,n K j,n P n − K j,n ≤ K ,n K j,n P n , j = 1 , . . . , r where the last bound follows from (5). This leads to λ ( n ) = r X j =1 µ j p j ≤ r X j =1 µ j K ,n K j,n P n ≤ K ,n K r,n P n (19)Combining (18) and (19) we get K ,n K r,n K ,n > P n nnα n for all n sufficiently large. Under (7) and (9), this immediatelyestablishes (17) since α n ≤ . Fact 5.3.
For any positive constants ℓ , ℓ , the function f ( x ) = x ℓ (1 − x ) n − ℓ , x ∈ (0 , (20) is monotone decreasing in x for all n sufficiently large.Proof: Differentiating f ( x ) with respect to x ∈ (0 , , weget ddx f ( x ) = ℓ x ℓ − (1 − x ) n − ℓ − ( n − ℓ ) x ℓ (1 − x ) n − ℓ − = x ℓ − (1 − x ) n − ℓ − ( ℓ (1 − x ) − ( n − ℓ ) x ) . The conclusion follows since ( ℓ (1 − x ) − ( n − ℓ ) x ) < forall n sufficiently large, for any positive ℓ , ℓ and x ∈ (0 , . Fact 5.4 ([19, Lemma 8]) . Given (5), If p i ( n ) = o (1) or K ,n K i,n P n = o (1) , then K ,n K i,n P n = p i ( n ) ± O (cid:16) ( p i ( n )) (cid:17) Fact 5.5 ([19, Fact 3]) . Let x and y be both positive functionsof n . If x = o (1) , and x y = o (1) hold, then (1 − x ) y ∼ e − xy Lemma 5.6 ([23, Lemma 7.1]) . For positive integers K , L ,and P such that K + L ≤ P , we have (cid:18) − LP − K (cid:19) K ≤ (cid:0) P − LK (cid:1)(cid:0) PK (cid:1) ≤ (cid:18) − LP (cid:19) K We will use several bounds given below throughout thepaper: (1 ± x ) ≤ e ± x , x ∈ (0 , (21) ( x + y ) p ≤ p − ( x p + y p ) (22) (cid:18) nℓ (cid:19) ≤ (cid:16) enℓ (cid:17) ℓ , ℓ = 1 , . . . , n, n = 1 , , . . . (23) ⌊ n ⌋ X ℓ =2 (cid:18) nℓ (cid:19) ≤ n (24) (cid:18) nℓ (cid:19) ≤ n ℓ , ℓ = 1 , . . . , n, n = 1 , , . . . (25)6. P ROOF OF T HEOREM
A. Establishing the one-law
The proof of Theorem 3.1 relies on the method of first andsecond moments applied to the number of nodes with degree ℓ in H ( n ; µµµ, ΘΘΘ n ) . Let X ℓ ( n ; µµµ, ΘΘΘ n ) denote the total numberof nodes with degree ℓ in H ( n ; µµµ, ΘΘΘ n ) , namely, X ℓ ( n ; µµµ, ΘΘΘ n ) = n X i =1
111 [ v i is of degree ℓ in H ( n ; µµµ, ΘΘΘ n )] The method of first moment [24, Eqn. (3.10), p. 55] gives P [ X ℓ ( n ; µµµ, ΘΘΘ n ) = 0] ≥ − E [ X ℓ ( n ; µµµ, ΘΘΘ n )] (26)The one-law states that the minimum node degree in H ( n ; µµµ, ΘΘΘ n ) is no less than k asymptotically almost surely(a.a.s.); i.e., lim n →∞ P [ X ℓ ( n ; µµµ, ΘΘΘ n ) = 0] = 1 , for all ℓ =0 , , . . . , k − . Thus, the one-law will follow if we show that lim n →∞ E [ X ℓ ( n ; µµµ, ΘΘΘ n )] = 0 , ℓ = 0 , , . . . , k − . (27)We let D i,ℓ ( n ; µµµ, ΘΘΘ n ) denote the event that node v i in H ( n ; µµµ, ΘΘΘ n ) has degree ℓ for each i = 1 , , . . . , n .Throughout, we simplify the notation by writing D i,ℓ insteadof D i,ℓ ( n ; µµµ, ΘΘΘ n ) . By definition, we have X ℓ ( n ; µµµ, ΘΘΘ n ) = P ni =1
111 [ D i,ℓ ] and it follows that E [ X ℓ ( n ; µµµ, ΘΘΘ n )] = n X i =1 P [ D i,ℓ ] = n P [ D x,ℓ ] (28)by the exchangeability of the indicator rvs {
111 [ D i,ℓ ] ; i = 1 , . . . , n } . In view of (26) and (28), we see that (27) and hence theone-law would follow upon showing lim n →∞ n P [ D x,ℓ ] = 0 , ℓ = 0 , , . . . , k − . (29)We start by deriving the probability of D x,ℓ . For any node v x , the events E x , E x , . . . , E ( x − x , E ( x +1) x , . . . , E nx aremutually independent conditionally on the type t x . It followsfrom (4) that the degree of a node v x , i.e., D x , is conditionallybinomial leading to D x = st Bin ( n − , Λ i ) , with probability µ i , i = 1 , . . . , r Thus, we get P [ D x,ℓ ] = r X i =1 µ i P [ D x,ℓ | t x = i ]= r X i =1 µ i (cid:18) n − ℓ (cid:19) (Λ i ( n )) ℓ (1 − Λ i ( n )) n − ℓ − ≤ r X i =1 µ i ( n Λ i ( n )) ℓ (1 − Λ i ( n )) n − ℓ − ! ≤ ( ℓ ! ) − ( n Λ ( n )) ℓ (1 − Λ ( n )) n − ℓ − ≤ ( ℓ ! ) − ( n Λ ( n )) ℓ e − ( n − ℓ − ( n ) for all n sufficiently large, as we invoke Fact 5.3 togetherwith (16), and note that ℓ is a non-negative integer constant.Combining (6) and (22), and using the fact that Λ ( n ) ≤ ,we see that n P [ D x,ℓ ] ≤ n ( ℓ ! ) − (log n + ( k −
1) log log n + γ n ) ℓ ·· e − log n − ( k −
1) log log n − γ n e ( ℓ +1)Λ ( n ) ≤ ℓ − (cid:16) (log n ) ℓ (1 + o (1)) ℓ + γ ℓn (cid:17) e − ( k −
1) log log n − γ n e O (1) = O (1) e − ( k − − ℓ ) log log n − γ n + O (1) γ ℓn e − ( k −
1) log log n − γ n . When lim n →∞ γ n = ∞ , we readily get the desired conclusion(29). This establishes the one-law. B. Establishing the zero-law
Our approach in establishing the zero-law relies on themethod of second moment applied to a variable that countsthe number of nodes in H ( n ; µµµ, ΘΘΘ n ) that are class- andwith degree ℓ . Similar to the discussion given before, we let Y ℓ ( n ; µµµ, ΘΘΘ n ) denote the total number of nodes that are class- and with degree ℓ in H ( n ; µµµ, ΘΘΘ n ) , namely, Y ℓ ( n ; µµµ, ΘΘΘ n ) (30) = n X i =1
111 [ v i is class and has degree ℓ in H ( n ; µµµ, ΘΘΘ n )] Clearly, if we can show that whp there exists at least one class- node with a degree strictly less than k under the enforcedassumptions (with lim n →∞ γ n = −∞ ) then the zero-lawimmediately follows.With a slight abuse of notations, we let D i,ℓ ( n ; µµµ, ΘΘΘ n ) denote the event that node v i in H ( n ; µµµ, ΘΘΘ n ) is class- andhas degree ℓ for each i = 1 , , . . . , n . Throughout, we simplify the notation by writing D i,ℓ instead of D i,ℓ ( n ; µµµ, ΘΘΘ n ) . Thus,we have Y ℓ ( n ; µµµ, ΘΘΘ n ) = P ni =1
111 [ D i,ℓ ] . The method of secondmoments [24, Remark 3.1, p. 55] gives P [ Y ℓ ( n ; µµµ, ΘΘΘ n ) = 0] ≤ − E [ Y ℓ ( n ; µµµ, ΘΘΘ n )] E [ Y ℓ ( n ; µµµ, ΘΘΘ n ) ] . (31)We have E [ Y ℓ ( n ; µµµ, ΘΘΘ n )] = n P [ D x,ℓ ] and E (cid:2) Y ℓ ( n ; µµµ, ΘΘΘ n ) (cid:3) = n P [ D x,ℓ ] + n ( n − P [ D x,ℓ ∩ D y,ℓ ] , whence E (cid:2) Y ℓ ( n ; µµµ, ΘΘΘ n ) (cid:3) E [ Y ℓ ( n ; µµµ, ΘΘΘ n )] = 1 n P [ D x,ℓ ] + n − n P [ D x,ℓ ∩ D y,ℓ ]( P [ D x,ℓ ]) . (32)From (31) and (32), we see that the zero-law will follow ifwe show that lim n →∞ n P [ D x,ℓ ] = ∞ , (33)and P [ D x,ℓ ∩ D y,ℓ ] ∼ ( P [ D x,ℓ ]) (34)for some ℓ = 0 , , . . . , k − under the enforced assumptions.The next two results will help establish (33) and (34). Lemma 6.1. If Λ ( n ) = o (cid:16) √ n (cid:17) , then for any non-negativeinteger constant ℓ and any node v x , we have P [ D x,ℓ ] ∼ µ ( ℓ ! ) − ( n Λ ( n )) ℓ e − n Λ ( n ) (35) Proof:
Considering any class- node v i ,and recalling (4), we know that the events E i , E i , . . . , E ( i − i , E ( i +1) i , . . . , E ni are mutuallyindependent. Thus, it follows that the degree of a givennode v i , conditioned on being class- , follows a Binomialdistribution Bin ( n − , Λ ( n )) . Thus, P [ D i,ℓ ] = µ P [ D i,ℓ | t i = 1]= µ (cid:18) n − ℓ (cid:19) Λ ( n ) ℓ (1 − Λ ( n )) n − ℓ − Next, given that Λ ( n ) = o (cid:16) √ n (cid:17) and ℓ is constant, itfollows that Λ ( n ) = o (1) and Λ ( n ) ( n − ℓ −
1) = o (1) .Invoking Fact 5.5, and the fact that (cid:0) n − ℓ (cid:1) ∼ ( ℓ ! ) − n ℓ , theconclusion (35) follows. Lemma 6.2.
Consider scalings K , . . . , K r , P : N → N r +10 and α : N → (0 , , such that λ ( n ) = o (1) and (6) holdswith lim n →∞ γ n = −∞ . The following two properties hold(a) If n Λ ( n ) = Ω(1) , then for any non-negative integerconstant ℓ and any two distinct nodes v x and v y , we have P [ D x,ℓ ∩ D y,ℓ ] ∼ µ ( ℓ ! ) − ( n Λ ( n )) ℓ e − n Λ ( n ) (36) (b) For any two distinct nodes v x and v y , we have P [ D x, ∩ D y, ] ∼ µ e − n Λ ( n ) (37)The proof of Lemma 6.2 is given in Appendix B. Wenow show why the zero-law follows from Lemma 6.1 andLemma 6.2 by means of establishing (33) and (34) for some ℓ = 0 , , . . . , k − . First, we see from (6) that Λ ( n ) ≤ log n +( k −
1) log log nn = o (cid:16) √ n (cid:17) when lim n →∞ γ n = −∞ .Invoking Lemma 6.1, this gives n P [ D x,ℓ ] ∼ nµ ( ℓ ! ) − ( n Λ ( n )) ℓ e − n Λ ( n ) (38)for each ℓ = 0 , , . . . . We will obtain (33) and (34) usingsubsubsequence principle [24, p. 12] and considering the caseswhere n Λ ( n ) = Ω(1) and n Λ ( n ) = o (1) separately.
1) The case where there exists an ǫ > such that n Λ ( n ) >ǫ for all n sufficiently large: In this case we will establish (33)and (34) for ℓ = k − . Setting ℓ = k − and substituting (6)into (38), we get n P [ D x,ℓ ] ∼ nµ [( k −
1) ! ] − ( n Λ ( n )) k − e − log n − ( k −
1) log log n − γ n = µ [( k −
1) ! ] − (log n + ( k −
1) log log n + γ n ) k − ·· e − ( k −
1) log log n − γ n (39)Let f n ( k ; γ n ):= (log n + ( k −
1) log log n + γ n ) k − e − ( k −
1) log log n − γ n , and note that (log n + ( k −
1) log log n + γ n ) ≥ ǫ for all n sufficiently large by virtue of the fact that n Λ ( n ) > ǫ . Fix n sufficiently large, pick ζ ∈ (0 , and consider the cases when γ n ≤ − (1 − ζ ) log n and γ n > − (1 − ζ ) log n , separately. Inthe former case, we get f n ( k ; γ n ) ≥ ǫe − ( k −
1) log log n +(1 − ζ ) log n , and in the latter case, we get f n ( k ; γ n ) ≥ ( ζ log n ) k − e − ( k −
1) log log n − γ n = ζ k − e − γ n . Thus, for all n sufficiently large, we have f n ( k ; γ n ) ≥ min n ǫe − ( k −
1) log log n +(1 − ζ ) log n , ζ k − e − γ n o . It is now clear that lim n →∞ f n ( k ; γ n ) = ∞ , (40)since ζ ∈ (0 , and lim n →∞ γ n = −∞ . Reporting (40) into(39), we establish (33). Furthermore, from Lemma 6.1 andLemma 6.2, it is clear that (34) follows for ℓ = k − .
2) The case where lim n →∞ n Λ ( n ) = 0 : In this case, wewill establish (33) and (34) for ℓ = 0 . Setting ℓ = 0 in (38),we obtain n P [ D x, ] ∼ nµ e n Λ ( n ) ∼ nµ by virtue of the fact that n Λ ( n ) = o (1) . This readilygives (33). Furthermore, from Lemma 6.1 (with ℓ = 0 ) andLemma 6.2, (34) immediately follows.The two cases considered cover all the possibilities for thelimit of n Λ ( n ) . By virtue of the subsubsequence principle[24, p. 12], we get (33) and (34) without any condition on thesequence n Λ ( n ) ; i.e., we obtain the zero-law even when thesequence n Λ ( n ) does not have a limit!
7. P
ROOF OF T HEOREM
A. Establishing the zero-law
Let κ denote the the vertex connectivity of H ( n, µµµ, ΘΘΘ n ) ,i.e., the minimum number of nodes to be deleted to makethe graph disconnected. Also, let δ denote the minimum nodedegree in H ( n, µµµ, ΘΘΘ n ) . It is clear that if a random graph is k -connected, meaning that κ ≥ k , then it does not have anynode with degree less than k . Thus [ κ ≥ k ] ⊆ [ δ ≥ k ] and theconclusion P [ κ ≥ k ] ≤ P [ δ ≥ k ] (41)immediately follows. In view of (41), we obtain the zero-lawfor k -connectivity, i.e., that lim n →∞ P [ H ( n ; µµµ, ΘΘΘ n ) is k -connected ] = 0 , when lim n →∞ γ n = −∞ from the zero-law part of Theo-rem 3.1. Put differently, the conditions that lead to the zero-law part of Theorem 3.1, i.e., λ ( n ) = o (1) and lim n →∞ γ n = −∞ , automatically lead to the zero-law part of Theorem 3.2. B. Establishing the one-law
An important step towards establishing the one-law ofTheorem 3.2 is presented in Appendix C. There, we showthat it suffices to establish the one law in Theorem 3.2under the additional condition that γ n = o (log n ) , whichleads to a number of useful consequences. Let a sequence β ℓ,n : N × N → R be defined through the relation Λ ( n ) = log n + ℓ log log n + β ℓ,n n (42)for each n ∈ N and ℓ ∈ N . In view of the arguments inAppendix C, the one-law (10) follows from the next result. Theorem 7.1.
Let ℓ be a non-negative constant integer.Under (7), (8), (9), and (42) with β ℓ,n = o (log n ) and lim n →∞ β ℓ,n = + ∞ , we have lim n →∞ P [ κ = ℓ ] = 0 . Before we give a formal proof, we first explain why theone-law (10) follows from Theorem 7.1. Comparing (42) with(6) and noting that γ n = o (log n ) , we get β ℓ,n = ( k − − ℓ ) log log n + γ n = o (log n ) (43)Moreover, for ℓ = 0 , , . . . , k − , we have lim n →∞ β ℓ,n = + ∞ (44)by recalling the fact that lim n →∞ γ n = + ∞ . Recalling (43)and (44), we notice that the conditions needed for Theo-rem 7.1 are met when ℓ = 0 , , . . . , k − ; thus, we have P [ κ = ℓ ] = o (1) for ℓ = 0 , , . . . , k − , which in turn impliesthat lim n →∞ P [ κ ≥ k ] = 1 , i.e., the one-law.We now give a road map to the proof of Theorem 7.1. Bya simple union bound, we get P [ κ = ℓ ] ≤ P [ δ ≤ ℓ ] + P [( κ = ℓ ) ∩ ( δ > ℓ )] . It is now immediate that Theorem 7.1 is established once weshow that lim n →∞ P [ δ ≤ ℓ ] = 0 (45)and lim n →∞ P [( κ = ℓ ) ∩ ( δ > ℓ )] = 0 (46)under the enforced assumptions of Theorem 7.1. We start byestablishing (45). Following the analysis of Section 6-A, it iseasy to see that n P [ D x,ℓ ] ≤ ℓ − (cid:16) (log n ) ℓ (1 + o (1)) ℓ + β ℓℓ,n (cid:17) ·· e − ℓ log log n − β ℓ,n e O (1) = O (1) e − β ℓ,n + O (1) β ℓℓ,n e − ℓ log log n − β ℓ,n , and it follows that lim n →∞ n P [ D x,ℓ ] = 0 as long as lim n →∞ β ℓ,n = + ∞ . From (26) and (28), this yields lim n →∞ P [ δ = ℓ ] = 0 when lim n →∞ β ℓ,n = + ∞ (47)However, from (42) it is easy to see that β ℓ,n is monotonicallydecreasing in ℓ . Thus, the fact that lim n →∞ β ℓ,n = + ∞ forsome ℓ implies lim n →∞ β ˆ ℓ,n = + ∞ , ˆ ℓ = 0 , , . . . , ℓ From (47) this in turn implies that P [ δ = ˆ ℓ ] = o (1) for ˆ ℓ =0 , , . . . , ℓ , or equivalently (45).We now focus on establishing (46) under the enforcedassumptions of Theorem 7.1. The proof is based on finding atight upper bound on the probability P [( κ = ℓ ) ∩ δ > ℓ ] andshowing that this bound goes to zero as n goes to infinity.Let N denote the collection of all non-empty subsets of { v , v , . . . , v n } . Define N ∗ = { T : T ∈ N , | T |≥ } and E ( JJJ ) = ∪ T ∈N ∗ (cid:2) |∪ v i ∈ T Σ i |≤ J | T | (cid:3) where JJJ = [ J , J , . . . , J n ] is an ( n − -dimensional integer-valued array. E ( JJJ ) encodes the event that for at least one | T | =2 , . . . , n , the total number of distinct keys held by at least oneset of | T | sensors is less than or equal to J | T | . Now, define m n := min (cid:18)(cid:22) P n K ,n (cid:23) , j n k(cid:19) (48)and let J i = ( max ( ⌊ (1 + ǫ ) K ,n ⌋ , ⌊ iζK ,n ⌋ ) i = 2 , . . . , m n ⌊ ψP n ⌋ i = m n + 1 , . . . , n (49)for some ζ, ψ in (0 , to be specified later at (50) and (51),respectively. A crude bounding argument gives P [( κ = ℓ ) ∩ δ > ℓ ] ≤ P [ E ( JJJ )] + P h ( κ = ℓ ) ∩ δ > ℓ ∩ E ( JJJ ) i Hence, establishing (46) consists of establishing the follow-ing two results.
Proposition 7.2.
Let ℓ be a non-negative constant integer.Assume that (42) holds with β ℓ,n > , and that we have (8)and (9). Also, assume that (7) holds such that P n ≥ σn for some σ > for all n sufficiently large. Then lim n →∞ P [ E ( JJJ )] = 0 , where JJJ is as defined in (49) with arbitrary ǫ ∈ (0 , , constant ζ ∈ (0 , ) selected small enough such that max ζσ, ζ (cid:18) e σ (cid:19) ζ − ζ ! < (50) and ψ ∈ (0 , ) selected small enough such that max p ψ (cid:18) eψ (cid:19) ψ ! σ , p ψ (cid:18) eψ (cid:19) ψ ! < (51) Proof:
The proof follows the same steps with [7, Propo-sition 7.2] to show that it suffices to establish Proposition 7.2for the homogenous case where all key rings are of the samesize K ,n . This is evident upon realizing that with U ℓ ( µµµ, θθθ ) = |∪ ℓi =1 Σ i | and U ℓ ( K ,n , P n ) = st U ℓ ( µµµ = { , , . . . , } , θθθ ) , wehave U ℓ ( K ,n , P n ) (cid:22) U ℓ ( µµµ, θθθ ) , where (cid:22) denotes the usual stochastic ordering. After thisreduction, the proof reduces to [19, Proposition 3]. Resultsonly require conditions (7), (17), and K ,n = o ( P n ) to hold.We note that K ,n = o ( P n ) follows from (8) and the factthat K ,n ≤ K r,n . Also, (17) follows under the enforcedassumptions as shown in Proposition 5.2. Proposition 7.3.
Let ℓ be a non-negative constant integer.Under (7), (8), (9), and (42) with β ℓ,n = o (log n ) and lim n →∞ β ℓ,n = + ∞ , we have lim n →∞ P h ( κ = ℓ ) ∩ ( δ > ℓ ) ∩ E ( JJJ ) i = 0 The proof of Proposition 7.3 is given in Section 8. Proposi-tion 7.2 and Proposition 7.3 establish (46) which, combinedwith (45), establish Theorem 7.1. We remark that Theorem 7.1establishes the one-law.8. P
ROOF OF P ROPOSITION H ( n ; µµµ, KKK, P, α ) by H .Let H ( U ) be a subgraph of H restricted to the vertex set U .For any subset of nodes U , define U c := { v , . . . , v n } \ U .We also let N U c denote the collection of all non-empty subsetsof { v , v , . . . , v n } \ U . We note that a subset T of N U c isisolated in H ( U c ) if there are no edges in H between nodesin T and nodes in U c \ T , i.e., E ij , v i ∈ T, v j ∈ U c \ T. Next, we present key observations that pave the way toestablishing Proposition 7.3. If κ = ℓ but δ > ℓ , then thereexists subsets U and T of nodes with U ∈ N , | U | = ℓ , T ∈ N U c , | T |≥ such that H ( T ) is connected while T isisolated in H ( U c ) . This ensures that H can be disconnectedby deleting a properly selected set of ℓ nodes, i.e., the set U .This would not be possible for sets T ∈ N U c with | T | = 1 since we have δ ≥ ℓ + 1 which implies that the single node in T is connected to at least one node in U c \ T . Finally, having κ = ℓ ensures that H remains connected after removing ( ℓ − nodes. Then, if there exists a subset U with | U | = ℓ such thatsome T ∈ N U c is isolated in H ( U c ) , each node in U must beconnected to at least one node in T and at least one node in U c \ T . This can be proved by contradiction. Consider subsets U ∈ N with | U | = ℓ , and T ∈ N U c with | T |≥ , such that T is isolated from U c \ T . Suppose there exists a node v i ∈ U such that v i is connected to at least one node in T but notconnected to any node in U c \ T . In this case, it is easy to seethat there are no edges between nodes in U c \ T and nodes in { v i } ∪ T . Thus, the graph could have been made disconnectedby removing nodes in U \ { v i } . But | U \ { v i }| = ℓ − , andthis contradicts the fact that κ = ℓ .We now present several events that characterize the afore-mentioned observations. For each non-empty subset T ⊆ U c ,we define C T as the event that H ( T ) is itself connected, and D U,T as the event that T is isolated in H ( U c ) , i.e., D U,T := \ v i ∈ Tv j ∈ U c \ T E ij , Moreover, we define B U,T as the event that each node in U has an edge with at least one node in T , i.e., B U,T := \ v i ∈ U [ v j ∈ T E ij , and finally, we let A U,T := B U,T ∩ D
U,T ∩ C T . It is clear that A U,T encodes the event that H ( T ) is itself connected, eachnode in U has an edge with at least one node in T , but T isisolated in H ( U c ) . The aforementioned observations enable usto express the event [( κ = ℓ ) ∩ ( δ > ℓ )] in terms of the eventsequence A U,T . In particular, we have [( κ = ℓ ) ∩ ( δ > ℓ )] ⊆ [ U ∈N n,ℓ ,T ∈N Uc , | T |≥ A U,T with N n,ℓ denoting the collection of all subsets of { v , . . . , v n } with exactly ℓ elements. We also note that the union need onlyto be taken over all subsets T with ≤ | T |≤ (cid:4) n − ℓ (cid:5) . This isbecause if the vertices in T form a component then so do thevertices in N U c \ T . Now, using a standard union bound, weobtain P h ( κ = ℓ ) ∩ ( δ > ℓ ) ∩ E ( JJJ ) i ≤ X U ∈N n,ℓ ,T ∈N Uc , ≤| T |≤ ⌊ n − ℓ ⌋ P h A U,T ∩ E ( JJJ ) i = ⌊ n − ℓ ⌋ X m =2 X U ∈N n,ℓ ,T ∈N Uc,m P h A U,T ∩ E ( JJJ ) i where N U c ,m denotes the collection of all subsets of U c with exactly m elements. Now, for each m = 1 , . . . , n − ℓ − , we simplify the notation by writing A ℓ,m := A { v ,...,v ℓ } , { v ℓ +1 ,...,v ℓ + m } , D ℓ,m := D { v ,...,v ℓ } , { v ℓ +1 ,...,v ℓ + m } , B ℓ,m := B { v ,...,v ℓ } , { v ℓ +1 ,...,v ℓ + m } , and C m := C { v ℓ +1 ,...,v ℓ + m } .From exchangeability, we get P [ A U,T ] = P [ A ℓ,m ] , U ∈ N n,ℓ , T ∈ N U c ,m and the key bound P h ( κ = ℓ ) ∩ ( δ > ℓ ) ∩ E ( JJJ ) i ≤ ⌊ n − ℓ ⌋ X m =2 (cid:18) nℓ (cid:19)(cid:18) n − ℓm (cid:19) P h A ℓ,m ∩ E ( JJJ ) i (52)is obtained readily upon noting that |N n,ℓ | = (cid:0) nℓ (cid:1) and |N U c ,m | = (cid:0) n − ℓm (cid:1) . Thus, Proposition 7.3 will be establishedif we show that lim n →∞ ⌊ n − ℓ ⌋ X m =2 (cid:18) nℓ (cid:19)(cid:18) n − ℓm (cid:19) P h A ℓ,m ∩ E ( JJJ ) i = 0 . (53)We now derive bounds for the probabilities P h A ℓ,m ∩ E ( JJJ ) i . First, for m = 2 , . . . , n − ℓ − , wehave D ℓ,m := n \ j = m + ℓ +1 (cid:2)(cid:0) ∪ i ∈ ν m,j Σ i (cid:1) ∩ Σ j = ∅ (cid:3) (54)where ν m,j is defined as ν m,j := { i = ℓ + 1 , . . . , ℓ + m : C ij } for each j = 1 , . . . , ℓ and j = m + ℓ +1 , . . . , n . Put differently, ν m,j is the set of indices in i = ℓ + 1 , . . . , ℓ + m for whichnodes v j and v i are adjacent in the ER graph G ( n ; α n ) . Then,(54) follows from the fact that for v j to be isolated from { v ℓ +1 , . . . , v ℓ + m } in H , Σ j needs to be disjoint from eachof the key rings { Σ i : i ∈ ν m,j } .Now, using the law of iterated expectation, we get P h D ℓ,m (cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m i = E h
111 [ D ℓ,m ] (cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m i = E " E (cid:20)
111 [ D ℓ,m ] (cid:12)(cid:12)(cid:12) Σ ℓ +1 ,..., Σ n C ij ,i = ℓ +1 ,...,ℓ + mj = ℓ + m +1 ,...,n (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m = E n Y j = ℓ + m +1 (cid:0) P −|∪ i ∈ νm,j Σ i || Σ j | (cid:1)(cid:0) P | Σ j | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m = E (cid:0) P −|∪ i ∈ νm Σ i || Σ | (cid:1)(cid:0) P | Σ | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m n − ℓ − m (55)by independence of the random variables ν m,j and | Σ j | for j = ℓ + m + 1 , . . . , n . Here we define ν m and | Σ | as genericrandom variables following the same distribution with any of { ν m,j , j = ℓ + m +1 , . . . , n } and {| Σ j | , j = ℓ + m +1 , . . . , n } ,respectively. Put differently, ν m is a Binomial rv with param-eters m and α , while | Σ | is a rv that takes the value K j withprobability µ j .Next, we bound the probabilities P [ B ℓ,m ] . We know that B ℓ,m := ∩ ℓi =1 ∪ mj = ℓ +1 E ij . Thus, P h B ℓ,m (cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m i = E h
111 [ B ℓ,m ] (cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m i = E " E (cid:20)
111 [ B ℓ,m ] (cid:12)(cid:12)(cid:12) Σ ,..., Σ ℓ + m C ij ,i = ℓ +1 ,...,ℓ + mj =1 ,...,ℓ (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m = E ℓ Y j =1 − (cid:0) P −|∪ i ∈ νm,j Σ i || Σ j | (cid:1)(cid:0) P | Σ j | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m = E − (cid:0) P −|∪ i ∈ νm Σ i || Σ | (cid:1)(cid:0) P | Σ | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ ℓ +1 , . . . , Σ ℓ + m ℓ (56)by independence of the random variables ν m,j and | Σ j | for j = 1 , . . . , ℓ .We note that, on the event E ( JJJ ) , we have |∪ i ∈ ν m Σ i |≥ (cid:0) J | ν m | + 1 (cid:1) | ν m | > and it is always the case that |∪ i ∈ ν m Σ i |≥ K | ν m | > and |∪ i ∈ ν m Σ i |≤ | ν m | K r . (57)Next, we define L ( ν m ) = max (cid:0) K | ν m | > , (cid:0) J | ν m | + 1 (cid:1) | ν m | > (cid:1) so that on E ( JJJ ) , we have |∪ i ∈ ν m Σ i |≥ L ( ν m ) . (58)Using (58) in (55) and (57) in (56), we get P h A ℓ,m ∩ E ( JJJ ) i (59) = E h
111 [ C m ] 111 [ B ℓ,m ] 111 h D ℓ,m ∩ E ( JJJ ) ii = E h E h
111 [ C m ] 111 [ B ℓ,m ] 111[ D ℓ,m ∩ E ( JJJ )] (cid:12)(cid:12)(cid:12) Σ ℓ +1 ,..., Σ ℓ + m C ij ,i,j = ℓ +1 ,...,ℓ + m ii ≤ P [ C m ] E " − (cid:0) P −| ν m | K r | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ℓ E " (cid:0) P − L ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) n − ℓ − m since C m is fully determined by the rvs Σ ℓ +1 , . . . , Σ ℓ + m and { C ij , i, j = ℓ + 1 , . . . , ℓ + m } while B ℓ,m , D ℓ,m , and E ( JJJ ) are independent from { C ij , i, j = ℓ + 1 , . . . , ℓ + m } . Here,we also used the fact that given { Σ ℓ +1 , . . . , Σ ℓ + m } , D ℓ,m isindependent from B ℓ,m .The following lemma provides upper bounds for (59). Lemma 8.1.
Let
JJJ be defined as in (49) for some ǫ ∈ (0 , , ζ ∈ (cid:0) , (cid:1) such that (50) holds, ψ ∈ (cid:0) , (cid:1) such that (51)holds. Assume that Λ ( n ) = o (1) and (7), (8), and (9) hold.Then for all n sufficiently large, and for each m = 2 , , . . . , n ,we have P h A ℓ,m ∩ E ( JJJ ) i (60) ≤ min n , m m − ( α n p rr ( n )) m − o (cid:20) m > (cid:22) P n − K r,n K r,n (cid:23)(cid:21) + 111 (cid:20) m ≤ (cid:22) P n − K r,n K r,n (cid:23)(cid:21) (cid:16) − e − mα n p rr ( n ) (cid:17) ℓ ! ·· min ( − Λ ( n ) , e − (cid:18) ǫ (cid:19) Λ ( n ) , e − ψK ,n
111 [ m > m n ] +min n − µ r + µ r e − α n p r ( n ) ζm , e − α n p ( n ) ζm o)! n − m − ℓ The proof of Lemma 8.1 is given in Appendix E. Now, theproof of Proposition 7.3 will be completed upon establishing(53) by means of Lemma 8.1. We devote Section 9 toestablishing (53). 9. E
STABLISHING (53)In this section, we make several use of the following lemma.
Lemma 9.1.
Consider a scaling K , K , . . . , K r , P : N → N r +10 and a scaling α : N → (0 , such that (42) holds with β ℓ,n = o (log n ) . We have
12 log nn ≤ α n p r ( n ) ≤ µ r log nn , (61) for all n sufficiently large, i.e., α n p r ( n ) = Θ (cid:16) log nn (cid:17) . If inaddition (9) holds, we have α n p rr ( n ) = o (log n ) α n p r ( n ) = o (cid:18) (log n ) n (cid:19) (62) and α n p r ( n ) = o (log n ) α n p ( n ) (63)The proof of Lemma 9.1 is given in Appendix D.We now proceed with establishing (53). We start by defining f n,ℓ,m as f n,ℓ,m = (cid:18) nℓ (cid:19)(cid:18) n − ℓm (cid:19) P h A ℓ,m ∩ E ( JJJ ) i Thus, establishing (53) becomes equivalent to showing lim n →∞ ⌊ n − ℓ ⌋ X m =2 f n,ℓ,m = 0 . (64)We will establish (64) in several steps with each step focusingon a specific range of the summation over m . Throughout,we consider scalings K , . . . , K r , P : N → N r +10 and α : N → (0 , such that (42) holds with lim n →∞ β ℓ,n = + ∞ and β ℓ,n = o (log n ) , and (7), (8), (9) hold. We will makerepeated use of the bounds (23), (24), (25), and (62).
1) The case where ≤ m ≤ M : This range considers fixedvalues of m . Pick an integer M to be specified later at (71).We note that on this range we have m ≤ ⌊ P n − K r,n K r,n ⌋ for all n sufficiently large by virtue of (8). On the same range we alsohave − e − mα n p rr ( n ) ≤ mα n p rr ( n ) (65)by virtue of (62), (21), and the fact that m is bounded.Using (25), (60), (62), and (65), and noting that Λ ( n ) = o (1) under (42) with β ℓ,n = o (log n ) , we get f n,ℓ,m ≤ n ℓ n m m m − ( α n p rr ( n )) m − (3 m ) ℓ ( α n p rr ( n )) ℓ ·· e − ( ǫ ) ( n − m − ℓ )Λ ( n ) = O (1) n ℓ + m ( α n p rr ( n )) ℓ + m − · e − ( ǫ ) ( n − m − ℓ )Λ ( n ) = o (1) n ℓ + m (cid:18) (log n ) n (cid:19) ℓ + m − e − ( ǫ ) (log n + ℓ log log n + β ℓ,n ) = o (1) n − ǫ (log n ) ℓ ( − ǫ ) +2( m − e − ( ǫ ) β ℓ,n = o (1) since ℓ is non-negative integer constant, m is bounded, and lim n →∞ β ℓ,n = + ∞ . This establishes lim n →∞ M X m =2 f n,ℓ,m = 0 .
2) The case where M +1 ≤ m ≤ min { m n , ⌊ µ r n ζ log n ⌋} : Ourgoal in this and the next subsubsection is to cover the range M + 1 ≤ m ≤ ⌊ µ r n ζ log n ⌋ . Since the bound given at (60) takesa different form when m > m n (with m n defined at (48)), wefirst consider the range M + 1 ≤ m ≤ min { m n , ⌊ µ r n ζ log n ⌋} ;we note from (8) and (5) that lim n →∞ m n = ∞ .On the range considered here, we have from (23), (25), and(60) that min { m n , ⌊ µrn ζ log n ⌋} X m = M +1 f n,ℓ,m ≤ min { m n , ⌊ µrn ζ log n ⌋} X m = M +1 n ℓ (cid:16) enm (cid:17) m m m − ( α n p rr ( n )) m − ·· (cid:16) − µ r (cid:16) − e − α n p r ( n ) ζm (cid:17)(cid:17) n − m − ℓ (66)From the upper bound in (61) and the fact that m ≤ µ r n ζ log n for all n sufficiently large, we have α n p r ( n ) ζm ≤ nµ r n ζ µ r n ζ log n = 1 . Using the fact that − e − x ≥ x for all ≤ x ≤ , we get − µ r (cid:16) − e − α n p r ( n ) ζm (cid:17) ≤ − µ r α n p r ( n ) ζm ≤ e − ζmµ r log n n (67)as we invoke the lower bound in (61). Reporting this last boundand (62) into (66), and noting that n − m − ℓ ≥ n − ℓ ≥ n , m = 2 , , . . . , (cid:22) n − ℓ (cid:23) , (68)we get min { m n , ⌊ µrn ζ log n ⌋} X m = M +1 f n,ℓ,m ≤ min { m n , ⌊ µrn ζ log n ⌋} X m = M +1 n ℓ + m e m (cid:18) (log n ) n (cid:19) m − e − ζmµ r log n n − m − ℓ n ≤ n ℓ +1 ∞ X m = M +1 (cid:16) e (log n ) e − ζ µr log n (cid:17) m (69)for all n sufficiently large. Given that ζ, µ r > we have e (log n ) e − ζ µr log n = o (1) . (70)Thus, the geometric series in (69) is summable, and we have min { m n , ⌊ µrn ζ log n ⌋} X m = M +1 f n,ℓ,m ≤ O (1) n ℓ +1 − ( M +1) ζ µr ( e log n ) M +1) and it follows that lim n →∞ min { m n , ⌊ µrn ζ log n ⌋} X m = M +1 f n,ℓ,m = 0 for any positive integer M with M > ℓ + 1) ζµ r . (71)This choice is permissible given that ζ, µ r > .
3) The case where min {⌊ µ r n ζ log n ⌋ , m n } < m ≤ ⌊ µ r n ζ log n ⌋ : Clearly, this range becomes obsolete if m n ≥ ⌊ µ r n ζ log n ⌋ . Thus,it suffices to consider the subsequences for which the range m n + 1 ≤ m ≤ ⌊ µ r n ζ log n ⌋ is non-empty. On this range,following the same arguments that lead to (66) and (69) gives ⌊ µrn ζ log n ⌋ X m = m n +1 f ℓ,n,m (72) ≤ ⌊ µrn ζ log n ⌋ X m = m n +1 n ℓ +1 (cid:0) e (log n ) (cid:1) m ·· (cid:16) − µ r (cid:16) − e − ζmα n p r ( n ) (cid:17) + e − ψK ,n (cid:17) n ≤ n ℓ +1 ⌊ µrn ζ log n ⌋ X m = m n +1 (cid:16) e (log n ) (cid:17) m (cid:16) e − ζmµ r log n n + e − ψK ,n (cid:17) n where in the last step we used (67) in view of m ≤ µ r n ζ log n .Next, we write e − ζmµ r log n n + e − ψK ,n = e − ζmµ r log n n (cid:16) e − ψK ,n + ζmµ r log n n (cid:17) ≤ exp (cid:26) − ζmµ r log n n + e − ψK ,n + ζmµ r log n n (cid:27) ≤ exp − ζmµ r log n n − e − ψK ,n + µ r ζmµ r log n n (73)where the last inequality is obtained from m ≤ µ r n ζ log n . Usingthe fact that m > m n = min {⌊ P n K ,n ⌋ , ⌊ n ⌋} and that P n ≥ σn for some σ > under (7), we have e − ψK ,n + µ r ζmµ r log n n ≤ max (cid:26) K ,n P n , n (cid:27) n e − ψK ,n ζµ r log n · e µ r ≤ max (cid:26) K ,n e − ψK ,n ζµ r σ log n , e − ψK ,n ζµ r log n (cid:27) · e µ r = o (1) by virtue of (17) and the facts that ζ, µ r , σ > . Reportingthis into (73), we see that for for any ε > , there exists afinite integer n ∗ ( ε ) such that (cid:16) e − ζmµ r log n n + e − ψK ,n (cid:17) ≤ e − ζmµ r log n n (1 − ε ) (74)for all n ≥ n ∗ ( ε ) . Using (74) in (72), we get ⌊ µrn ζ log n ⌋ X m = m n +1 f ℓ,n,m ≤ n ℓ +1 ⌊ µrn ζ log n ⌋ X m = m n +1 (cid:16) e (log n ) (cid:17) m (cid:16) e − ζmµ r log n n (1 − ε ) (cid:17) n ≤ n ℓ +1 ∞ X m = m n +1 (cid:16) e (log n ) e − ζµ r log n (1 − ε ) (cid:17) m (75)Similar to (70), we have e (log n ) e − ζµ r log n (1 − ε ) = o (1) sothat the sum in (75) converges. Following a similar approachto that in Section 9-2, we then see that ⌊ µrn ζ log n ⌋ X m = m n +1 f n,ℓ,m = O (1) n ℓ +1 − m nζµr (1 − ε )12 ( e log n ) m n +1) = o (1) since lim n →∞ m n = ∞ under the enforced assumptions.
4) The case where ⌊ µ r n ζ log n ⌋ +1 ≤ m ≤ ⌊ νn ⌋ : We consider ⌊ µ r n ζ log n ⌋ + 1 ≤ m ≤ ⌊ νn ⌋ for some ν ∈ (0 , ) to be specifiedlater at (77). Recalling (23), (25), (60), (61), and (68), andnoting that (cid:0) nm (cid:1) is monotone increasing in m when ≤ m ≤ (cid:4) n (cid:5) , we get ⌊ νn ⌋ X m = ⌊ µrn ζ log n ⌋ +1 f n,ℓ,m ≤ ⌊ νn ⌋ X m = ⌊ µrn ζ log n ⌋ +1 n ℓ (cid:18) n ⌊ νn ⌋ (cid:19) ·· (cid:16) − µ r + µ r e − ζmα n p r ( n ) + e − ψK ,n (cid:17) n ≤ n ℓ ⌊ νn ⌋ X m = ⌊ µrn ζ log n ⌋ +1 (cid:16) eν (cid:17) νn ·· (cid:18) − µ r + µ r e − ζ µrn ζ log n log n n + e − ψK ,n (cid:19) n ≤ n ℓ (cid:16) eν (cid:17) νn (cid:16) − µ r + µ r e − µr + e − ψK ,n (cid:17) n = n ℓ (cid:18)(cid:16) eν (cid:17) ν (cid:16) − µ r + µ r e − µr + e − ψK ,n (cid:17)(cid:19) n (76)for all n sufficiently large.We have − µ r + µ r e − µr < from µ r > and e − ψK ,n = o (1) from (17). Also, it holds that lim ν → (cid:0) eν (cid:1) ν = 1 . Thus,if we pick ν small enough to ensure that (cid:16) eν (cid:17) ν (cid:16) − µ r + µ r e − µr (cid:17) < , (77)then for any < ε < − ( e/ν ) ν (cid:0) − µ r + µ r e − µr (cid:1) thereexists a finite integer n ⋆ ( ε ) such that (cid:16) eν (cid:17) ν (cid:16) − µ r + µ r e − µr + e − ψK ,n (cid:17) ≤ − ε, ∀ n ≥ n ⋆ ( ε ) . Reporting this into (76), we get lim n →∞ ⌊ νn ⌋ X m = ⌊ µrn ζ log n ⌋ +1 f n,ℓ,m = 0 since lim n →∞ n ℓ (1 − ε ) n/ = 0 for any positive integer ℓ .
5) The case where ⌊ νn ⌋ + 1 ≤ m ≤ ⌊ n − ℓ ⌋ : In this range,we use (24), (25), (60), and (68) to get ⌊ n − ℓ ⌋ X m = ⌊ νn ⌋ +1 f n,ℓ,m ≤ n ℓ ⌊ n − ℓ ⌋ X m = ⌊ νn ⌋ +1 (cid:18) nm (cid:19) (cid:16) e − ζmα n p ( n ) + e − ψK ,n (cid:17) n ≤ n ℓ ⌊ n − ℓ ⌋ X m = ⌊ νn ⌋ +1 (cid:18) nm (cid:19) (cid:16) e − ζνnα n p ( n ) + e − ψK ,n (cid:17) n ≤ n ℓ (cid:16) e − ζνnα n p ( n ) + 8 e − ψK ,n (cid:17) n Noting that ζ, ν, ψ > and recalling (63) and the lowerbound of (61), we get e − ζνnα n p ( n ) = e − ζνn wn log n α n p r ( n ) ≤ e − ζνwn for some sequence w n satisfying lim n →∞ w n = + ∞ . It isnow obvious that e − ζνnα n p ( n ) = o (1) . Moreover, we have e − ψK ,n = o (1) from (17). The conclusion lim n →∞ ⌊ n − ℓ ⌋ X m = ⌊ νn ⌋ +1 f n,ℓ,m = 0 immediately follows and the proof of one-law is completed.A CKNOWLEDGMENT
This work has been supported in part by National ScienceFoundation through grants CCF-1617934 and CCF-1422165and in part by the start-up funds from the Department ofElectrical and Computer Engineering at Carnegie MellonUniversity (CMU). R
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Proposition A.1 ([7, Proposition 4.4]) . For any set of positiveintegers K , . . . , K r , P and any scalar a ≥ , we have (cid:0) P −⌈ aK i ⌉ K j (cid:1)(cid:0) PK j (cid:1) ≤ (cid:0) P − K i K j (cid:1)(cid:0) PK j (cid:1) ! a , i, j = 1 , . . . , r (A.1) Proposition A.2.
Consider a random variable Z defined as Z = 1 − p i = (cid:0) P − K K i (cid:1)(cid:0) PK i (cid:1) , with probability µ i , i = 1 , . . . , r. We have var [ Z ] ≤ ( p r ) . Proof:
Recalling (15), we see that p ij increases with both i and j , and it follows that − p r ≤ Z ≤ − p , From Popoviciu’s inequality [25, pp. 9], we see thatvar [ Z ] ≤
14 ( Z max − Z min ) = 14 ( p r − p ) ≤
14 ( p r ) since p r ≥ p ≥ . Fact A.3. If λ ( n ) = o (1) , then p i ( n ) = o (1) , i = 1 , . . . , r Proof:
Recalling (3), we obtain p i ( n ) ≤ (cid:18) µ i (cid:19) λ ( n ) = O ( λ ( n )) = o (1) under the given assumption that λ ( n ) = o (1) . Fact A.4.
For ≤ x ≤ , the following properties hold.(a) [19, Fact 2] If < y < , then (1 − x ) y ≤ − xy .(b) Let a > . Then, − x a ≤ a (1 − x ) .Proof: By a crude bounding, we have − x a = Z x at a − dt ≤ Z x a dt = a (1 − x ) . Fact A.5 ([19, Fact 5]) . Let a , x , and y be positive integerssatisfying y ≥ (2 a + 1) x . Then, (cid:0) y − axx (cid:1)(cid:0) yx (cid:1) ≥ " (cid:0) y − xx (cid:1)(cid:0) yx (cid:1) a Fact A.6.
Let x ∈ (0 , and a > . Then, − x a ≤ a (1 − x ) Lemma A.7.
Consider a scaling K , . . . , K r , P : N → N r +10 such that (5) holds, a scaling α : N → (0 , , and Λ ( n ) = log n +( k −
1) log log n + γ n n . The following properties hold for anythree distinct nodes v x , v y , and v j . (a) We have P (cid:2) ( K xj ∩ K yj ) | K xy , t x = 1 , t y = 1 (cid:3) ≤ (cid:18) µ r (cid:19) λ ( n ) (A.2) (b) If λ ( n ) = o (1) , then for any u = 0 , , . . . , K ,n , wehave P [( K xj ∩ K yj ) | ( | S xy | = u ) , t x = 1 , t y = 1]= uK ,n λ ( n ) ± O (cid:16) ( λ ( n )) (cid:17) , and P [ E xj ∪ yj | ( | S xy | = u ) , t x = 1 , t y = 1]= 2Λ ( n ) − α n uK ,n Λ ( n ) ± O (cid:16) (Λ ( n )) (cid:17) (A.3) Proof:
We know that P (cid:20) ( K xj ∩ K yj ) (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = 1 − P (cid:20)(cid:0) K xj ∪ K yj (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = 1 − P (cid:20) K xj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) (A.4) − P (cid:20) K yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) + P (cid:20)(cid:0) K xj ∩ K yj (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) It is easy to see that P (cid:20) K xj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = P h K xj (cid:12)(cid:12)(cid:12) t x = 1 i = r X i =1 µ i (1 − p i ( n ))= 1 − λ ( n ) (A.5)Similarly, it is easy to see that P (cid:20) K yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = 1 − λ ( n ) (A.6)Next, by recalling (A.1), we observe that P (cid:20)(cid:0) K xj ∩ K yj (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 (cid:21) = P h Σ j ∈ P \ { Σ x ∪ Σ y } (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i = r X i =1 µ i (cid:0) P n − K ,n K i,n (cid:1)(cid:0) P n K i ,n (cid:1) ≤ r X i =1 µ i (cid:0) P n − K ,n K i,n (cid:1)(cid:0) P n K i,n (cid:1) ! = E h Z n ( µµµ, θθθ n ) i = ( E [ Z n ( µµµ, θθθ n )]) + var [ Z n ( µµµ, θθθ n )] (A.7) where Z n ( µµµ, θθθ n ) is a rv that takes the value − p i ( n ) withprobability µ i for i = 1 , . . . , r . Note that E [ Z n ( µµµ, θθθ n )] = r X i =1 µ i (1 − p i ) = 1 − λ ( n ) , (A.8)and λ ( n ) = r X i =1 µ i p i ( n ) ≥ µ r p r (A.9)for positive µµµ . Recalling Proposition A.2, and using (A.8) and(A.9) in (A.7), we get P (cid:20)(cid:0) K xj ∩ K yj (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 (cid:21) ≤ (1 − λ ( n )) + 14 λ ( n ) µ r = 1 − λ ( n ) + λ ( n ) (cid:18) µ r (cid:19) (A.10)The desired conclusion (A.2) follows from (A.4) in view of(A.5), (A.6), and (A.10).Next, we establish part (b) of the lemma under the as-sumption that λ ( n ) = o (1) . Conditioning on | S xy | = u andrecalling (A.7), we see that P (cid:20)(cid:0) K xj ∩ K yj (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = r X i =1 µ i (cid:0) P n − K ,n + uK i (cid:1)(cid:0) PK i (cid:1) (A.11)Invoking Lemma 5.6 and Fact A.4, we observe that (cid:0) P n − K ,n + uK i,n (cid:1)(cid:0) PK i,n (cid:1) ≤ (cid:18) − K ,n − uP n (cid:19) K i,n ≤ − K i,n (2 K ,n − u ) P n + 12 (cid:18) K i,n (2 K ,n − u ) P n (cid:19) = 1 − K i,n (2 K ,n − u ) P n + O (cid:18) K ,n K i,n P n (cid:19) ! (A.12)and (cid:0) P n − K ,n + uK i,n (cid:1)(cid:0) PK i,n (cid:1) ≥ (cid:18) − K ,n − uP n − K ,n (cid:19) K i,n ≥ − K i,n (2 K ,n − u ) P n − K ,n = 1 − (cid:18) K i,n (2 K ,n − u ) P n − K ,n − K i,n (2 K ,n − u ) P n (cid:19) − K i,n (2 K ,n − u ) P n = 1 − K i,n (2 K ,n − u ) P n − O (cid:18) K ,n K i,n P n (cid:19) ! (A.13) Combining (A.11), (A.12), and (A.13), we notice that P (cid:20)(cid:0) K xj ∩ K yj (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = r X i =1 µ i − K i,n (2 K ,n − u ) P n ± O (cid:18) K ,n K i,n P n (cid:19) !! Recalling Fact A.3 and Fact 5.4, we observe that under theenforced assumption λ ( n ) = o (1) , we have P (cid:20)(cid:0) K xj ∩ K yj (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = r X i =1 µ i − K i,n K ,n P n + uK i,n P n ± O (cid:18) K ,n K i,n P n (cid:19) !! = r X i =1 µ i − K i,n K ,n P n + uK ,n K i,n K ,n P n ± O (cid:18) K ,n K i,n P n (cid:19) ! ! = r X i =1 µ i − (cid:0) p i ( n ) ± O (cid:0) p i ( n ) (cid:1)(cid:1) + uK ,n (cid:0) p i ( n ) ± O (cid:0) p i ( n ) (cid:1)(cid:1) ± O ( p i ( n )) ! = r X i =1 µ i (cid:18) − p i ( n ) + uK ,n p i ( n ) ± O ( p i ( n )) (cid:19) = 1 − λ ( n ) + uK ,n λ ( n ) ± O r X i =1 µ i ( p i ( n )) ! (A.14)Next, we note that E h Z n ( µµµ, θθθ n ) i = r X i =1 µ i (1 − p i ( n )) = r X i =1 µ i (cid:16) − p i ( n ) + ( p i ( n )) (cid:17) = 1 − λ ( n ) + r X i =1 µ i ( p i ( n )) Now, we recall from (A.10) that E h Z n ( µµµ, θθθ n ) i ≤ − λ ( n ) + λ ( n ) (cid:18) µ r (cid:19) , and it follows that r X i =1 µ i ( p i ( n )) ≤ λ ( n ) (cid:18) µ r (cid:19) = O (cid:0) λ ( n ) (cid:1) (A.15)Combining (A.5), (A.6), (A.14), and (A.15), the conclusion P (cid:20) ( K xj ∩ K yj ) (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = uK ,n λ ( n ) ± O (cid:0) λ ( n ) (cid:1) , (A.16)follows. Next, we establish (A.3). We know that P (cid:20) E xj ∪ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = P (cid:20) E xj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) + P (cid:20) E yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) − P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) . (A.17)Now, since E xj = C xj ∩ K xj and E yj = C yj ∩ K yj , itis clear that E xj and E yj are each independent of the event | S xy | = u . It follows that P (cid:20) E xj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = P [ E xj | t x = 1]= Λ ( n ) , (A.18)and similarly P (cid:20) E yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = Λ ( n ) . (A.19)Finally, P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = P [ C xj ∩ C yj ] ·· P (cid:20) K xj ∩ K yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = α n P (cid:20) K xj ∩ K yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = α n uK ,n Λ ( n ) ± O (cid:0) Λ ( n ) (cid:1) (A.20)by virtue of (A.16). Combining (A.17), (A.18), (A.19), and(A.20), the conclusion (A.3) follows. Lemma A.8.
Consider a scaling K , . . . , K r , P : N → N r +10 such that (5) holds, a scaling α : N → (0 , , Λ ( n ) = log n +( k −
1) log log n + γ n n , with lim n →∞ γ n = −∞ . Let m , m ,and m be non-negative integer constants. We define event F as follows. F := [ | N xy | = m ] ∩ [ | N xy | = m ] ∩ [ | N xy | = m ] . (A.21) Then, given u in { , , . . . , K ,n } and Λ ( n ) = o ( √ n ) under lim n →∞ γ n = −∞ , we have P (cid:20) F (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) ∼ n m + m + m m ! m ! m ! e − n Λ ( n )+ uαnK ,n n Λ ( n ) ·· (cid:18) P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21)(cid:19) m · (cid:18) P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21)(cid:19) m · (cid:18) P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21)(cid:19) m with j distinct from x and y . Proof: The proof of Lemma A.8 is very similar with [19,Lemma 4]; in fact, it would follow directly from [19, Eq.(212)-(213)] if we show that (cid:16) P h E xj ∩ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i(cid:17) n − m − m − m − ∼ e − n Λ ( n )+ uαnK ,n n Λ ( n ) . (A.22)Recalling Lemma A.7 and the fact that Λ ( n ) ≤ log n +( k −
1) log log nn for all n sufficiently large under lim n →∞ γ n = −∞ , we get P h E xj ∩ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i = 1 − P h E xj ∪ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i (A.23) = 1 − (cid:18) ( n ) − α n uK ,n Λ ( n ) ± O (cid:16) (Λ ( n )) (cid:17)(cid:19) = 1 − O (cid:18) log nn (cid:19) = 1 − o (1) . (A.24)Also, ( n − m − m − m − ·· (cid:16) P h E xj ∪ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i(cid:17) = ( n − m − m − m − (cid:20) O (cid:18) log nn (cid:19)(cid:21) = o (1) (A.25)Invoking Fact 5.5 for (A.23), and using (A.24) and (A.25),we get (cid:16) P h E xj ∩ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i(cid:17) n − m − m − m − ∼ e ( n − m − m − m ) P [ E xj ∪ yj | ( | S xy | = u ) ,t x =1 ,t y =1 ] ∼ e − n h ( n ) − αnuK ,n Λ ( n ) ± o ( n ) i e ( m + m + m +2) o (1) ∼ e − n Λ ( n )+ uαnK ,n n Λ ( n ) . (A.26)This gives (A.22) and Lemma A.8 is established in view of[19, Lemma 4]. Lemma A.9 ([19, Lemma 10]) . If P n ≥ K ,n , we have P h | S xy | = u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ≤ u ! K ,n P n − K ,n ! u Lemma A.10.
With m ≥ and Λ ( n ) = o (1) , we have E " (cid:0) P n − Q ( ν m ) | Σ | (cid:1)(cid:0) P n | Σ | (cid:1) ≤ e − ( ǫ ) Λ ( n ) , for all n sufficiently large and any ǫ ∈ (0 , , where we define Q ( ν m ) = K ,n
111 [ | ν m | = 1]+( ⌊ (1 + ǫ ) K ,n ⌋ + 1) 111 [ | ν m | > . Proof:
Consider fixed
KKK, P . We have Q ( ν m ) ≥ K (111 [ | ν m | = 1] + (1 + ǫ )111 [ | ν m | > Thus, by recalling (A.1), we get E " (cid:0) P − Q ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ≤ E (cid:0) P − K | Σ | (cid:1)(cid:0) P | Σ | (cid:1) | ν m | =1]+(1+ ǫ )111[ | ν m | > = E h Z | ν m | =1]+(1+ ǫ )111[ | ν m | > i where Z = ( P − K | Σ | )( P | Σ | ) . Taking the expectation over | ν m | , we get E " (cid:0) P − Q ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ≤ E h (1 − α ) m + mα (1 − α ) m − Z + (cid:16) − (1 − α ) m − mα (1 − α ) m − (cid:17) Z ǫ i ≤ E h (1 − α ) + 2 α (1 − α ) Z + (cid:16) − (1 − α ) − α (1 − α ) (cid:17) Z ǫ i = (1 − α ) + 2 α (1 − α ) E [ Z ] + α E (cid:2) Z ǫ (cid:3) by virtue of the fact that (1 − α ) m + mα (1 − α ) m − T + (cid:16) − (1 − α ) m − mα (1 − α ) m − (cid:17) T ǫ is monotonically decreasing in m (see [19, Lemma 12]).Next, we have E [ Z ] = r X j =1 µ j (cid:0) P − K K j (cid:1)(cid:0) PK j (cid:1) = 1 − λ Also by recalling Fact A.4, we get E (cid:2) Z ǫ (cid:3) = E (cid:0) P − K | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ! ǫ = r X j =1 µ j (cid:0) P − K K j (cid:1)(cid:0) PK j (cid:1) ! ǫ = r X j =1 µ j (1 − p j )(1 − p j ) ǫ ≤ r X j =1 µ j (1 − p j )(1 − ǫp j )= 1 − λ (1 + ǫ ) + ǫ r X j =1 µ j p j . Note that r X j =1 µ j (1 − p j ) = 1 − λ + r X j =1 µ j p j and we have from (A.7) and (A.10) that r X j =1 µ j (1 − p j ) ≤ − λ + λ (cid:18) µ r (cid:19) This gives r X j =1 µ j p j ≤ λ (cid:18) µ r (cid:19) and we get E " (cid:0) P − Q ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ≤ (1 − α ) + 2 α (1 − α ) (1 − λ )+ α (cid:18) − λ (1 + ǫ ) + ǫλ (cid:18) µ r (cid:19)(cid:19) = 1 − Λ (cid:18) − (1 − ǫ ) α − ǫ (cid:18) µ r (cid:19) Λ (cid:19) Now, consider a scaling such that Λ ( n ) = o (1) . We have Λ ( n ) ≤ µ r µ r +1) for all n sufficiently large. Given also that α n ≤ , we get E " (cid:0) P n − Q ( ν m ) | Σ | (cid:1)(cid:0) P n | Σ | (cid:1) ≤ − Λ (cid:16) − (1 − ǫ ) − ǫ (cid:17) ≤ e − ( ǫ ) Λ ( n ) for all n sufficiently large. This completes the proof.A PPENDIX BP ROOF OF L EMMA P [ D x,ℓ ∩ D y,ℓ ]= P (cid:2) D x,ℓ ∩ D y,ℓ ∩ E xy (cid:3) + P [ D x,ℓ ∩ D y,ℓ ∩ E xy ] . (B.1)Thus, Lemma 6.2 will be established upon showing the nexttwo results. Proposition B.1.
Consider scalings K , . . . , K r , P : N → N r +10 and α : N → (0 , , such that λ ( n ) = o (1) and (6)holds with lim n →∞ γ n = −∞ . The following hold(a) If n Λ ( n ) = Ω(1) , then for any non-negative integerconstant ℓ and any two distinct nodes v x and v y , we have P (cid:2) D x,ℓ ∩ D y,ℓ ∩ E xy (cid:3) ∼ µ ( ℓ ! ) − ( n Λ ( n )) ℓ e − n Λ ( n ) (B.2) (b) For any two distinct nodes v x and v y , we have P (cid:2) D x, ∩ D y, ∩ E xy (cid:3) ∼ µ e − n Λ ( n ) (B.3) Proposition B.2.
Consider scalings K , . . . , K r , P : N → N r +10 and α : N → (0 , , such that λ ( n ) = o (1) and (6)holds with lim n →∞ γ n = −∞ . If n Λ ( n ) = Ω(1) , then forany non-negative integer ℓ and any distinct nodes v x and v y ,we have P [ D x,ℓ ∩ D y,ℓ ∩ E xy ] = o (cid:0) P (cid:2) D x,ℓ ∩ D y,ℓ ∩ E xy (cid:3)(cid:1) (B.4)We establish Propositions B.1 and B.2 in the followingtwo subsections respectively. Next, we show why Lemma 6.2follows from Propositions B.1 and B.2. If n Λ ( n ) = Ω(1) ,then for any non-negative integer constant ℓ , we observe that (36) follows from (B.2) and (B.4) in view of (B.1).Now, considering the case when ℓ = 0 , we see that (B.3)directly implies (37) by virtue of (B.1) and the fact that P [ D x, ∩ D y, ∩ E xy ] = 0 since it is impossible for nodes v x and v y to be adjacent to each other (i.e., under E xy ) whenboth nodes have zero degree. A. Proof of Proposition B.1
Consider the vertex set V = { v , . . . , v n } . For each node v i ∈ V , we define N i as the set of neighbors of node v i . Also,for any pair of vertices v x , v y , we let N xy be the set of nodesin V \ { v x , v y } that are neighbors of both v x and v y ; i.e., N xy = N x ∩ N y . We also let N xy denote the set of nodes in V \ { v x , v y } that are neighbors of v x , but are not neighbors of v y . Similarly, N xy is defined as the set of nodes in V \{ v x , v y } that are not neighbors of v x , but are neighbors of v y . Finally, N xy is the set of nodes in V \ { v x , v y } that are not connectedto either v x or v y . We also define S xy = Σ x ∩ Σ y .We start by defining the series of events A h as follows A h := [ | N xy | = h ] ∩ [ | N xy | = ℓ − h ] ∩ [ | N xy | = ℓ − h ] . It is simple to see that D x,ℓ ∩ D y,ℓ ∩ E xy = ℓ [ h =0 (cid:0) A h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1] (cid:1) , whence we get P (cid:2) D x,ℓ ∩ D y,ℓ ∩ E xy (cid:3) = ℓ X h =0 P (cid:2) A h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1] (cid:3) (B.5)since the events { A h , h = 0 , . . . , ℓ } are mutually exclusive.Furthermore, since E xy = K xy ∪ C xy = K xy ∪ (cid:0) K xy ∩ C xy (cid:1) and K xy ∩ [ t x = 1] ∩ [ t y = 1] = ∪ K ,n u =1 ( | S xy | = u ) we have under t x = t y = 1 that E xy = K xy ∪ K ,n [ u =1 ( | S xy | = u ) ∩ C xy = K xy ∪ K ,n [ u =1 X u (B.6)where we define the event X u as X u = ( | S xy | = u ) ∩ C xy , u = 1 , . . . , K ,n (B.7)Now, we get P (cid:2) A h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1] (cid:3) = P (cid:2) A h ∩ K xy ∩ [ t x = 1] ∩ [ t y = 1] (cid:3) + K ,n X u =1 P [ A h ∩ X u ∩ [ t x = 1] ∩ [ t y = 1]] , (B.8) by virtue of (B.6) and the fact that the events K xy , X , X , . . . , X K ,n are mutually disjoint. Combining(B.5) and (B.8) we obtain P (cid:2) D x,ℓ ∩ D y,ℓ ∩ E xy (cid:3) = µ ℓ X h =0 P h A h ∩ K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i + µ ℓ X h =0 K ,n X u =1 P h A h ∩ X u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i . (B.9)Proposition B.1 is established by virtue of (B.9) and thefollowing two results. Proposition B.3.
Consider scalings K , . . . , K r , P : N → N r +10 and α : N → (0 , , such that λ ( n ) = o (1) and (6)holds with lim n →∞ γ n = −∞ . Then for any non-negativeinteger ℓ , we have ℓ X h =0 P (cid:2) A h ∩ K xy | t x = t y = 1 (cid:3) ∼ ( ℓ ! ) − ( n Λ ( n )) ℓ e − n Λ ( n ) (B.10) Proposition B.4.
Consider scalings K , . . . , K r , P : N → N r +10 and α : N → (0 , , such that λ ( n ) = o (1) and (6)holds with lim n →∞ γ n = −∞ . If n Λ ( n ) = Ω(1) , then ℓ X h =0 K ,n X u =1 P h A h ∩ X u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i = o ℓ X h =0 P h A h ∩ K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i! (B.11) for any ℓ = 0 , , . . . . Furthermore, we have (B.11) for ℓ = 0 without requiring the condition n Λ ( n ) = Ω(1) . Before we prove Propositions B.3 and B.4, we explain whyProposition B.1 follows from these two results. Combining(B.10) and (B.11) we establish (B.2) in view of (B.9). Fur-thermore, by using (B.10) and (B.11) with ℓ = 0 , we readilyobtain (B.3) in view of (B.9). This establishes Proposition B.1.
1) Proof for Proposition B.3:
We write ℓ X h =0 P h A h ∩ K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i = ℓ X h =0 P h A h (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i P h K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i , where P h K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i = 1 − p ( n ) ∼ (B.12)under the assumption λ ( n ) = o (1) and Fact A.3. Also, usingLemma A.8 with u = 0 , m = h , and m = m = ℓ − h , wesee that P h A h (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i ∼ n ℓ − h h ! (( ℓ − h ) ! ) e − n Λ ( n ) ·· (cid:16) P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i(cid:17) h · (cid:16) P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i(cid:17) ℓ − h · (cid:16) P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i(cid:17) ℓ − h . (B.13)Next, we evaluate the three probability terms appearing in(B.13). We know that P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i = P [ C xj ∩ C yj ] · P h K xj ∩ K yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i = α n P h K xj ∩ K yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i ≤ (cid:18) µ r (cid:19) Λ ( n ) (B.14)by virtue of Lemma A.7. We also see that P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i = P h E xj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i − P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i = P h E xj (cid:12)(cid:12)(cid:12) t x = 1 i − P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i = Λ ( n ) − O (cid:0) Λ ( n ) (cid:1) ∼ Λ ( n ) (B.15)as we invoke (B.14) and use the fact that Λ ( n ) = o (1) under lim n →∞ γ n = −∞ . It is also easy to see that P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i ∼ Λ ( n ) (B.16)via similar arguments.For h = 1 , , . . . , ℓ , we observe from (B.13), (B.14), (B.15),and (B.16) that P h A h (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i P h A (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i ∼ n − h ( ℓ ! ) h ! (( ℓ − h ) ! ) P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i ·· P h E xj ∩ yj (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i ! h ≤ n − h ( ℓ ! ) h ! (( ℓ − h ) ! ) (cid:16) µ r (cid:17) Λ ( n ) Λ ( n ) (1 − o (1)) h = o (1) (B.17)Similarly, setting h = 0 , we obtain P h A (cid:12)(cid:12)(cid:12) K xy , t x = 1 , t y = 1 i ∼ ( ℓ ! ) − ( n Λ ( n )) ℓ e − n Λ ( n ) (B.18)The conclusion (B.10) follows by combining (B.12), (B.17),(B.18), and noting that ℓ is constant.
2) Proof of Proposition B.4:
Our approach is to find anupper bound to the left hand side of (B.11) and show that thisupper bound is o (cid:16)P ℓh =0 P h A h ∩ K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i(cid:17) . Itwill be clear that the condition n Λ ( n ) = Ω(1) needed toestablish (B.11) is not needed for the case when ℓ = 0 .We know that P h A h ∩ X u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i = P h A h ∩ | S xy | = u ∩ C xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ≤ P h A h ∩ | S xy | = u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i Thus, ℓ X h =0 K ,n X u =1 P h A h ∩ X u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ≤ ℓ X h =0 K ,n X u =1 P h A h ∩ | S xy | = u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i = K ,n X u =1 P h | S xy | = u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ·· ℓ X h =0 P h A h (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i Now, since E xj = C xj ∩ K xj and E yj = C yj ∩ K yj , itis clear that E xj and E yj are each independent of the event | S xy | = u . It follows that P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) ≤ P (cid:20) E xj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) = Λ ( n ) . (B.19)Similarly, we have P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) ≤ Λ ( n ) (B.20)and P (cid:20) E xj ∩ yj (cid:12)(cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 (cid:21) ≤ Λ ( n ) (B.21)Now, using Lemma A.8 with m = h , and m = m = ℓ − h , (B.19), (B.20), and (B.21), it follows that P h A h (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i ≤ n ℓ − h e − n Λ ( n )+ uαnK ,n n Λ ( n ) (Λ ( n )) ℓ − h (B.22)for all n sufficiently large. Thus, we get ℓ X h =0 K ,n X u =1 P h A h ∩ X u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ≤ K ,n X u =1 P h | S xy | = u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ·· e − n Λ ( n )+ uαnK ,n n Λ ( n ) ℓ X h =0 ( n Λ ( n )) ℓ − h ! (B.23) Now, if n Λ ( n ) = Ω(1) it follows that ℓ X h =0 ( n Λ ( n )) ℓ − h = O (cid:16) ( n Λ ( n )) ℓ (cid:17) . (B.24)Note that (B.24) follows trivially for ℓ = 0 with no conditionon n Λ ( n ) . Combining (B.23), (B.24) and Lemma A.9, weget ℓ X h =0 K ,n X u =1 P h A h ∩ X u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i (B.25) ≤ O (cid:16) ( n Λ ( n )) ℓ e − n Λ ( n ) (cid:17) K ,n X u =1 K ,n P n − K ,n e αnK ,n n Λ ( n ) ! u In view of Proposition B.3 (and the fact that ℓ is constant),we will immediately establish the desired result (B.11) from(B.25) if we show that K ,n P n − K ,n e αnK ,n n Λ ( n ) = o (1) . (B.26)Next, we establish (B.26). From (5), we get for all n sufficiently large that K ,n P n − K ,n ≤ K ,n P n ≤ p ( n ) where the last bound used the fact that K ,n P n ∼ p ( n ) when p ( n ) = o (1) (e.g., see [7, Lemma 4.2]); this in turn followsfrom the assumption that λ ( n ) = o (1) in view of Fact A.3. Itis also clear from the definition λ ( n ) = P ri =1 µ i p i ( n ) that p ( n ) ≤ µ λ ( n ) . Thus, for all n large, we get K ,n P n − K ,n ≤ µ λ ( n ) . (B.27)Now, with Λ ( n ) ≤ log n +( k −
1) log log nn for all n sufficientlylarge under lim n →∞ γ n = −∞ , we see that n Λ ( n ) = nα n λ ( n ) ≤
32 log n (B.28)for all n sufficiently large. Combining (B.27) and (B.28) andthe fact that K ,n ≥ , we obtain K ,n P n − K ,n e αnK ,n n Λ ( n ) = O (1) λ ( n ) e α n log n . (B.29)Next, we define F ( n ) = λ ( n ) e α n log n . Fix n sufficientlylarge such that (B.27) and (B.28). We consider the cases when α n ≤ n and α n > n . In the former case, F ( n ) ≤ λ ( n ) e / follows directly. In the latter case we use (B.28) toget F ( n ) ≤
32 log nnα n e α n log n ≤
32 (log n ) n n by virtue of the fact that α n log n ≤ log n . Combining the twobounds, we have F ( n ) ≤ max n λ ( n ) e . , . n − . (log n ) o for all n sufficiently large. In view of λ ( n ) = o (1) thisimmediately gives lim n →∞ F ( n ) = 0 , and the conclusion (B.26) follows in view of (B.29). The desired result (B.11)is now established from (B.25) and (B.26) for constant ℓ .Note that for ℓ = 0 , we have (B.11) without requiring n Λ ( n ) = Ω(1) , since that extra condition is used only oncein obtaining (B.24) which holds trivially for ℓ = 0 . Thisestablishes Proposition B.4. B. Proof of Proposition B.2
Recalling Proposition B.4 and (B.9), Proposition B.2 willfollow if we show that P [ D x,ℓ ∩ D y,ℓ ∩ E xy ]= o ℓ X h =0 P h A h ∩ K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i! , (B.30)for each ℓ = 1 , . . . . To establish (B.30), we define the seriesof events B h as follows B h := [ | N xy | = h ] ∩ [ | N xy | = ℓ − h − ∩ [ | N xy | = ℓ − h − , for each h = 0 , , . . . , ℓ − . Now, it is easy to see that D x,ℓ ∩ D y,ℓ ∩ E xy = ℓ − [ h =0 ( B h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1]) . (B.31)Note that h varies from to ℓ − in (B.31) because given theevent E xy , nodes x and y are adjacent; thus, they could haveat most ℓ − nodes in common when their degrees are ℓ . Sincethe events B h are mutually exclusive for h = 0 , . . . , ℓ − , weget P [ D x,ℓ ∩ D y,ℓ ∩ E xy ]= ℓ − X h =0 P [ B h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1]] Thus, the proof of Proposition B.2 will be completed uponshowing ℓ − X h =0 P [ B h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1]]= o ℓ X h =0 P h A h ∩ K xy (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i! (B.32)under the enforced assumptions of Proposition B.2, namely,with lim n →∞ γ n = −∞ , and n Λ ( n ) = Ω(1) . Proceeding asbefore, and noting that P [ E xy ] = α P [ K xy ] we write ℓ − X h =0 P [ B h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1]] (B.33) = µ α ℓ − X h =0 K ,n X u =1 P h B h ∩ ( | S xy | = u ) (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ≤ µ K ,n X u =1 P [( | S xy | = u )] ℓ − X h =0 P h B h (cid:12)(cid:12)(cid:12) | S xy | = u, t x = t y = 1 i Next, by recalling Lemma A.8 with m = h , m = m = ℓ − h − , we get P h B h (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i ∼ n ℓ − h − h ! (( ℓ − h −
1) ! ) e − n Λ ( n )+ uαn Λ1( n ) K ,n n × n P h E xj ∩ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 io h × n P h E xj ∩ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 io ℓ − h − × n P h E xj ∩ yj (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 io ℓ − h − . Recalling (B.19), (B.20), and (B.21), we get P h B h (cid:12)(cid:12)(cid:12) ( | S xy | = u ) , t x = 1 , t y = 1 i ≤ e − n Λ ( n )+ uαnK ,n n Λ ( n ) ( n Λ ( n )) ℓ − h − (B.34)for all n sufficiently large. Using (B.34) in (B.33), we get forall n sufficiently large that ℓ − X h =0 P [ B h ∩ E xy ∩ [ t x = 1] ∩ [ t y = 1]] ≤ µ K ,n X u =1 P h | S xy | = u (cid:12)(cid:12)(cid:12) t x = 1 , t y = 1 i ·· e − n Λ ( n )+ uαnK ,n n Λ ( n ) ℓ X h =0 ( n Λ ( n )) ℓ − h − ! = µ ( n Λ ( n )) − × right hand side of ( B. O ( right hand side of ( B. (B.35)since n Λ ( n ) = Ω(1) . We have shown in the proof ofProposition B.4 thatright hand side of ( B. o ℓ X h =0 P (cid:2) A h ∩ K xy | t x = t y = 1 (cid:3)! Together with (B.35) this establishes (B.32) and the proof ofProposition B.2 is complete.A
PPENDIX CC ONFINING γ n In this section, we show that establishing the one-law ofTheorem 3.2 under the additional constraint γ n = o (log n ) (C.1)establishes the one-law for the case when that additionalconstraint is not present. Namely, we will show that for anyscaling that satisfies conditions (7), (8), (9), and (6) with lim n →∞ γ n = + ∞ , there exists a scaling that satisfies thesame conditions with lim n →∞ γ n = + ∞ and γ n = o (log n ) ,such that the probability of k -connectivity under the latterscaling (with γ n = o (log n ) ) is less than or equal to that underthe former scaling.Firstly, consider a probability distribution µµµ = { µ , . . . , µ r } with µ i > for i = 1 , . . . , r , a scaling K ∗ , K ∗ , . . . , K ∗ r , P ∗ : N → N r +10 , and a scaling α ∗ : N → (0 , such that Λ ∗ ( n ) = α ∗ n λ ∗ ( n ) = log n + ( k −
1) log log n + γ ∗ n n , (C.2) for each n = 1 , , . . . . Assume that P ∗ n = Ω( n ) , K ∗ r,n P ∗ n = o (1) , and K ∗ r,n K ∗ ,n = o (log n ) (C.3)and that we have lim n →∞ γ ∗ n = + ∞ ; i.e., the ∗ -scalingsatisfies all conditions enforced by part (b) of Theorem 3.2.Now, with the same distribution µµµ , consider a scaling ˆ K , ˆ K , . . . , ˆ K r , ˆ P : N → N r +10 and a scaling ˆ α : N → (0 , such that ˆ P n = P ∗ n and ˆ KKK n = KKK ∗ n . Obviously, we have ˆ λ ( n ) = λ ∗ ( n ) by recalling (2) and (3) and also that ˆ P n = Ω( n ) , ˆ K r,n ˆ P n = o (1) , and ˆ K r,n ˆ K ,n = o (log n ) . Next, let ˆ γ n := min ( γ ∗ n , log log n ) and define ˆ α n through ˆ α n ˆ λ ( n ) = log n + ( k −
1) log log n + ˆ γ n n . (C.4)Clearly, we have ˆ γ n = o (log n ) and lim n →∞ ˆ γ n = + ∞ .This establishes that for any scaling satisfying the conditionsof part (b) of of Theorem 3.2, there exists another scaling(with the same µµµ, KKK n , and P n ) that satisfies all of the sameconditions and (C.1). In addition, this latter scaling has asmaller probability of a channel being on than the originalscaling; i.e., we have ˆ α n ≤ α ∗ n , n = 2 , , . . . (C.5)by virtue of the fact that ˆ γ n ≤ γ ∗ n for all n .In view of the above, we will establish that part (b) ofTheorem 3.2 under γ n = o (log n ) implies Theorem 3.2 ifwe show that P " H ( n ; µµµ, KKK ∗ n , P ∗ n , α ∗ n ) is k − connected ≥ P " H ( n ; µµµ, ˆ KKK n , ˆ P n , ˆ α n ) is k − connected (C.6)This is clear since (C.6) would ensure that if H ( n ; µµµ, ˆ KKK n , ˆ P n , ˆ α n ) is k -connected asymptotically almostsurely (as would be deduced from Theorem 3.2 under γ n = o (log n ) ), then so would H ( n ; µµµ, KKK ∗ n , P ∗ n , α ∗ n ) .In view of (C.5), we get (C.6) by means of an easy couplingargument showing that H ( n ; µµµ, ˆ KKK n , ˆ P n , ˆ α n ) is a spanningsubgraph of H ( n ; µµµ, KKK ∗ n , P ∗ n , α n ) . This follows from the factthat under (C.5) the corresponding ER graphs satisfy G ( n ; ˆ α n ) ⊆ G ( n ; α ∗ n ) meaning that for any monotone increasing graph property P (e.g., k -connectivity), the probability of that G ( n ; α ∗ n ) has P islarger than that of G ( n ; ˆ α n ) ; see [19, Section V.B] for details.A PPENDIX DP ROOF OF L EMMA β ℓ,n = o (log n ) , we clearly have
12 log nn ≤ Λ ( n ) ≤ nn (D.1)for all n sufficiently large. We also have Λ ( n ) = α n r X j =1 µ j p j ≥ µ r α n p r ( n ) Now, since p j is monotone increasing in j = 1 , . . . , r byvirtue of (15), we also see that Λ ( n ) = α n r X j =1 µ j p j ( n ) ≤ α n p r ( n ) r X j =1 µ j = α n p r ( n ) Thus, we obtain that Λ ≤ α n p r ( n ) ≤ µ r Λ and the conclusion (61) immediately follows by virtue of (D.1)for all n sufficiently large.Next, we establish (62). Here this will be established byshowing that p rr ( n ) ≤ max (cid:18) , nw n (cid:19) p r ( n ) , n = 2 , , . . . (D.2)for some sequence w n such that lim n →∞ w n = ∞ . Fix n =2 , , . . . . We have either p r ( n ) > , or p r ( n ) ≤ . In theformer case, it automatically holds that p rr ( n ) ≤ p r ( n ) (D.3)by virtue of the fact that p rr ( n ) ≤ .Assume now that p r ( n ) ≤ . We know from [23, Lem-mas 7.1-7.2] that − e − Kj,nKr,nPn ≤ p jr ( n ) ≤ K j,n K r,n P n − K j,n , j = 1 , . . . , r (D.4)and it follows that K ,n K r,n P n ≤ log (cid:18) − p r ( n ) (cid:19) ≤ log 2 < . (D.5)Using the fact that − e − x ≥ x with x in (0 , , we then get p r ( n ) ≥ K ,n K r,n P n . (D.6)In addition, using the upper bound in (D.4) with j = r gives p rr ( n ) ≤ K r,n P n − K r,n ≤ K r,n P n as we invoke (5). Combining the last two bounds we obtain p rr ( n ) p r ( n ) ≤ K r,n K ,n (D.7)Next, combining (9) and (D.7), we get p rr ( n ) ≤ nw n p r ( n ) (D.8)for some sequence w n such that lim n →∞ w n = ∞ . Combining(D.3) and (D.8), we readily obtain (D.2).It is easy to see that (63) can be established using the samesteps with the proof of (D.2). A PPENDIX EP ROOF OF L EMMA P [ C m ] ≤ m m − ( α n p rr ( n )) m − Next, we derive upper bounds on the terms E (cid:20) − ( P −| νm | Kr | Σ | )( P | Σ | ) (cid:21) and E (cid:20) ( P − L ( νm ) | Σ | )( P | Σ | ) (cid:21) , respectively. Itis clear that Lemma 8.1 will follow if we show that E − (cid:0) P n −| ν m | K r,n | Σ | (cid:1)(cid:0) P n | Σ | (cid:1) ≤ − e − α n p rr ( n ) m (E.1)for all m ≤ ⌊ P − K r,n K r,n ⌋ and that E " (cid:0) P n − L ( ν m ) | Σ | (cid:1)(cid:0) P n | Σ | (cid:1) (E.2) ≤ min − Λ ( n ) , e − ( ǫ ) Λ ( n ) , min (cid:16) − µ r + µ r e − α n p r ( n ) ζm , e − α n p ( n ) ζm (cid:17) + e − ψK ,n
111 [ m > m n ] ! . We establish (E.1) and (E.2) in turn in the next two sections.
A. Establishing (E.1)
First, with m ≤ P − K r K r , we have | ν m |≤ m ≤ P − K r K r andusing Fact A.5 we get E (cid:20) − (cid:0) P −| ν m | K r | Σ | (cid:1)(cid:0) P | Σ | (cid:1) (cid:21) ≤ E (cid:20) − (cid:0) P − K r | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ! | ν m | (cid:21) = 1 − E h W | ν m | i (E.3)where we set W = ( P − Kr | Σ | )( P | Σ | ) . We also have E h W | ν m | i = E m X j =0 (cid:18) mj (cid:19) α j (1 − α ) m − j W j = E h(cid:0) − α (cid:0) − W (cid:1)(cid:1) m i ≥ E [(1 − α (1 − W )) m ] (E.4)using Fact A.4 in the last step. We also know that W = (cid:0) P − K r | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ≥ (cid:0) P − K r K r (cid:1)(cid:0) PK r (cid:1) = 1 − p rr (E.5)Thus, α n (1 − W n ) ≤ α n p rr ( n ) ≤ for all n sufficiently large by virtue of (62) and that β ℓ,n = o (log n ) . Using the fact that − x ≥ e − x for all ≤ x ≤ ,we then get from (E.4) and (E.5) that E h W | ν m | n i ≥ E h e − α n (1 − W n ) m i ≥ e − α n p rr ( n ) m for all n sufficiently large. The desired conclusion (E.1) nowfollows immediately by means of (E.3). B. Establishing (E.2)
Let
YYY be defined as follows Y i = ( ⌊ iζK ,n ⌋ i = 2 , . . . , m n ⌊ ψP n ⌋ i = m n + 1 , . . . , n where ζ ∈ (0 , ) selected small enough such that (50) holds,and ψ ∈ (0 , ) selected small enough such that (51) holds.Recalling (49), we see that J i = ( max ( ⌊ (1 + ǫ ) K ,n ⌋ , Y i ) i = 2 , . . . , m n Y i i = m n + 1 , . . . , n Next, we let M ( ν m )= K ,n
111 [ | ν m | = 1] + max (cid:0) K ,n , Y | ν m | + 1 (cid:1)
111 [ | ν m | > , and Q ( ν m ) = K ,n
111 [ | ν m | = 1]+( ⌊ (1 + ǫ ) K ,n ⌋ + 1) 111 [ | ν m | > . We also recall that L ( ν m ) = max (cid:0) K ,n
111 [ | ν m | > , (cid:0) J | ν m | + 1 (cid:1)
111 [ | ν m | > (cid:1) Let’s consider the following three cases1) | ν m | = 0 : In this case we have L ( ν m ) = M ( ν m ) = Q ( ν m ) = 0 .2) | ν m | = 1 : In this case we have L ( ν m ) = M ( ν m ) = Q ( ν m ) = K ,n .3) | ν m |≥ : In this case we have– L ( ν m ) = max (cid:0) K ,n , J | ν m | + 1 (cid:1) .– M ( ν m ) = max (cid:0) K ,n , Y | ν m | + 1 (cid:1) .– Q ( ν m ) = ⌊ (1 + ǫ ) K ,n ⌋ + 1 .More specifically, considering the case when | ν m | =2 , , . . . , m n , we have J | ν m | = max (cid:0) (1 + ǫ ) K ,n , Y | ν m | (cid:1) and it follows that L ( ν m ) = max (cid:0) K ,n , ⌊ (1 + ǫ ) K ,n ⌋ + 1 , Y | ν m | + 1 (cid:1) = max ( ⌊ (1 + ǫ ) K ,n ⌋ + 1 , M ( ν m ))= max ( Q ( ν m ) , M ( ν m )) Also, when | ν m | = m n + 1 , . . . , n , we clearly have J | ν m | = Y | ν m | , and thus L ( ν m ) = M ( ν m ) = max ( K ,n , ⌊ ψP n ⌋ + 1) . Since K ,n ≤ K r,n = o ( P n ) in view of (8), we have ⌊ ψP n ⌋ ≥ ⌊ (1 + ǫ ) K ,n ⌋ for all n sufficiently large. Thus, we can rewrite L ( ν m ) as L ( ν m ) = max ( K ,n , ⌊ ψP n ⌋ + 1 , ⌊ (1 + ǫ ) K ,n ⌋ + 1)= max ( Q ( ν m ) , M ( ν m )) . Combining, we conclude that it always holds that L ( ν m ) =max ( Q ( ν m ) , M ( ν m )) , whence E " (cid:0) P − L ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ≤ min E " (cid:0) P − M ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) , E " (cid:0) P − Q ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) (E.6) Note that it was shown in [9, Lemma 7.2] that E " (cid:0) P − M ( ν m ) | Σ | (cid:1)(cid:0) P | Σ | (cid:1) ≤ min (cid:16) − Λ ( n ) , min (cid:16) − µ r + µ r e − α n p r ( n ) ζm , e − α n p ( n ) ζm (cid:17) + e − ψK ,n m > m n ] (cid:17) for all n sufficiently large. On the same range, we also getfrom Lemma A.10 that E " (cid:0) P n − Q ( ν m ) | Σ | (cid:1)(cid:0) P n | Σ | (cid:1) ≤ e − ( ǫ ) Λ ( n ) upon noting that Λ ( n ) = o (1) under (42) with β ℓ,n = o (log n ))