k-Connectivity of Random Key Graphs

# k-Connectivity of Random Key Graphs

Jun Zhao CyLab and Dept. of ECE
Carnegie Mellon University
junzhao@cmu.edu
Osman Yağan CyLab and Dept. of ECE
Carnegie Mellon University
oyagan@ece.cmu.edu
Virgil Gligor CyLab and Dept. of ECE
Carnegie Mellon University
gligor@cmu.edu
###### Abstract

Random key graphs represent topologies of secure wireless sensor networks that apply the seminal Eschenauer–Gligor random key predistribution scheme to secure communication between sensors. These graphs have received much attention and also been used in diverse application areas beyond secure sensor networks; e.g., cryptanalysis, social networks, and recommender systems. Formally, a random key graph with nodes is constructed by assigning each node keys selected uniformly at random from a pool of keys and then putting an undirected edge between any two nodes sharing at least one key. Considerable progress has been made in the literature to analyze connectivity and -connectivity of random key graphs; e.g., Yağan and Makowski [ISIT ’09, Trans. IT’12] on connectivity under , Rybarczyk [Discrete Mathematics ’11] on connectivity under , and our recent work [CDC ’14] on -connectivity under , where -connectivity of a graph ensures connectivity even after the removal of nodes or edges. Yet, it still remains an open question for -connectivity in random key graphs under and (the case of is trivial). In this paper, we answer the above problem by providing an exact analysis of -connectivity in random key graphs under .

Wireless sensor networks, key predistribution, random key graphs, -connectivity, minimum degree.

## I Introduction

Random key graphs, also known as homogeneous random intersection graphs, have been investigated widely in the literature [2, 6, 7, 9, 10, 12, 13, 1, 5]. The notion of random key graph results from the seminal Eschenauer–Gligor (EG) random key predistribution scheme [4], which is the most recognized solution to secure communication using cryptographic keys in wireless sensor networks [10]. The definition of a random key graph can also be generalized beyond cryptographic keys. Consider a random key graph that is constructed on a set of nodes as follows. Each node is independently assigned a set of distinct objects, selected uniformly at random from a pool of objects, where and are both functions of . An undirected edge exists between two nodes if and only if they possess at least one common object. An object is a cryptographic key in the application of random key graphs to the Eschenauer–Gligor random key predistribution scheme. In addition to the area of secure sensor networks, random key graphs have also been used in various applications including cryptanalysis [1], social networks [12], and recommender systems [5].

(-)Connectivity of a random key graph has received much interest [2, 6, 7, 9, 10, 12, 13]. A graph is said to be -connected if it remains connected despite the deletion of at most nodes or edges111-connectivity given here is equivalent to -vertex-connectivity, which can also be defined when only node failure is considered; i.e., the ability of the graph remaining connected in spite of the removal of at most nodes. -edge-connectivity is defined similarly for graphs that are still connected despite the failure of any edges. It is plain to prove that -vertex-connectivity implies -edge-connectivity [3].; an equivalent definition of -connectivity is that for each pair of nodes there exist at least mutually disjoint paths connecting them [3]. In the case of being 1, -connectivity becomes connectivity, meaning that each node in the graph can find at least one path to any other node, either directly or with the help of other relaying nodes. A graph property related to and implied by -connectivity is that the minimum degree of the the graph is at least (i.e., each node is directly connected to no less than other nodes), where the minimum degree refers to the minimum among the numbers of neighbors that nodes have.

We investigate -connectivity of random key graphs. Our contribution is, for a random key graph, to derive the asymptotically exact probabilities for -connectivity and the property that the minimum degree is at least .

The rest of the paper is organized as follows. Section II presents the results. We elaborate the proof of Theorem 1 in Section III. Section IV provides numerical findings to support the theoretical results. Section V surveys related work; and Section VI concludes the paper.

## Ii The Results

For a random key graph , Theorem 1 and Corollary 1 below present the asymptotically exact probabilities for -connectivity and the property of the minimum degree being at least , where is a positive integer and does not scale with . The term stands for the natural logarithm function, and is its base. We use the standard asymptotic notation ; in particular, for two positive sequences and , the relation means . All asymptotic statements are understood with . Also, denotes the probability that event occurs.

###### Theorem 1.

For a random key graph , let be the probability that there exists an edge between two nodes. With a sequence defined by

 qn =lnn+(k−1)lnlnn+αnn, (1)

then under , it follows that

 limn→∞P[G(n,Xn,Yn) is k-connected.] =limn→∞P[G(n,Xn,Yn) has a minimum degree at least k.] =⎧⎪ ⎪⎨⎪ ⎪⎩0, if limn→∞αn=−∞,1, if limn→∞αn=∞,e−e−α∗(k−1)!, if limn→∞αn=α∗∈(−∞,∞). (2)

We have the following corollary by replacing the condition (1) on the edge probability with a condition on the asymptotics of (formally, holds under ; see [12, Lemma 8].)

###### Corollary 1.

For a random key graph , with a sequence defined by

 Xn2Yn =lnn+(k−1)lnlnn+βnn, (3)

then under , it follows that

 limn→∞P[G(n,Xn,Yn) is k-connected.] =limn→∞P[G(n,Xn,Yn) has a minimum degree at least k.] =⎧⎪ ⎪⎨⎪ ⎪⎩0, if limn→∞βn=−∞,1, if limn→∞βn=∞,e−e−β∗(k−1)!, if limn→∞βn=β∗∈(−∞,∞).
###### Remark 1.

From Lemma 4 (resp., Lemma 5) in the Appendix, we can introduce an extra condition (resp., ) in proving Theorem 1 (resp., Corollary 1).

###### Remark 2.

In Theorem 1 and Corollary 1, since the results are in the asymptotic sense, the conditions only need to hold for all sufficiently large.

Establishing Corollary 1 given Theorem 1 is straightforward and is given in the Appendix. Below we explain how to obtain Theorem 1. Since a necessary condition for a graph to be -connected is that the minimum degree is at least , the proof of Theorem 1 will be completed once we have the following two lemmas. Lemma 1 is from our prior work [12]. Lemma 2 simply reproduces the result on the minimum degree in Theorem 1.

###### Lemma 1 (Our work [12, Lemma 5]).

For a random key graph under (1) and , it follows that

 =0. (4)
###### Lemma 2.

For a random key graph under (1) and , it follows that

 limn→∞P[G(n,Xn,Yn) has a minimum degree at least k.] =right hand side of (???).

By [11, Lemma 2], Lemma 2 will follow once we show Lemma 3 below, where we let be the set of nodes in a random key graph .

###### Lemma 3.

For a random key graph under (1) and , it follows for integers and that

 P[Nodes v1,v2,…,vm have degree h] ∼(h!)−m(nqn)hme−mnqn. (5)

We detail the proof of Lemma 3 in the next section.

## Iii The Proof of Lemma 3

In a random key graph , recalling that is the set of nodes, we let be the set of distinct objects assigned to node . We further define as and as . Among nodes in , we denote by the set of nodes neighboring to for . We denote by , and by .

We have the following two observations:

• If node has degree , then , where the equal sign holds if and only if is directly connected to none of nodes in ; i.e., if and only if event happens.

• If for any , then

 ∣∣∣⋃1≤i≤mNi∣∣∣≤∑1≤i≤mNi≤hm, (6)

where the two equal signs in (6) both hold if and only if

 (7)

From i) and ii) above, if nodes have degree , we have either of the following two cases:

• Any two of have no edge in between (namely, ); and event (7) happens.

• .

In addition, if case (a) happens, then nodes have degree . However, if case (b) occurs, there is no such conclusion. With (resp., ) denoting the probability of case (a) (resp., case (b)), we obtain

 Pa≤P[Nodes v1,v2,…,vm % have degree h]≤Pa+Pb,

where

 Pa=P[(⋂1≤i

and

 Pb =P[∣∣∣⋃1≤i≤mNi∣∣∣≤hm−1].

Hence, (5) holds after we prove the following (8) and (9):

 Pb =o((nqn)hme−mnqn). (8)

and

 Pa ∼(h!)−m(nqn)hme−mnqn⋅[1+o(1)], (9)

We will prove (8) and (9) below. We let denote the tuple . The expression “” means “given ”, where with being arbitrary -size subsets of the object pool. Note that . For two different nodes and in the graph , we use to denote the event that there is an edge between and ; i.e., the symbol “” means “is directly connected to”.

### Iii-a The Proof of (8)

Let be an arbitrary node in . We have

 P[∣∣∣⋃1≤i≤mNi∣∣∣=t∣Sm=S∗m] (10) =(n−m)!t!(n−m−t)! ×{P[w↔ at least one of % nodes in Vm∣Sm=S∗m]}t ×{P[w↔none of nodes in Vm∣Sm=S∗m]}n−m−t. (11)

By the union bound, it holds that

 P[w↔at least one of nodes in Vm∣Sm=S∗m] ≤∑1≤i≤mP[w↔vi∣Sm=S∗m]=mqn, (12)

which yields

 P[w↔none of nodes in Vm∣Sm=S∗m]≥1−mqn. (13)

 P[w↔none of nodes in Vm∣Sm=S∗m] =(Yn−|⋃1≤i≤mS∗i|Xn)(YnXn) ≤(1−qn)Xn−1|⋃1≤i≤mS∗i|(by \@@cite[cite]{[\@@bibref{}{yagan_onoff}{}{}, Lemma 5.1]}) ≤e−Xn−1qn|⋃1≤i≤mS∗i|(by 1+x≤ex for any real x). (14)

We will prove

 ∑S∗m{P[Sm=S∗m] ×{P[w↔none of nodes% in Vm∣Sm=S∗m]}n−m−hm} (15) ≤e−mnqn⋅[1+o(1)]. (16)

From (11) (12) and (16), we derive

 Pb =P[∣∣∣⋃1≤i≤mNi∣∣∣≤hm−1] =hm−1∑t=0∑S∗m{P[Sm=S∗m]⋅(???)} ≤hm−1∑t=0[nt⋅(mqn)t⋅(???)] ≤(nqn)hme−mnqn⋅[1+o(1)]⋅mhmhm−1∑t=0(mnqn)t−hm. (17)

As noted in Remark 1, we can introduce an extra condition in establishing Theorem 1. From and (1), we obtain

 qn ∼lnnn. (18)

Applying (18) to (17), we obtain (8). Hence, we complete the proof of (8) once showing (16), whose proof is detailed below.

From (13) (14) and (18), we have

 (???) ≤(1−mqn)−m−hm ×∑S∗m{P[Sm=S∗m]⋅e−Xn−1nqn|⋃1≤i≤mS∗i|} ≤[1+o(1)]⋅∑S∗m{P[Sm=S∗m]⋅e−Xn−1nqn|⋃1≤i≤mS∗i|}, (19)

so (16) holds once we demonstrate

 ∑S∗m{P[Sm=S∗m]⋅e−Xn−1nqn|⋃1≤i≤mS∗i|} ≤e−mnqn⋅[1+o(1)]. (20)

We denote the left hand side of (20) by . Dividing into two parts and , we derive

 Zm,n =∑S∗m−1S∗m{P[(Sm−1=S∗m−1)∩(Sm=S∗m)] ×e−Xn−1nqn|⋃1≤i≤mS∗i|} =∑S∗m−1P[Sm−1=S∗m−1]{e−Xn−1nqn|⋃1≤i≤m−1S∗i| (21)

where

 ∑S∗mP[Sm=S∗m]e−Xn−1nqn|S∗m∖⋃1≤i≤m−1S∗i| =e−nqnXn∑r=0P[∣∣∣Sm∩(m−1⋃i=1S∗i)∣∣∣=r]eXn−1nqnr. (22)

For satisfying

 0 ≤r≤|Sm|=Xn

and

 r =|Sm|+∣∣∣m−1⋃i=1S∗i∣∣∣−∣∣∣Sm∪(m−1⋃i=1S∗i)∣∣∣ ≥Xn+∣∣∣m−1⋃i=1S∗i∣∣∣−Yn,

as given in [11, Eq. (36)], we have

 P[∣∣∣Sm∩(m−1⋃i=1S∗i)∣∣∣=r] ≤1r!(mXn2Yn−Xn)r. (23)

Applying (23) to (22), we establish

 ∑S∗mP[Sm=S∗m]e−Xn−1nqn|S∗m∖⋃1≤i≤m−1S∗i| ≤e−nqnXn∑r=01r!(mXn2Yn−Xn)r⋅eXn−1nqnr ≤e−nqn⋅emXn2Yn−Xn⋅eXn−1nqn. (24)

From (18) and (46), it holds that , resulting in

 mXn2Yn−Xn ∼mXn2Yn∼mlnnn. (25)

For an arbitrary , from (18), we obtain for all sufficiently large, which with condition yields that for all sufficiently large,

 eXn−1nqn ≤e12(1+ϵ)lnn=n12(1+ϵ). (26)

From (25) and (26), we get

 mXn2Yn−Xn⋅eXn−1nqn ≤mlnnn⋅[1+o(1)]⋅n12(1+ϵ) ≤mlnn⋅n12(ϵ−1)⋅[1+o(1)]. (27)

Since is arbitrary, it follows from (27) that for arbitrary , then for all sufficiently large, it is clear that

 mXn2Yn−Xn⋅eXn−1nqn ≤n−c. (28)

Using (28) in (24), for all sufficiently large, it follows that

 ∑S∗mP[Sm=S∗m]e−Xn−1nqn|S∗m∖⋃1≤i≤m−1S∗i| ≤e−nqn⋅en−c. (29)

Substituting (29) into (21), for all sufficiently large, we obtain

 Zm,n ≤e−nqn⋅en−c⋅∑S∗m−1P[Sm−1=S∗m−1]e−Xn−1nqn|⋃1≤i≤m−1S∗i| ≤e−nqn⋅en−c⋅Zm−1,n. (30)

We then evaluate . By (20), it holds that

 Z2,n =∑S∗1P[S1=S∗1]∑S∗2P[S2=S∗2]e−Xn−1nqn|S∗1∪S∗2|. (31)

Setting in (29), for all sufficiently large, we derive

 ∑S∗2P[S2=S∗2]e−Xn−1nqn|S∗2∖S∗1| ≤e−nqn⋅en−c.

Then for all sufficiently large, it follows that

 ∑S∗2P[S2=S∗2]e−Xn−1nqn|S∗1∪S∗2| =e−nqn∑S∗2P[S2=S∗2]e−Xn−1nqn|S∗2∖S∗1| ≤e−2nqn⋅en−c. (32)

From (31) and (32), for all sufficiently large, we obtain

 Zm,n ≤(e−nqn⋅en−c)m−2⋅Z2,n ≤(e−nqn⋅en−c)m−2⋅e−2nqn⋅en−c ≤e−mnqn⋅e(m−1)n−c. (33)

Letting , we finally establish

 Zm,n ≤e−mnqn⋅[1+o(1)];

i.e., (20) is proved. Then as explained above, (16) holds; and then (8) follows.

### Iii-B The Proof of (9)

Again let be an arbitrary node in . We have

 P[(⋂1≤i

and

 Pa =∑S∗m:⋂1≤i

where .

For , under , we have

 P[w↔vi, but none of nodes in Vm∖{vi}∣Sm=S∗m] −∑1≤j≤mj≠iP[w↔both vi and vj∣Sm=S∗m] ≥qn−(m−1)⋅2qn2(by \@@cite[cite]{[% \@@bibref{}{mobihocQ1}{}{}, Lemma 3]}). (37)

Substituting (13) and (37) to (35), and then from (36), we obtain

 Pa ≥(n−m−hm)hm(h!)m⋅[qn−2(m−1)q2n]hm

Then from (18), it further hold that

 Pa ≥nhm(h!)m⋅(qn)hm⋅e−mnqn ×[1−o(1)]⋅P[⋂1≤i

From (14), under , it holds that

 P[w↔none of nodes in Vm∣Sm=S∗m] ≤e−mqn. (39)

For each , we have

 P[w↔vi, but w↔none of nodes in Vm∖{vi}∣Sm=S∗m] (40)

Substituting (40) and (39) to (35), and then from (36), we obtain

 Pa =nhm(h!)m⋅(qn)hm⋅e−mnqn⋅P[⋂1≤i

From (38) and (41), we obtain

 Pa ∼nhm(h!)m⋅(qn)hm⋅e−mnqn⋅P[⋂1≤i

By the union bound, it is clear that

 P[⋂1≤i

From (18) and (43), since a probability is at most , we get

 limn→∞P[⋂1≤i

Using (44) in (42), we establish (9).

## Iv Numerical Experiments

We present numerical experiments to back up our theoretical results. Figure 1 depicts the probability that graph is -connected. We let vary, with other parameters fixed at , and . The empirical probabilities corresponding to the experimental curves are obtained as follows: we count the times of -connectivity out of independent samples of , and derive the empirical probability through dividing its corresponding count by . For the theoretical curves, we first compute by setting the edge probability (viz., (45) in the Appendix) as and then use as the theoretical value for the probability of -connectivity. Figure 1 confirms our analytical results as the experimental and theoretical curves are close.

## V Related Work

For a random key graph , Rybarczyk [6] derives the asymptotically exact probabilities of connectivity and of the property that the minimum node degree is no less than , covering a weaker form of the results – the zero-one laws which are also obtained in [2, 10]. As demonstrated in [6], in with , and , the probability of connectivity and that of the minimum degree being at least both approach to as . Rybarczyk [7] implicitly obtains zero-one laws (but not the asymptotically exact probabilities) for -connectivity and for the property that the minimum degree is at least . The implicit result is that if for some and , graph has (resp., does not have) the two properties with probability approaching to , given that tends to (resp., ) as . Our Corollary 1 significantly improves her result [7] in the following two aspects: (i) we cover the wide range of all sufficiently large, instead of the much stronger condition in [7] (note that the analysis under is trivial), and (ii) we establish not only zero–one laws for -connectivity and the minimum degree, but also the asymptotically exact probabilities. The latter results are not given by Rybarczyk [7]. Recently, we [14] give the asymptotically exact probability of -connectivity in graph under through a rather involved proof. We improve this result to cover through a simpler proof and fill the gap where is at least , but is not . This improvement is of technical interest as well as of practical importance since random key graphs have been used in diverse applications including modeling the Eschenauer–Gligor random key predistribution scheme (the most recognized solution to secure communication in wireless sensor networks).

## Vi Conclusion

In this paper, for a random key graph, we derive the asymptotically exact probabilities for two properties with an arbitrary : (i) the graph is -connected; and (ii) each node has at least neighboring nodes. Numerical experiments are in accordance with our analytical results.

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### -a Establishing Corollary 1 given Theorem 1:

As noted in Remark 1, we can use an extra condition in establishing Corollary 1.

With denoting the probability that there exists an edge between two nodes in graph , as shown in previous work [2, 6, 10], we have

 qn =1−(Yn−XnXn)(YnXn). (45)

Further, it holds from [12, Lemma 8] that

 qn =Xn2Yn±O((X