On the Strengths of Connectivity and Robustnessin General Random Intersection Graphs

On the Strengths of Connectivity and Robustness in General Random Intersection Graphs

Abstract

Random intersection graphs have received much attention for nearly two decades, and currently have a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. In this paper, we investigate the strengths of connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero–one laws for -connectivity and -robustness, as well as the asymptotically exact probability of -connectivity, for any positive integer . The -connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures. On the other hand, -robustness measures the effectiveness of local diffusion strategies (that do not use global graph topology information) in spreading information over the graph in the presence of misbehaving nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomial random intersection graph and a uniform random intersection graph, which both have numerous applications as well. For these two specialized graphs, our results on asymptotically exact probabilities of -connectivity and asymptotic zero–one laws for -robustness are also novel in the literature.

{keywords}

Connectivity, consensus, random graph, random intersection graph, random key graph, robustness.

I Introduction

I-a Graph Models

Random intersection graphs have been introduced by Singer-Cohen [1] and received considerable attention [2, 7, 4, 5, 6, 3, 10, 8, 11, 12, 13, 14, 15, 16, 9, 17] for nearly two decades. In these graphs, each node is assigned a set of objects selected by some random mechanism. An undirected edge exists between any two nodes that have at least one object in common. Random intersection graphs have proved useful in modeling and analyzing real-world networks in a wide variety of application areas. Examples include secure wireless sensor networks [2, 7, 4, 5, 6, 3], frequency hopping spread spectrum [3], spread of epidemics [10, 8], and social and information networks [9, 8, 7] including collaboration networks [9, 8] and common-interest networks [7]. Several classes of random intersection graphs have been analyzed, and results concerning various graph properties such as clustering [9], component evolution [11, 2] and degree distribution [12] have been obtained.

The model considered in this paper, hereafter referred to as a general random intersection graph, represents a generalization [9, 12, 2] of random intersection graphs. It is defined on a node set as follows. Each node () is assigned an object set from an object pool consisting of distinct objects, where is a function of . Each object is constructed using the following two-step procedure: First, the size of , , is determined according to some probability distribution . Of course, we have , with denoting the probability that event occurs. Next, is formed by selecting distinct objects uniformly at random from the object pool . In other words, conditioning on , set is chosen uniformly among all -size subsets of . This process is repeated independently for all object sets . Finally, an undirected edge is assigned between two nodes if and only if their corresponding object sets have at least one object in common; namely, distinct nodes and have an edge in between if and only if . The graph defined through this adjacency notion is denoted by .

A specific case of the general model , known as the binomial random intersection graph, has been widely explored to date [9]–[14]. Under this model, each object set is constructed by a Bernoulli-like mechanism; i.e., by adding each object to independently with probability . Like integer , probability is also a function of . The term “binomial” accounts for the fact that now follows a binomial distribution with as the number of trials and as the success probability in each trial. We denote the binomial random intersection graph by , where subscript “b” stands for “binomial”.

Another well-known special case of the general model is the uniform random intersection graph [15, 4, 16, 6, 17, 5]. Under the uniform model, the probability distribution concentrates on a single integer , where ; i.e., for each node , the object set size equals with probability . and are both integer functions of . We denote by the uniform random intersection graph, with “u” meaning “uniform”.

A concrete example for the application of random intersection graphs can be given in the context of secure wireless sensor networks. As explained in detail in numerous other places [2, 7, 4, 5, 3, 10, 8, 11], the uniform random intersection graph model is induced naturally by the Eschenauer–Gligor (EG) random key predistribution scheme [6], which is a typical solution to ensure secure communications in wireless sensor networks. In particular, let the set of nodes in graph stand for the sensors in the wireless network. Also, let the object pool (with size ) represent the set of cryptographic keys available to the network and let be the number of keys assigned to each sensor (selected uniformly at random from the key pool ). Then, the edges in represent pairs of sensors that share at least one cryptographic key and thus that can securely communicate over existing wireless links in the EG scheme. In the above application, objects that nodes have are cryptographic keys, so uniform random intersection graphs are also referred to as random key graphs [17, 4, 3].

In the secure sensor network area, the general random intersection graph model in this paper captures the differences that may exist among the number of keys possessed by each sensor. This may occur for various reasons that include (a) the assigned numbers of keys on sensors may vary prior to deployment given the heterogeneity in available sensor memory [2]; (b) the number of keys available to a sensor may decrease after deployment due to revocation of compromised keys [7]; and (c) the number of keys on a sensor may increase due to the path key establishment phase of the EG scheme [6], where new path keys are generated and distributed to participating sensors.

I-B (-)Connectivity and (-)Robustness

We now introduce the graph properties that we are interested in. First, a graph is connected if there exists at least a path of edges between any two nodes [18]. A graph is said to be -connected if each pair of nodes has at least internally node-disjoint path(s) in between [14]; equivalently, a graph is -connected if it can not be made disconnected by deleting at most nodes or edges.1 In this manner, -connectivity quantifies the resiliency of graph connectivity against node or edge failures. In addition, it enables multi-path routing, and is also useful to achieve consensus in the graph [7]. In particular, to achieve consensus in the presence of adversarial nodes in a large-scale graph (with node size greater than ), a necessary and sufficient condition is that the graph is -connected [21].

Many algorithms have been proposed to achieve consensus [27, 28, 29, 30, 31, 32, 33] in graphs with sufficient connectivity. However, these algorithms typically assume that nodes have full knowledge of the graph topology, which is impractical in some cases [27]. To this end, Zhang and Sundaram [27] introduce the notion of graph robustness. They show that when nodes are limited to local information instead of the global graph topology, consensus can be reached in a sufficiently robust graph in the presence of adversarial/misbehaving nodes, but not in a sufficiently connected and insufficiently robust graph. Therefore, graph robustness quantifies the effectiveness and resiliency of local-information-based consensus algorithms in the presence of adversarial/misbehaving nodes. Robustness is an important property with broad relevance in graph processes beyond consensus; e.g., robustness plays a key role in information cascades and contagion processes [27]. It is worth noting that robustness is a stronger property than connectivity in the sense that any -robust graph is also -connected, whereas a -connected graph is not necessarily -robust [27].

Formally, a graph with a node set is -robust if at least one of (a) and (b) below hold for any non-empty and strict subset of : (a) there exists at least a node such that has no less than neighbors inside ; and (b) there exists at least a node such that has no less than neighbors inside .

I-C Contributions and Organization

With various applications of random intersection graphs, and -connectivity and -robustness graph properties in mind, a natural question to ask is whether random intersection graphs are -connected or -robust under certain conditions? Our paper answers this question. We summarize our contributions as follows:

  1. We derive sharp zero–one laws and asymptotically exact probabilities for -connectivity in general random intersection graphs.

  2. We establish sharp zero–one laws for -robustness in general random intersection graphs.

  3. For the two specific instances of the general graph model, a binomial random intersection graph and a uniform random intersection graph, we provide the first results on the asymptotically exact probabilities of -connectivity and zero–one laws for -robustness.

The rest of the paper is organized as follows. Section II presents the main results as Theorems 16. Then, we introduce some auxiliary facts and lemmas in Section III, before establishing the main results in Sections IV and V. Section VI details the proofs of the lemmas. We provide numerical experiments in Section VII. Section VIII reviews related work; and Section IX concludes the paper.

Ii The Results

Our main results are presented in Theorems 16 below. We defer the proofs of all theorems to Sections IV and V. Throughout the paper, is a positive integer and does not scale with ; and is the base of the natural logarithm function, . All limits are understood with . We use the standard Landau asymptotic notation and ; in particular, for two positive functions and , the relation signifies . For a random variable , the terms and stand for its expected value and variance, respectively.

Ii-a Zero–One Laws and Exact Probabilities for Asymptotic -Connectivity

We provide zero–one laws and exact probabilities for asymptotic -connectivity in different graphs below.

-Connectivity in General Random Intersection Graphs

Theorem 1 below presents a zero–one law and the exact probability for asymptotic -connectivity in a general random intersection graph.

Theorem 1

Consider a general random intersection graph . Let be a random variable following probability distribution . With a sequence for all defined through

(1)

if , and , then

-Connectivity in Binomial Random Intersection Graphs

Theorem 2 below presents a zero–one law and the exact probability for asymptotic -connectivity in a binomial random intersection graph.

Theorem 2

For a binomial random intersection graph , with a sequence for all defined through

(2)

if and , then

Remark 1

As we will explain in Section IV-B within the proof of Theorem 2, for the zero–one law, the condition can be weakened as , while we enforce for the asymptotically exact probability result.

-Connectivity in Uniform Random Intersection Graphs

Theorem 3 below presents a zero–one law and the exact probability for asymptotic -connectivity in a uniform random intersection graph.

Theorem 3

For a uniform random intersection graph , with a sequence for all defined through

(3)

if and , then

Ii-B Zero–One Laws for Asymptotic -Robustness

We provide zero–one laws for asymptotic -robustness in different graphs below.

-Robustness in General Random Intersection Graphs

Theorem 4 as follows gives a zero–one law for asymptotic -robustness in a general random intersection graph.

Theorem 4

Consider a general random intersection graph . Let be a random variable following probability distribution . With a sequence for all defined through

(4)

if , and , then

-Robustness in Binomial Random Intersection Graphs

Theorem 5 below gives a zero–one law for asymptotic -robustness in a binomial random intersection graph.

Theorem 5

For a binomial random intersection graph , with a sequence for all defined through

(5)

if and , then

-Robustness in Uniform Random Intersection Graphs

Theorem 6 below gives a zero–one law for asymptotic -robustness in a uniform random intersection graph.

Theorem 6

For a uniform random intersection graph , with a sequence for all defined through

(6)

if and , then

In view of Theorems 16, for each general/binomial/uniform random intersection graph, its -connectivity and -robustness asymptotically obey the same zero–one laws. Moreover, these zero–one laws are all sharp since can be much smaller compared to ; e.g., even with an arbitrary positive constant satisfies .

Iii Auxiliary Facts and Lemmas

We present a few facts and lemmas which are used to establish the theorems. To begin with, recalling that does not scale with , we obtain Facts 1 and 2 below, whose proofs are straightforward and thus omitted here.

Fact 1

For , it holds that

Theorem 3

Theorem 2

Lemma 1

Lemma 3

Theorem 5

Lemma 5

Theorem 6

Theorem 4

Lemma 2

Lemma 4

Theorem 1

Fig. 1: A plot illustrating the steps of deriving the theorems from the lemmas, with the arrows indicating the directions. For example, Theorem 3 and Lemma 2 are used to prove Theorem 1.
Fact 2

For , we have

and

Lemma 1 below presents the result on -robustness of an Erdős-Rényi graph. An Erdős–Rényi graph [18] is defined on a set of nodes such that any two nodes have an edge in between independently with probability .

Lemma 1

For an Erdős–Rényi graph , with a sequence for all through

(7)

then it holds that

(8)

To prove Lemma 1, we note the following three facts. (a) The desired result (8) with is demonstrated in [27, Theorem 3]. (b) By [14, Facts 3 and 7], for any monotone increasing graph property , the probability that graph has property is non-decreasing as increases. (c) -robustness is a monotone increasing graph property according to [34, Lemma 3]. In view of (a) (b) and (c) above, we obtain Lemma 1.

Throughout Lemmas 25 below, is an arbitrary monotone increasing graph property, where a graph property is called monotone increasing if it holds under the addition of edges. Except Lemma 4 which is from [2, Lemma 4], the proofs of Lemmas 2, 3 and 5 are deferred to Section VI.

Lemma 2

Let be a random variable with probability distribution . If , then there exists such that

and

Lemma 3

If and , then there exists such that

(9)

Lemma 4 ([2, Lemma 4])

If , and for all sufficiently large,

then

Lemma 5

If and , then

Figure 1 on the next page illustrates the steps of using the lemmas to prove the theorems. Note that the facts used in deriving the theorems are not shown in the plot for brevity.

Iv Establishing Theorems 13

Theorems 13 describe results on -connectivity for various random intersection graphs.

Iv-a The Proof of Theorem 1

We demonstrate Theorem 1 with the help of Theorem 3, the proof of which is detailed in Section IV-C.

For any , it is clear that

(10)

We recall conditions (1) and , which together with (10) and Fact 2 yields

(11)

With and , it follows that , which along with (11) and enables the use of Theorem 3 to derive

(12)

Since -connectivity is a monotone increasing graph property [14], Theorem 1 is proved by (12) and Lemma 2.

Iv-B The Proof of Theorem 2

From Lemma 4 and Theorem 3, the proof of Theorem 2 is completed once we show that with defined by

(13)

under conditions of Theorem 2, we have and with defined by

(14)

then

(15)

From conditions (2) and , and Fact 1, it is clear that

(16)

Substituting (16) and condition into (13), it holds that

(17)

and

(18)

Then from (2) (14) (18) and Fact 2, we obtain (15). As explained before, with (14) (15) and (17), Theorem 2 is proved from Lemma 4 and Theorem 3.

As noted in Remark 1, to prove only the zero–one law but not the asymptotically exact probability result in Theorem 2, condition can be weakened as . This can be seen by the argument that under , (15) can be weakened as , which can still used to establish the zero–one law.

Iv-C The Proof of Theorem 3

We derive in [35] the asymptotically exact probability and an asymptotic zero–one law for -connectivity in graph , which is the superposition of an Erdős–Rényi graph on a uniform random intersection graph . Setting , graph becomes . Then with , we obtain from [35, Theorem 1] that if and

(19)

then

(20)

Note that if , then (i) exists if and only if exists; and (ii) when they both exist, . Therefore, Theorem 3 is proved once we show and (19) with given conditions , and (3).

From , (3) and Fact 1, it holds that

(21)

which along with yields

We derive in [7, Lemma 8] that

(22)

Applying (21) to (22),

which together with (3) and Fact 2 leads to (19) with condition . Since we have proved and (19) with , Theorem 3 follows from (20).

V Establishing Theorems 46

Theorems 46 present results on -robustness for various random intersection graphs.

V-a The Proof of Theorem 4

Similar to the process of proving Theorem 1 with the help of Theorem 3, we demonstrate Theorem 4 using Theorem 6, the proof of which is given in Section V-C.

Note that condition (4) is the same as (1), and condition holds. Then as shown in Theorem 1, for any , from (1) (10), and Fact 2, we obtain (11) here. From and , it follows that , which along with (11) enables the use of Theorem 6 to yield that for and any , we have

(23)

Since -robustness is a monotone increasing graph property according to [34, Lemma 3], Theorem 4 is proved by (23) and Lemma 2.

V-B The Proof of Theorem 5

Since -robustness implies -connectivity by [27, Lemma 1], the zero law of Theorem 5 is clear from Theorem 2 and Remark 1 in view that under conditions of Theorem 5, if ,

(24)

Below we prove the one law of Theorem 5. Note that (5) is the same as (2), and we have condition . Then as proved in Theorem 2, given (2) and , we obtain (16), which together with condition leads to

(25)

Noting that (25) implies condition in Lemma 3, we apply Lemmas 1 and 3, and condition (5) to derive the following: there exists such that if ,

(26)

The proof of Theorem 5 is completed via (24) and (26).

V-C The Proof of Theorem 6

The zero law of Theorem 6 is proved below by an approach similar to that of Theorem 5. Since -robustness implies -connectivity by [27, Lemma 1], the zero law of Theorem 6 is clear from Theorem 3 in view that under conditions of Theorem 6, if ,

(27)

Below we establish the one law of Theorem 6 with the help of Theorem 5. Given , we use Lemma 5 to obtain that with set by

(28)

it holds that

(29)

Note that (6) is the same as (3); and holds as a condition. Then as shown in Theorem 3, from (3), and Fact 2, we obtain (21) here, which together with results in

(30)

From and (28), it follows that

(31)

By (6) (31) and Fact 2, it is clear that

(32)

Given (30) (32) and , we use Theorem 5 and (29) to get that if ,

(33)

The proof of Theorem 6 is completed via (27) and (33).

Vi Establishing Lemmas in Section Iii

Lemmas 1 and 4 are clear in Section III. Below we prove Lemmas 2, 3 and 5.

Vi-a The Proof of Lemma 2

According to [2, Lemma 3], for any monotone increasing graph property and any ,

(34)

and

(35)

By (34) (35) and the fact that for (this can be proved by a simple Taylor series expansion as in [7, Fact 2]), the proof of Lemma 2 is completed once we demonstrate that with , there exists such that

(36)

and

(37)

To prove (36) and (37), Chebyshev’s inequality yields

(38)

We set by