On the Robustness, Connectivity and Giant Component Size of Random K-out Graphs

In recent years, the rapid proliferation of affordable sensing and computing devices has led to rapid growth in technologies powered by the IoT (Internet of Things). A key challenge in this space is to develop network models for generating a securely connected ad-hoc network in a distributed fashion while minimizing operational costs.

With its unique connectivity properties, a class of random graph models known as the random $K$-out graphs has found many applications in the design of ad-hoc networks. A random K-out graph [3, 4, 5], denoted as $\mathbb{H}(n;K)$, is an undirected graph with $n$ nodes where each node forms an edge with $K$ distinct nodes chosen uniformly at random. Random K-out graphs are known to achieve connectivity easily, i.e., with far fewer edges $(O(n))$ as compared to classical random graph models including Erdős-Rényi (ER) graphs [6, 3], random geometric graphs [7], and random key graphs [8], which all require $O(n\log n)$ edges for connectivity. In particular, it is known [5, 4] that random K-out graphs are connected with high probability (whp) when $K\geq 2$. This had led to their deployment in several applications including random key predistribution schemes for secure communication in sensor networks [9, 10, 11], differentially-private distributed averaging algorithms [12], anonymity preserving cryptocurrency networks [13], and distributed secure mapping of network addresses in next-generation internet architectures [14].

In the context of sensor networks, random K-out graphs have been used [5, 11, 15] to analyze the performance of the random pairwise key predistribution scheme and its heterogeneous variants [16, 17]. The random pairwise scheme works as follows. Before deployment, each sensor chooses $K$ others uniformly at random. A unique pairwise key is given to each node pair where at least one of them selects the other. After deployment, two sensors can securely communicate if they share a pairwise key. The topology of the sensor network can thus be represented by a random K-out graph; each edge of the random K-out represents a secure communication link between two sensors. Consequently, random K-out graphs have been analyzed to answer key questions on the values of the parameters $n$ and $K$ needed to achieve certain desired properties, including connectivity at the time of deployment [4, 5], connectivity under link removals [11, 15], and unassailability [18]. Despite many prior works on random K-out graphs, very little is known about its connectivity properties when some of its nodes are removed. This is an increasingly relevant problem since many IoT networks are deployed in remote and hostile environments where nodes may be captured by an adversary, or fail due to harsh conditions.

Another application of random K-out graphs is in distributed learning, where a key goal is to perform computations on user data without compromising the privacy of the users. Random K-out graphs have recently been used to construct the communication graph in a differentially-private federated averaging scheme called the GOPA (GOssip Noise for Private Averaging) protocol [12, Algorithm 1]. According to the GOPA protocol, a random K-out graph is constructed on a set of nodes, of which an unknown subset is dishonest. It was shown in [12, Theorem 3] that the privacy-utility trade-offs achieved by the GOPA protocol are tightly dependent on the subgraph on honest nodes being connected. When the subgraph on honest nodes is not connected, it was shown that the performance of GOPA is tied to the size of the connected components of honest nodes. Since dishonest users can be modeled as randomly deleted nodes, analyzing the connectivity and giant component size of random K-out graphs under node deletions is the key in understanding the performance of the GOPA protocol.

In the context of applications discussed in the previous section, we can identify several key properties of random K-out graphs that need to be well-understood for performance evaluation and efficient design of the underlying systems. We believe that the graph properties discussed here can also be useful in facilitating new applications of random K-out graphs in different fields, akin to our recent work [19] paving the way to new applications by establishing connectivity guarantees in the finite node regime.

A key metric in quantifying the utility of a network is connectivity which is defined as the existence of a path of edges between every node pair [20]. Connectivity ensures that all agents in the network can communicate with one another and no node is isolated from the network. In practice, resource constraints limit the number of links that can be established in the network. Thus, a key goal is to design a resiliently connected network while keeping the number of links to be established within operational constraints. Depending on the resource constraints and mission requirements of the application at hand, it may suffice to ensure a weaker notion of connectivity or in cases where agents may routinely fail or compromise, we may even need a stronger notion of connectivity.

In resource-constrained environments, preserving connectivity despite node failures may not be feasible and the goal instead might be to ensure that there is a large enough, connected sub-network of users, also known as a giant component. For example, it may suffice to aggregate the temperature readings of the majority of sensors deployed in a field to get an estimate of the true temperature. Another example is the power grid network where it is essential to ensure supply to the majority of the users in the event of failures.

In addition to ensuring that the network remains resiliently connected in the event of node failures, it is often desirable to ensure that consensus can be achieved even in the presence of adversarial agents. In [21], it was shown that network connectivity is not sufficient to characterize consensus when nodes use a certain class of local filtering rules. In particular, it was shown that consensus can be reached in graphs that have the property of being sufficiently robust. This is formally quantified by the property of $r$-robustness, which was introduced in [21]. A graph is said to be $r$-robust if, for every disjoint subset pair that partitions the graph, at least one node in one of these subsets is adjacent to at least $r$ nodes in the other set.

The $r$-robustness property is especially useful in applications of consensus dynamics, where parameters of several agents get aligned after a sufficiently long period of local interactions. In another example, it was shown [22] that if the network is $(2F+1)$-robust (for some non-negative integer $F$), then the nodes in the network can reach consensus even when there are up to $F$ malicious nodes in the neighborhood of every correctly-behaving node. Thus, $r$-robustness is particularly important for applications based on consensus dynamics in adversarial environments. Moreover, $r$-robustness is known [22] to be a stronger property than $r$-connectivity and thus can provide guarantees on the connectivity of the graph when up to $r-1$ nodes in the graph are removed. Due to this importance, $r$-robustness has been studied in various graph models in the random graph literature, such as the ER graph and the Barabási-Albert model [23]. $r$-robustness has been studied in [24] in our prior work; and the results presented here improve upon those results by providing a sharper threshold for $r$-robustness.

With these motivations in mind, this paper aims to fill the gaps in the literature on the connectivity and robustness properties of random K-out graphs. We provide a comprehensive set of results on the connectivity and size of the giant component of the random K-out graph when some of its nodes are dishonest, have failed, or have been captured. We further analyze the conditions required for ensuring $r$-robustness of the random K-out graph. Our main contributions are summarized below:

1. 1.

   Let $\mathbb{H}(n;K_{n},\gamma_{n})$ denote the random graph obtained after removing $\gamma_{n}$ nodes, selected uniformly at random, from the random K-out graph $\mathbb{H}(n;K_{n})$. We provide a set of conditions for $K_{n}$, $n$, and $\gamma_{n}$ under which $\mathbb{H}(n;K_{n},\gamma_{n})$ is connected with high probability (whp). This is done for both cases where $\gamma_{n}=\Omega(n)$ and $\gamma_{n}=o(n)$, respectively. Our result for $\gamma_{n}=\Omega(n)$ (see Theorem III.1) significantly improves a prior result [25] on the same problem and leads to a sharp zero-one law for the connectivity of the random K-out graph under node deletions. Our result for the case $\gamma_{n}=o({n})$ (see Theorem III.2) expands the existing threshold of $K_{n}\geq 2$ required for connectivity by showing that the graph is connected whp for $K_{n}\geq 2$ even when $o(\sqrt{n})$ nodes are deleted.

2. 2.

   We derive conditions on $K_{n}$, $n$, $\gamma_{n}$ that lead to a giant component in $\mathbb{H}(n;K_{n},\gamma_{n})$ whp and provide an upper bound on the number of nodes not contained in the giant component. This is also done for both cases $\gamma_{n}=\Omega(n)$ and $\gamma_{n}=o(n)$; see Theorem III.3 and Theorem III.4, respectively. An important consequence of this result is to establish $K_{n}\geq 2$ as a sufficient condition to ensure whp the existence of a giant component in the random K-out graph despite the removal of $o(n)$ nodes in the network.

3. 3.

   Using a novel proof technique, we show that $K\geq 2r$ when $r\geq 2$ is sufficient to ensure that the random K-out graph $\mathbb{H}(n;K)$ is $r$-robust whp (see Theorem III.5). Since it is already known that $\mathbb{H}(n;K)$ is not $r$-robust whp when $K<r$, this result is tight up to at most a multiplicative factor of two (and it is is much tighter than the condition established in [24]).

4. 4.

   We also provide a comparison of random K-out graphs with ER graphs. We determine that random K-out graphs are much more robust in terms of $r$-robustness property, and also attain connectivity and admit a giant component with fewer edges compared to ER graphs.

5. 5.

   Combining our theoretical results with numerical simulations, we highlight the usefulness of random K-out graphs as a topology design tool for efficient design of secure, resilient and robust distributed networks.

The rest of this article is organized as follows. In Section II, we introduce the notation used across this article and the network model, the random K-out graph, and extend this model to account for node deletions. In Section III, we present the main results along with the simulation results and provide a detailed discussion. In Section IV, we provide the proof of all Theorems presented in Section III. Conclusions are provided in Section V.

Let $P(n,K_{n},\gamma_{n})=\mathbb{P}\left[\mathbb{H}(n;K_{n},\gamma_{n})\text{ is connected}\right]$.

The proof of the one-law in (3), i.e., that $\lim_{n\to\infty}P(n,K_{n},\gamma_{n})=1$ if $c>1$, is given in Section IV. The zero-law of (3), i.e., that $\lim_{n\to\infty}P(n,K_{n},\gamma_{n})=0$ if $c<1$, was established previously in [25, Corollary 3.3]. There, a one-law was also provided: under (2), it was shown that $\lim_{n\to\infty}P(n,K_{n},\gamma_{n})$ if $c>\frac{1}{1-\alpha}$, leaving a gap between the thresholds of the zero-law and the one-law. Theorem III.1 presented here fills this gap by establishing a tighter one-law, and constitutes a sharp zero-one law; e.g., when $\alpha=0.5$, the one-law in [25] is given with $c>2$, while we show that it suffices to have $c>1$.

We remind that random K-out graph is known [4, 5] to be connected whp when $K_{n}\geq 2$. Part $(a)$ of Theorem III.2 extends this result by showing that having $K_{n}\geq 2$ is sufficient for the random K-out graph to remain connected whp even when $o(\sqrt{n})$ of its nodes (selected randomly) are deleted. We believe that this result will further facilitate the application of random K-out graphs in a wide range of applications where connectivity despite node failures is crucial.

Let $C_{max}(n,K_{n},\gamma_{n})$ denote the set of nodes in the largest connected component of $\mathbb{H}(n;K_{n},\gamma_{n})$ and let $P_{G}(n,K_{n},\gamma_{n},\lambda_{n}):=\mathbb{P}[|C_{max}(n,K_{n},\gamma_{n})|>n-\gamma_{n}-\lambda_{n}]$. Namely, $P_{G}(n,K_{n},\gamma_{n},\lambda_{n})$ is the probability that less than $\lambda_{n}$ nodes are outside the largest component of $\mathbb{H}(n;K_{n},\gamma_{n})$.

We remark that if $\lambda_{n}=\beta n$ with $0<\beta<1/3$, then $r_{3}(\gamma_{n},\lambda_{n})=1+o(1)$. This shows that when $\gamma_{n}=o(n)$, it suffices to have $K_{n}\geq 2$ for $\mathbb{H}(n;K_{n},\gamma_{n})$ to have a giant component containing $(1-\beta)n$ nodes for arbitrary $0<\beta<1/3$. Put differently, by choosing $K_{n}\geq 2$, we ensure that even when $\gamma_{n}=o(n)$ nodes are removed, the rest of the network contains a connected component whose fractional size is arbitrarily close to 1.

It can be seen from this result that $K_{n}$ needs to scale as $K_{n}\sim\log(\frac{\alpha n}{\lambda_{n}})$ for a random K-out graph to have a giant component of size $n-\lambda_{n}$ when $\alpha n$ of its nodes are removed (or, if each node is independently removed with probability $0<\alpha<1$). We also remark that the threshold $r_{4}(\alpha,\lambda_{n})$ is finite when $\lambda_{n}=\Omega(n)$. This shows that even when a positive fraction of the nodes of the random K-out graph are removed, a finite $K_{n}$ is still sufficient to have a giant component of size $\Omega(n)$ in the graph. This result can be useful in applications where it is required to maintain a giant component as efficiently (i.e., with as fewest edges) as possible even when large scale node failures take place.

In Theorem III.5, we establish a threshold for one-law of $r$-robustness in random K-out graphs, and find that $r^{\star}(K)\geq\lfloor K/2\rfloor$. In other words, we find that with high probability, a random K-out graph is $r$-robust when $K\geq 2r,r\geq 2,r\in\mathbb{Z}^{+}$, and $n\to\infty$. This threshold is much smaller than the previously established threshold [24] of $K>\frac{2r\left(\log(r)+\log(\log(r)+1\right)}{\log(2)+1/2-\log\left(1+\frac{\log(2)+1/2}{2\log(r)+5/2+\log(2)}\right)}$ which scales with $r\log r$. Hence, Theorem III.5 constitutes a sharper one-law for $r$-robustness.

In Theorem III.1, we improve the results given in [25], and with this, we close the gap between the zero law and the one law, and hence establish a sharp zero-one law for connectivity when $\gamma_{n}=\Omega(n)$ nodes are deleted from $\mathbb{H}(n;K_{n},\gamma_{n})$. In Theorem III.2, we establish that the graph $\mathbb{H}(n;K_{n},\gamma_{n})$ with $\gamma_{n}=o(n)$ is connected whp when $K_{n}\sim\log(\gamma_{n})$; and when $\gamma_{n}=o(\sqrt{n})$, $K_{n}\geq 2$ is sufficient for connectivity. The latter result is especially important, since $K_{n}\geq 2$ is the previously established threshold for connectivity [4]. We improve this result by showing that the graph is still connected with $K_{n}\geq 2$ even after $o(\sqrt{n})$ nodes (selected randomly) are deleted.

Since most distributed systems require connectivity in the event of node failures, our results can be useful in many applications of distributed systems, particularly when the resources on each node is limited and it is critical to achieve desired connectivity and robustness properties using as few edges as possible. For example, in wireless sensor networks, knowing the minimum conditions needed for connectivity or giant component size under such failures is crucial as it enables designing them with fewest edges possible per node [18, 28], which reduces the communication overhead and potentially the cost of the hardware on each node.

We also note that Theorems III.3 - III.4 constitute the first results concerning the giant component size of random K-out graphs under randomly deleted nodes. In particular, these results help choose the value of $K_{n}$ for any anticipated level of node failure and for any given giant component size required, enabling the designs of distributed systems to compromise between efficiency, robustness, and the required giant component size. Thus, we expect these results to be useful in applications where connectivity is not a stringent condition under node failures, and instead having a certain giant component size is sufficient to continue the operation of the system.

In Theorem III.5, we establish that a random K-out graph is $r$-robust whp when $K_{n}\geq 2r$ for any $r\in\mathbb{Z}^{+}$. This is a much sharper one-law than the previous result given in [24] where it was shown that $K_{n}$ needs to scale as $K_{n}\sim r\log(r)$ for $r$-robustness. This tighter result was made possible through several novel steps introduced here. While the proofs in prior work [29, 24] also rely on finding upper bounds on the probability of having at least one subset that is not $r$-reachable, they tend to utilize standard upper bounds for the binomial coefficients ${n\choose k}\leq\left(\frac{en}{k}\right)^{k}$ and a union bound to establish them. Instead, our proof uses extensively the Beta function $B(a,b)$ and its properties to obtain tighter upper bounds on such probabilities, which then enables us to establish a much sharper one-law for $r$-robustness of random K-out graphs. We believe this result will pave the way for further applications of random K-out graphs in distributed computing applications such as the design of consensus networks in the presence of adversaries.

It is also of interest to compare the threshold of $r$-robustness and $r$-connectivity in random K-out graphs. For Erdős-Rényi graphs, the threshold for $r$-connectivity and $r$-robustness have been shown [29] to coincide with each other. For random K-out graphs, we know from [4] that $\mathbb{H}(n;K_{n})$ is $r$-connected whp whenever $K_{n}\geq r$, and it is not $r$-connected whp if $K_{n}<r$. This leaves a factor of 2 difference between the condition $K_{n}\geq 2r$ we established for $r$-robustness here and the threshold of $r$-connectivity. Put differently, we know from [4] and Theorem III.5 that for any $r=2,3,\ldots$

For $r=1$, it is instead known that $\lim_{n\to\infty}\mathbb{P}\left[\mathbb{H}(n;K_{n})\text{ is $r$-robust}\right]=1$ if only if $K_{n}\geq 2r=2$. Since the currently established conditions for the zero-law and one-law of $r$-robustness are not the same for random K-out graphs (unlike ER graphs where the two thresholds coincide), there is a question as to whether our threshold of $2r$ is the tightest possible for $r$-robustness. This is currently an open problem and would be an interesting direction for future work, e.g., by establishing a tighter zero-law for $r$-robustness that coincides with the one-law of Theorem III.5. Comparison

To put all these results in perspective, we provide comparisons of our results with the results from an Erdős-Rényi graph $G(n,p)$, which is one of the most commonly used random graph models. Firstly, in terms of $r$-robustness, it was shown in [29] that ER graph G(n; p) is $r$-robust whp if $p_{n}=\frac{\log(n)+(r-1)\log(\log(n))+\omega(1)}{n}$, which translates to a mean node degree of $\langle d\rangle\sim\log(n)+(r-1)\log(\log(n))$. Since the mean node degree required for random K-out graphs scales as $\langle d\rangle\sim 2r$, we can conclude that for large $n$, random K-out graphs can be ensured to be $r$-robust whp at a mean node degree significantly smaller than the mean node degree required for an ER graph. In terms of connectivity, an ER graph becomes connected whp if $p>\log n/n$ [30], and this translates to having a mean node degree of $\langle d\rangle\sim\log n$. Similarly, when $o(\sqrt{n})$ nodes are removed, the mean node degree required for connectivity scales with $\langle d\rangle\sim\log n$ [6]. The $\langle d\rangle$ required for the random K-out graph to be connected whp is much lower, with $\langle d\rangle=O(1)$ when $o(\sqrt{n})$ nodes are removed, and $\langle d\rangle\sim\log(\gamma_{n})$ when $\gamma_{n}=\Omega(\sqrt{n})$ nodes are removed.

Next, since we are not aware of any theoretical results on the giant component size of ER graphs under node removals, we compare the size of the giant component in random K-out graphs and ER graphs under node removals through simulations in the finite node regime. We examine the maximum number of nodes outside the giant component out of 1000 experiments of a random K-out graph $\mathbb{H}(n;K_{n},\gamma_{n})$ and an ER graph $G(n,p)$ with the same mean node degree when $\gamma_{n}$ nodes are removed from both graphs. To ensure that both graphs have the same mean node degree, $p$ in the ER graph is selected as $p=2K_{n}/n$. The results are given in Fig. 2 for $n=5000$, $\gamma_{n}=0.4n$ on (Left), and $n=50,000$, $\gamma_{n}=500$ on (Right). As can be seen, the random K-out graph tends to have fewer nodes outside of the giant component than the ER graph and this difference is more pronounced when $\gamma_{n}$ is smaller.

In conclusion, we see that when both graphs have the same mean node degree, random K-out graphs are more robust than ER graphs in terms of connectivity and giant component size under random node removal, and also in terms of the $r$-robustness property. This reinforces the efficiency of the K-out construction in many distributed computing applications where connectivity in the event of node failures or adversarial capture of nodes is crucial. Similarly, the fact that random K-out graphs tend to achieve $r$-robustness with fewer edges per node than ER graphs (for any $r=1,2,\ldots$), makes it more suitable in applications based on distributed consensus.

Since our results are asymptotic in nature, i.e., they have been established in the limit $n\to\infty$, an important question is whether they can also be useful in practical settings where the number $n$ of nodes is finite. We check the usefulness to validate Theorems III.1 - III.4 under practical settings, To answer this, we examine the probability of connectivity and the number of nodes outside the giant component for the graph $\mathbb{H}(n;K_{n},\gamma_{n})$ (random K-out graph with deleted nodes) through computer simulations in two different setups¹¹1Determining whether a graph is $r$-robust is a co-NP-complete problem [29] making it not feasible to check the usefulness of Theorem III.5 through computer simulations..

In the first setup, we consider the case where the number of deleted nodes, $\gamma_{n}=\alpha n$, with $\alpha$ in $(0,1)$. We generate instantiations of the random graph $\mathbb{H}(n;K_{n},\gamma_{n})$ with $n=5000$, varying $K_{n}$ in the interval $[1,25]$ and consider several $\alpha$ values in the interval $[0.1,0.8]$. Then, we record the empirical probability of connectivity of the graph $\mathbb{H}(n;K_{n},\gamma_{n})$ and $\lambda_{n}$ from 1000 independent experiments for each $(K_{n},\alpha)$ pair. The results of this experiment are shown in Fig. 3 and Fig. 4. Fig. 3 (Left) depicts the empirical probability of connectivity of $\mathbb{H}(n;K_{n},\gamma_{n})$. The vertical lines stand for the critical threshold of connectivity obtained from Theorem III.1. In each curve, $P(n,K_{n},\gamma_{n})$ exhibits a threshold behaviour as $K_{n}$ increases, and the transition from $P(n,K_{n},\gamma_{n})=0$ to $P(n,K_{n},\gamma_{n})=1$ takes place around $K_{n}=\frac{\log n}{1-\alpha-\log\alpha}$, the threshold established in (2), reinforcing the usefulness of Theorem III.1 under practical settings.

In Fig. 4, the maximum number of nodes outside the giant component in 1000 experiments is plotted for each parameter pair. For comparison, we also plot the upper bound on $n-\gamma_{n}-|C_{max}|$ obtained from Theorem III.4 by taking the maximum $\gamma_{n}$ value that gives a threshold less than or equal to the $K_{n}$ value tested in the simulation. As can be seen, for any $K_{n}$ and $\gamma_{n}$ value, the experimental maximum number of nodes outside the giant component is smaller than the upper bound obtained from Theorem III.4, validating the usefulness of this result in the finite node regime.

The goal of the second experimental setup is to examine the case where the number of deleted nodes is $\gamma_{n}=o(n)$. As before, we generate instantiations of the random graph $\mathbb{H}(n;K_{n},\gamma_{n})$, with $n=50,000$, varying $K_{n}$ in $[2,5]$, and varying $\lambda_{n}$ in $[10,2000]$. For each $(K_{n},\gamma_{n})$ pair, the maximum number of nodes outside the giant component in 1000 experiments is recorded; if no node is outside the giant component, then it is understood that the graph is connected. In Fig. 3 (Right), the maximum number of nodes outside the giant component observed in $1000$ experiments is depicted as a function of $K_{n}$. The plots for $\gamma_{n}=10$ and $\gamma_{n}=100$ are considered to represent the case $\gamma_{n}=o(\sqrt{n})$ in Theorem III.2(a). We see that there is at most one node outside the giant component for $\gamma_{n}=10$ and $\gamma_{n}=100$, even when $K_{n}=2$. This shows that the asymptotic behavior given in Theorem III.2(a), i.e., that random K-out graph remains connected if $o(\sqrt{n}$ nodes are deleted, already appears when $n=50,000$. The plots for $\gamma_{n}=1000$ and $\gamma_{n}=2000$ are used to check the case $\gamma_{n}=w(\sqrt{n})$ and $\gamma_{n}=o(n)$ in Theorem III.2(b). The thresholds on $K_{n}$ for these $\gamma_{n}$ values, obtained using Theorem III.2(b) are $r_{2}(1000)=6.79$ and $r_{2}(2000)=7.37$, rounded to two digits after decimal (the $\omega(1)$ term in Theorem III.2(b) is ignored due to $n$ having a finite value in the simulations). It is clear from the plot that when $K_{n}\geq 4$, the graph with $\gamma_{n}=1000$ is connected, while $K_{n}\geq 5$ suffices to ensure connectivity when $\gamma_{n}=2000$. Thus, selecting $K_{n}$ above the theoretical thresholds given in III.2(b) is seen to ensure the connectivity of the graph in the finite node regime as well, supporting the usefulness of Theorem III.2(b) in practical cases.

In Fig. 5, the maximum number of nodes outside the giant component in 1000 experiments is plotted as a function of $K_{n}$. For comparison, we also plot the upper bound on $n-|C_{max}|$ obtained from Theorem III.4. In particular, for each Theorem, the maximum $\gamma_{n}$ value that gives a threshold less than or equal to the $K_{n}$ value tested in the simulation is found. Then, the lowest of these maximum $\gamma_{n}$ values is used as the theoretical $n-|C_{max}|$ value. As can be seen, for any $K_{n}$ and $\gamma_{n}$ value, the experimental maximum number of nodes outside the giant component is smaller than the upper bounds obtained from Theorem III.4, reinforcing the usefulness of our results in the finite node regime.

Since the preliminary steps related to the proofs of III.1 - III.4 are the same, in this Section we present these steps. First, recall from Section II that the metrics connectivity and the size of the giant component under node removals are defined for the graph $\mathbb{H}(n;K_{n},\gamma_{n})$, where the set $D$ of nodes is removed from the graph $\mathbb{H}(n;K_{n})$; also recall that $R=\mathrm{V}\setminus D$. Let $\mathcal{N}_{R}$ denote the set of labels of the vertex set of $\mathbb{H}(n;K_{n},\gamma_{n})$ and let $\mathcal{E}_{n}(K_{n},\gamma_{n};S)$ denote the event that $S\subset\mathcal{N}_{R}$ is a cut in $\mathbb{H}(n;K_{n},\gamma_{n})$ as per Definition II.2. The event $\mathcal{E}_{n}(K_{n},\gamma_{n};S)$ occurs if no nodes in $S$ pick neighbors in $S^{\rm c}$, and no nodes in $S$ pick neighbors in $S^{\rm c}$. Note that nodes in $S$ or $S^{\rm c}$ can still pick neighbors in the set $\mathcal{N}_{D}$. Thus, we have

Let $\mathcal{Z}(\lambda_{n};K_{n},\gamma_{n})$ denote the event that $\mathbb{H}(n;K_{n},\gamma_{n})$ has no cut $S\subset\mathcal{N}_{R}$ with size $\lambda_{n}\leq|S|\leq n-\gamma_{n}-\lambda_{n}$ where $x:\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$ is a sequence such that $\lambda_{n}\leq(n-\gamma_{n})/{2}\ \forall n$. In other words, $\mathcal{Z}(\lambda_{n};K_{n},\gamma_{n})$ is the event that there are no cuts in $\mathbb{H}(n;K_{n},\gamma_{n})$ whose size falls in the range $[\lambda_{n},n-\gamma_{n}-\lambda_{n}]$.

Lemma IV.1 states that if the event $\mathcal{Z}(\lambda_{n};K_{n},\gamma_{n})$ holds, then the size of the largest connected component of $\mathbb{H}(n;K_{n},\gamma_{n})$ is greater than $n-\gamma_{n}-\lambda_{n}$; i.e., there are less than $\lambda_{n}$ nodes outside of the giant component of $\mathbb{H}(n;K_{n},\gamma_{n})$. Also note that $\mathbb{H}(n;K_{n},\gamma_{n})$ is connected if $\mathcal{Z}(\lambda_{n};K_{n},\gamma_{n})$ takes place with $\lambda_{n}=1$, since a graph is connected if no node is outside the giant component. In order to establish the Theorems III.1-III.4., we need to show that $\lim_{n\to\infty}\mathbb{P}[\mathcal{Z}(\lambda_{n};K_{n},\gamma_{n})^{\rm c}]=0$ with $\lambda_{n}$, $K_{n}$ and $\gamma_{n}$ values as stated in each Theorem. From the definition of $\mathcal{Z}(\lambda_{n};K_{n},\gamma_{n})$, we have

where $\mathcal{P}_{n}$ is the collection of all non-empty subsets of $\mathcal{N}_{R}$. Complementing both sides and using the union bound, we get

where $\mathcal{P}_{n,r}$ denotes the collection of all subsets of $\mathcal{N}_{R}$ with exactly $r$ elements. For each $r=1,\ldots,\left\lfloor(n-\gamma_{n})/2\right\rfloor$, we can simplify the notation by denoting $\mathcal{E}_{n,r}({K}_{n},{\gamma_{n}})=\mathcal{E}_{n}({K}_{n},{\gamma_{n}};\{1,\ldots,r\})$. From the exchangeability of the node labels and associated random variables, we have

$|\mathcal{P}_{n,r}|={n-\gamma_{n}\choose r}$, since there are ${n-\gamma_{n}\choose r}$ subsets of $\mathcal{N}_{R}$ with r elements. Thus, we have

Substituting this into (8), we obtain

Remember that $\mathcal{E}_{n,r}({K}_{n},{\gamma_{n}})$ is the event that the $n-\gamma_{n}-r$ nodes in $S$ and $r$ nodes in $S^{\rm c}$ do not pick each other, but they can pick nodes from the set $\mathcal{N}_{D}$. Thus, we have:

Abbreviating $\mathbb{P}\left[\mathcal{Z}(1;K_{n},\gamma_{n})^{\rm c}\right]$ as $P_{Z}$, we get from (9) that

Using the upper bound on binomials (14) again, we have

In order to establish the Theorems, we need to show that (12) goes to zero in the limit of large n for $\lambda_{n}$, $\gamma_{n}$ and $K_{n}$ values as specified in each Theorem.

Since they will be referred to frequently throughout the proofs, we also include here the following standard bounds.

Recall that in Theorem III.1, we have $\gamma_{n}=\alpha n$ with $0<\alpha<1$ and that we need $\lambda_{n}=1$ for connectivity. Using (13) in (12), we have

We will show that the right side of the above expression goes to zero as $n$ goes to infinity. Let

We write