1 Existence and Size of the Giant Component in Inhomogeneous Random K-out Graphs Mansi Sood and Osman Yagan˘ Abstract—Random K-out graphs are receiving attention as nodes, denoted by H(n; K), is constructed as follows. Each a model to construct sparse yet well-connected topologies in of the n nodes draws K edges towards K distinct nodes distributed systems including sensor networks, federated learn- chosen uniformly at random from among all other nodes. ing, and cryptocurrency networks. In response to the growing heterogeneity in emerging real-world networks, where nodes The orientation of the edges is then ignored, yielding an differ in resources and requirements, inhomogeneous random undirected graph. Random K-out graphs achieve connectivity K-out graphs were proposed recently. In this model, first, each with probability approaching one in the limit of large n for all of the n nodes is classified as type-1 (respectively, type-2) with K ≥ 2 [5], [7], with the probability of connectivity growing 2 probability µ (respectively, 1−µ) independently from the others, as 1 − Θ(1=nK −1) for all K ≥ 2 [8]. Consequently, random where 0 < µ < 1. Next, each type-1 (respectively, type-2) node K-out graphs get connected very easily, i.e., with far fewer draws 1 arc towards a node (respectively, Kn arcs towards Kn distinct nodes) selected uniformly at random. The orientation of edges (O(n)) as compared to classical random graph models the arcs is ignored yielding the inhomogeneous random K-out including Erdos-R˝ enyi´ (ER) random graphs (O(n log n)). Ow- graph, denoted by H(n; µ, Kn). It was recently established that ing to the simplicity of graph construction and their unique H(n; µ, Kn) is connected with high probability (whp) if and only connectivity properties, random K-out graphs are receiving if K = !(1). Motivated by practical settings where establishing n increasing attention as a model to construct sparse, yet well- links is costly and only a bounded choice of Kn is feasible (Kn = O(1)), we study the size of the largest connected sub- connected topologies in a fully distributed fashion in ad- network of H(n; µ, Kn). We first show that the trivial condition hoc networks. Random K-out graphs have received renewed of Kn ≥ 2 for all n is sufficient to ensure that H(n; µ, Kn) interest for designing securely connected sensor networks contains a giant component of size n − O(1) whp. Next, to model [7], anonymity preserving cryptocurrency networks [9], and settings where nodes can fail or get compromised, we investigate fully decentralized learning networks with differential privacy the size of the largest connected sub-network in H(n; µ, Kn) when dn nodes are selected uniformly at random and removed from guarantees [10]. the network. We show that if dn = O(1), a giant component In the context of wireless sensor networks (WSNs), random of size n − O(1) persists for all Kn ≥ 2 whp. Further, when K-out graphs have been studied [7], [11]–[13] to model dn = o(n) nodes are removed from H(n; µ, Kn), the remaining the random pairwise key predistribution scheme [14]. Along nodes contain a giant component of size n(1 − o(1)) whp for all with the original key predistribution scheme proposed by Kn ≥ 2. We present numerical results to demonstrate the size of the largest connected component when the number of nodes Escheanuer and Gligor [15], the random pairwise scheme is finite. is one of the most widely recognized security protocols for Index Terms—Network analysis and control, random graphs, sensor networks. The random pairwise scheme is implemented connectivity, ad-hoc networks in two phases. In the first phase, each sensor node is paired offline with K distinct nodes chosen uniformly at random among all other sensor nodes. Next, a unique pairwise key I. INTRODUCTION is inserted in the memory of each of the paired sensors. After A. Background deployment, two sensor nodes can communicate securely only arXiv:2009.01610v2 [cs.IT] 29 Jul 2021 The field of random graphs lays the mathematical foun- if they have at least one key in common. The random pairwise dations for modeling and analyzing the complex patterns of key predistribution scheme induces a random K-out graph on interconnections typical to real-world networks including com- the set of the participating sensor nodes [7], and therefore the munication networks [2], social networks [3], and biological connectivity properties of random K-out graphs are of interest networks [4]. A class of random graphs called the random in key management schemes for establishing secure WSNs. K-out graph is one of the earliest known models in the In recent years, the rapid proliferation of affordable sensing literature [5], [6]. The random K-out graph comprising n and mobile devices has led to advances in applications such as remote-sensing, environmental surveillance [16], and decen- This work has been supported in part by the National Science Foundation through grant CCF #1617934, David H. Barakat and LaVerne Owen-Barakat tralized learning [17]. Such emerging applications increasingly Carnegie Institute of Technology Dean’s Fellowship, Pennsylvania Infras- rely on integrating heterogeneous agents that have different tructure Technology Alliance (PITA), CyLab@IoT, and Cylab Presidential capabilities and (security and connectivity) requirements. Con- Fellowship at Carnegie Mellon University. M. Sood and O. Yagan˘ are with Department of Electrical and sequently, the analysis of heterogeneous variants of classical Computer Engineering and CyLab, Carnegie Mellon University, random graph models has emerged as an active area of Pittsburgh, PA, 15213 USA. Email: [email protected], research. [18]–[23], [24]–[28]. A heterogeneous variant of [email protected] A preliminary version of this work was presented at the 59th IEEE random K-out graph known as the inhomogeneous random Conference of Decision and Control (CDC) 2020 [1]. K-out graph was proposed recently to model heterogeneous 2 B C B C instance, if the power available for transmission is limited, picks C picks B, D, F the underlying physical network may not be dense enough to guarantee global connectivity with key predistribution schemes A A [30]. In such resource-constrained environments where it is F picks B F picks B, E, F infeasible to establish a connected network, the goal is to establish a large connected sub-network spanning almost the E E picks B, D, F picks F D D entire network as efficiently as possible, i.e., by using the least amount of resources (links). [31]. For example, if a sensor Fig. 1. An inhomogeneous random K-out graph with 6 nodes. Nodes A; C and E are type-2 and the rest (B; D; F ) are type-1. Each type-1 node selects 1 node network is designed to monitor the temperature of a field, it uniformly at random and each type-2 node selects Kn = 3 nodes uniformly at may suffice to aggregate readings from a majority of sensors in random. An undirected edge is drawn between two nodes if at least one selects the field [32]. Formally, the notion of connected sub-networks the other. is described by connected components; see Definition 3.1 for a precise characterization. With this in mind, the first networks secured by random pairwise key predistribution question which we address is when Kn is bounded (i.e., schemes [19]. Kn = O(1)), how many nodes are contained in the largest The inhomogeneous random K-out graph is constructed on connected component of H(n; µ, Kn)? In the literature on a set of n nodes as follows. First, each node is independently random graphs, this is often studied in terms of the existence classified as type-1 with probability µ and type-2 with proba- and size of the giant component, defined as a connected sub- bility 1−µ, where 0 < µ < 1. The type of the node determines network comprising Ω(n) nodes; see [33] for a classical study the number of selections made by it, viz., each type-1 node on the size of the giant component of Erdos-R˝ enyi´ graphs. selects one node uniformly at random from all other nodes and Not only is it important to design networks that bear large each type-2 node selects Kn ≥ 2 distinct nodes uniformly connected components, but it is also crucial to ensure these at random from all other nodes; see Figure 1. The notation large components are robust to node failures. Sensor networks Kn indicates that the number of selections made by type-2 are often deployed in hostile environments e.g., battlefield nodes scales as a function of the number of nodes (n). Since monitoring and environmental surveillance, which makes the homogeneous random K-out graphs, where all nodes make the nodes susceptible to operational failures and adversarial cap- same number of selections, achieve connectivity whp for any tures. Therefore, it is of interest to analyze the properties of K ≥ 2 [5], [7], the interesting case arises when we allow networks induced by random K-out graphs when some nodes type-1 nodes to select just 1 edge; see Section II. Further, are dishonest, have failed, or have been captured from the the analysis in this paper also generalizes to inhomogeneous network. From a modeling perspective, this translates to ana- random K-out graphs with an arbitrary number of node types; lyzing the connectivity properties of inhomogeneous random see Appendix. Throughout our discussion, we let H(n; µ, Kn) K-out graphs when a subset of nodes are selected uniformly denote the inhomogeneous random K-out graph on n nodes at random and deleted.