Figure 3 - uploaded by Martin Haenggi
Content may be subject to copyright.
Source publication
Secrecy graphs model the connectivity of wireless networks under secrecy constraints. Directed edges in the graph are present whenever a node can talk to another node securely in the presence of eavesdroppers. In the case of infinite networks, a critical parameter is the maximum density of eavesdroppers that can be accommodated while still guarante...
Similar publications
We establish a sufficient condition to obtain continuum percolation on a class of proximity graphs when their vertices are distributed under a stationary Pois-son point process with unit intensity in the plane. We apply this result on a family of graphs which generalizes β-skeleton graphs.
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster...
We consider soft random geometric graphs, constructed by distributing points (nodes) randomly according to a Poisson Point Process, and forming links between pairs of nodes with a probability that depends on their mutual distance, the "connection function." Each node has a probability of being isolated depending on the locations of the other nodes;...
Random geometric networks are graphs consisting of a set of nodes placed
according to a point process in some domain
$\mathcal{{V}}\subseteq\mathbb{R}^{d}$ mutually coupled with a probability
dependent on their Euclidean separation. We consider the well established
`betweenness' centrality measure, which quantifies how often the shortest paths
in t...
We define a family of random trees in the plane. Their nodes of level k, k = 0,...,m are the points of a homogeneous Poisson point process Πk, whereas their arcs connect nodes of level k and k + 1, according to the least distance principle: If V denotes the Voronoi cell w.r.t. Πk+1 with nucleus x, where x is a point of Πk+1, then there is an arc co...
Citations
... In the study of physical layer security, geometric secrecy graphs have been introduced in [4][5][6] to model large-scale wireless networks in the presence of eavesdroppers, in order to analyse the secrecy connection. However, existing works mainly use secrecy graphs for analysing the security properties of networks, rather than for the network design motivating security improvement, which is one of the objectives of this contribution. ...
This study proposes a security-oriented cooperation (SOC) scheme, which first determines one of the three possible cooperation scenarios, namely the jammer only, relay only and the relay-jammer pair, according to the security requirement and network's operational conditions. After determining a cooperation scenario, then, the selection of a relay or/and a jammer as well as their transmit power are jointly optimised with the objective to attain a good trade-off between security performance and energy consumption. The performance of the networks employing the proposed SOC is investigated here over finite-state Markov channels. The studies and results explain that the proposed SOC scheme constitutes one of the promising SOC schemes. It belongs to a generalised cooperative security scheme, which adapts according to the specific security requirement and communication environments.
... Motivation for considering continuum AB percolation is discussed in detail in [3]; the main motivation comes from wireless communications networks with two types of transmitter. Another type of continuum percolation model with two types of particle is the secrecy random graph [9] in which the type B particles (representing eavesdroppers) inhibit percolation; each type A particle may send a message to every other type A particle lying closer than its nearest neighbour of type B. See also [7]. Such models are not considered here but are complementary to ours. ...
Consider a bipartite random geometric graph on the union of two independent
homogeneous Poisson point processes in $d$-space, with distance parameter $r$
and intensities $\lambda,\mu$. We show for $d \geq 2$ that if $\lambda$ is
supercritical for the one-type random geometric graph with distance parameter
$2r$, there exists $\mu$ such that $(\lambda,\mu)$ is supercritical (this was
previously known for $d=2$). For $d=2$ we also consider the restriction of this
graph to points in the unit square. Taking $\mu = \tau \lambda$ for fixed
$\tau$, we give a strong law of large numbers as $\lambda \to \infty$, for the
connectivity threshold of this graph.
... In the second model, a fixed number of points are independently and uniformly distributed (IUD) in a certain region of the plane, characterised by a single parameter, the fixed number of points. These two models are widely used in the literature on information-theoretic secrecy [12][13][14][15], the reason being twofold. First, they provide a good first-order approximation for the spatial distribution of communication nodes in real networks. ...
Information-theoretic secrecy is combined with cryptographic secrecy to create a secret-key exchange protocol for wireless
networks. A network of transmitters, which already have cryptographically secured channels between them, cooperate to exchange a secret key with a new receiver at a random location, in the presence of passive eavesdroppers at unknown locations. Two spatial point processes, homogeneous Poisson process and independent uniformly distributed points, are used for the spatial distributions of transmitters and eavesdroppers. We analyse the impact of the number of cooperating transmitters
and the number of eavesdroppers on the area fraction where secure communication is possible. Upper bounds on the probability of existence of positive secrecy between the cooperating transmitters and the receiver are derived. The closeness of the upper bounds to the real value is then estimated by means of numerical simulations. Simulations also indicate that a deterministic spatial distribution for the transmitters, for example, hexagonal and square lattices, increases the probability of existence of positive secrecy capacity compared to the random spatial distributions. For the same number of friendly nodes, cooperative transmitting provides a dramatically larger secrecy region than cooperative jamming and cooperative relaying.
... Of recent interest is the problem of percolation in wireless networks in the presence of eavesdroppers [5,8,9]. In these models, referred to as the information theoretic secure models, a legitimate node i is connected (has an edge) to node j provided node j is closer to node i than its nearest eavesdropper. ...
... Assuming that the legitimate and eavesdropper nodes are distributed according to independent Poisson point processes in R 2 of intensities λ, and λ E , respectively, existence of phase transition of percolation in these graphs was established in [5,8,9]. ...
... Using a branching process argument [5,9] show that if the ratio λ/λ E < 1, then ...
We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λ
E
, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λ
E
of eavesdropper nodes, there exists a critical intensity λ
c
< ∞ such that for all λ > λ
c
the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λ
c
a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.
... In recent years, information-theoretic security has re-gained popularity since it is a powerful tool to study security of wireless networks. In [5] and [6] Haenggi and Sarkar introduced the concept of a secrecy graph. Nodes in a wireless network are represented as vertices, and edges between vertices exist only if the secrecy capacity between the corresponding nodes in the wireless network is positive. ...
... We use directed weighted graphs to represent the existence of positive secrecy capacity between any pair of communicating nodes. Such graphs are called directed secrecy graphs in [5] and [6]. Each communicating node is represented as a vertex. ...
The use of cooperation to increase the information-theoretic secrecy in a decentralized ad-hoc wireless network is investigated. In particular, four cases of cooperative relaying are analyzed and compared. These cases include the no-cooperation case, the case with single-hop cooperation with multiple relays, the case with single-hop cooperation with the strongest relay, and finally the case of multi-hop cooperation with the strongest relay. From the results, it is seen that cooperation increases the probability of a positive secrecy rate between two nodes in a network with friendly (cooperative) nodes and eavesdropping nodes. The improved information-theoretic secrecy increases the probability that two nodes can share a secret message with perfect secrecy using a multi-hop route of trusted nodes. Cooperative jamming is also studied, and it is observed that very often it is more beneficial to use friendly nodes for cooperative jamming than cooperative relaying. Finally, a combination of cooperative relaying and jamming is considered.
Connectivity is an important key performance indicator and a focal point of research in large-scale wireless networks. Due to path-loss attenuation of electromagnetic waves, direct wireless connectivity is limited to proximate devices. Nevertheless, connectivity among distant devices can still be attained through a sequence of consecutive multi-hop communication links, which enables routing and disseminating legitimate information across wireless ad hoc networks. Multi-hop connectivity is also foundational for data aggregation in the Internet of things (IoT) and cyberphysical systems (CPS). On the downside, multi-hop wireless transmissions increase susceptibility to eavesdropping and enable malicious network attacks. Hence, security-aware network connectivity is required to maintain communication privacy, detect and isolate malicious devices, and thwart the spreading of illegitimate traffic (e.g., viruses, worms, falsified data, illegitimate control, etc.). In 5G and beyond networks, an intricate balance between connectivity, privacy, and security is a necessity due to the proliferating IoT and CPS, which are featured with massive number of wireless devices that can directly communicate together (e.g., device-to-device, machine-to-machine, and vehicle-to-vehicle communication). In this regards, graph theory represents a foundational mathematical tool to model the network physical topology. In particular, random geometric graphs (RGGs) capture the inherently random locations and wireless interconnections among the spatially distributed devices. Percolation theory is then utilized to characterize and control distant multi-hop connectivity on network graphs. Recently, percolation theory over RGGs has been widely utilized to study connectivity, privacy, and security of several types of wireless networks. The impact and utilization of percolation theory are expected to further increase in the IoT/CPS era, which motivates this tutorial. Towards this end, we first introduce the preliminaries of graph and percolation theories in the context of wireless networks. Next, we overview and explain their application to various types of wireless networks.
A random graph model on a host graph H$$ H $$ is said to be 1‐independent if for every pair of vertex‐disjoint subsets A,B$$ A,B $$ of E(H)$$ E(H) $$, the state of edges (absent or present) in A$$ A $$ is independent of the state of edges in B$$ B $$. For an infinite connected graph H$$ H $$, the 1‐independent critical percolation probability p1,c(H)$$ {p}_{1,c}(H) $$ is the infimum of the p∈[0,1]$$ p\in \left[0,1\right] $$ such that every 1‐independent random graph model on H$$ H $$ in which each edge is present with probability at least p$$ p $$ almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that p1,c(ℤd)$$ {p}_{1,c}\left({\mathbb{Z}}^d\right) $$ tends to a limit in [12,1]$$ \left[\frac{1}{2},1\right] $$ as d→∞$$ d\to \infty $$, and they asked for the value of this limit. We make progress on a related problem by showing that limn→∞p1,c(ℤ2×Kn)=4−23=0.5358….$$ \underset{n\to \infty }{\lim }{p}_{1,c}\left({\mathbb{Z}}^2\times {K}_n\right)=4-2\sqrt{3}=0.5358\dots . $$In fact, we show that the equality above remains true if the sequence of complete graphs Kn$$ {K}_n $$ is replaced by a sequence of weakly pseudorandom graphs on n$$ n $$ vertices with average degree ω(logn)$$ \omega \left(\log n\right) $$. We conjecture the answer to Balister and Bollobás's question is also 4−23$$ 4-2\sqrt{3} $$.
A probability measure on the subsets of the edge set of a graph G is a 1‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , denote by the associated random graph model. Let denote the collection of 1‐ipms on G for which each edge is included in with probability at least p. For , Balister and Bollobás asked for the value of the least p⋆ such that for all p > p⋆ and all , almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p⋆. We also determine the 1‐independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f1, G(p), the infimum over all of the probability that is connected. We determine f1, G(p) exactly when G is a path, a complete graph and a cycle of length at most 5.
Working in the infinite plane R2, consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration Aλ, which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability pλ(n) that this disc is covered by Aλ? We prove that if λ3nlogn = y then, for sufficiently large n, e-8π2
y
≤ pλ(n) ≤ e-2π2
y/3. The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.
The connectivity graph of wireless networks, under many models as well as in practice, may contain unidirectional links. The simplifying assumption that such links are useless is often made, mainly because most wireless protocols use per-hop acknowledgments. However, two-way communication between a pair of nodes can be established as soon as there exists paths in both directions between them. Therefore, instead of discarding unidirectional links, one might be interested in studying the strongly connected components of the connectivity graph. In this paper, we look at the percolation phenomenon in some directed random geometric graphs that can be used to model wireless networks. We show that among the nodes that can be reached from the origin, a non-zero fraction can also reach the origin. In other words, the percolation threshold for strong connectivity is equal to the threshold for one-way connectivity.
