More Benefits of Adding Sparse Random Links to Wireless Networks: Yet Another Case for Hybrid Networks

We theoretically and experimentally analyze the process of adding sparse random links to random wireless networks modeled as a random geometric graph. While this process has been previously proposed, we are the first to prove theoretical bounds on the improvement to the graph diameter and random walk properties of the resulting graph as a function of the frequency of wires used, where this frequency is diminishingly small. In particular, given a parameter k controlling sparsity, any node has a probability of 1 / k 2 n r 2 for being a wired link station. Amongst the wired link stations, we consider creating a random 3-regular graph superimposed upon the random wireless network to create model G 1 , and alternatively we consider a sparser model G 2 as well, which is a random 1-out graph of the wired links superimposed upon the random wireless network. We prove that the diameter for G 1 is O ( k + log     ( n ) ) with high probability and the diameter for G 2 is O ( k log     ( n ) ) with high probability, both of which exponentially improve the Θ ( n / log n ) diameter of the random geometric graph around the connectivity threshold, thus also inducing small-world characteristics as the high clustering remains unchanged. Further, we theoretically demonstrate that as long as k is polylogarithmic in the network size, G 1 has rapidly mixing random walks with high probability, which also exponentially improves upon the mixing time of the purely wireless random geometric graph, which yields direct improvement to the performance of distributed gossip algorithms as well as normalized edge connectivity. Finally, we experimentally confirm that the algebraic connectivities of both G 1 and G 2 exhibit significant asymptotic improvement over that of the underlying random geometric graph. These results further motivate future hybrid networks and advances in the use of directional antennas.


Introduction
Ever since the first observation of "six degrees of separation" by Stanley Milgram [1], small-world phenomenon have been noted in numerous diverse network domains, from the World Wide Web to scientific co-author graphs [2].The pleasant aspect of the small-world observations is that, despite the high clustering characteristic of relationships with "locality", these various real world networks nonetheless also exhibit short average path lengths as well.This is surprising because purely localized graphs such as low dimensional lattices have very high average path lengths and diameter, whereas purely non-localized graphs such as random edge graph models of Erdos and Renyi [3] exhibit very low clustering coefficient.With intuition consolidating these two extremal graph types, the first theoretical and generative model of small world networks was proposed by Watts and Strogatz [4]: Start with a one dimensional k-lattice, and re-wire every edge to a new uniformly at random neighbor with a small constant probability.They showed that even for a very small but constant re-wiring probability, the resulting graph has small average path lengths while still retaining significant clustering.
Despite the prevalence of small world phenomenon in many real-world networks, wireless networks, in particular ad-hoc and sensor networks, do not exhibit the small average path lengths required of small-world networks despite the evident locality arising from the connectivity of geographically nearby nodes.Although taking a high enough broadcast radius r clearly can generate a completely connected graph of diameter one, this is a non-realistic scenario because energy and interference also grow with r.Rather, from a network design and optimization perspective, one must take the smallest reasonable radius from which routing is still guaranteed.To discuss such a radius in the first place, we must employ a formalization which is common in all theoretical work on wireless networks [5,6], namely we fix the random geometric graph model of wireless networks.Given parameters n, the number of nodes, and r, the broadcast radius, the random geometric graph G(n, r) is formed by uniformly at random dispersing the n nodes into the unit square (which is a normalized view of the actual space in which the nodes reside), and then connecting any two nodes iff they are within distance r of each other.Note that due to the normalization of the space, r is naturally viewed as a function of n.Given such a model, it is a seminal result of Gupta and Kumar [6] that the connectivity property exhibits a sharp threshold for G(n, r) at critical radius r con = log n πn , which also corresponds to an average degree of log n.As connectivity is a minimal requirement for routing, r con is the reference point to take for analysis of G(n, r), and yet, as we shall see, such a radius still yields average path lengths of Θ( πn log n ) with high probability.Note that a result stated with high probability means probability approaching 1 as n approaches infinity.This serves as a first motivation for the question: In the spirit of small-world generative models [4] that procured short average path lengths from a geographically defined lattice by adding random "long" edges, can we obtain significant reduction in path lengths by adding random "short cut" wired links to a wireless network?The first to ask this question in the wireless context was Ahmed Helmy [7] who experimentally observed that even using a small amount of wires (in comparison to network size n) that are of length at most a quarter of the physical diameter of the network yields significant average path lengths reduction.Another seminal work on this question is that of Cavalcanti et al. [8] which showed that introducing a fraction f of special nodes equipped with two radios, one for short-range transmission and the other for long-range transmission, improves the connectivity of the network, where this property is seen to exhibit a sharp threshold dependent on both the fraction f and the radius r.Other work yet include an optimization approach with a specified sink, in which the placement of wired links is calculated to decrease average path lengths in the resulting topology [9].The existing body of literature authored by practitioners in the field of wireless networks on inducing small-world characteristics (particularly shortened average path lengths) into wireless networks by either introducing wired links or nodes with special long range radios or directional antennas yields that such hybrid scenarios are eminently reasonable to consider for real networks.Within this small world hybrid wireless networks literature, the closest in spirit to our work, and indeed the only theoretical work to our knowledge on hybrid wireless networks is that of [10].In [10], both deterministic and randomized wiring schemes are given, and bounds are proven on path lengths and energy efficiency under a model in which (i) a designated sink is specified, (ii) routing is based on greedy geographic forwarding only, and (iii) the frequency of wires can be controlled with a parameter l(n).In contrast, in this work, whereas we do allow the wiring frequency to be controlled by a sparsity parameter k, we do not assume a designated sink, nor that routing is necessarily greedy geographic forwarding.As such, we obtain very contrasting results to that of [10], in that we find the benefits of totally random wiring, while the totally random wiring exhibited the worst performance under their model and assumptions.Having said this, we now introduce our precise model and assumptions.
In particular, we consider the following models of adding new wired edges: Divide the normalized space into bins of length k r 2 , given that the radius is on the order required to guarantee asymptotic connectivity.For each bin, choose a bin-leader.Let the G 1 new wiring be such that we form a random cubic graph amongst the bin-leaders and superimpose this upon the random geometric graph.Let the G 2 new wiring be such that we form a random 1-out graph amongst the bin-leaders and superimpose this upon the random geometric graph.We prove that the diameter for G 1 is O(k + log(n)) with high probability, and the diameter for G 2 is O(k log(n)) with high probability, both of which exponentially improve the Θ( n log n ) diameter of the random geometric graph, thus also inducing small-world characteristics as the high clustering remains unchanged.Our results on resulting average path lengths are also stable in comparison to using a constant fraction of wire lengths, as that in the work of Ahmed Helmy [7].To see this note that, for example, using a maximum wire length of onequarter the maximum distance can be simulated by subdividing the unit square into 16 parts and applying results to the parts separately, then combining into a maximum average path length that is still at most 16 of that within each part.
Whereas the first part of this work concerns bounding the average shortest path lengths of modifications of random geometric graphs, the second part concerns bounding the efficacy of random walks on such graphs.When speaking of a random walk, we connote the natural random walk process which is formed by starting from an arbitrary vertex and continuing each step by picking a neighbor uniformly at random from the set of neighbors of the current vertex.If shortest paths may be viewed as optimizing routes under global information, the trace of a random walk can be viewed as a path the node takes under total uncertainty and only local information.Whereas this may not be an optimal source-destination routing method, it can prove useful for general information collection, sampling, gossiping, and discovery of alternate paths when the optimal ones suffer failure [11][12][13][14].The usefulness of random walk based methods depends entirely on the properties of the underlying graph, and can be measured via different metrics dependings on the intent of the method.Two such metrics are the cover time, which is the expected time (as in number of steps) in which the random walk visits all nodes of the network, and mixing time, which is the maximum time (measured in number of steps starting from an arbitrary node) in which the random walk is within ǫ distance to the stationary distribution [15].
To give a perspective on what constitutes good cover time properties and what constitutes good mixing time properties, consider that the optimal values for these two properties are exhibited on a clique, and the worst case asymptotic cover time is exhibited on a lollipop graphs and the worst case asymptotic mixing time is exhibited on a barbell graph.The clique has cover time Θ(n log n) and constant mixing time, whereas a lollipop graph has Θ(n 3 ) cover time, and the barbell graph has polynomial mixing time.Accounting for the degree of a graph, as the clique has maximal degree, for graphs whose degrees are O(poly log(n)), the optimal mixing time is also Θ(poly log(n)) whereas the optimal cover time remains Θ(n log n).Therefore, graphs with poly-logarithmic mixing time are referred to as rapid mixing [16,17].Previous work [14] showed that whereas the cover time of a random geometric graph about the connectivity threshold is optimal, such graphs far from being rapid mixing.In fact, it was shown that only for radius r = Ω( 1 poly log n ), which is exponentially larger than the critical radius required for connectivity r con , can the random geometric graph be rapid mixing w.h.p. [18,19,14].
In this work, in addition to establishing bounds on resultant path lengths upon sparse random edge additions in the first part, we are the first to consider both theoretically and experimentally the improvement in the resultant mixing time and algebraic connectivity, in comparison to that of the random geometric graph.Although short average path lengths is necessary for a graph to exhibit optimal random walk sampling properties, it is far from sufficient (the barbell graph being a notable counterexample).As a strange omission, the small world literature thus far has primarily ignored spectral gap as a measure in their analyses and generative models despite the known expansion of random edge graph models [20,21].It is well established that the mixing time is intrinsically related to the node expansion, edge expansion [22], algebraic connectivity, and random walk properties of the given graph [16,17,15].
Our motivation is as follows: Yet another limitation of random geometric graphs in comparison to random edge graph models that is especially problematic for oblivious routing, sampling, and gossiping applications [11][12][13] is that, whereas sparse random regular graphs as well as random connected Erdos-Renyi graphs are expanders with excellent mixing properties, connected random geometric graphs G(n, Θ(r con )) are far from being rapidly mixing.In general, additional edges need not improve the mixing time of the resulting graph.Fortunately, in this work, we are able to show that sparse additional edge additions, when done randomly as in models G 1 and G 2 , do indeed yield exponentially improved mixing time.We show these results for G 1 and G 2 using a conductance argument and confirm with experimental calculation of the resultant spectral gap of the normalized Laplacian, which is a normalized measure of algebraic connectivity [23].More recently, algebraic connectivity has been noted by network scientists to be an intrinsic measure of the robustness of a complex network to node and link failures [24], thus giving even stronger motivation for our present study.
In terms of related work, we must note the work [25] of Abraham Flaxman, which is an excellent related work in which the spectra of randomly perturbed graphs have been considered in a generality that already encompasses major small-world models thus far.[25] demonstrates that, no matter what is the starting graph G 0 , adding random 1-out edges at every node of G 0 will result in a graph with constant spectral gap (the best possible asymptotically).The work also presents a condition in which a random Erdos-Renyi graph superimposed upon the nodes of G 0 would yield good expansion, whereas without that condition the resulting graph may have poor expansion.Despite the apparent generality of the work in terms of the arbitrariness of the underlying graph that is considered, unfortunately the results do not generalize to situations in which not all edges are involved in a wired linkage.Notably, the small world models thus far also require such a high probability of new random links.In contrast, in this work, we focus on adding sparse random wires and presenting general bounds on mixing time dependent on the frequency of wired link stations.In particular, the fraction of nodes involved in a wired link will be no more than O( 1 log n ), and in general shall be O( 1 k 2 log n ), both of which are asymptotically diminishing fractions.
Finally, we note that this work is a significant extension to the author's conference paper [26].

Theoretical Preliminaries
The results can be divided logically into those concerning average path lengths and those concerning random walk sampling properties.Therefore, the preliminaries are also so divided.

Random Geometric Graph Preliminaries
Random geometric graphs above the connectivity threshold exhibit certain "smooth lattice-like" properties including uniformity of node distribution and regularity of node degree, that are useful in their analysis.As introduced in [14], we utilize the notion of a geo-dense graph to characterize such properties, that is, a geometric graph (random or deterministic) with uniform node density across the unit square.It was shown that random geometric graphs are geo-dense and for radius r reg = Θ(r con ) all nodes have the same order degree [14].We formally present the relevant results from [14] in this section, as well as the notion of bins, namely the equal size areas that partition the unit square.Such "bins" are the concrete link between lattices and random geometric graphs, essentially forming the lattice backbone of such graphs.
Formally, a geometric graph is a graph G(n, r) = (V, E) with n = |V | such that the nodes of V are embedded into the unit square with the property that e = (u, v) ∈ E if and only if d(u, v) ≤ r (where d(u, v) is the Euclidean distance between points u and v).In wireless networks, r naturally corresponds to the broadcast radius of each node.The following formalizes geo-denseness for geometric graphs: Definition 21 [14] Let G(n, r(n)) be a geometric graphs 1 .For a constant µ ≥ 1 we say that such a class is µ-geo-dense if every square bin of size A ≥ r2 /µ (in the unit square) has Θ(nA) nodes. 2   The following states the almost regularity of geo-dense geometric graphs [14]: Lemma 22 Let G(n, r) be a 2-geo-dense geometric graph with V the set of nodes and E the set of edges.Let δ(v) denote the degree (i.e number of neighbors) of v ∈ V .Then: Recall that the critical radius for connectivity r con is s.t πr 2 con = log n n [6].The following is the relevant lemma that states that random geometric graphs with radius at least on the order of that required for connectivity are indeed geo-dense [14]: That is, w.h.p. (i) any bin area of size r 2 /µ in G(n, r) has Θ(log n) nodes, and (ii) ∀v ∈ G(n, r), δ(v) = Θ(nr 2 ) and m = |E| = Θ(n 2 r 2 ).Further, note that increasing the radius r can only smoothen the distribution further, maintaining regularity.
Both geo-denseness and other results both in this paper and on previous work on random geometric graphs follows from a "folk theorem" often referred to as Coupon Collection (due to the example process given), so we state this before continuing the characterization of random geometric graphs [15]: Theorem 24 (Coupon Collection) Assume that there are a total of n types of coupons, and one attempts to collect all types by picking m coupons independently and uniformly at random.Upon this process, let x i denote the number of coupons of type i that have been collected.Then if m = Ω(n log n), for any types i and j, x i = Θ(x j ) with high probability.In particular, the probability concerned is very high as 1 − O( 1n ).This process and the corresponding folk theorem is also alternatively referred to as "Balls in Bins".
Given that geo-denseness of connected random geometric graphs is established, we wish to utilize the "binning" directly for its lattice-like properties.As such, for the sake of notational convenience, we shall introduce the notion of a lattice skeleton for geo-dense geometric graphs, including random geometric graphs above connectivity: bin-points, where each vertex set B i,j represents the set of nodes of G(n, r) that lie in lattice location {i, j} of L. Further, for a node v ∈ G(n, r) with Cartesian coordinates (x, y) ∈ [0, 1] 2 , denote by B(v) the lattice-bin containing v, namely the bin B i,j such that i = ceiling( We note, before proceeding, that such ideas of geometric bins in representing random geometric graphs are not new to this work (hence, they appear as preliminaries), but rather have arisen naturally in a number of theoretical works on wireless networks above the connectivity regime.The idea of the random geometric graph as a global lattice skeleton composed with local cliques in particular as appears here has been formalized via the global-local decomposition representation of such graphs introduced the author's thesis [27].
What is directly clear by geo-denseness is that there is not much variance in the sizes of the bins: Remark 26 Let LS(G(n, r)) = (L, B i,j ) be the µ-lattice skeleton of a µ-geodense geometric graph G(n, r).Then, ∀i, j, |B i,j | = Θ(nr 2 ).
Further, utilizing the choice of µ ≥ 5, we may make the stronger statement that the connectivity of the lattice is inherited in the nodes of the overall graph.The justification is simply that r becomes the length of the diagonal connecting the farthest points of adjacent bins, and we formalize combining with Remark 26 and Lemma 23: π r 2 con then the 5-lattice skeleton LS(G(n, r) = (L, B i,j ) of random geometric graph G(n, r) satisfies the following: From (ii) it is clear that each bin B i,j forms a clique (namely all pairs of nodes within are connected directly by length one paths).From (iii) it follows that a path in the lattice L yields a path in the graph G(n, r) as well, while (iv) bounds the converse situation in that nodes that lie in bins at least 4 lattice-hops away cannot be directly connected in the graph G(n, r) either.In particular, (iii) and (iv) yield that pairwise distances between points in the graph G(n, r) inherit the shortest paths (Manhattan) distances in the corresponding lattice-bins of the lattice-skeleton, up to constant factors.We formalize with the following corollary: π r 2 con then the 5-lattice skeleton LS(G(n, r)) = (L, B i,j ) of random geometric graph G(n, r) satisfies the following w.h.p.: where the function dist G indicates shortest paths distances in graph G.
Having established that connectivity and distances for G(n, r) with radius at least a small constant times r con roughly preserve connectivity and distances in the Such fractions represent the probability that a node is at distance d away from a given node v. Thus, we may calculate the average path length AP L which is the expectation of that probability distribution on the very function d itself: Thus, the average path length for random geometric graphs above the connectivity threshold is the same as the order of the diameter (maximum shortest path lengths) for such graphs, which is Θ( 1 r ).While the dependence on the radius r in that term may seem optimistic at first, noting that r should be kept as low as possible to reduce energy overhead and interference of the ad-hoc network represented, a realistic constraint on r becomes r = Θ(r con ) = Θ( log n n ), namely that achieving degree Θ(log n).Thus, AP L of reasonable random geometric graphs (of minimal radius guaranteeing connectivity) scales quite badly as Θ( n log n ).

Random Walk and Connectivity Preliminaries
When speaking of a random walk, we connote in particular this natural process: If the random walk is currently at node q, then the simplest probabilistic rule by which to choose the next node is simply to choose a node uniformly at random from among the set of neighbors of q.And, the Markov chain M = (Ω, P ) corresponding to such a random walk on a graph G = (V, E) is the simple random walk on G.For such G, for any node v ∈ V , let δ(v) denote the degree of v, that is the number of neighbors of v in G and let P (v, u) = 1 δ(v) for (v, u) ∈ E and 0 otherwise.In linear algebraic terms, the process is an application of P to the current distribution vector v t of step t, where the initial distibution vector v 0 is concentrated completely at an arbitrary node: v t = v t−1 P = v 0 P t .
In such terms, the stationary distribution of M, if such exists, is the unique probability vector π such that πP = π The stationary distribution being a fixed point vector that remains unchanged upon operator P is also the distribution to which the random walk converges, regardless of the starting point, given that G is connected and non-bi-partite (which is guaranteed by any odd length cycle): Moreover, when the underlying graph G is regular, then the stationary distribution is the uniform distribution [28], and this statement remain true asymptotically when G is almost-regular as well (namely, when the degree of every node is Θ(f (n)) for the same function f ).Therefore, for almost regular graphs, it is clear that the random walk samples efficiently at stationarity, and the faster the random walk on a regular graph converges to stationarity, the greater its loadbalancing qualities.This rate of convergence to stationarity is called the mixing time.
To define mixing time, we must first introduce the relevant notion of distance over time.Let x be the state at time t = 0 and denote by P t (x, •) the distribution of the states at time t.The variation distance at time t with respect to the initial state x is defined to be [16] Note that when the state space Ω is finite it can be verified that [14]: Now we may formally define the mixing time as the following function [16] Clearly, as the name indicates, for a random walk to be used for efficient sampling (according to its stationary distribution), it should be rapidly mixing.Now, on the way towards proving the rapid mixing property of a random walk, we shall make use of a number of beautiful connections amongst mixing time, the eigenvalues of the Markov chain (in particular the spectral gap, namely the difference between the first and second eigenvalues), and connectivity properties of the underlying graph as encapsulated by notions called conductance which is a normalized form of expansion.In introducing the connection between expansion and rapid mixing, we note that intuitively graphs with minimal "bottlenecks" have also a lower the probability of getting stuck in any particular set of states, and thus a faster mixing time as well.We shall see that the graphconnectivity based property of "no bottlenecks" is formalized in a continuous manner with the notion of conductance and in a combinatorial manner with expansion.And, then we shall make the relationship between conductance and mixing time precise.
In fact, one of the motivations we have in considering random edge additions to random geometric graphs is precisely based on the nice connectivity properties that random d-regular graphs possess, which we shall see are very much not possessed by random geometric graphs: Random d-regular graphs are expanders w.h.p. for d ≥ 3. [20,21].The combinatorial meaning of this statement is as follows: W.h.p., every subset S ⊂ V has many edges separating Cut(S, S), particularly |Cut(S, S)| = αd|S| for a constant α > 0 [15].In general, the expansion of a graph is thus the ratio of the worst case cut divided by the size of the set, and an expander is a graph with constant expansion.Note that the property of a graph being an expander is a much stronger notion than k-connectivity in that it clearly implies an edge connectivity that is at least on the same asymptotic order as the minimum degree, but it further requires that the density of edges separating any set from the rest of the graph is proportional to the size of the set.In fact, being an expander is an extremal property and also much stronger than both the properties of having logarithmic diameter and being rapidly mixing.As such, unsurprisingly, we will not be able to prove that our graphs resulting from random edge additions are expanders.Nonetheless, we will be able to prove sufficient expansion so as to guarantee that the random walk is rapid mixing.We will do so by bounding the conductance.
The conductance of a reversible Markov chain M is defined by [17] where S = Ω − S, π(S) is the probability density of S under the stationary distribution π, and In graph-theoretic terms, the conductance of M is the minimum over all subsets S ⊂ Ω of the ratio of the weighted flow across the cut Cut(S, S) to the weighted capacity of S, and as such is clearly a continuous measure of "the degree of no bottlenecks" property.For almost regular graphs of degree Θ(d), we may simplify the expression for conductance as follows: And, for this case, it is clear that conductance is a type of normalized measure of expansion where the degree is taken into account as well.Now that we have defined expansion and conductance, we must soon relate these measures to the rapid mixing property.We do this by connecting both conductance and mixing time to the spectral gap.
As the stationary distribution π is defined to be such that πP = π, it corresponds to the eigenvalue λ 0 = 1 of P .Let the rest of the eigenvalues of P in decreasing order of absolute value be: For a finite, connected, non-bipartite Markov chain as the type in this work, the rate of convergence to π, which as you may recall is captured by the mixing time, is governed by the difference between the first and second eigenvalues, namely the spectral gap which is 1 − λ 1 [16].And, here are the theorems establishing these relationships: Theorem 29 For an ergodic Markov chain3 , the quantity τ x (ǫ) And, to relate conductance explicitly to mixing time, it thus suffices to bound the spectral gap with the conductance: Theorem 210 ( [16]) The second eigenvalue λ 1 of a reversible Markov chain M satisfies Combining, as in [14]: Corollary 211 ( [17]) Let M be a finite, reversible, ergodic Markov chain with loop probabilities P (x, x) ≥ 1 2 for all states x.Let Φ be the conductance of M.Then, for any initial state x, the mixing time of M satisfies In particular, the following is immediate too: Remark 1.For a random walk to be rapid mixing, it is necessary and sufficent that the conductance be inverse poly-logarithmic.
Finally, we must speak of the mixing properties of the random geometric graph above the connectivity threshold as shown in [18,19,14]: In particular at Θ(r con ), λ 2 (G(n, r)) is Ω( log n n ) and O( ( log n n ).This gives a mixing time of Ω(n ǫ ) for ǫ > 0. Compare to a rapid mixing Markov chain which requires only O(polylog(n)) steps: Possible in wireless network only for very large radius r = Θ( 1 polylog(n) ), exponentially larger than r con .Re-stating, from [14]: Corollary 213 (Mixing Time of RGG) Radius r = Ω(1/poly(log n)) is w.h.p. necessary and sufficient for G(n, r) to be rapidly mixing.
On the other hand, recall: Even sparse random regular graphs are rapid mixing: Remark 214 It is well-known that the random 3-regular graph G R,1 (k) is an expander with high probability [22,20].Therefore, G R,1 (k) also exhibits diameter and average path lengths asymptotically at most logarithmic in its vertex set |V R (k)|, with high probability.

Models of Random Edge Additions
As we start to consider the business of adding random edges to a given initial graph G 0 = (V 0 , E 0 ), note that the set of additional edges E R and the existing nodes connected by them 4 .That is, it is also convenient to view the additional random edges as a new graph G R superimposed upon the original graph G 0 .
Given such a characterization, let us be given a 5-geo-dense geometric graph G 0 = (V 0 , E 0 ) = G(n, r) with 5-lattice-skeleton (L, B i,j ).In particular, note from Lemma 23 that results apply to any G 0 = G(n, Ω(r con )).Given parameter k ≥ 1, let vertex set V R (k) be generated as follows: For any i, j ≤ √ 5 √ kr pick a node v i,j uniformly at random from the nodes in the set of bins B i,j = ∪ ki≤i ′ ≤(k+1)i,kj≤j ′ ≤(k+1)j B i ′ ,j ′ 5 , and set v i,j ∈ V R (k).For the case k = 0, let V R (0) = V 0 .We now define the various types of random edge sets E R,i for graphs G R,i (k) = (V R (k), E R,i ) whose superimpositions upon G 0 we shall consider in this work: Let E R,1 be generated as follows: For every node v ∈ V R (k) pick 3 neighbors in V R (k) uniformly at random, discarding situations in which any node has degree greater than 3. Thus, the resulting graph G R, is the random 3-regular Erdos-Renyi graph defined on vertex set V R (k).Let E R,2 be generated as follows: For every node v ∈ V R (k) pick 1 neighbor in V R uniformly at random.Thus, the resulting graph G R,2 (k) = (V R (k), E R,2 ) is the random 1-out graph defined on vertex set V R (k).Similarly to the above, let us define the resulting graphs as follows: Let Essentially, k controls the frequency of special nodes which shall serve as wired link stations.For k = Θ(1), the frequency is in line exactly with the bins, and thus the occurence of such wired link stations is 1 in every Θ(nr 2 ).For r = Θ(r con ) that frequency becomes Θ( 1 log n ), and for larger broadcast radius it is sparser: Before proceeding to prove results on average path lengths for G 1 = G 0 + G R,i , we note that the manner in which V R (k) is generated can be simulated approximately by simply choosing a total of logarithmically more wired link stations uniformly at randomly from the original set V 0 .This too follows from Coupon Collection: Remark 32 For any k, if every v ∈ V 0 is chosen to be a wired link station with probability Θ( nk 2 r 2 ), then, with high probability, for every k 2 -bin B i,j = ∪ ki≤i ′ ≤(k+1)i,kj≤j ′ ≤(k+1)j B i ′ ,j ′ there exists a vertex v ′ ∈ B i,j such that v ′ is a wired link station.Moreover, all of the vertices in any k 2 bin are almostequiprobable and almost-independent whp.Now, we note that the maximum distance of any node in a k 2 bin to the corresponding wired link station in the k 2 bin is simply bounded by the hopdiameter of the k 2 bin: Remark 33 Every node is within Θ(k) hops of the wired link station in its k 2 -bin since the k 2 -bin simply stretches the Manhattan distances of the original 5-lattice-skeleton by k.
This remark shall prove relevant in relating inter-node distances in the graph Combining Remarks 31, 33, and 214, we obtain our first bounds on the resulting average and worst-case path lengths:

Experimental Bounds on the Algebraic Connectivity
Experiments were conducted for networks of 100 to 1620 nodes.The networks were constructed in a way that is consistent with the models G 1 and G 2 of the theoretical section, the parameter k was chosen to be 2, and the radius was chosen to be r con exactly, with nodes thrown uniformly at random into the unit square and the edge selections generated in accordance with the described random models.Disconnected G(n, r) were discarded from consideration.A caveat in our simulations is that we guaranteed a node in the exact center of each bin, because otherwise there were too many discarded geometric graphs due to lack of connectivity.This problem would not be an issue for sufficiently large networks due to the asymptotic theoretical connectivity guarantee, and anyway comparative results are dominated by how edges are chosen rather than precise node locations.
The results can be seen in Figure 1, where the Y -axis is the spectral gap of the normalized Laplacian, namely the normalized algebraic connectivity.Notably, the spectral gap for the random geometric graph approaches zero quickly, whereas the spectral gap for G 1 and G 2 appear to diminish very slowly after 500 nodes.Moreover, note that the number of wired-nodes in comparison to the network size n for n values of 100, 300, 800, 1000, 1300, and 1620 are respectively as follows: 36, 64, 196, 256, 256, 324.The fraction of wired nodes for the network size of 1620 was just 1 5 .

Observations on Experimental Results
Since we are considering the normalized Laplacian spectral gap, automatically all results must be between 0 and 1.Note that we considered the normalized spectral gap because we want to avoid the scaling problem that could arise if we compared the non-normalized spectra for graphs with very different degrees.
Further, we consider the Laplacian instead of simply taking the adjacency matrix, because the Laplacian is symmetric, making for faster computations while giving comparable bounds.That all results are less than 0.1 in particular should not be bothersome as well, for two reasons: First, even if we were taking the strong property of being an expander into account, a graph family whose normalized spectral gap never falls below a given constant (e.g.0.01) would be satisfactory, regardless of the constant.But, we are not attempting to show such a strong property anyway.We are concerned with sufficient expansion, in terms of rapid mixing, which does not even require a constant lower bound, but merely that the rate at which the spectral gap falls is slow (in particular, inverse poly-logarithmic).
In fact, as theoretical result Theorem 51 already demonstate that G 1 and G 2 are indeed rapid mixing, we also notice in the experimental results is that the (yellow) pattern for G 2 is extremely similar to the (red) pattern of G 1 .On the other hand, the spectral gap for the underlying random geometric graph G(n, r), which is theoretically known to have bad expansion [14], falls to zero far more quickly.As we discarded disconnected cases, it is notable that the spectral gap for G(n, r) cannot be zero exactly, although it clearly gets arbitrarily close to zero quickly.The spectral gap patterns for G 1 and G 2 , however, fall much more slowly.Thus, the experimental results confirm our theoretical bounds by showing that the spectral gap for G 1 and G 2 is exponentially larger than that of the underlying random geometric graph.

Conclusion
We have presented theoretical bounds on the diameter, APL, conductance, and mixing time of sparse random edge additions onto random wireless networks around the connectivity regime, where our bounds are expressed as functions of the wiring frequency.We have also shown experimental results comparing the normalized algebraic connectivities of the underlying random geometric graph to the hybrid models.In particular, we have shown that when the wiring frequency is at least inverse poly-logarithmic, then the subsequent hybrid network exhibits polylogarithmic diameter and mixing time, both of which are exponential improvements to the wireless network about the connectivity regime.We have also shown that there is correspondingly significant asymptotic improvement to the normalized algebraic connectivity which is known to govern network robustness.Taking broadcast radius on the order of the connectivity threshold is particularly important in the case of sensor networks where energy must be preserved and interference diminished, and thus comprises the relevant base graph model as used in much theoretical work on such networks.Nonetheless, the results regarding bounds on diameter and APL are also expressed for general broadcast radii.The mixing time bounds in particular are relevant for distributed gossip applications where it is well established that the performance is dominated by this value.Taken as a whole, this work provides a strong support for hybrid sensor networks.
From a practical standpoint, one may ask how such random wired links should be established atop a wireless sensor network in order that the hybrid model presented be of true relevance.In this regard, we note that the analyses presented is sufficiently general to include an existing sparse wired network atop which a wireless network resides.A benefit of theoretical bounds is precisely this lack of restriction of how the network details are established.In fact, the random edges of the superimposed links need not even be wired, but may be generated via a sufficiently sharply angled and long ranged directional antenna model as well, as long as the problem of the side lobe may be solved.We point the reader to the existing literature on small worlds for hybrid networks stated in the introduction, as many proposals are given towards the practical aspects of hybrid network creation.
We reiterate that this is the first graph theoretic work to establish solid theoretical foundations of the improvement to graph diameter and mixing time of sparse totally random links (of a non-broadcast nature) upon a random wireless network above and around the connectivity threshold.The two most relevant works with which to compare and contrast results would be those of [10] and [25].The results of [25] take the base graph to be arbitrary but the wiring probability to so high as to re-wire every edge on average.Moreover, that work is concerned with the extremal property of the resulting graph being an expander or not, rather than general expressions of the degree of expansion or mixing time.The arguments used there are beautiful and tight, but are neither sufficiently general to take diminishing random wiring probability, nor sufficiently relevant for the base graph being the random wireless domain in particular.The work of [10], in contrast to [25] does consider asymptotically diminishing wiring probability, and restricts to the relevant model of wireless base graph around connectivity.However, in that case, base station is fixed and a particular type of greedy forwarding is assumed for the routing protocol, so that they actually obtained the worst results for the case of totally random links.Moreover, they do not discuss the mixing time at all.Thus, our work may be considered to be a positive complementation to that work in the sense that of the positive results we obtain when routing is both shortest paths based and random.

r
lattice skeleton, let us then consider the number of lattice-nodes N d,L (v) that are at lattice distance exactly d away from v in the lattice L: Clearly, N d,L (v) grows linearly in d by a simple induction on upper and lower bounds.And, the maximum distance to consider is d = Θ( µ r 2 ).Moreover, due to the smooth distribution of random geometric graph nodes in the lattice bins, we must have that the fraction f d,L (B(v)) of lattice bins at lattice-distance exactly d away from B(v) must be on the same order as the fraction of random geometric graph nodes f d,G(n,r) (v) at hop-distance exactly d away from v. Thus,

4. 1
APL and Diameter bounds for G 1 Thus, we now proceed concerning inter-node distances for the first model G R,1 :