Applications

In this section we are going to apply our proposed methodology to a number of examples, starting with the three cases in the literature we have discussed.

Example 1

In the Gupta-Kumar model [9] the nodes are static and transmit within a fixed radius. The transmission policy is such that Γ = Θ(n), and is inde-pendent of the scheduling policy. For a given realization of node placement random process, with high probability the associated connectivity graph will have a large fraction of nodes with constant degreec.

Consider ac-regular graph where the weight associated with each edge is at most 1/c. For each (s_i, t_i) pair there existcmax-flow paths, each carrying throughput at most 1/c, of average length Θ(√

n). Thus, from Lemma 3.5 we get that

λ(n)≤Θ( 1

√n). (3.20)

We can get the same result from the sparsity cut bound as follows. The nodes are placed uniformly at random on the area of a unit circle. Take a cut across a diameter of this circle. This cut will separate order Θ(n) source-destination pairs. However, the capacity of the cut will be of order Θ(√

n) (determined by the number of nodes within a small distance from the cut), which leads again to Eq. (3.20).

3.6. Applications 65 If we start from a random geometric graph, where nodes transmit at the same range to their closest neighbors, the corresponding connectivity graph will have the structure of a planar graph. For planar graphs, the sparsity bound can be achieved within a constant factor O(1) [46].

Example 2

In the Grossglausser and Tse model [26], uniform mobility implies that the connectivity graph is a complete graph with uniform weight associated with every edge. That is, each node has degreen−1, and the weight associated with each edge is Θ(1/n) (corresponding to the probability that two nodes are nearest neighbors and the probability that a feasible sender receiver pair is scheduled). This is depicted in Fig.3.2.

We are going to apply the sufficient conditions in Theorem 3.1.

• From each source to each destination node, there exist n−2 edge-disjoint paths of length two, and one path of length one. Indeed, since the graph is complete, there exists an edge from node s_i to all other nodes in the graph (including the destination t_i) and an edge fromti to every other node in the graph, which put together form the described max-flow paths. Using these paths, we can route throughput Θ(1) = (n−1)¹_n. This shows that Condition 1 is verified.

• We are going to show that if Θ(n) source-destination pairs share the network and use the paths previously described, each edge is used a finite number of times. Consider edge (k, l) between nodes k and l.

This edge is going to be used only if nodekis a source, or if nodel is a destination - so at the most it is going to be used two times. This verifies Condition 2.

Example 3

In the Diggavi, Grossglauser and Tse model [42], mobility is restricted since the nodes are only allowed to move along great circles in the surface of a sphere, and constant throughput is still achieved by employing the same two-phases policy. In [42], it is argued that the two phases policy achieves a constant throughput mainly for two reasons. First, each node spends the same order amount of time as the nearest neighbor to every other node.

This ensures that each source spreads its information units uniformly across

S

D

Θ(1 /n) Θ(1/n)

Θ(1/n)

Θ(1 /n) Θ(1/n)

Θ(1/n) Θ(1/n)

Θ(1/n) Θ(1/n)

1

Phase 2

Phase

S

D

Θ(1 /n) Θ(1/n)

Θ(1/n)

Θ(1 /n) Θ(1/n)

Θ(1/n) Θ(1/n)

Θ(1/n) Θ(1/n)

1

Phase 2

Phase

Figure 3.2: The connectivity graph is a complete graph in the case of uniform mobility pattern.

all other nodes and the traffic is equally distributed among all relays. Sec-ondly, similarly to [9], communications are constrained to nearest neighbors.

Hence, the capture probability is not vanishingly small even in a large sys-tem, despite the fact that there are O(n) interfering nodes transmitting simultaneously. These observations correspond to having the connectivity graph be a complete graph, where the degree for every node is of the order Θ(n) and the weights associated with each edge are Θ(1/n). Thus, the anal-ysis is the same as in the previous example. Having a connectivity graph to be a complete graph with equal capacity edges is a sufficient, but not nec-essary condition, for constant throughput. Moreover, the two-phase routing

3.6. Applications 67 strategy, is tied to the fact that the graph is complete, and thus it is also not necessary for constant throughput, as the next example demonstrates.

Example 4

Consider the example of geographic gossip over a sensor network as described in [55]. The n sensors collect a measurement and it is of interest that all nodes can compute the average of all n sensor measurements. Traditional gossip algorithms solve this problem by having each node randomly pick one of their one-hop neighbors and exchange their current values. The pair of nodes compute the pairwise average which becomes the new value for both nodes, and this process is iterated until all nodes converge to the global average. [55] introduces the notion of geographic gossip where geographic routing is used to exchange information with random nodes who are far away in the network. The wireless sensor network is then modeled as a random geometric graph where the transmission range scales as Θ(

qlogn

n ) and the number of hops between a source and a randomly uniformly distributed destination scales as Θ(q _n

logn). This random geometric graph could be represented by our connectivity graph again.

Example 5

Consider the rectangular grid depicted in Fig. 3.3 that has 2dlines. Assume that the nodes are uniformly distributed on the lines of the grid. Thus, each line will contain n/2d nodes. Assume that our transmission policy is such that, nodes in the same line or in intersecting line can communicate with each other if they are within a certain range (protocol model), but nodes in parallel lines cannot communicate. Since the min-cut between parallel lines is at most equal to d, a necessary condition to have constant throughput is thatn/2 =O(d).

Now consider the case where n= 2d, that is, each line contains exactly one node. Then the corresponding connectivity graph is a complete bipartite graph, since from assumption nodes in parallel lines do not communicate.

It is easy to see that, for a complete bipartite as depicted in Fig. 3.4, if the source-destination pairs belong to parallel lines, there exist n/2 non-intersecting paths of length two, while if they belong to non-intersecting lines, there exist n/2−1 non-intersecting paths of length three and one path of

d lines

d lines

Figure 3.3: Mobility on a rectangular grid.

length one. Thus, from Theorem 3.1, if each edge has capacity Θ(1/n), it is possible to achieve constant throughput.

If nodes are allowed to uniformly randomly move on their line, then ig-noring edge-effects, the capacity of each edge is upper bounded by Θ(1/n²).

However, it is possible to construct a mobility pattern that leads to edge capacities Θ(1/n). For example, to avoid edge-effects we can assume that the square grid envelopes the surface of a torus. Thus the nodes move on parallel (horizontal and vertical) “circles” instead of lines, i.e.., the end points of the line segments are connected. We can then construct a mo-bility pattern, where nodes move clockwise on their circle, where at time slots (i) mod (d) = k, the node at the horizontal circle i is within range (and successfully communicates) with the node at the vertical circle (i+k) mod (d).

Example 6

This example is a direct application from the multi-commodity flow litera-ture. Consider a connectivity graph, that has constant degree, and is an ex-pander graph [53]. Exex-pander graphs have been the subject of much study in combinatorics and computer science. Broadly speaking, an expander graph

3.6. Applications 69

S1 S2 D2

D1

...

horizontal lines

vertical lines

Figure 3.4: Bipartite graph.

should be a graph in which every reasonably small set of vertices has many neighbors. Then, the sparsity bound gives thatλ(n)≤Θ(1). However, the best achievable throughput equalsλ(n) = Θ(_log¹_n). This example is used to demonstrate that the sparsity bound cannot always be achieved.

Example 7

We consider the example studied in [51] where nodes are placed randomly with n^α,0 ≤ α ≤ 1 overlapping neighborhoods. The n mobile nodes are restricted to move within their assigned neighborhood. In the associated graph of the model described above, each node has degree n^1−α and the weight associated with each edge is at most n^α−1. Moreover, the average path length in this model is Θ(n^α2). Thus, for each (s_i, t_i) pair there exist n^1−α max-flow paths, each carrying throughput at most n^α−1, of average path length Θ(n^α2). From Lemma 3.5 we get that the per source destination throughput is:

λ(n) =O(n^−α2) (3.21)

which corresponds to an aggregate throughput of O(n^1−α2). The connec-tivity graph paradigm takes into account the restricted mobility model in terms of the average path length and the node degree, which motivates the next section.