Highway Preferential Attachment Models for Geographic Routing

Abstract

In the 1960 s, the world-renowned social psychologist Stanley Milgram conducted experiments that showed that not only do there exist “short chains” of acquaintances between any two arbitrary people, but that these arbitrary strangers are able to find these short chains. This phenomenon, known as the small-world phenomenon, is explained in part by any model that has a low diameter, such as the Barabási and Albert’s preferential attachment model, but these models do not display the same efficient routing that Milgram’s experiments showed. In the year 2000, Kleinberg proposed a model with an efficient $\mathcal {O}(\log ^2{n})$ greedy routing algorithm. In 2004, Martel and Nguyen showed that Kleinberg’s analysis was tight, while also showing that Kleinberg’s model had an expected diameter of only $\varTheta (\log {n})$—a much smaller value than the greedy routing algorithm’s path lengths. In 2022, Goodrich and Ozel proposed the neighborhood preferential attachment model (NPA), combining elements from Barabási and Albert’s model with Kleinberg’s model, and experimentally showed that the resulting model outperformed Kleinberg’s greedy routing performance on U.S. road networks. While they displayed impressive empirical results, they did not provide any theoretical analysis of their model. In this paper, we first provide a theoretical analysis of a generalization of Kleinberg’s original model and show that it can achieve expected $\mathcal {O}(\log {n})$ routing, a much better result than Kleinberg’s model. We then propose a new model, windowed NPA, that is similar to the neighborhood preferential attachment model but has provable theoretical guarantees w.h.p. We show that this model is able to achieve $\mathcal {O}(\log ^{1 + \epsilon }{n})$ greedy routing for any $\epsilon > 0$.

7 Appendix

1.1 7.1 Experimental Analysis

Goodrich and Ozel’s paper on the neighborhood preferential model [8] was able to show that a hybrid model combining elements from Kleinberg’s model with preferential attachment is able to outperform both individual models for decentralized greedy routing on road networks by showing many experimental results. In the previous sections, we provided some theoretical justification for their results, by proving asymptotically better greedy routing times for a similar model. In this section, we complete our comparisons by reproducing their key experimental results with our new model. Our experimental framework is nearly identical to theirs, except that we implement directed versions of each algorithm, i.e. where each long-range connection is directed (local connections are by definition always undirected). This allows us to run experiments much more efficiently—we sample between 30,000 to 200,000 source/target pairs for each data point, as compared to their 1,000 pairs—but results in all algorithms having a worse performance. For our experiments we picked $\epsilon = 0.5$ and $A = 1.01$. It is possible that other parameters would yield better results.

Key Results. Our main key result is that our windowed NPA model outperforms Kleinberg’s model for road networks by a factor of 2, as shown in Fig. 4. This result is directly in line with Goodrich and Ozel’s experimental results with their similar model [8]. It is worth mentioning that our directed version of the model is worse than the undirected version from Goodrich and Ozel’s paper by roughly a factor of 2.

Fig. 4.

Comparison of greedy routing times for Kleinberg’s model and the windowed NPA model when $Q = 1, \epsilon = 0.5, A = 1.01$. The right plot is in log scale.

Similarly, we show that by increasing the degree density to 32 we can achieve a result of less than 20 degrees of separation, which again is roughly twice the results from Goodrich and Ozel’s paper (see Fig. 5), which we attribute primarily to the directed implementation of the models for our experiments.

Fig. 5.

The greedy routing times for the windowed NPA model on the 50 US states when $Q = 32$, $\epsilon = 0.5$, and $A = 1.01$.

1.2 7.2 Kleinberg Highway Proofs

In this section, we prove Theorem 1 by proving upper bounds on each of the three steps of the greedy routing algorithm: routing from s to the highway using local connections, within the highway towards t using standard Kleinberg routing, and finally from the highway to t again using local connections.

Lemma 6

It is possible to route from any node $s \in \mathcal {G}$ to a highway node $h \in \mathcal {G}_H$ in at most $\sqrt{k}$ hops, if the location of h is known, or in at most $k - 1$ hops, if the location of h is not known.

Proof

Without loss of generality, let’s assume highway nodes are located wherever $\bmod (x, \sqrt{k}) = 0$ and $\bmod (y, \sqrt{k}) = 0$. Then, the maximum distance in the x dimension to a highway node is $\delta _x = \min (\bmod (x, \sqrt{k}), \sqrt{k} - \bmod (x, \sqrt{k})) = \left\lfloor \frac{\sqrt{k}}{2} \right\rfloor$, and an equivalent result holds for $\delta _y$. Therefore, the maximum lattice distance to a highway node is the sum of both, or at most $2 \left\lfloor \frac{\sqrt{k}}{2} \right\rfloor \le \sqrt{k}$. If the location of h is known, then we can route to it directly taking a number of hops equal to the lattice distance to h. If the location of h is not known, we can visit every node in a $\sqrt{k} \times \sqrt{k}$ square, guaranteeing that we will encounter a highway node h, in $k - 1$ hops.

After we reach the highway subgraph $\mathcal {G}_H$, we can use the standard Kleinberg routing algorithm towards t. As in Kleinberg’s original analysis, we first prove a lower bound on the probability that a long-range connection exists between two arbitrary highway nodes.

Lemma 7

The normalization constant z for $\mathcal {G}_H$ is upper bounded by $z \le 4 \ln (6 n_H) \le 4 \ln (6 n)$. As such, the probability of any two highway nodes u and v being connected is at least $[4 \ln (6 n) d_H(u, v)^2]^{-1}$, where $d_H(u, v)$ is the lattice distance between u and v in $\mathcal {G}_H$.

Proof

This result follows directly from Kleinberg’s original analysis on the highway subgraph $\mathcal {G}_H$.

In Kleinberg’s analysis, the probability that a node u has a long-range connection to a node v that halves its distance to the destination is proportional to $[\log {n}]^{-1}$, when a node has a constant number of long-range connections Q. In our case, each highway node has $Q \times k$ long-range connections, where k does not need to be constant. This gives us improved distance-halving probabilities:

Lemma 8

In the Kleinberg highway model, the probability that a node u has a long-range connection to a node v that halves its distance to the destination is proportional to at most $k/\log {n}$ for $k \in \mathcal {O}(\log {n})$ and is constant for $k \in \varOmega (\log {n})$.

Proof

Following Kleinberg’s analysis, the probability that a single long-range connection from u halves its distance to the destination is still proportional to $[\log {n}]^{-1}$. Therefore, the probability that a single long-range connection does not halve its distance to the destination is proportional to $1 - [\log {n}]^{-1}$. The probability that all Qk long-range connections do not halve the distance is therefore proportional to $\left( 1 - [\log {n}]^{-1} \right) ^{Qk} = \left[ \left( 1 - [\log {n}]^{-1} \right) ^{\log {n}} \right] ^{\frac{Qk}{\log {n}}} \le e^{-\frac{Qk}{\log {n}}}$. Finally, the probability that any one of the Qk succeed in halving the distance is therefore proportional to $1 - e^{-\frac{Qk}{\log {n}}}$. When $k \in \omega (\log {n})$, the exponential term tends towards zero, and the probability tends towards one. For smaller values of k, a Taylor expansion of $e^{-\frac{Qk}{\log {n}}}$ shows that this probability is proportional to at least $1 - \left[ 1 - \frac{Qk}{\log {n}} + \mathcal {O}\left( \left[ \frac{Qk}{\log {n}} \right] ^2 \right) \right] = \frac{Qk}{\log {n}} - \mathcal {O}\left( \left[ \frac{Qk}{\log {n}} \right] ^2 \right)$. When $k \in o(\log {n})$, the lower order terms become asymptotically negligible, and we are left with a probability proportional to $\frac{Qk}{\log {n}} = \mathcal {O}(k/\log {n})$. When $k = \varTheta (\log {n})$, we are left with a constant dependent on Q.

Importantly, this result reproduces Kleinberg’s original result when k is constant, since we are left with a probability proportional to $1/\log {n}$. Finally, we can prove the main result of this section:

Proof

(of Theorem 1). It is possible to describe the greedy routing path in terms of at most $\log {n}$ phases, where a node u in phase j if it is at a lattice distance between $2^j$ and $2^{j+1}$ from the destination t. It is easy to see that halving the distance to the destination results in reducing what phase a node is in by one. The expected amount of hops spent in each phase is therefore $1 / \Pr (\text {distance halving}) = \mathcal {O}(\log (n)/k)$. Note that importantly, when no long-range connections halve the distance, we take local connections on the highway graph towards t, as in the original Kleinberg model. Since there are at most $\log {n}$ phases, we expect to spend at most $\mathcal {O}(\log {n}(\log (n)/k + 1))$ hops on the highway⁴. Finally, the final highway node is known to be at most $\sqrt{k}$ hops away from the destination t. The theorem follows from these results along with the results from Lemma 6.

1.3 7.3 Randomized Highway Proofs

We now present proofs of theorems and lemmas discussed in Sect. 4.2.

The Nested Lattice Construction. For our proofs, similarly to the Kleinberg highway model, we will conceptually subdivide the highway into a lattice of balls of various sizes (see Fig. 6 for an example nested lattice structure), and show upper and lower bounds on the number of highway nodes within each ball with varying degrees of probability bounds. Specifically we will prove:

Lemma 9

Results from the nested lattice structure:

• 1. All balls of radius $3 \sqrt{k \log {n}}$, centered around any of the $n^2$ nodes, contain at least $9 \log {n}$ highway nodes with high probability in n.

• 2. All balls of radius $3 \sqrt{k \log {n}}$, centered around any of the $n^2$ nodes, contain fewer than $41 \log {n}$ highway nodes with high probability in n.

• 3. $\mathcal {O}(\log ^2{n})$ balls of radius $3 \sqrt{k \log {\log {n}}}$, centered

  around any $\mathcal {O}(\log ^2{n})$ nodes, contain fewer than $41 \log {\log {n}}$ highway nodes with high probability in $\log {n}$.

• 4. Any arbitrary ball of radius $2 \sqrt{k}$ has at most 18 highway nodes with probability at least 1/2. This result is not a high probability bound, and is only independent for balls centered around nodes with lattice distance greater than $4 \sqrt{k}$ between them.

Fig. 6.

The nested lattice construction showing balls of radius 3, centered around an orange node. The central ball is depicted in solid light green, while the 8 adjacent balls are shown in dashed yellow.

Proof

Consider balls of radius $a \sqrt{k \log {n}}$ for some constant a. There are at least $2 a^2 k \log {n}$-many nodes within each ball of radius $a \sqrt{k \log {n}}$. The probability that any node is a highway node is 1/k, so the expected number of highway nodes within each ball is $\mu \ge 2 a^2 \log {n}$. We can lower bound the number of highway nodes within each ball by using a Chernoff bound. Letting X be the number of highway nodes within each ball, we have:

$$\begin{aligned} \Pr (X \le (1 - \delta )\mu ) \le e^{-\frac{\delta ^2 \mu }{2}} = e^{-a^2 \delta ^2 \log {n}} = n^{-\frac{a^2 \delta ^2}{\ln {2}}} \end{aligned}$$

By union bound, the probability this fails for a ball centered at any of the $n^2$ vertices is at most $n^{2 - \frac{a^2 \delta ^2}{\ln {2}}}$. Setting $\delta = 1/2$ and $a = 3$, we obtain that all balls with radius $3 \sqrt{k \log {n}}$ have at least $9 \log {n}$ highway nodes with probability at least $1 - n^{-1.24}$, which is w.h.p. For an upper bound, we first note that there are fewer than $3 a^2 k \log {n}$-many nodes within each ball of radius $a \sqrt{k \log {n}}$ for radii of at least 3. Using another Chernoff bound:

$$\begin{aligned} \Pr (X \ge (1 + \delta )\mu ) \le e^{-\frac{\delta ^2 \mu }{2 + \delta }} = e^{-\frac{2 a^2 \delta ^2 \log {n}}{2 + \delta }} = n^{-\frac{3 a^2 \delta ^2}{\ln {2}(2 + \delta )}} \end{aligned}$$

By setting $\delta = 1/2$ and $a = 3$, we obtain that all balls with radius $3 \sqrt{k \log {n}}$ have fewer than $41 \log {n}$ highway nodes w.h.p. (with probability at least $1 - n^{-1.89}$). We can obtain similar bounds for smaller balls, although with worse probabilities. For example, for balls of radius $a \sqrt{k \log {\log {n}}}$, we expect $\mu < 3 a^2 \log {\log {n}}$ highway nodes for radii of at least 3. Using another Chernoff bound with $\delta = 1/2$ and $a = 3$, we obtain that any given ball with radius $3 \sqrt{k \log {\log {n}}}$ has more than $41 \log {\log {n}}$ highway nodes with probability less than $\log ^{-3.89}{n}$. Assuming we will only invoke this bound at most $\mathcal {O}(\log ^2{n})$ times, the probability that any of the invocations fail is negligible (at most $\mathcal {O}(\log ^{-1.89}{n})$). Finally, we consider balls of radius only $2 \sqrt{k}$, which have at most 18 highway nodes with probability at least 1/2.

Finding the Normalization Constant. The probability that highway node u picks highway node v as a long-range connection is $d(u, v)^{-2}/ \left[ \sum _{w \ne u}{d(u, w)^{-2}} \right]$, where each w in the summation is a highway node. In order to lower bound this probability, we must upper bound the denominator, known as the normalization constant z.

Proof

(of Lemma 1). Let’s consider a lattice of balls centered around an arbitrary highway node u. Let’s define a notion of “ball distance” b to measure the distance between two balls in this ball lattice. Let $\mathcal {B}_b(u)$ be the set of all balls at ball distance b from a ball centered at u. There is 1 ball at ball distance 0 ($|\mathcal {B}_0(u)| = 1$), 8 balls at ball distance 1, and in general at most 8b balls at distance b for $b > 0$ (see Fig. 6). The minimum distance between u to a node in another ball at distance b is $2b - 1$ times the ball radius for $b > 0$. Let’s consider a lattice of balls with radius $3 \sqrt{k \log {n}}$. From Lemma 9.2 we know that there are at most $41 \log {n}$ highway nodes within this ball w.h.p. Let’s also find the normalization constant in two parts, first due to highway nodes in $b > 0$ ($z_{>0}$), and then due to highway nodes within the same ball ($z_0$).

Note that any two balls are separated by ball distance at most 2n/twice the ball radius, or $\frac{n}{3 \sqrt{k \log {n}}}$.

$$\begin{aligned} z_{>0} &\le \sum _{b = 1}^{\frac{n}{3 \sqrt{k \log {n}}}}{ \frac{(\text {max } \# \text { highway nodes in } \mathcal {B}_b(u))}{(\text {min distance to node in }\mathcal {B}_b(u))^2}} \\ &\le \sum _{b = 1}^{\frac{n}{3 \sqrt{k \log {n}}}}{ \frac{41 \log {n} \times 8b}{(2b - 1)^2 \times 9 k \log {n}}} < \frac{37}{k} \sum _{b = 1}^{\frac{n}{3 \sqrt{k \log {n}}}}{ \frac{b}{(2b - 1)^2}} \\ &\le \frac{37}{k} \sum _{b = 1}^{\frac{n}{3 \sqrt{k \log {n}}}}{ \frac{1}{b}} = \frac{37}{k} \mathcal {H}\left( \frac{n}{3 \sqrt{k \log {n}}}\right) \\ &\le \frac{37}{k} \mathcal {H}\left( \frac{n}{3 \sqrt{\log {n}}}\right) < 26\frac{\log {n}}{k} \text { for } n > 2 \end{aligned}$$

Now that we showed the contribution of highway nodes in different balls from u, let’s bound the contribution due to highway nodes within the same ball. We are only interested in the normalization constant for nodes that we visit along the highway, which we will show is at most $\mathcal {O}(\log ^2{n})$ nodes. Knowing this, we can use the improved bound for balls of radius $3 \sqrt{k \log {\log {n}}}$, which from Lemma 9.3 we know contain fewer than $41 \log {\log {n}}$ highway nodes w.h.p. Let’s consider the worst case where they are all bunched up around u. Let’s denote their contribution $z_{0, \text {inner}}$.

$$\begin{aligned} z_{0, \text {inner}} &\le \sum _{j = 1}^{\lceil \sqrt{41 \log {\log {n}}} \rceil }{ \frac{4j}{j^2}} < 4 \mathcal {H}\left( \sqrt{41 \log {\log {n}}} + 1 \right) \\ &< 25 \log {\log {\log {n}}} \text { for } n > 5 \end{aligned}$$

Recall that we can still have up to $41 \log {n}$ highway nodes in in the same (large) ball as u. Let’s assume they are all as close as possible, meaning that they are all at the edge of the inner ball. Let’s denote their contribution $z_{0, \text {outer}}$.

$$\begin{aligned} z_{0, \text {outer}} < \frac{41 \log {n}}{(3 \sqrt{k \log {\log {n}}})^2} = \frac{41}{9} \frac{\log {n}}{k \log {\log {n}}} \end{aligned}$$

Combining these results, we obtain:

$$\begin{aligned} z < 25 \log {\log {\log {n}}} + \frac{41}{9} \frac{\log {n}}{k \log {\log {n}}} + 26\frac{\log {n}}{k} \text { for } n > 5 \end{aligned}$$

w.h.p., for at most $\mathcal {O}(\log ^2{n})$ invocations.

We provide a tighter bound for the normalization constant, $z'$, in a similar fashion:

Proof

(of Lemma 2). Recall from Lemma 9.4 that balls of radius $2 \sqrt{k}$ have at most 18 highway nodes with probability at least 1/2. When this occurs, $z_{0, \text {inner}}$ can be improved to:

$$\begin{aligned} z_{0, \text {inner}} < \sum _{j = 1}^{5}{\frac{4j}{j^2}} = 4 \mathcal {H}(5) < 10 \end{aligned}$$

Meanwhile, $z_{0, \text {outer}}$ changes to:

$$\begin{aligned} z_{0, \text {outer}} < \frac{41 \log {n}}{(2 \sqrt{k})^2} = \frac{41}{4} \frac{\log {n}}{k} \end{aligned}$$

Overall, with probability at least 1/2, we obtain the improved bounds on the normalization constant:

$$\begin{aligned} z' < 10 + 37 \frac{\log {n}}{k} \text { for } n > 2 \end{aligned}$$

Probability of Distance Halving. As explained before, the first step is to show that we can use the improved bounds on the normalization constant by incurring only an increase in a constant factor to the probability of halving the distance:

Proof

(of Lemma 3). The probability of the improved normalization constant bound $z'$ applying is at least 1/2, and this probability is independent for any nodes a distance of at least $4 \sqrt{k}$ apart (see Lemma 9.4). For values of $k \in o\left( \frac{\log {n}}{\log {\log {\log {n}}}}\right)$, the improved normalization constant bound is already only a constant factor better. For values of $k \in \varOmega \left( \frac{\log {n}}{\log {\log {\log {n}}}}\right)$ we will show that we can always visit highway nodes that are at least $4 \sqrt{k}$ apart, so that we have independence. All our routing algorithms expect to take $\mathcal {O}(\log {n})$ hops on the highway, or $a \log {n}$ hops for some constant a. We expect at least $\frac{1}{2} a \log {n}$ of the highway nodes visited to have the improved bounds apply. By Chernoff bound, we visit at least $\frac{1}{4} a \log {n}$ highway nodes with the improved bounds w.h.p. (with probability at least $1 - n^{-\frac{a}{16 \ln {2}}}$). Since a can be picked arbitrarily large, then with high probability we will visit $\mathcal {O}(\log {n})$-many nodes with the improved bounds along our path, which is the same as our original expectation of how many nodes we will visit, meaning our results are the same up to a constant hidden by the asymptotic notation. Note that a similar reasoning works for smaller values of k as well.

Next, we need to prove a lower bound on how many nodes are in a better phase than us w.h.p.:

Proof

(of Lemma 4). Kleinberg showed that there are more than $2^{2j - 1}$ nodes within lattice distance $2^j$ of t [9], for $\log {\log {n}} \le j < \log {n}$. Within this range, we expect there to be at least $2^{2j - 1}/k$ highway nodes. Since we are only considering the case where $j \ge \log (c(k + \log {n}))$, we can use this to create a Chernoff bound (with $\delta = 1/2$). Letting X be the number of highway nodes:

$$\begin{aligned} \Pr (X \le \mu /2) &\le e^{-\frac{\mu }{8}} = e^{-\frac{2^{2j - 1}}{8k}} \le e^{-\frac{2^{2\log (c(k + \log {n}))}}{16k}} \\ &= e^{-\frac{[c(k + \log {n})]^2}{16k}} < e^{-\frac{c^2(2k\log {n})}{16k}} = n^{-\frac{c^2}{8 \ln {2}}} \\ &< n^{-0.18c^2} \end{aligned}$$

In summary, since we picked $\delta = 1/2$, we expect at least $2^{2j - 2}/k$ highway nodes, to be within lattice distance $2^j$ of t w.h.p. (with probability at least $1 - n^{-0.18c^2}$).

Finally, we use these results to prove the main lemma of this section, the probability of halving the distance:

Proof

(of Lemma 5). From our previous results, we

know we can use the improved bounds for the normalization constant, $z' = 10 + 37\frac{\log {n}}{k}$, with at most a constant factor increase in the probability of halving the distance. Furthermore, we know that there exist at least $2^{2j - 2}/k$ highway nodes in better phases w.h.p. Since they are in phase j or better, they are each within lattice distance $< 2^{j + 1} + 2^{j} < 2^{j + 2}$ from u. Using this, and letting v be an arbitrary long-range connection of u, we obtain:

$$\begin{aligned} \Pr (v \in B_{2^j}(u)) > [64 k z']^{-1} > [64k \times 37(1 + \log (n)/k)]^{-1} \end{aligned}$$

The probability of v not being in a better phase is similarly $1 - \Pr (v \in B_{2^j}(u))$. Recalling that each highway node has Qk independently chosen random long-range connections, the probability of none of them being connected to a better phase is therefore $(1 - \Pr (v \in B_{2^j}(u)))^{Qk} \le e^{-Qk \Pr (v \in B_{2^j}(u))}$. The probability of any one of them being connected is therefore:

$$\begin{aligned} \Pr (\exists v \in B_{2^j}(u)) \ge 1 - e^{-Qk \Pr (v \in B_{2^j}(u))} > 1 - e^{-\frac{Qk}{2368(k + \log {n})}} \end{aligned}$$

When $k \in o(\log {n})$, the $\log {n}$ term in the denominator dominates, and we obtain similar asymptotic results to Lemma 8. When $k \in \varOmega (\log {n})$, the k term in the denominator dominates, cancelling out the k term in the numerator, and leaving us with a constant term dependent on Q. It is worth noting that the constant factors in this analysis are very loose, and also considerably decrease for larger values of n. In any case, we obtain that the probability of halving the distance is at least in $\mathcal {O}(k/\log {n})$ for $k \in o(\log {n})$, and at least $f(Q) = \mathcal {O}(1)$ for $k \in \varOmega (\log {n})$.

1.4 7.4 Removing Local Contact Dependence

In this section, we complete the proof of Theorem 2 by removing the dependence on local connections. The results of the theorem directly follow.

If we do find a long-range connection that takes us to the next phase, we can just take it, but what do we do when there aren’t any? To continue the Kleinberg analogy, we would just keep taking local connections to keep re-rolling the dice, and as long as we never traverse any space twice and never traverse any space that is within $4 \sqrt{k}$ of previous spaces (because of Lemma 9.4), we can assume each step taken is independent of other steps. The obvious problem here is that there is no notion of “local connections” in this randomly selected highway. We could either greedily take local connections in the entire graph until we happen to reach a highway node again (in expected $\mathcal {O}(k)$ time), or we can simply pick any long-range connection that takes us closer to the destination by at least $4 \sqrt{k}$. For values of $k \in o(\log {n})$, we will use the first method (greedily taking local connections), and for values of $k \in \varOmega (\log {n})$, we will use the second.

Values of $\boldsymbol{k \in o\left( \frac{\log {n}}{\log {\log {\log {n}}}}\right) .}$ For these smaller values of k, from Lemma 5, we expect to take $\mathcal {O}(\log (n)/k)$ hops on highway nodes to reach the next phase, and since there are at most $\log {n}$ total phases, we expect to visit at most $\mathcal {O}(\log ^2(n)/k)$ highway nodes throughout the entire routing process w.h.p. In the worst case, whenever we can’t halve the distance, we never have any closer long-range connections, so we would need to greedily move along local contacts towards t until reaching another highway node. Recalling that each node has probability 1/k of being a highway node, and that we expect to visit a highway node every k independent hops. In order to avoid visiting highway nodes within $4 \sqrt{k}$ of each other, we can first walk $4 \sqrt{k}$ hops before checking for highway nodes, which we will expect to find after $4 \sqrt{k} + k \in \mathcal {O}(k)$ hops. Over the entire duration of the routing, we expect to spend $\mathcal {O}(\log ^2(n)/k \times k) = \mathcal {O}(\log ^2{n})$ hops using local connections to reach highway nodes w.h.p.

Values of $\boldsymbol{k \in \varOmega \left( \frac{\log {n}}{\log {\log {\log {n}}}}\right) }$ For these larger values of k, we will prove that we can find a long-range connection to an arbitrary highway node u in phase $\log (c(k + \log {n})) \le j < \log {n}$ that is at least $4 \sqrt{k}$ closer to the destination t, w.h.p. Recall that long-range connections are always only between highway nodes, so taking them will always keep us on the highway. To find the probability of one of these connections existing, we consider a ball of radius $d - 4 \sqrt{k}$ centered on the destination t ($B_{d - 4 \sqrt{k}}(t)$), where d is the distance from u to t ($d = d(u, t)$). Let’s lower bound the probability of an arbitrary long-range connection of u going into this ball. We can assume w.l.o.g. that u shares either an x or a y coordinate with t (see Lemma 13). As before, let’s consider the nested lattice construct, where this time u sits at the edge of one such ball. There are exactly $2b - 1$ balls closer to t than u is at ball distance b, for $1 \le b \le \frac{2d - 2}{6\sqrt{k \log {n}}}$. In order to enforce the condition that we improve the distance by at least $4 \sqrt{k}$, we can dismiss the outer layer of balls, leaving us with $2b - 3$ balls for $2 \le b \le \frac{d - 1}{3 \sqrt{k \log {n}}} - 1$. The maximum distance from u to any node in one of these balls is $2b \times 3 \sqrt{k \log {n}}$. From Lemma 9.1, we know that each ball of radius $3 \sqrt{k \log {n}}$ has at least $9 \log {n}$ highway nodes w.h.p. This lower bound must apply w.h.p. for any highway node along our path, so we must use the looser normalization constant bound, z. We can now lower bound the probability that v is in one of these closer balls:

$$\begin{aligned} \Pr (v \in B_{d - 4\sqrt{k}}) &\ge \sum _{b = 2}^{\frac{d - 1}{3 \sqrt{k \log {n}}} - 1}{ \frac{(\text {min }\# \text { dist } b \text { highway nodes})}{z (\text {max dist to node at dist } b)^2}} \\ &\ge \sum _{b = 2}^{\frac{d - 1}{3 \sqrt{k \log {n}}} - 1}{ \frac{(2b - 3) \times 9 \log {n}}{z(2b \times 3 \sqrt{k \log {n}})^2}} \\ &= \frac{2}{9kz} \sum _{b = 2}^{\frac{d - 1}{3 \sqrt{k \log {n}}} - 1}{ \frac{2b - 3}{b^2}} \\ &> \frac{2}{9kz} \left[ \ln \left( \frac{d - 1}{3 \sqrt{k \log {n}}} - 1 \right) \right] \\ &> \frac{\ln \left( \frac{d}{3 \sqrt{k \log {n}}} \right) }{9kz} \end{aligned}$$

Note that this result holds for $d \ge c(k + \log {n})$ for large enough constant c.

This result holds for a single long-range connection of u. The probability that none of u’s long-range connections are closer is:

$$\begin{aligned} \Pr (\text {none closer}) &< \left[ 1 - \frac{\ln \left( \frac{d}{3 \sqrt{k \log {n}}} \right) }{9kz} \right] ^{Q k} \\ &= \left( \left[ 1 - \frac{\ln \left( \frac{d}{3 \sqrt{k \log {n}}} \right) }{9kz} \right] ^{k z} \right) ^{\frac{Q}{z}} \\ &< e^{-\frac{Q}{9z} \ln \left( \frac{d}{3 \sqrt{k \log {n}}} \right) } \\ &< e^{-\frac{Q \ln {d}}{9 z}} = d^{-\frac{Q}{9z}} \end{aligned}$$

again, holding for large enough constant c.

With this probability established, let’s try seeing how many hops we can take before we hit a dead end. Let’s do this in two parts. First, let’s see if we can get to within a distance of $(a \log {n})^{b z}$ from t for some constants a and b. Since the probability of hitting a dead end only increases as we get closer, the probability of hitting a dead end while in this range is always going to be $< (a \log {n})^{-\frac{b Q}{9}}$. This gives us an expected number of hops of $\varOmega \left( (a \log {n})^\frac{b Q}{9} \right)$ w.h.p. When setting b large enough, we can get this to be $\varOmega (\log ^2{n})$, which is more than the maximum number of steps we expect to spend in routing.

In the second part, we are within distance $(a \log {n})^{b z} \ge d \ge c(k + \log {n})$ of t. From Lemma 1, we know that our normalization constant z is at most $\mathcal {O}(\log {\log {\log {n}}})$ for $k \in \varOmega \left( \frac{\log {n}}{\log {\log {\log {n}}}}\right)$ w.h.p., so $z < w \log {\log {\log {n}}}$ for some constant w. This gives us probability of hitting a dead end of less than $(c (k + \log {n}))^{-\frac{b Q}{9 w \log {\log {\log {n}}}}}$. Setting constant c large enough, we can expect to take at least $\varOmega \left( \log {n}^\frac{Q}{9 w \log {\log {\log {n}}}} \right)$ hops on the highway within this range before hitting a dead end w.h.p. Let’s call this our “allowance”. While this is less than the maximum number of steps we expect to spend while routing, we only have at most $b z \log (a \log {n})$ phases left in this second part, while we spend at most $\mathcal {O}(\log {\log {\log {n}}})$ highway hops per phase. Putting this together, we expect to take at most $f(\log {\log {\log {n}}})^2 \log {\log {n}}$ hops in this second part of the routing for some large enough constant f. Let’s determine if our allowance is enough to get us to t, by considering the ratio r between our allowance and the number of remaining highway hops:

$$\begin{aligned} r &= \lim _{n \rightarrow \infty }{\frac{\log {n}^\frac{Q}{9 w \log {\log {\log {n}}}}}{f(\log {\log {\log {n}}})^2 \log {\log {n}}}} \\ \log {r} &= \lim _{n \rightarrow \infty }{\frac{Q \log {\log {n}}}{9 w \log {\log {\log {n}}}} - \log (f(\log {\log {\log {n}}})^2 \log {\log {n}}) } \\ &= \lim _{n \rightarrow \infty }{\frac{\log {\log {n}}}{\log {\log {\log {n}}}} - \log ((\log {\log {n}})^3) } = \infty \end{aligned}$$

Since $\log {r}$ tends towards infinity, r tends towards infinity, meaning that for a large enough constant c, our allowance is enough to get us to t w.h.p. for arbitrarily large n. Combining these results, we can conclude that we can reach a highway node within distance $c(k + \log {n})$ of t w.h.p. while only taking long-range connections that improve our distance by at least $4 \sqrt{k}$, thus eliminating the need for local connections.

1.5 7.5 Randomized Highway Variant

If it is desired to improve the greedy decentralized routing time of the randomized highway model for smaller values of k to be inline with the Kleinberg highway model, it is possible to reintroduce local connections within the highway nodes, despite the fact that nodes are picked arbitrarily. One straightforward way to do so is to add a local connection between each highway node to an arbitrary highway node in each of the 8 adjacent balls of radius $3\sqrt{k \log {n}}$ (see Fig. 6). From Lemma 9.1 we know that at least one highway node will exist in each of those balls w.h.p. At least one of these adjacent highway nodes will be at least $3 \sqrt{k \log {n}}$ closer to the destination. With this variant, the routing time for smaller values of k is improved to $\log ^2(n)/k$, while only increasing the average degree by a constant, inline with the randomized highway model. However, this model is not as clean as the original, and still maintains the same optimal parameter k of $\varTheta (\log {n})$ with the same result of $\varTheta (\log {n})$ hops, so we will not consider it further.

1.6 7.6 Windowed NPA Proofs

In this section, we prove that the windowed NPA model maintains a constant average degree while having a greedy, decentralized routing algorithm taking at most $\mathcal {O}(\log ^{1 + \epsilon }{n})$ hops w.h.p. Specifically, we will define the routing algorithm as follows: define the subgraph made of nodes with popularity $\log {n} \le k \le A \log {n}$ as the highway, ignoring any long-range connections that do not connect two “highway” nodes. We expect to have $\mathcal {O}(1 / \log ^{1 + \epsilon }{n})$ highway nodes. Using the results from the previous section, we are able to route in $\mathcal {O}(\log ^{1 + \epsilon }{n})$ hops w.h.p.

First, we prove the expected constant average degree:

Lemma 10

The average node degree in the windowed NPA model is Q.

Proof

$$\begin{aligned} \int _{k = 1}^\infty {\epsilon Q k / k^{2+ \epsilon } dk} = \epsilon Q \int _{k = 1}^\infty {1 / k^{1 + \epsilon } dk} = \epsilon Q \times 1 / \epsilon = Q \end{aligned}$$

Where the normalization constant to pick k is:

$$\begin{aligned} \int _{k = 1}^\infty {1 / k^{2 + \epsilon } dk} =\frac{1}{1 + \epsilon } \end{aligned}$$

Next, we show that there are an expected $\mathcal {O}(1 / \log ^{1 + \epsilon }{n})$ highway.

Lemma 11

There are $\varTheta (\log ^{1 + \epsilon }{n})$ highway nodes w.h.p.

Proof

Now, let’s find the probability that a node has popularity between $\log n$ and $A \log n$:

$$\begin{aligned} \Pr (\log n \le k \le A \log n) &= \int _{k = \log {n}}^{A \log {n}}{\Pr (k) dk} \\ &= \int _{k = \log {n}}^{A \log {n}}{1/k^{2+\epsilon } dk} \\ &= \frac{(A^{1 + \epsilon } - 1) \ln ^{1+ \epsilon }(2)}{(1 + \epsilon )A^{1 + \epsilon }} \frac{1}{\log ^{1 + \epsilon }{n}} \end{aligned}$$

Since A and $\epsilon$ are predetermined constants, the probability that a node has a popularity in this range is $\propto \log ^{-(1 + \epsilon )}(n)$.

Importantly, each node within this range of popularities considers all other points within this range of popularities as long-distance node candidates with equal likelihoods, a requirement important for the analysis of the randomized highway model. Next we must prove:

Lemma 12

Each highway node expects to connect a constant fraction of its connections to other highway nodes, where the constant is at least $[1 + A^{1 + \epsilon }]^{-1}$.

Proof

The case where there is the least probability of overlap is when $k = \log {n}$. Let’s consider v, an arbitrary long-range connection of node u, where $k_u = \log {n}$. The probability that v is part of the highway is:

$$\begin{aligned} \Pr (v \in \text {highway}) = \frac{\int _{k = \log {n}}^{A \log {n}}{k^{-2-\epsilon } dk}}{\int _{k = \log {n}/A}^{A \log {n}}{k^{-2-\epsilon } dk}} = [1 + A^{1 + \epsilon }]^{-1} \end{aligned}$$

This is enough to set up an instance of the randomized highway model. An $(N, P, Q, \epsilon , A)$ instance of the windowed NPA model corresponds with an $(N' = N, P' = P, Q' = \epsilon Q [1 + A^{1 + \epsilon }]^{-1}, k' = \log ^{1 + \epsilon }{n})$ instance with a few minor modifications. The highway graph, instead of consisting of nodes with degrees k, consists of nodes with degrees $\log {n} \le k \le A \log {n}$.

A little nuance applies since while $k = \log ^{1 + \epsilon }{n}$, each of the nodes has fewer connections, only $\mathcal {O}(\log {n})$. However, the constant probability of halving the distance analysis still holds, and this algorithm achieves $\mathcal {O}(\log ^{1 + \epsilon }{n})$ expected total greedy-routing steps. This concludes the proof for Theorem 3.

1.7 7.7 Miscellaneous Proofs

Lemma 13

Let $S_d(w)$ denote the set of vertices at lattice distance d away from any vertex w. Let u be any vertex, and let v be any vertex such that $v \in S_d(u)$, and let $B = B_d(u)$. Then $|S_j(v) \cap B|$ is $\varTheta {(j)}$ for all $1 \le j \le 2d$.

Proof

Consider the ratio $R_{j, v} = \frac{|S_j(v) \cap B|}{|S_j(v)|}$ at each $1 \le j \le 2d$. It is clear that no matter where v is located in $S_j(u)$, $R_{j, v}$ always grows smaller as j increases. The value of j that minimizes $R_{j, v}$ for a particular $v\in S_d(u)$ is then 2d, and we can achieve $\min _v(R_{v, 2d})$ when v is a non-corner vertex in $S_d(u)$, in which case $R_{v, 2d}=\frac{d}{8d} = 1/8$. Therefore at every $1 \le j \le 2d$, we have that $\frac{1}{8} \le \frac{|S_j(v) \cap B|}{4j}$, and therefore $|S_j(v) \cap B| \ge j/2$. Since we already have that $|S_j(v) \cap B|\le |S_j(v)| \le 4j$, the lemma follows.