Temporal Reachability Graphs

Note that if $\delta<\tau$ then $\mathcal{R}_{\delta}$ is empty, as even one-hop journeys do not have enough time to arrive before the maximum delay.

This follows naturally from the definition of a reachability graph. Note that the reverse is not true. For example let $\mathcal{G}$ be a TVG with edge traversal time $\tau>0$. Its derived reachability graphs $\mathcal{R}_{\tau/2}$ and $\mathcal{R}_{\tau/3}$ are both empty as their maximum delays are smaller than the time it takes to cross one edge. Therefore, as per definition 2, $\mathcal{R}_{\tau/2}\subseteq\mathcal{R}_{\tau/3}$ even though $\frac{\tau}{2}>\frac{\tau}{3}$.

While not yet in a calculable form, this theorem is at the heart of our approach. Intuitively, it states that if all the reachability graphs with delays close to $\delta$ and $\mu$ are known, then one can calculate the reachability graph with delay $\delta+\mu$. A full formal proof is provided in the appendix (Section A.1), but intuitively, it relies on the following simple idea. Any arc in $\mathcal{R}_{\delta+\mu}$ corresponds to a journey in $\mathcal{G}$ starting after time $t$ and arriving before $t+\delta+\mu$. Either it can be neatly divided into a (possibly empty) journey in $\mathcal{R}_{\delta}$ starting after $t$ and arriving before $t+\delta$ and another (possibly empty) journey in $\mathcal{R}_{\mu}$ starting after $t+\delta$ and arriving before $t+\delta+\mu$, or it starts crossing an edge before $t+\delta$ but after $t+\delta-\tau$ that straddles both time intervals. In the latter case, since the edge traversal time is $\tau$, one can incorporate the straddling edge into a journey arriving before $t+\delta+\epsilon\tau$ and the rest into another journey leaving after $t+\delta+\epsilon\tau$ of temporal length less than $\mu-\epsilon\tau$.

Real-life datasets all have a maximum resolution (e.g., a second or a millisecond) that corresponds to the precision with which they were measured. While it is tempting to map their time domain $\mathcal{T}$ to $\mathbb{N}$, in reality one must account for $0$-second edge durations. For example, if the edge traversal time $\tau$ is null, then even ephemeral edges that last $0$ seconds may be part of a journey. In the more common situation where $\tau$ is strictly positive, one may still encounter $0$-duration arcs in a reachability graph. For example, consider an edge $(u,v)$ that is present for only one second. If $\tau$ and $\delta$ are both also equal to one second, then a one-hop journey from $u$ to $v$ using that edge only exists at precisely the instant $t$ that the edge appears. This, in turn, corresponds to two $0$-second arcs in $\mathcal{R}_{\delta}(t)$: $(u,v)$ and $(v,u)$. Note that both of these arcs will become one-second-long arcs in the reachability graph when $\delta=2s$. These observations lead to the definition of regular time-varying graphs.

Without loss of generality, this definition assumes that the first epoch starts at $t=0$. Intuitively, an $\eta$-regular TVG is one whose instantaneous graph topology cannot change arbitrarily quickly, as, during each epoch, the graph topology remains constant. Ephemeral $0$-second edges or arcs may exist at the start of an epoch, but the TVG then remains constant until the start of the next epoch. Not only are regular TVGs a natural fit for real-life traces, but, as we will see, they also represent a class of TVGs whose reachability graphs are calculable.

This theorem ensures that, as long as $\delta$ is a “multiple” of the resolution $\eta$, $\mathcal{R}_{\delta}$ carries on the $\eta$-regularity of $\mathcal{G}$. The full formal proof is a little technical (Section A.2 in the appendix), but it relies on the following idea. Let us consider a one-hop journey in an $\eta$-regular TVG $\mathcal{G}$ leaving in the middle of an epoch at time $t$. Let $k$ be the integer such that $k\eta<t<(k+1)\eta$. It arrives in the next epoch at time $(k+1)\eta<t+\tau<(k+2)\eta$ (the theorem assumes that $\tau\geq\eta$). Because $\mathcal{G}$ is $\eta$-regular, the departure time of this one-journey can be nudged forwards or backwards as long as it remains with the same epoch. Hence all one-hop journeys with a departure time $t^{\prime}$ such that $k\eta\leq t^{\prime}\leq(k+1)\eta$ are also valid journeys in $\mathcal{G}$. This “nudging” can be extended, though not trivially, to multi-hop journeys which proves the theorem.

Furthermore, the $\tau\geq\eta$ hypothesis guarantees that an arc present in a reachability graph during an epoch is present not only at the start of the epoch like in any regular graphs both also at the start of the next epoch. This property will be leveraged later in this section. However $\mathcal{R}_{\delta}(t)\neq\mathcal{R}_{\delta}(k\eta)\cap\mathcal{R}_{\delta}\left((k+1)\eta\right)$. An example of such a situation is discussed in Section 3.3.

This theorem is the adaptation of Theorem 1 to regular TVGs. The formal proof is in the appendix (Section A.3) but, as previously, it relies on the idea that within an epoch, there exists a little freedom to “nudge” journey departure and arrival times backwards or forwards. In a sense, if both the minimum departure time and maximal arrival time of a journey fall exactly at the start of a time epoch, then this journey can be divided into two sub-journeys whose departure and arrival times also map exactly to the start of epochs. Under this form, the value of a reachability graph at the start of each epoch can be exactly calculated from the values, at the start of each epoch, of the proper set of $\tau$ pairs of reachability graphs.

Unfortunately, this exact formula does not extend to the time spent strictly within an epoch. In a way, the state of the reachability graph during epochs is more important than its state at their starting times. Indeed, the time spent at exactly the start of epochs is infinitesimally small. Therefore any metric averaged over time (e.g. average density) will depend only on the states strictly within epochs. In the next section, we propose an upper and a lower bound on the reachability graph during epochs. As we will later see in Section 5, these upper and lower bounds are in fact nearly always equal and therefore achieve an excellent approximation of the real reachability graph.

We define for every regular reachability graph $\mathcal{R}_{\delta}$ an upper and lower bound, i.e., two TVGs $\mathcal{U}_{\delta}$ and $\mathcal{L}_{\delta}$ such that $\mathcal{L}_{\delta}\subseteq\mathcal{R}_{\delta}\subseteq\mathcal{U}_{\delta}$. Both of these bounds are equal to the reachability graph at the start of each epoch but only differ slightly during the epoch. The upper bound is a straightforward application of Theorem 2.

The lower bound is a little more complicated and involves a modified version of the composition operator (Definition 6).

By definition, at the start of each epoch the approximate composition is equal to the regular composition of the reachability graphs. During an epoch, the condition $(u,v)\in\mathcal{R}_{\delta}\left((a+1)\eta\right)$ is easier to meet than the more intuitive $(u,v)\in\mathcal{L}_{\delta}(t)$. Indeed, with Theorem 2, $\mathcal{L}_{\delta}(t)\subseteq\mathcal{R}_{\delta}(t)\subseteq\mathcal{R}_{\delta}\left((a+1)\eta\right)$, and this composition operator will catch the arcs (including the $0$-second ones) occurring at the start of the next epoch. The same is true for the $(u,v)\in\mathcal{R}_{\mu}(a\eta+\delta)$ condition for arcs ending at the start of the current ($\delta$-shifted) epoch. Combined, these more inclusive conditions make for a tighter lower bound.

The proof, detailed in Section A.4 of the appendix, is fairly straightforward and involves setting the value of $\epsilon$ in Theorem 1 as a function of $\eta$, $t$, $k$, and $n$.

This recursive definition means that the exact value of $\mathcal{L}_{\delta}$ during a time epoch depends on the sequence of compositions that was used to calculate it. For example, in the simple situation where $\tau=\eta=1$, the lower bounds $\mathcal{L}_{4}=\mathcal{L}_{2}\odot\mathcal{L}_{2}$ and $\mathcal{L}_{4}^{\prime}=\mathcal{L}_{3}\odot\mathcal{L}_{1}$ may be different. Proposition 3 only guarantees that they are both included in the real reachability graph $\mathcal{R}_{4}$. However, as we will see, this is not a concern because this lower bound is tight regardless of how it is calculated.

Fig. 3 is an example of a situation with four vertices where the upper bound has an arc during an epoch that is not in the reachability graph. Here $\tau=\eta=1$s. $\mathcal{R}_{1}$ is exactly derived from $\mathcal{G}$ by shortening the end time of each edge by $\tau$. For example, this leads to an ephemeral arc from $a$ to $b$ and a two-second-long arc from $b$ to $d$ (down from a three-second-long edge $(b,d)$ in $\mathcal{G}$). Obviously, not all arcs are represented on Fig. 3. In $\mathcal{R}_{2}$, there are two ephemeral arcs from $a$ to $d$, corresponding to the $a\leadsto b\leadsto d$ journey, and the $a\leadsto c\leadsto d$ journey an epoch later. In between those instants, it is both too late to use the first journey and too early to use the second. Since this arc exists at the start of two successive epochs, $\mathcal{U}_{2}$ errs in considering that it exists during the entire epoch. In this example, the lower bound $\mathcal{L}_{2}$ matches the reachability graph.

Conversely, Fig. 4 is an example of a situation with three vertices where the lower bound misses an arc during an epoch that is present in the reachability graph. Here again $\tau=\eta=1$. The one-second-long edges in $\mathcal{G}$ all become ephemeral $0$-second arcs in $\mathcal{R}_{1}$ (not shown on the figure), that in turn become one-second-long arcs in $\mathcal{R}_{2}$. Furthermore, $\mathcal{R}_{2}$ also contains two ephemeral arcs $(a,c)$ and $(b,d)$ that correspond to the $a\leadsto b\leadsto c$ and $b\leadsto c\leadsto d$ journeys respectively. Here $\mathcal{R}_{2}=\mathcal{L}_{2}=\mathcal{U}_{2}$. Finally, $\mathcal{R}_{4}$ contains an arc $(a,d)$ that corresponds to the existence of the three-hop journey $a\leadsto b\leadsto c\leadsto d$. $\mathcal{L}_{4}$ correctly identifies this journey at two instants $k$ and $k+1$. At time $k$, it composes arc $(a,b)$ from $\mathcal{R}_{2}(k)$ with the ephemeral arc $(b,d)$ in $\mathcal{R}_{2}(k+2)$. At time $k+1$ it composes the ephemeral arc $(a,c)$ in $\mathcal{R}_{2}(k+1)$ with the arc $(c,d)$ in $\mathcal{R}_{2}(k+3)$. However, for $k<t<k+1$, $\mathcal{L}_{4}$ has no way of finding the journey from $a$ to $b$. Note that in this case, $\mathcal{U}_{4}$ is equal to $\mathcal{R}_{4}$.

The example of Fig. 4 also helps us understand the absence of error propagation when calculating successive lower approximations from previous lower approximations. In this example, thanks to the conditions in Definition 8 that refer to the starting times of epochs, $\mathcal{L}_{8}$ will “bridge” over the missing $(a,d)$ arc in $\mathcal{L}_{4}$. However it may miss composed arcs that use the $(a,d)$ arc in $\mathcal{R}_{4}$. These missed arcs will be, in-turn, “bridged” in, for instance, a $\mathcal{L}_{16}$ approximation. Roughly speaking, errors during epochs do not propagate beyond two compositions. Combined with the fact that the lower approximation matches the reachability graph at the start of each epoch, this leads, as we will verify empirically, to a tight lower approximation or the reachability graph.

In this section, we examine how the theoretical results above apply to some simple situations.

Another interesting property of lower approximations of reachability graphs is that an upper approximation may be derived from it. Indeed, since it contains the exact values of the reachability graph at the start of each epoch, it is trivial to calculate their intersection during each epoch. While the method described in this section focuses on the efficient computation of the lower approximations of reachability graphs, it simultaneously computes the upper approximations.

At a high level, the algorithm presented in this section is a binary exponentiation on families of lower bounds of reachability graphs using a special additive operator. To simplify notations, we will consider in this section that $\eta=1$ and $\tau\in\mathbb{N}^{*}$.

We now have our self-sufficient elements, the lower bound families, and an additive operation $\oplus$ between them. For illustration, Fig. 5 details the process of adding $\bar{\mathcal{L}}_{4}$ and $\bar{\mathcal{L}}_{8}$ families to obtain an $\bar{\mathcal{L}}_{4}\oplus\bar{\mathcal{L}}_{8}=\bar{\mathcal{L}}_{12}$ family. The $\mathcal{L}_{3}$, $\mathcal{L}_{4}$, $\mathcal{L}_{5}$, $\mathcal{L}_{7}$, $\mathcal{L}_{8}$, and $\mathcal{L}_{9}$ lower bounds are combined as inputs for three applications of Proposition 3 that yield $\mathcal{L}_{11}$, $\mathcal{L}_{12}$, and $\mathcal{L}_{13}$. Viewed as a black box, this operation combines the $\bar{\mathcal{L}}_{4}$ and $\bar{\mathcal{L}}_{8}$ families into an $\bar{\mathcal{L}}_{12}$ family.

The inner workings of the $\oplus$ operator are embarrassingly parallel. Indeed, each application of Proposition 3 can be run completely independently of the others. This opens the way for highly distributed implementations, whose speed will be determined by that of the composition of $\tau$ pairs of lower bounds of reachability graphs. Accordingly, the next section proposes an efficient algorithm for this composition operation.

A time-varying graph $\mathcal{G}$ may be stored as a time-indexed sequence of edge UP and DOWN events. For example, if at time $t$ an event $\{(u,v),\text{{DOWN}}\}$ occurs, then the edge $(u,v)$ disappears at time $t$ in $\mathcal{G}(t)$. Such a representation is well suited for algorithms that sequentially examine all states of the TVG.

We present a streaming algorithm for composing $\tau$ reachability graph lower bounds as in Proposition 3. Streaming algorithms are well suited to TVGs as their memory requirements do not depend on the duration of the trace but only on the number of vertices [1]. Memory is indeed a limited resource, as reachability graphs can become fully connected cliques and data structures such as adjacency matrices cannot therefore be considered sparse. Furthermore, in our case, a streaming algorithm facilitates a parallel implementation of the $\oplus$ operator as one process can read the input families, duplicate their events, and dispatch these to various workers, each calculating one composition operation.

Algorithm 1 reads its input from $\tau$ pairs of event streams $\{\mathcal{L}_{d+k},\mathcal{L}_{m-k}\}$ and writes $\mathcal{L}_{d+m}=\bigcup_{0\leq k<\tau}\mathcal{L}_{d+k}\odot\mathcal{L}_{m-k}$ to its output stream. It makes use of an arc counter that tracks how many of the conditions in the definitions of $\mathcal{L}_{d+k}\odot\mathcal{L}_{m-k}$ (see Definition 8) are verified by each arc. When a previously down link fulfills one of these conditions, its counter is initialized to $1$ and an UP event is written to the output stream (Line 1). When this counter goes to $0$ the arc is removed from $\mathcal{L}_{d+m}$ and a DOWN event is written to the output stream (Line 1). Note that an arc may be brought up and down at the same time $i$ if its counter goes to $0$ right after it appears (i.e., an ephemeral arc). In this case, the arc triggers both lines 1 and 1.

In more detail, this algorithm directly maps to the three conditions in the definition of the approximate composition $\odot$ (Definition 8):

Algorithm 1 is then used as a building block to implement the $\oplus$ addition of lower bound families. Starting from $\mathcal{L}_{\tau}$’s family, for any $n\in\mathbb{N}^{*}$, $\mathcal{L}_{n\tau}$ is obtained in $\log(n)$ applications of the $\oplus$ operation using a binary exponentiation process.

An implementation of Algorithm 1, fully integrated into a binary exponentiation algorithm over lower bound families, is available as a part of our dynamic trace library (DiTL [28]). This package also contains the code for transforming a lower bound TVG into an upper bound TVG, as well as the time-varying dominating set computation used in Section 5.

The worst-case memory requirements for this one-pass streaming algorithm are straightforward. Its local memory contains a non-sparse arc counter that requires up to $O(N^{2})$ space. Furthermore, for each $0\leq k<\tau$, it maintains two non-sparse adjacency matrices and an arc event list thereby requiring $O(N^{2})$ space. Adding everything together yields a worst-case space complexity of $O(\tau N^{2})$ that is independent of the duration of the trace.

Before examining the worst-case time complexity, a word must be said about the implementation of the adjacency matrices (the arc counter is backed by an adjacency matrix). In our implementation, these are backed by per-vertex hash tables. Insertion and removal are therefore constant time operations but this approach may not scale to TVGs with much greater number of vertices than those considered in this paper. In this analysis we will consider that insertion and removal cost $O(\log(N))$ (e.g., by using binary trees).

At each epoch, arcs are brought up and/or down. For each $0\leq k<\tau$, we note $M_{d+k}$ and $M_{m-k}$ the number of UP and DOWN events in the entire TVGs $\mathcal{L}_{d+k}$ and $\mathcal{L}_{m-k}$, respectively. Let $M=\max_{k}\left\{\max\{M_{d+k},M_{m-k}\}\right\}$ be the maximum number of events in all the involved TVGs. In the worst case scenario where all possible arcs are updated at each epoch, we have $M=O(TN^{2})$ events, where $T$ is the number of epochs. Each processed event costs a modification of the arc counter ($O(\log(N))$), an adjacency matrix ($O(\log(N)$), and up to $N$ counters for the composed paths ($O(N\log(N))$). All of these must be performed for the $\tau$ pairs of input TRGs. Adding everything together yields a worst-case time complexity of $O\left(\tau MN\log(N)\right)$, or $O\left(\tau TN^{3}\log(N)\right)$ in terms of $N$ exclusively. In practice however $M\ll TN^{2}$, so the first formulation is more accurate.

While we have calculated reachability graphs on many publicly available datasets, in this paper we present results based on two real-life contact traces selected for their short beaconing periods:

For comparison purposes, we also provide results based on two well understood synthetic mobility models:

The characteristics of these four $\eta$-regular traces are summarized in Table 2.

For each reachability graph, we compute its time-varying dominating set (TVDS), i.e., a mutable set such that, at all times, there exists an incoming arc to any vertex in the TVG from a member of the TVDS. Thus, at any given time $t$, the TVDS is a traditional dominating set on the directed graph at time $t$. We note this TVDS $\mathcal{D}_{\delta}$ if it is derived from a reachability graph $\mathcal{R}_{\delta}$.

Even on static graphs, calculating a minimal dominating set is a classic NP-complete problem. Here we adapt the well-known greedy algorithm for choosing multipoint relays for broadcasting in a wireless network to the time-varying context [17]. The size of the dominating set calculated by this algorithm is within a factor $\log(N)$ of the optimal, where $N$ is the number of nodes. Our TVDS calculation algorithm reacts to arc UP/DOWN events as follows.

1. 1.

   If, after the arc event is processed, the previous dominating set is no longer a dominating set, then proceed to step 2. Otherwise do nothing.

2. 2.

   Iteratively build the new dominating set, starting from an empty set, by greedily adding the node with the greatest outgoing degree to nodes not yet covered by the new dominating set. If two nodes have equal outgoing degrees, pick the one that is in the previous dominating set.

The time-varying dominating set thus calculated attempts to be both reasonably stable over time and close the optimal.

The rest of this section focuses on the following metrics. Their formulas and average values are found in Table 3.

• •

  Density. The ratio of the number of arcs in a reachability graph at time $t$ over the total number of possible arcs.

• •

  Dominating set size. The normalized number of vertices in the dominating set at time $t$.

• •

  Asymmetry. Here we no longer consider directed arcs but undirected pairs of vertices. At time $t$, the asymmetry is the ratio of asymmetric pairs among pairs that have at least one edge between them.

The approximation algorithm detailed in Section 4 is extremely accurate. Indeed, thanks to Theorem 3 and Definition 8, the approximation calculates the exact value for all times $t\in\eta\mathbb{N}$. Furthermore, the upper and lower bounds are nearly identical during epochs ($t\notin\eta\mathbb{N}$). Fig. 6 plots the density over time for both the upper and lower bounds of the Rollernet reachability graph for $\tau=5$s and $\delta=1$min. The plot has been zoomed in to show only a small span of the $y$-axis for 30 seconds. The upper and lower bounds on density are nearly equal at all times. So are the values of the dominating set size computed from the upper and lower approximations that only disagree in the circled area on Fig. 6. These observations hold for all the reachability graphs computed in this paper. Indeed, looking at the values of average density and average dominating set of the over 5,000 pairs of upper/lower bound TVGs calculated in this paper, the maximum disagreement is $8.10^{-3}$ for the former and $4.10^{-2}$ for the latter. In practice, the difference is smaller than the width of the line in plots. Therefore, in the rest of this paper, we only plot the values based on the lower approximation.

The dynamics of reachability graphs highlight temporal structural properties of the original connectivity trace that are not otherwise accessible. Fig. 7 shows snapshots of a subset of the reachability graph in the high school network captured in the Stanford dataset. When the delay is 20 minutes, the classroom structure of the trace is clearly visible. In the original contact trace, edges within a classroom are unstable and lead to merges and splits of small connected components. In the reachability graph, a classroom is a stable complete subgraph with one teacher. A group of interacting teachers are also visible at the top of Fig. 7a. When some teachers and students later change classrooms, they create strong asymmetries in higher-delay reachability graphs such as the 40-minute one depicted in Fig. 7b.

Fig. 8 shows snapshots of the reachability graph captured in the Rollernet dataset. These correspond to an acceleration phase, where the head of the tour (right on Fig. 8) pulls ahead of the rest of the rollerbladers. Due to the accordion phenomenon, the tail of the tour does not react immediately [25]. This completely prevents short delay communications between these two groups ($\delta=10$s on Fig. 8a). With a longer delay (e.g., $\delta=60$s on Fig. 8b), backward communications towards the rear of the tour become possible (in particular through organizers who stop on the side of the road and let the tour pass them), thereby creating strong asymmetry in the reachability graph.

A more systematic study of dynamic properties over time is shown on Fig. 9. It plots both the proportion of connected pairs of vertices (left axis) and the size of the dominating set (right axis). The pairs of vertices are further divided into symmetric pairs (yellow) and asymmetric pairs (red). At a given time, if the red histogram reaches 1 then a journey exists in at least one direction between all pairs of nodes. If the yellow histogram reaches 1, then a journey exist in both direction between all pairs of nodes.

For a 1-minute delay, the Rollernet reachability graph alternates, sometimes very rapidly, between fully connected states and highly asymmetric partially connected states (Fig. 9a). Therefore, any opportunistic communication system aiming for latencies under a minute will be strongly impacted by the accordion phenomenon. However, if a communication system can tolerate up to three minute delays, then it should be possible to smooth out the dynamic mobility. Indeed, the 3-minute reachability graph is fully connected for the entire duration of the trace, and the size of its dominating set is almost always equal to one.

The Stanford trace alternates static phases in classroom (the valleys on Fig. 9b) and dynamic phases moving between classrooms or around the food court (the peaks on Fig 9b). The progressive “shrinking” of the valleys illustrates how reachability graphs “grow backwards” with increasing delays. Indeed, if there exist a journey from $a$ to $b$ within 20 minutes at time $t$, then there exists a journey from $a$ to $b$ within 40 minutes at time $t$ minus $20$ minutes. Of course, after the increase in delays exceeds the width of the valleys, the reachability graph eventually reaches it maximum density. Note that in this case, an incompressible amount of asymmetry subsists throughout the trace.

Due to space constraints, we have only shown, in this section, results for specific values of $\tau$ but, as a general rule, the smaller the value of $\tau$, the faster the TRG becomes a clique. When analysing a network with a specific application in mind, the value of $\tau$ can be set to a realistic value (e.g. message size over bit-rate), and the plots in this section give an immediate visual understanding of the communication possibilites. The next section examines quantitatively the importance the delay ($\delta$) and the edge traversal time ($\tau$) parameters.

Reachability graphs give straightforward bounds on communication capabilities. Indeed, the density at a given moment is exactly equal to the maximum delivery ratio expectancy of a perfect opportunistic routing protocol whose delay-tolerance is equal to $\delta$ and whose message size over bit-rate ratio is equal to $\tau$. Furthermore, the size of the dominating set indicates the achievable offload ratio in a scenario where the opportunistic network is assisting an infrastructure (e.g., 3G) for disseminating content to all nodes in the network [12].

Given real-world system requirements, i.e., wireless bit-rate estimates, messages sizes, target delivery ratio, and delay-tolerance, reachability graphs provide an immediate answer to the following question: can an opportunistic network support this service? If not, can it effectively supplement an infrastructure in an offloading scheme?

Fig. 10 plots the average density against the delay tolerance for all datasets and for increasing edge traversal times, while Fig. 11 plots the average dominating set size. Rollernet and Random Waypoint share similar characteristics. Both are very sensitive to increasing edge traversal times. When these are close to one second, near 100% density is achievable with a couple of minutes of delay tolerance (Figs. 10a, 10c, 11a, and 11c). In this case, they can support pure opportunistic communications. However, when edge traversal times are longer, tight delay constraints are impossible unless as part of an offloading scenario (e.g., $\tau=10$s and $\delta=60$s for Rollernet, Fig 10a). For $\tau=20$s, Random Waypoint cannot even provide offloading for reasonable delays (Fig. 11c).

Similarly, Stanford and Community share similar features. Regardless of the delay-tolerance and message size, no pure opportunistic routing protocol can provide anything near 100% delivery ratio ¹¹1The $\tau=0$ results for Stanford are an artifact of the 20-second resolution. (Figs. 10b and 10d). Despite this, they are both good offloading scenarios as the size of their dominating sets is consistently below 20% of the total number of nodes, thereby offering potential offload ratios of around 80%. Indeed, in order to disseminate content to the entire network, pushing one copy per classroom in the Stanford case, or one copy per community in the Community case, plus copies to single nodes is an obvious strategy. However, for larger values of $\tau$ (e.g., 20), Community is no longer able to offload content with reasonable delays (Fig. 11d).

Asymmetric communications are a fundamental aspect of opportunistic networks. Too strong focus on inter-contact times may lead to overlook asymmetry, but reachability graphs provide a natural way of studying and quantifying it.

Fig. 12 plots for all datasets the average asymmetry against the maximum delay for increasing values of the edge traversal time. The asymmetry for the $\mathcal{R}_{\tau}$ graph directly derived from the original contact trace is always 0 as it only contains symmetric arcs (see Section 3.4). Asymmetry in Random Waypoint follows a regular pattern: a bell-shaped curve of constant max value, whose width and center increase with $\tau$ (Fig. 12c). In this scenario, depending on the delay constraint, up to 80% on average of connected pairs of vertices can only communicate in one direction. This is due to nodes traveling long straight distances creating asymmetric reachability to nodes they meet from nodes they had met earlier. As the maximum delay increases, return journeys using different intermediate nodes appear, the reachability becomes complete, and the asymmetry returns to 0.

As previously, Rollernet and Random Waypoint show similar behavior. However, for Rollernet the maximum asymmetry also increases with $\tau$ (Fig. 12a). Indeed, as fewer edges are traversable, journeys become less likely between the front and the rear of the rollerblading tour. When such a journey is possible, it creates a longer-lasting asymmetry. In the Stanford trace, asymmetry follows the same increasing bell shape as in Rollernet, but never returns to 0 (Fig. 12b). In fact, this incompressible minimum asymmetry also increases with $\tau$.