Preface
In a world increasingly interconnected by advanced technologies, we have become reliant on the ubiquitous networks that bind our devices, our services, and, indeed, our lives. As technology grows more pervasive and complex, the expectations for network performance grow concurrently. Simultaneously, advancements in mobile computing, the Internet of Things (IoT), and the rapid emergence of smart cities have transformed the connectivity landscape, unveiling an era of unprecedented opportunities and challenges. In this evolving landscape, one burgeoning concept stands out: Opportunistic Networks.
Opportunistic networks (OppNets) represent a distinctive category of networks, pioneering unconventional approaches for facilitating communication in exceptionally challenging conditions characterized by high node mobility, sporadic connectivity, and resource constraints. These networks are typically marked by recurrent network partitioning and extensive delays due to the dynamic mobility of the nodes and sporadic or intermittent connectivity.
The concept of opportunistic networking stems from the insight that human contacts often follow patterns and that these patterns can be harnessed to transport data. OppNets do not assume an existing end-to-end path between a source and a destination. Instead, they propagate data by leveraging sporadic and transient contacts between network nodes. Opportunistic routing is facilitated by mobile nodes that act as data carriers, forwarding packets to their destinations by capitalizing on their movement and temporary connections to other nodes.
One of the most crucial aspects of an OppNet is the concept of store, carry, and forward, where each node stores the message, carries it around, and forwards it to other nodes or the destination node when the opportunity arises. Opportunistic networking has been used successfully in emergency response and wildlife tracking scenarios where traditional networking methods do not suffice.
There are other various similar or at least related types of connectivity forms:
• Delay-tolerant networking (DTN): DTN is a network architecture designed for environments where network connectivity is intermittent or unreliable. DTN allows nodes to communicate with each other even when there is no direct con-
nection using store-and-forward protocols to pass messages through intermediate nodes.
• Mobile ad hoc networking (MANET): MANET is a type of wireless network in which nodes communicate directly without needing a centralized infrastructure. This allows for dynamic and flexible networking, making it suitable for opportunistic networks.
• Content-centric networking (CCN): CCN is a networking paradigm based on content rather than hosts or locations. In CCN, content is identified by its name, and nodes store and forward content based on its name. This makes it suitable for opportunistic networks, as content can be cached and delivered even without a direct connection between nodes.
• Mesh networking: Mesh networking is a type of network architecture in which nodes communicate with each other in a self-organizing and decentralized manner. In an opportunistic network, mesh networking could be used to create a network of nodes that can communicate with each other, even when there is no centralized infrastructure available.
This book delves into various aspects of Opportunistic Networks, offering an indepth analysis of their structure, function, and application in various scenarios. Each chapter of this book is designed to provide a comprehensive understanding of the intricate concepts related to OppNets.
The first part of the book, titled Foundations of Opportunistic Networks, explores the fundamental principles and metrics associated with Opportunistic Networks. Chapter 2 delves into the mobility of opportunistic networks, discussing the process of collecting mobility traces, converting these traces into location and contact information, and examining the impact of mobility on OppNets. This chapter also explores the critical aspects of large-scale mobility characteristics and the role of scale and density in these networks.
Chapter 3 provides an extensive exploration of data dissemination in opportunistic networks. Here, we discuss optimal dissemination, flooding, mobility, social awareness, and network coding-based protocols. We also discuss the role of caching in OppNets and summarize each chapter to consolidate the concepts discussed.
Chapter 4 focuses on security and trust issues in opportunistic networks. The chapter delves into various attacks that opportunistic networks can experience and provides strategies to prevent these attacks. This includes methods to prevent selective forwarding, sabotage, manipulation of complex metadata, and data forgery.
The second part of the book, titled Evaluation of Opportunistic Networks, provides a comprehensive exploration of the techniques for evaluating opportunistic networks. Chapter 5 provides an overview of the methods and metrics used in assessing OppNets, offering an understanding of how these networks can be optimized and improved based on these evaluations.
Chapter 6 discusses theoretical models that offer a conceptual understanding of opportunistic networks. This includes general assumptions and their validity, Markov chain, and compartmental models. We also explore other methods that offer insights into the functioning of OppNets.
Chapter 7 delves into the simulation models for opportunistic networks, providing a comprehensive understanding of application models and link abstraction. This chapter also explores mobility models and contact traces, thoroughly exploring these models’ terminology, properties, and applications.
Chapter 8 discusses the simulation tools used to understand and develop opportunistic networks. This includes a brief introduction to discrete event-based simulation and exploring various simulation tools such as OMNeT++, The ONE, ns-3, and others. We also provide a comparative analysis of these simulation tools to help readers choose the best fit for their needs.
The final chapter of this part delves into the concept of benchmarking OppNets. Here, we focus on the various benchmarks and metrics used for comparing different opportunistic protocols. This section provides the reader with the required understanding to assess the performance and effectiveness of OppNet protocols.
In the third part of the book, titled Implementation of Opportunistic Networks, we look at the technologies that enable connectivity in OppNets. We discuss a range of technologies from Mobile Ad Hoc Networks (MANETs) to LoRa, Bluetooth Low Energy (BLE), Wi-Fi Direct, ZigBee, and even Satellite communication. We also explore the integration of OppNets with existing and upcoming cellular technologies like 4G, 5G, and 6G. Each technology is discussed in detail, discussing its advantages, limitations, and potential uses in enabling OppNets.
This technological understanding sets the stage for exploring device characterization in Chap. 11, where we discuss factors like power availability, cache restrictions, and the processing and memory requirements in devices used in OppNets.
Building upon this foundation, the book moves towards practical scenarios in Chap. 12, providing numerous case studies. These cases cover a variety of applications for OppNets, ranging from disaster management, communication in remote areas, challenged IoT scenarios, and satellite networks to Smart Cities.
As we turn the final page of this fascinating journey exploring opportunistic networks, we can reflect on all the intricate aspects we have encountered. Starting from the basics and reaching out to real-world applications, our journey has encapsulated the vast landscape that is OppNets. In the pages of this book, we have dived deep into the structure and operation of OppNets. We have unraveled the complexities behind these innovative networks, providing you with the knowledge to understand what makes them tick. Having a grasp on the nuts and bolts of OppNets is crucial. It allows you to analyze critically how these networks perform, understand their strengths and weaknesses, and evaluate their adaptability in various situations. Moreover, we have ventured into the practical aspects of implementing OppNets. This is where the rubber meets the road. Understanding how to bring OppNets to life in the real world is key. To do this, we have covered the enabling technologies, those essential bits and pieces that make OppNets function. These technologies are the backbone of OppNets, and understanding them provides insight into how we can apply OppNets effectively. We have also delved into device characterization. In other words, we have explored how different devices can use OppNets. This helps us understand the capabilities of different devices, how they interact with OppNets,
and the role they play within these networks. By understanding the role of devices, we gain a deeper appreciation of how all the parts of OppNets come together.
But the understanding of OppNets does not stop at the technical details. We have also journeyed into the myriad ways these networks can be used. We have looked at OppNets’ potential across a spectrum of applications, from disaster management to communication in remote places.
Disaster management, for instance, poses many challenges where conventional networks might fail. However, OppNets can make a huge difference, providing critical communication capabilities when they are needed most. We have delved into how OppNets can contribute to managing disasters, giving you an idea of their potential in such high-stakes situations. Remote communication is another fascinating application. In areas where conventional networks can not reach, OppNets can bridge the gap. They can provide communication capabilities in areas where it would otherwise be impossible, from deep wilderness locations to high-sea marine environments. Moreover, the emergence of the Internet of Things (IoT) has revolutionized how devices communicate. But it also brings its own challenges. This is where OppNets can step in, offering solutions to keep IoT devices communicating effectively even in challenging conditions. We’ve also explored how OppNets can play a role in the future of smart cities. OppNets offer a way to manage and facilitate this complex web of communication as cities become more connected. This is a glimpse into the future, a future where OppNets play an integral role in our daily lives.
In addition to all of these, we have touched on the exciting realm of satellite networks. Here, OppNets can offer significant benefits, helping to manage the complex task of maintaining communication across vast distances of space.
So, as we close this book, you now have a wealth of knowledge about OppNets. You understand their structure, operation, implementation, and wide-ranging applications. With this knowledge, the reader is now equipped to understand and appreciate the potential of OppNets. Whether thinking about how they could be used in new situations, understanding their role in different technologies, or even exploring the possibility of working with these networks in your future career, you are now ready to take on the world of OppNets. This is just the beginning, and there is still so much to explore.
Bremen, GermanyAnna Förster Valencia, SpainPietro Manzoni Valencia, SpainEnrique Hernández Orallo Hamburg, GermanyKoojana Kuladinithi Bremen, GermanyAsanga Udugama July 2023
Chapter 1 Mobility of Opportunistic Networks
Opportunistic networks are mostly driven by their mobile, volatile nature. They form spontaneously and fall apart as quickly again. In this chapter, we will explore the mobility, scale, and density of opportunistic networks independently from any forwarding or data dissemination protocols.
Mobility is probably the most important and most well-studied property of opportunistic networks. Three different directions of research can be observed: the study of human mobility in general and its properties; the development of mobility models which mimic human mobility and can be used in simulations; and the impact of mobility on the performance of opportunistic networks. We will explore the first and third properties here in this chapter and will leave the simulation mobility models to Chap. 6.
As discussed in the preface, opportunistic networks refer to applications on Earth and do not include other delay-tolerant applications, such as satellite and space probe communications. Differently from applications on Earth, space communications are very well predictable—one knows exactly when and where the satellite or probe is and what communication quality to expect. In fact, it is exactly humanborne mobility that drives the development of opportunistic networks since it is not well predictable and very complex. Human-borne mobility includes walking patterns but also vehicular patterns from bikes, scooters, cars, trains, etc.
In the following sections, we will first discuss two main properties of opportunistic networks closely related to their mobility: scale and density. Then, we will dive deeper into how to gather realistic mobility data and how to characterize it to better understand the behavior and nature of opportunistic data dissemination.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
A. Förster et al., Opportunistic Networks, https://doi.org/10.1007/978- 3- 031- 47866- 6_1
1.1 Scale and Density
Scale is all about the number of involved nodes, while density discusses how many nodes are there per given area. These two properties, even if seemingly very similar to each other, are, in fact, independent and very important for opportunistic networks.
The scale of opportunistic networks has been discussed in almost every paper published in the area. The goal (more of a wish) of all researchers is to enable very-large networks, meaning millions of devices. For example, such a network will easily emerge from a smart city scenario, where all cars, people, and their personal devices, sensors on traffic lights, street lights, etc., will connect to each other. However, such a scenario is also quite unrealistic to cover with a purely opportunistic network.
In Chap. 11, we will discuss in depth which applications assume which scales. We have seen many different applications, with most of them rather small to middle scale (i.e., hundreds to thousands of nodes). Here, we would like to discuss how the scale of the network impacts its data dissemination properties.
The first and most important statement we can make is that the more nodes we have, the more traffic we have. This is plausible if we assume that the nodes themselves produce the traffic. It is also straightforward to realize that it quickly becomes unrealistic to assume that all nodes can receive all information. This is also typically not necessary since information is more relevant in its direct surroundings (e.g., traffic information in the city) than in further away areas. At the same time, as we will discuss in detail in Chap. 2, many data dissemination protocols are floodingbased, irrespective of the fact which data is relevant or not.
With Increasing Scale, the Delay Increases This is also straightforward since delay in opportunistic networks is not only based on the delay of the communication technology but mainly on the mobility delay of the nodes. Thus, it can take very long to get one piece of data from, let’s say, one village to the next if the data needs to wait for the local public bus to transport it.
One interesting question is how large can an opportunistic network grow and still be useful. The answer is simple: it depends. If this is the only existing network in a post-disaster scenario, it will always be useful. If it supports a smart city scenario with real-time information, it will quickly become non-useful, as most of the data will arrive too late. This also depends on the density of the nodes. Still, in general, the delay is always higher for opportunistic networks than for infrastructure-based networks and will never be real-time in a multi-hop scenario. For each individual application, a maximum data delay needs to be identified. If the opportunistic network does not support it, infrastructural elements must be considered, such as drones, roadside units, etc.
The density of the network is defined as the number of nodes per area. The impact of this property on data dissemination in opportunistic networks is quite straightforward: the more nodes we have, the easier it is to disseminate data.
Fig. 1.1 The denser the network is, the better the performance, and the less mobility plays a role. Two experiments are presented: with 50 nodes and with 1000 nodes. With more nodes, the mobility model (CMM, ORBIT, or SWIM) is less important to achieve better performance metrics
Density has a slightly different impact on opportunistic networks compared to scale. On one side, density also increases the traffic. The impact is the same: at some point, on-time delivery becomes unrealistic because the traffic exceeds the capacity of the network. However, density decreases the delivery delay. This is due to the fact that data dissemination has many more options to forward the data, the number of contacts increases [133] and the mobility delay decreases. This can be observed in Fig. 1.1. Here, we have depicted the delivery ratio and the delivery delay for two densities (50 nodes vs. 1000 nodes in the same area), three mobility models (models to simulate mobility, explained in detail in Chap. 6), and for the well-known Epidemic data dissemination protocol (explained in Chap. 2). The two metrics are depicted over time, so we see how the network develops over time.
Let us first analyze the delivery delay alone (the bottom graph in Fig. 1.1). We see two clusters of lines: the lines with lower delays correspond to the 1000 nodes scenarios. It is interesting to note that there is almost no difference between the different mobility models. The lines with higher delays correspond to a significantly lower density of 50 nodes. Here, the delay is larger by several orders of magnitude. Additionally, we can observe that the mobility model matters: different mobility models induce lower and higher delays, depending on how mobility is modeled
(refer to Chap. 6). A similar observation can be made from the delivery ratio (the upper graph). The same cluster of lines (this time, the lines with lower delivery ratios) exhibits the same behavior: the mobility model matters. For the denser scenario (the upper cluster of lines for delivery ratios), this does not hold: the mobility model does not matter.
This leads us to a very important observation: With increasing density, mobility plays a less significant role. However, one has to be careful about the average against density variability in the network. While a large network like a smart city might have a high average density, individual areas might differ in density significantly and might affect the rest of the network. For example, if there is a large park with only pedestrians allowed inside and closing during the night, this area might develop into a blind spot for the network.
In the next sections, we will turn to the problem of how to gather realistic mobility data, how to characterize it and use the newly acquired knowledge to better understand the nature of opportunistic networks.
1.2 Collecting Mobility Traces
In order to study human-borne mobility, we first need to obtain large-scale, highquality data. This is usually done by collecting traces from devices carried by humans or by vehicles. Two different types of traces can be differentiated: GPS traces and contact traces. GPS traces are simple logs of GPS coordinates by devices carried by people or vehicles, such as smartphones or sensor nodes. These GPS traces can be used to understand the mobility of individual people—how far they move, how often they repeat particular routes, how fast they move, etc. Such traces can also be used to simulate OppNets by calculating which devices would have contact with each other considering a particular communication technology— details about this process will be provided in Chap. 6.
However, such contacts between people are also interesting from the research perspective—how often do people meet each other? How many people do you mean on average, and for how long? In order to study these properties, researchers have two options: either convert GPS traces into contacts (by assuming some concrete communication technology, as discussed above, for example, Bluetooth) or gather contract traces directly. This is done, for example, by using a particular communication technology like Bluetooth on a smartphone and regularly scanning for neighbors. This option has one big advantage: data privacy is better. While GPS traces show exactly how, where, and when a person moves, contact traces only scan for Bluetooth IDs and do not reveal movement information.
Collecting traces of either alternative is expensive and tedious work. One has to implement an application or even a dedicated device, approach people to carry these devices, manage errors and problems while collecting the information, and finally, process and store the data. Additionally, high-quality traces require high-scale and long-term experiments. Many researchers who have done such data collection
Table 1.1 Some of the most popular mobility traces of opportunistic networks
Trace and citation
San Francisco [170]
Participant Experiment
Device carriers number duration
Taxis 536 30days
Rome [30] Taxis 320 30days
Fire Department of Asturias, Spain [34] Vehicles and firemen 229 1year
Rio de Janeiro [58] Buses 12,000 few days
ZebraNet [224] Zebras 11 10days
SUVNet [142] Taxis 4000 Few days
MIT [60] People 97 196days
Haggle Infocom [191] People 44 few days
Milano [150] People 12 6days
Gowalla [104] People 978 4months
Microsoft [122] Cars 40 2weeks
UCSD [147] People 275 3months
ETH [212] People 3000 75days
Lake Geneva [127] People 200 365days
campaigns luckily share their data in various databases. Some well-known large datasets are listed in Table 1.1.
For example, the San Francisco trace is very popular—it is large (536 taxis), it is relevant for both opportunistic and vehicular networking applications, and it is long (30days). The GPS trace is depicted in Fig. 1.2
1.3 From Mobility Traces to Location Information
Raw GPS traces are not very useful per se. Let us consider the example from Fig. 1.3. The example is taken from our publication on a GPS-based mobility model
Fig. 1.2 GPS trace sample (San Francisco cabs)
Fig. 1.3 An example of a raw mobility trace with extracted points-of-interest (POIs). Depending on the parameters of the extraction model, more or less POIs are identified
called TRAILS [69], which will be detailed further in Chap. 6. On the left, the raw data is presented, separated by color for different times (e.g., the red color represents one continuous route on a particular day. In fact, this simple separation by time is already a pre-processing of the data, as it reveals which GPS points belong to which route. One can clearly see that there are places where many points overlap or are close to each other or that the distances between the individual points are different, depending on how often the GPS measurements were taken and how fast the travel was. Much information is still hidden: did the user take a vehicle? Which one? Did the user stop somewhere? And what about that point: was it the actual target of the user (e.g., her office or the supermarket)? or was it only a necessary involuntarily stop such as a traffic light? This information can be partially revealed by extracting points of interest (POIs).
Identifying POIs is not trivial at all. Let us start with a simple example and just define a POI as a place of diameter D , where the user has spent at least time T . However, what is an appropriate value of T ? If the user went to the post office to drop a letter, and that went really fast, is this a proper POI? Context-wise, it is because this was the ultimate reason for the user to move. However, if we fix the value of T to something as small as to serve the post office example, then many other POIs will pop up, such as stopping at the traffic lights (which is clearly not of interest) or even stopping to tie your shoe! This can also be seen in the example of Fig. 1.3 in the center and on the right—depending on the values of D and T , the number of POIs grow or shrink, always leaving lots of false positives or false negatives. The value of D is not any better: a shopping center is large, while a post office is small. If we set the value of D to be too small, many large POIs will be missed altogether and vice versa.
FurtherchallengesinidentifyingPOIsinGPStracesisthefactthatGPSlocations are costly to obtain and noisy. In order to save energy, GPS fixes (another term for GPS measurements) are obtained not very often, for example, every minute or even every hour. Depending on the time one spends in a particular POI, it might be missed altogether. This is illustrated in Fig. 1.4. When many regular GPS fixes are taken, the route looks smooth, and POIs can easily be identified. The example on the left assumes GPS fixes are taken every 30s (very often for a realistic case!) On the right,
Fig. 1.4 An example of GPS traces with measurements taken every 30s and every 5min. Pay attention to the POI denoted by a circle—on the right, it almost disappears
we have reduced the GPS fixes to only every 5min (ten times less, quite a realistic value for real traces). We can clearly observe how important properties of the data are lost forever—which route did I take? How long did I spend in a particular place? Pay special attention to the circled area—on the left, it is clearly a POI, while on the right, it is not as clear anymore. To make things worse, sometimes a GPS fix is not available when you try to obtain it, and a point is missing altogether. To make things worse than that, the precision of GPS fixes is not ideal, and a point can as well be on the next street or in the middle of a lake.
There are several ways of addressing this problem, which go beyond the scope of this book. Good overviews of existing algorithms are provided in [126, 139], and [209].
1.4 From Mobility Traces to Contact Information
As we already discussed shortly above, mobility traces can also be converted into contact information. This is, how long was an individual node connected to another node? Generally speaking, there are two ways of collecting this data—either directly or indirectly by converting GPS traces into contact traces. Let us explore these two ways.
1.4.1 Converting GPS Traces Into Contact Traces
GPS traces are usually lists of coordinates of each individual node at discrete time intervals, e.g. every second or every minute. Thus, at each time interval, we can
calculate the distance between any two nodes in the network. After that, a function can be applied to the distance to decide whether particular two nodes are in contact or not. The simplest such function is:
Thus, if the distance between two nodes is less or equal to a predefined nominal distance X , then the nodes are in contact; otherwise, they are not. This is, of course, an oversimplification. In reality, the nominal distance itself depends on the communication technology used, theproperties of theenvironment, theinterference, etc. Complex models, which take some or all of these into consideration, are typically called link models, and we will discuss them in greater detail in Chap. 6 For our purposes here, it is sufficient to understand that the contact is strongly correlated with the distance between two nodes and that it varies over time. Unless the mobility of two nodes can be predicted perfectly (which is only possible if we have full control over that—like in satellite networks), there is no way to anticipate the contact either. In other words, if you cannot predict mobility, you will not know whether two nodes will be in contact in the future. The contact will come and go, and this is the main reason to call them opportunistic.
1.4.2 Direct Gathering of Contact Traces
Contact traces can also be gathered directly. This means that not the coordinates of the nodes are taken, but their current contacts. For example, we could note down all Bluetooth contacts of our devices regularly. This approach has many advantages:
• The real coordinates arenot taken, and thus higher dataprivacy isachieved. Many people feel uncomfortable giving away their coordinates, as this reveals their habits and locations (e.g., one could deduce that a person lives at a particular address and is out of home every Tuesday morning). Contact information does not reveal your location, and the devices to which contact is available are simply numbers and not people. This increases the acceptance of data gathering campaigns.
• Since coordinates are not taken, a GPS interface is not needed, and this makes the experiment cheaper. This is especially true when no smartphones are used but dedicated devices, like in [191]or[71].
• A real communication technology in a real environment is used. Thus, we do not need a complex model to compute the contact, and all environmental properties are intrinsically taken into account.
However, it also has one significant disadvantage: The data is not symmetric. Let us explore this problem in detail.
Let us assume only two nodes, A and B. At any time interval, both nodes use somecommunication technology, forexample, Bluetooth,toscantheirsurroundings and to note down which devices are in range. We already know that the success of this scan depends not only on the technology itself but also on interference, the environment itself, etc. Thus, it can happen that node A “sees” node B but not vice versa. This is normal, and for other applications, like pairing, the devices will simply repeat the process for some time until they both see each other and can exchange data. However, for us, it is a single trial, which simply repeats regularly. This is a problem later when we want to use this contact (for example, in simulation) to exchange data—can we assume the contact exists or not? There are two main strategies to resolve this issue: (1) If one node “sees” the other, then we have contact; and (2) only if both nodes “see” each other we have a contact. Usually, the first one is preferred, as it is less restrictive and assumes that communication between both nodes is somehow possible.
An example of this problem is illustratedinFig. 1.5. The two nodes have different perspectives on whether the contact is there or not. These perspectives need to be combined in some way to decide finally whether this pair of nodes are in contact or not.
Of course, we could also use the raw asymmetric data in simulation studies. This would mean that we assume one node can send something to the other, but not always vice versa. For example, when node A “sees” node B, this actually means that node B was able to send something to node A. On one side, this would make the simulation even more realistic. However, it also introduces a lot of complexity to the simulation study, which makes it less transparent and understandable. For most simulation purposes, this is simply too much, and the simpler method is sufficient.
Fig. 1.5 Example of asymmetric contact traces. Top: The Y axis shows whether the node “sees” the other node or not. Thus, the contact is different from the perspective of Node A and B. Bottom: Two different interpretations of the contact are implemented: a more restrictive one (orange line, two-way communication is required) and a less restrictive one (red line, one-way communication is sufficient)
1.5
Mobility Metrics
Once mobility traces are available, one can start exploring their properties. Various metrics can be considered, typically organized into spatial, social, and temporal ones [38, 56]. The purpose of computing such metrics is twofold. First, they can be used to understand and classify the mobility and contact patterns of users or vehicles in order to develop better data dissemination strategies, design better infrastructures or provision the right amount of resources. However, the same metrics can also be used directly in the data dissemination process. For example, suppose a vehicle would like to distribute a piece of information quickly. In that case, it can use some mobility metrics of the other vehicles to decide which vehicles to hand over this information.
Spatial and temporal metrics are usually applied to mobility traces (GPS coordinates), while social metrics are usually applied to contact traces. Spatial metrics describe the movement of a particular user or group of users and include:
• Spatial variability is defined as the variability of locations a user visits. Given auser u and her set of visited locations L, we can compute first the probability P(L) of each location for this user (in other words: how probable is for this user to visit a particular location again). Then, we can compute Shannon’s entropy [196] for this set of locations. Explaining Shannon’s theory will go well beyond the scope of this book. We recommend reading this intuitive introduction by S. Vajapeyam [219]. The spatial variability of a population can be easily computed as the mean of individual variabilities.
The larger the variability is (individual or of a group), the more different and unpredictable locations are visited. For example, taxis that obey the requests of their passengers will have a very high variability, while a human being will have a low variability because she visits mostly the same few places (home, office, school, supermarket). However, recall the definition of location from Sect. 1.3: spatial variability does not mean that a person gets very far; it only means that she visits many different locations, which could be anywhere.
• Radius of gyration is exactly what we were missing as information in the spatial variability. It describes how far a person moves around. It can be defined in two ways: either starting from a particular location (e.g., home) or without a center location. The first one is very practical and easy to understand for people of vehicles: the center is simply their most often visited location. Most researchers define the radius of gyration as the maximum distance from their home to any other visited location. However, it also makes sense to define it as the mean or median distance from their home to any other location to account also for the probability of visiting remote locations.
Calculating the radius of gyration without a predefined center is also possible but somewhat impractical, more complicated, and less intuitive. One would need to simply calculate the maximum distance between any two locations in the trace of a node and then divide it by two. This would make sense for nodes which do
not have the notion of a home—for example, shared cars or scooters, which are never returned to a particular starting point.
• Travel distance is the distance between two consecutive locations in the trace of a user. It is different from the radius of gyration as it reveals more about the pattern of movement of a particular node. A person could have a very large maximum radius of gyration and very short mean travel distance at the same time because she only rarely goes to a very remote place and mostly stays around her home.
Temporal metrics explore time patterns in mobility traces. The most important ones are:
• Visit time is simply the time spent by a user in each location. It is useful to predict how long a particular user will spend there next time, but also to predict how long any user will spend in a particular location.
• Travel time is the temporal property of the travel distance and is defined as the time between two locations. This can be used to deduce the mean of transportation, the transport infrastructure condition, etc.
Social metrics explore the contact information of users and attempt to deduce their social interactions. One has to be really careful in interpreting these metrics since the data only says that someone has been close to someone else for some time. This does not mean at all that they know each other or that they intentionally meet. It could be that I “meet” the security guide in a parking lot every day without any further intention but parking my car. At the same time, for the purpose of data dissemination (which is the ultimate goal of exploring opportunistic networks in this book), this information is sufficient. The most important ones include:
• Contact entropy is the variability equivalent for contacts and is defined as the variability of different contacts of a particular user. It is defined and computed similarly to the spatial variability above. The larger the contact entropy, the more contacts a user has. The most probable contacts are often interpreted as friends (recall the warning about interpreting results!).
• Inter-contact time (ICT) is normally defined as the time interval between two consecutive contacts of the same pair of nodes. From inter-contact times we can derive the contact rate, which is the average number of contacts per time unit between pair of nodes. We can also consider the contacts between the same node and the others. For data dissemination, this value is interesting in case I am looking for this node in particular; I could wait until I meet it directly, or I could forward the data to someone else if I know that I will meet it only in a few days.
• Contact duration is very similar to visit time but defined as the duration of a contact. This is very useful for predicting how much time we have to exchange data.
• Encounter regularity is defined over a particular time window, usually a day. It is calculated as the time two nodes stay in contact during the time window.
There exist many more metrics, which are typically variations or combinations of the above. Now that we know how to evaluate and describe mobility and contact traces let us explore their characteristics for different types of nodes.
1.6 Large-Scale Mobility Characteristics in OppNets
At the beginning of research in opportunistic networks, it quickly became evident that the mobility of opportunistic devices is not purely random. This was supported by plausibility (people do not move randomly, but rather periodically and predictably), by data gathering campaigns like the ones discussed above, and last but not least by the results of data dissemination schemes, which were worse than expected.
Early research on mobility characteristics focused on inter-contact times, i.e., the time between two consecutive contacts of the same pair of devices (see also the definition in Sect. 1.5). This was due to the assumption that inter-contact times drive the performance of opportunistic networks. Several works agree on the following findings:
The Inter-Contact Time Distribution Follows a Power-Law Let us consider the graph from [114] and copied in Fig. 1.6. It depicts the CCDF (complementary cumulative distribution function) of inter-contact times of various mobility traces. It can be clearly seen that in the beginning (until the vertical line, which we will explain next), all traces follow a straight line with not-so-different slopes and, thus, apower law.
Fig. 1.6 Figure 1 from [114] (copied). It shows the CCDF (complementary cumulative distribution function) of inter-contact times of various mobility traces
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She looked at him now with clear, resigned eyes. She said quietly, "Is there any use of it?"
"Not if you—couldn't love me. If you don't believe in my love, or that I could make you happy."
She replied slowly, "How I wish I could say to you, or put clearly to myself, all that is in my mind. I wish I dared listen to you. But it will be easier for both of us—the less there is to remember. Please let me go."
Despair crept into his voice as he answered her, "Perhaps you have condemned me already. Is that what you mean?"
She said proudly, "You don't understand. If I was sure I loved a man, and wanted to marry him, it would be for what he meant to me, not what he had meant to other women." He started eagerly to interrupt, but she held up her hand. "But even if you were that man," she said firmly, "I wouldn't say 'yes.' It would only mean unhappiness for both of us—in the end. We are not meant for each other."
"But why?—why?" he cried.
She replied, "I cannot tell you."
"It's unjust. Unfair! You're denying me—and perhaps yourself—the greatest happiness in the world, and giving no reason for it."
"Please!" she cried, as he seemed about to sweep her into his arms, to crush out all of her doubts and questionings. "There is so much unnecessary suffering in the world. Let us spare ourselves any needless pain. I mean what I have said—and please believe that I am sorry—for both of us."
He followed her with stricken, beaten eyes as she slowly walked into her office and took her hat and coat.
"Good-bye, Mary," he said as calmly as he could.
"Good-bye," she said. "I shall be in in the the morning—as usual."
As usual! When she had gone, he flung himself into John's chair and put his head into his arms, pressing his fingers into his forehead to crush out the pain that was there. He remained thus for half an hour, unable to think, to move, aching in body and soul. Then, gradually, a reaction set in. Why had he to suffer so? Why had the only pure love he had ever had in his life been cast aside as if it were something presumptuous, unclean?
He forced himself to his feet and walked into his own office, hardly knowing what he was doing, and, spreading the papers from his brief case out upon his desk, he tried to work on the new estimates for Dorning. He was starting to pity himself now, and gradually a fierce resentment, not against Mary—for he still loved Mary—but against the whole scheme of things, the world with its petty moral code, seized him. He laughed aloud, and it echoed very unpleasantly through the vacant office. Bah! What was the use of burying his past, when the past had arisen from its coffin to mock him at the critical moment. Bah! Why deny one's self pleasures, why fight against women like Elise, why try to change the leopard's spots when the world chose to think them blacker than they really were?
He tossed his pencil down impatiently and took to pacing the office. A mad, reckless mood was coming upon him that he could not control. It was after nine o'clock when finally he forced himself back into his chair, his mind having been wrenched into a semblance of order, and resumed his labors upon the report.
He did not hear the light tap upon his door. It was not until the door softly opened and quietly closed again that he became aware that a second person was in the room.
CHAPTER XV
Elise was standing just inside the door, smiling at him from the shadows.
She had changed her costume. A gauzy black wrap covered her black evening gown, from which her white shoulders could be seen rising. Her small, tight-fitting black hat was draped with black lace that shaded her eyes almost with the effect of a mask. She was quite aware of the impression she made as she stood there silently. The tired man raised his eyes and stared at her. Then he pulled his long, handsome body out of the chair and arose questioningly.
At length she approached him and said lightly, "Ah, you are a man of your word after all. I was afraid the work was just a bluff."
"Why have you come back?" he asked almost sharply from behind the table. "You must go at once."
"Oh, don't be annoyed, Rodrigo. No one saw me come in. I have a private key to the place, you know. Please don't begrudge me the innocent pleasure of doing something slightly clandestine." She came near to the chair opposite him, and went on, "And now that I am here, won't you be a polite host and ask me to sit down?" Without waiting for the invitation, she took the chair, laying aside her wrap as she did so.
He sank into the chair opposite and rustled the papers uneasily.
"I haven't had dinner yet, Rodrigo," she suggested. "I thought that if I came back you might relent and take me to dinner. In fact, I was so sure of it that I dashed up to Aunt Helen's and changed my gown."
He was silent for a moment, and then he raised his head suddenly and, looking her in the eyes, said, "I'm sorry you came back, and I won't take you to dinner. But now that you are here, it's as good a time as any to talk certain things out that are bound to come up sooner or later."
She made a wry little face, cupped her finely ovaled chin in her hands, and smiled at him. "Heavens, Rodrigo, I believe you, of all men, are about to preach to me. Please don't, I beg of you! Remember that I am the wife of a man who is somewhat of a preacher."
"Leave John out of it," he returned. "He's altogether too good for either of us to discuss. You and I understand each other, Elise. I am quite aware of
the game you have been playing. I——"
She cut in with an unpleasant little laugh. She rather enjoyed his violent seriousness.
"So—I have been playing a game," she encouraged him.
"Yes. You married John for his money. You wanted to marry me until you discovered that I was poor."
She was not offended. "Why didn't you warn your best friend then, if you saw through my purpose so clearly?" she asked calmly.
"What chance did I have to warn him? He was head over heels in love with you at once, and it was too late. It would have half killed the poor chap to tell him the truth. I had to let him make his mistake."
Her smile left her face and her eyes darkened. "No, I will not let you say that," she replied earnestly. "I will grant you that I married John chiefly for his money. I admire him—but I have never loved him. And I did outwit you, my friend. You made a very clever attempt to show John what an error he was making in falling in love with me, but I forestalled you. And that was quite a feat, Rodrigo, for you are usually rather keen in matters of that sort.
"But I have made John happy. He tells me that fifty times a day—fifty times too many! One does not enjoy having it drummed into one that one is an angel. I have taken his money and his love, but I have been faithful to him and I have made him a good wife. Now I think I have earned the right to a little something for myself."
He stared at her tensely. "What do you mean, Elise?" he asked.
Her manner lost its hardness, and she leaned toward him.
"I had to marry the wrong man, Rodrigo," she said softly. "My whole existence has had its foundation in money. But it hasn't made me happy. I have been miserable ever since my marriage. It hasn't made me stop—loving you!"
"Elise—no! You mustn't say that," he cried, and rose swiftly to his feet.
She had risen too as she said, "I won't believe that you are as surprised as you pretend to be. You must have guessed it plainly enough many times. I think that we were meant for each other and that a few words spoken by a minister is not going to keep us apart. I will ask John for a divorce if you wish, and marry you. I will do anything—but I will not go on living a lie."
He stared at her, fascinated, wondering if he had heard her aright. Had some malignant fate brought her and her confession to him at the precise moment when Mary had abandoned him?
He said slowly, "You will have to go on. You are mad to think of anything else. I will go away at once, home to Italy. I had planned to go over there this fall anyway. But I will change my plans, and not return."
She laughed. "And you think that I am innocent enough to believe that would be the only reason for your departure. Mary Drake has evidently told you that she is leaving here also."
He started, suspicion dawning in his eyes. He asked, "Mary has told you that? You must have spoken to her about me first. What did you say to her?"
"Yes, I spoke to her about you. I told her the truth. I told her that she could never be happy with you, that you are not the sort to stay contented with such a wife. And I think she agreed with me. She is a very sensible girl. There are certain traits in your character, Rodrigo, that a girl like Mary could never reconcile herself to."
He returned fiercely, "That's rot! If there's anything about me that Mary could object to, it's long since passed. I'm through with my past forever. No woman in the world means a thing to me but Mary."
Her answer was to come close to him and say softly, "You have lost Mary forever, Rodrigo. In your heart you know it. And in your heart you know that what you have just said is nonsense. I dare you to test it. I dare you to hold me in your arms and repeat it. I dare you to kiss me—and then let me go!"
He caught her hand. "Elise, are you crazy!" But he did not relinquish her hand, until her arms had slipped slowly around his neck and her lips were very close to his.
"Why do we pretend any longer, you and I?" she murmured. "You are not like John or Mary. You are only chasing thin, white shadows when you try to fashion yourself after them." And with a swift movement of her head she had kissed him.
He cried, "No—no! You must go—please."
"I love you." And she clung to him.
Fascinated, harassed, he did not resist her any longer. He took her into his arms and buried her face in his kisses.
When at last he let her go, she still held him close and said almost in a whisper, "We will go to the Van Clair, Rodrigo, and dine high up on the roof, near the stars. You can go to your appointment afterward. No one will know —or care."
"Except
our own souls."
"I don't believe we have any," she said, with a queer little note of solemnity. "I think this life, this happiness, is all—and we must take it while we may."
He kissed her again, completely under her spell, and then he said tensely, "Go now, Elise. I will finish here, and I will meet you in fifteen minutes at the Van Clair."
Obediently she secured her wrap from the table, flung it about her and started for the door.
She smiled back at him, whispering, "In fifteen minutes, Rodrigo." And just as she closed the door, "I love you."
He turned, his emotions running riot within him, back to the papers on his desk. For ten minutes he tried in vain to work. Then, with a gesture of
helplessness, he started tossing the papers into his brief case. He had risen and taken his hat when the telephone rang.
It was John.
"Did you line up the stuff, Rodrigo?" came Dorning's distant voice. "I took a chance on finding you at the office. I wanted to make sure to-night that everything was all right and you were coming down here, because Hodge and Story's representative just got in and is all set to take the business away from us."
John! It was like the voice of a rescuing angel. Rodrigo with an effort composed himself and replied, "Everything is fine, John, and I'm taking a train in half an hour."
"Great," said the voice on the wire. "And Elise—did you have dinner with her? Have you seen her?"
Rodrigo replied, "No."
By the time he had hung up the receiver he had made his decision. Elise's spell was broken, broken by that trusting voice on the wire. He would not even telephone her that he was not coming to her. He could not trust himself to talk with her. If she were desperately offended, so much the better!
He seized his brief case and hat and made for the door and a taxi-cab.
CHAPTER XVI
Rodrigo flung himself into his berth on the midnight train to Philadelphia with no idea of sleep. One resolve kept pounding in his head. He would tell John Dorning everything when he saw him, and then he would clear out. He heard people shuffling in the aisle outside of his curtained resting place. They were addressing the porter and each other in that hoarse penetrating
whisper that passengers affect on sleeping cars with the mistaken idea that it does not disturb the sleepers. He became conscious of the train getting under way with clanging bell and dashing about of human feet on the cement platform. For half the journey across the flats of New Jersey he was awake. Then, emotionally and physically exhausted, he fell into a doze.
Crisp, sunshiny weather greeted him as he stepped out into Broad Street, Philadelphia, some hours later. It had the effect of clearing his brain. The world was rolling along cheerfully, unconcerned, after all. Ferris and the other members of the library committee were already in session with John when Rodrigo appeared at Ferris's office.
John had the opportunity for only a word or two with his partner privately before the conference went into session. "Did you go to dinner and the concert with Elise?" Dorning asked eagerly. When Rodrigo shook his head in the negative, John frowned a little and went on dolefully, "Gad, how I miss her. Whatever the consequence, I'm not going to leave her again. I'll bring her along, no matter how bored she gets."
It seemed to Rodrigo in that instant that it would be nothing short of murder to shatter this man's dream. He simply couldn't do it, at least not for the present.
Nor was he any nearer to his confession that evening as John sat opposite him in the dining car on the way back to New York. John was elated. They had closed the contract successfully and he was going back to Elise. He chided Rodrigo several times with being so preoccupied. They parted at Grand Central Station, John having two minutes in which to catch the Greenwich-bound train.
Mary Drake was putting flowers in a vase on his desk when Rodrigo arrived at his office the next morning. She frequently did this, but considering the circumstances surrounding their last conversation, he was a little surprised to find her there. Nevertheless he greeted her gravely and stood standing until she would have finished her task and departed. But he became gradually aware that she was using the flowers as a subterfuge, that she did not intend to leave until she had spoken to him.
Mary said, with the air of a person who has been thinking something over for some time and is having some difficulty in expressing exactly what she means, "Rodrigo—there is something I should like to say." And, though he offered her no encouragement, she continued. "I have come to the conclusion that I was not as wise the last time I spoke to you as I thought I was. I have been thinking it over ever since. I was unjust to you. I belittled my feelings toward you. And I said there was a reason why we could never marry, and I didn't do you the justice to tell you what it was."
"I don't think telling me now will help either of us," he replied, striving to keep the bitterness out of his voice. "Things have changed for me since then. I over-estimated myself. I told you I was a better man—than I am. Today I see clearly that I was a fool."
She asked, suddenly apprehensive, "Something—that has taken away your love for me?"
His reply was bitter. "No, my faith in myself. Night before last, I weakened so that I don't deserve anybody's love, least of all, yours."
She recovered, smiled and came nearer to him, bravely intending to comfort him. "You are too hard on yourself, Rodrigo. You are angry and bitter. And that is my fault, I know."
"No, you have nothing to do with it," he said almost brutally. "I am going away from here too, as soon as I can. I shall stay away, forever."
He was surprised at the response in her face. She seemed glad, relieved. She hastened to explain. "Oh, Rodrigo, don't you see that that clears things up for us, for you and me? That eliminates the barrier that stood between us? I did not have the heart to tell you I could never say I loved you as long as you remained with Dorning and Son, as long as you and John were so closely associated. And I did not dare suggest breaking off your friendship."
"John?" he asked, mystified. "What has John got to do with you and me?"
"Not John, but——"
"Elise?"
She hesitated, then, "Yes. If I admitted to you that I loved you, I would always have had her to fight. And I couldn't. She spoke to me about you day before yesterday, and I saw that she would do anything to prevent us loving each other. I did not believe what she said about you. But it showed me to what lengths she would go, and I was afraid. Fighting her would mean the end of your friendship with John, of your connection with Dorning and Son. Oh, I realize the grip she has upon John. If it came to a choice between you and her, you know which he would keep. And I was not sure what your feeling for me might turn to if I were the cause of a break between you and John. Mrs. Dorning is clever, fascinating, and, I am afraid, quite relentless. I know her feelings toward you and how hard she has tried to——"
He cut in savagely, "Have I ever given you any reason to suppose that Elise and I——"
"No. Not you," she interrupted quietly. "I have overheard you talking to her on the telephone several times. I know how you have sought to avoid her. I can speak frankly about her to you, I think. You will know that I am not moved by jealousy or a desire to gossip or anything petty. But she has called John's office several times from the Van Clair Hotel, for instance, on occasions when she knew he was not here and was to meet her somewhere later. She has given me messages over the 'phone for him, and each time I heard voices laughing and shouting near her. One evening when I passed the Van Clair on the way to the subway, she got out of a taxi with a strange man and went in. That place had a bad reputation, you know. It is just as well for New York that it has burned down."
He stared at her, startled, and, striving to make the question casual. "There was a fire at the Van Clair? When?"
"Why, night before last, just after midnight. It was in all the papers. It burned to the ground."
Dismay gripped him, and he turned away quickly so that she could not see his face. At once Mary read that it had something to do with her, and she laid her hand upon his shoulder, her face flushed and smiling.
She said softly, "Perhaps it was that fire, the feeling it brought that we never know what will happen, never realize how short a time we may have to rectify a mistake, that showed me how wrong I was day before yesterday. I love you, Rodrigo. I will be your wife—if you still want me."
He turned a stricken face to her. He was held in a sudden fear and foreboding. He had hardly heard what she had said. And he had no time to answer her, for the door of his office was flung violently open and John Dorning, excited, disheveled, burst upon them.
"Rodrigo!" he cried from the door. Then, coming forward, "Thank God. I found you here."
He looked so badly that Mary asked in alarm, "You're ill, John. Can I do anything for you?"
"Thank you, Mary—no," he answered, and gathering from his tone that he wished to be alone with his friend, she left quietly.
He almost ran up to Rodrigo. "Elise was not there when I got home, Rodrigo! She left no word of any kind. I've called up everybody. I can't find her."
Rodrigo sagged against the desk, as if struck a blow. He repeated dully, "Can't find—Elise?"
"No. Rodrigo, do you know where she is—do you? I'm worried to death. Anything might happen to her in this town. Accidents—anything."
By this time, with a great effort, Rodrigo had recovered a semblance of control over himself. He spoke soothingly. "Oh, that's nonsense, John. Have you called her friends?"
"Everybody. I've been to the police. I've traced all the ambulance calls. I've found out about fatal fires, and there haven't been any, except one in some hotel. I've driven and telephoned all over town. People must think I'm crazy. I 'phoned Warren down at his place, and he's helping me search too." He ran his hands nervously through his damp, blond hair. He cried, "And I will go crazy, Rodrigo, if I don't get on track of her soon." He seized his
friend's lapel and fixed wild eyes upon him. "You don't think she could have run away from me, left me without a word, do you? No, of course not. Not that. We loved each other too much." He fell to pacing the floor rapidly.
"There's probably some very obvious explanation of her absence," Rodrigo strove to soothe him, and himself. "There usually is. Have you called Mrs. Palmer?"
John turned abruptly, his whole expression changing to one of intense relief. "I'm an idiot!" he cried. "I never thought of her. My car's outside. I'll drive up there at once. Mrs. Palmer is ill, as a matter of fact. Perhaps she's taken a turn for the worse and Elise was called there suddenly. I'll run right up." He snatched up his hat and was gone.
Hardly had the door closed when Rodrigo bounded to the clothes-tree and took the unread morning paper from his overcoat pocket. He sank into his chair and spread the sheet eagerly on the desk in front of him. There, in screaming black headlines, it leaped out at him:
VAN CLAIR FIRE VICTIM IS STILL UNIDENTIFIED
B W G T H
P H T H N B F
Up to an early hour this morning, the woman occupying the room on the ninth floor of the ill-fated Hotel Van Clair, which burned to the ground shortly after midnight Wednesday, remained unidentified, and no trace of her charred body had been found in the still smoking ruins. The hotel register, the only direct means of identification, has evidently burned and—
With a sudden cry of anguish, he crushed the paper violently between his hands, as if to destroy the devastating news it brought him. The sheet fell to the floor as he stretched his arms out in a gesture of hopelessness.
After a while he became aware of a hand upon his shoulder, and Mary's voice was saying gently, "I heard John leave, so I came back. What is wrong!"
He felt himself crumpling. He leaned against her, raising his fear-stricken eyes to her. "Elise! She's gone. John has been looking for her. He's half crazy. But he'll never find her. I know." And, as her face remained questioning, "The paper says a woman has been burned. The woman was Elise—and I—I sent her there. She came back here that night and—well, she fascinated me. I forgot everything. I was to meet her later at the Van Clair. She left me to meet me there later. Then John telephoned long distance, about the business, and I came to my senses. I didn't go to the hotel. She must have stayed there. The fire broke out half an hour after she left me. So you see, Mary—I sent her there—I killed Elise! And I can never tell John— never!"
Growing horror gathered in her eyes. She whispered, "It is—horrible."
"I sent John away on a fool's errand. I had to have time to think."
She said tensely, "But you say she came back to you, here? It was her idea, your going to the hotel? I know—the fascination of her, Rodrigo. And she went there alone—"
"What difference does that make?" he said wildly. "What good that I came to my senses? I sent her there. And now John! Counting on me to see him through—me!"
"You must tell John," she said firmly.
"I can't!"
"It would be kinder than to let him live not knowing, always wondering and hoping. It's cowardly not to tell him."
"Tell him—that, because of me, his wife, his wife, whom he adored, is dead?"
"Not because of you—in spite of you."
Rodrigo answered her, calmer, now reasoning. "You don't realize how he loved her, set her up as a saint upon an altar. I could not tell him the truth. It would blacken her forever before the whole world. I think he would prefer suffering any torture rather than that."
"That is a compromise, Rodrigo, and, therefore, wrong."
He said excitedly: "Call it what you please—I can't tell him!"
"Not even if I promise to help you with all the love I am capable of? Don't you see, Rodrigo?—I feel guilty with you. If I had not been so blind before, this might not have happened." She held out her hands, pleading with him, "Oh, Rodrigo, I love you. I did not realize how much until now. I can forgive everything in you—but cowardice. I will stand by you—but please, please tell John and ask him to forgive you. You can't see him through with that guilt always before you. It's impossible."
But he reiterated stubbornly. "No, I cannot tell him. I cannot kill him too. I would rather kill myself."
She asked quietly, "Not even if it means my love for you? Will you kill that too?"
He replied slowly, "There's nothing—could make me tell him." His voice was unsteady, hie eyes blinded with tears as he turned away from her, her whole body drooping.
The telephone shrilled like a crack of doom, and he fumblingly lifted the receiver as she waited.
"She hasn't been here, Rodrigo!" came John's anguished voice. "What am I to do? I don't know——"
"Don't lose your nerve, old man," Rodrigo replied, and his tones were weak, almost unrecognizable.
"I'm at my wits' end. I've questioned her aunt, the servants here, everybody."
"Come on back down here then, old man," urged Rodrigo. "We'll workout a plan. Don't worry. I'll be here waiting. Come right down."
He hung up the receiver, staring ahead of him, seeming unconscious that Mary was still there. When he became aware of her, he said as steadily as his trembling body allowed, "We'll all be upset terribly—for a while. Please— you will carry on temporarily, try to keep the place going, help us, won't you?"
She answered, "Yes, I will carry on. Don't worry about business. It will be all right." And her eyes too were full of tears.
CHAPTER XVII
Rodrigo sat on the edge of a chair in the living room of Henry Dorning's house at Greenwich. Near him, his frail body sunk deeply in the cushions of a large chair especially comfortably upholstered for his benefit, rested Henry Dorning. The attitude of both was one of nervous expectancy. Had you, however, been unacquainted personally with the two men and been told that one of them was a semi-invalid, you might have been excused for choosing Rodrigo as the ailing one. His lean face had grown thinner and his eyes were dark-ringed from the ordeal he was passing through. His clothing showed little trace of his usual sartorial fastidiousness. He fidgeted in his chair, and when he attempted to light a cigarette the match was held so unsteadily that the tobacco with difficulty caught fire. Henry Dorning, on the other hand, though affected very deeply by the plight of his son, maintained a surface calm that belied the turmoil within him.
Indeed, Henry Dorning was at somewhat of a loss to understand the extreme havoc which the disappearance of Elise had wrought in Rodrigo Torriani. He knew that the friendship between John and Rodrigo was so close that the catastrophe which had befallen his son would be shared by his son's friend. But, after all, Rodrigo was a man of the world, of considerable experience in emotional crises. Why had another's tragedy now broken him up so savagely that he seemed upon the verge of a breakdown? Had not more vital matters been pressing, Henry Dorning would have liked to discover the answer to this question.
As for John Dorning, his mad search for his missing wife had, in the physical sense, terminated for the time being. It had now been two weeks since the fatal fire in the Van Clair. The wild rushing about and pursuit of false clews, the almost total loss of sleep and food had caused John's frail body and almost his strong mind, to snap definitely that morning. For days Rodrigo had been warning him, urging him to abandon the search temporarily, tried with everything in his power, except the uttering of the truth about Elise, to prevent John from becoming a second victim. That morning John had collapsed in Rodrigo's arms and lain in the latter's apartment unconscious. Rodrigo had summoned a doctor and revived his friend. On the physician's advice, he had brought the stricken man at top speed in his car to Henry Dorning's home in Greenwich.
"DO YOU REALLY WANT TO KNOW WHAT I THINK OF YOU?" MARY ASKED SOFTLY.
John had slumped, apparently more dead than alive, in the seat beside Rodrigo throughout that rapid ride. John's father and sister Alice, apprised in advance by telephone, had been awaiting their arrival. The tortured man, at last too weak to protest further, had been put to bed and the Dorning's family physician summoned.
The latter was now up with the sick man, as was Alice Pritchard. Henry Dorning and Rodrigo were at present waiting for the report upon John's condition.
"Hotchkiss is a long time about his examination," Henry Dorning said finally, breaking a long silence. He had been observing Rodrigo narrowly,
and he thought perhaps occupying the Italian's mind with conversation might allay some of Rodrigo's evident nervousness. Otherwise he feared Dr. Hotchkiss might have another patient on his hands when he came downstairs.
Rodrigo nodded shortly.
"It is a blessing, in a way, that John has given out at last," Mr. Dorning went on seriously. "This break was bound to come. I could not stop his frantic search. Neither could you, I suppose. Now he will be kept quiet and will have a chance to recover." He was silent a moment, and then he asked suddenly, "Rodrigo, do you think Elise will ever be found?"
Rodrigo turned his tired eyes quickly to his questioner. "Why—I don't know," he faltered. "But I don't believe—she will."
"Nor do I," said Henry Dorning. "I think she ran away from my boy, and, in doing so, met with a fatal accident somewhere, probably in a motor car. That is my own theory, but of course I have nothing on which to base it. It is merely intuition."
"But she and John were so happy together, loved each other so dearly. Why should she run—"
"Nonsense," the elder Dorning said shortly. "John loved her with all his heart. But I have thought from the first that she had no especial regard for him. I diagnosed her at once as a selfish, frivolous woman. She married John for his money, after carefully sizing the situation up and deciding that I probably would not live very long. Oh, I know that is a brutal way of talking about a woman who is probably dead now. But I cannot help it. I always distrusted her and feared for what she would do to John. A number of her actions confirmed my first suspicions. I was never one to interfere in the private affairs of my children—both Alice and John will tell you that. But I could not help but notice how, for example, Elise would disappear the moment John had left on a business trip and not come back to Greenwich until a few hours before his return. And the type of people she brought into his house—riff-raff is the only word for them.
"Yes, Rodrigo—I may be a terrible old ogre for saying so—but I am glad that woman has gone. I do not, of course, wish her dead. I am afraid, however, that is what has happened. Otherwise she would certainly have communicated with John in some way by this time. You will remember that I had Warren ask you once what you knew of Elise. He said that you told him nothing. I am not going to question you, Rodrigo—now. But I will say that I believe you knew what sort of woman she really was and that you were afraid to tell John, because he was so infatuated with her that it would hurt him. I respect you for that and think you did wisely. I also respect you for trying to protect her when Warren questioned you. Any gentleman would have done the same, and I was foolish and a little caddish for having the question asked. However——"
But Rodrigo was never to know into what deep waters Henry Dorning's line of thought might have led them, for at that moment Dr. Hotchkiss appeared on the stairs and both men turned expectantly. The doctor was a splendid figure of a man, tall, gray and distinguished looking. He was a personal friend of Henry Dorning's as well as his medical advisor. His face now bore a grave expression that confirmed the fears of the patient's two best male friends.
Dr. Hotchkiss approached Rodrigo, who had risen and taken a step or two forward in his anxiety, and the still seated Henry Dorning, whose condition made it imperative that he walk only when necessary. The doctor said quietly, "There is no use in minimizing things. John is in a very serious condition. He is physically and mentally exhausted. I have telephoned for a nurse. It is too big a job for Alice, willing as she is. I don't want either of you to disturb John. I don't want anybody to go near him, except the nurse, until further instructions from me. To speak frankly, any kind of a shock now would bring on—well, something I don't want to contemplate. It will be a long hard pull, I can tell you, to bring him around. And I want you both— and Alice too—to cooperate with me by assuring John absolute quiet during the next weeks and months."
The two listeners nodded. There was a faint feeling of relief in their minds that Dr. Hotchkiss had not pronounced matters hopeless and had even implied that with good fortune and care John might come through satisfactorily.
When the medical man had left, Rodrigo prepared to follow him. He shook hands with Henry Dorning and received the latter's promise to inform him at once if there was any decided change in John's condition.
"As for continuing the search for Elise—you may use your own judgment about that," said Henry Dorning. "I suppose John would wish it pursued with the same zeal. But I leave it to you."
"Very well," Rodrigo replied softly. "I will use—my own judgment."
He drove back to New York at a snail's pace, the speed of his car in harmony with his thoughts of the long, dreary months of remorse ahead of him.
The next day Rodrigo tried hard to submerge himself in the numerous details of business that made up his work and John's with Dorning and Son. It was the only way now that he could stand by his stricken friend. Mary Drake was his able lieutenant—a silent, rather impersonal sort of lieutenant, to be sure, but he could expect nothing different now, he grimly told himself.
An alarming week followed at the Dorning home in Greenwich. For two or three days John's condition was very bad. There were periods in which he alternately raved in hysteric delirium and then sank into a coma, recognizing nobody and sustained by a scarcely detectable heart-beat. In his periods of delirium he called loudly upon Elise, upon Rodrigo, upon the mother who had died in his childhood, while the nurse, Alice Pritchard, and Doctor Hotchkiss labored with physical strength and opiates to quiet him. In that week, Rodrigo lived through a hundred hells, calling on the telephone every few hours to receive bulletins that sank his heart anew each time.
At the end of the week he learned from the doctor that John's physical condition had taken a slight turn for the better. Mentally, however, he was very bad. Dr. Hotchkiss indicated his fears that, unless the strain were in some way removed, his young patient's mind might go.
In disposing of the increased business worries placed upon his shoulders by the absence of John, Rodrigo found unexpectedly efficient assistance in the person of Max Rosner. For Rodrigo had taken a practical means of
