Alice has bits of viral information – so-called vitbits – to share. Despite her best efforts, she coughs up a strong signal and emits it using directional breathforming. Luckily, the line-of-sight is masked. Nearby Bob is unconcerned, he relies on an outage. After all, he is not in the near field, and he wouldn’t touch any of the non-intelligently reflecting surfaces in the room.

However, masking the main lobe of transmission leads to multi-path propagation and diversity. Waves of vitbits sitting on droplets (scientifically known as votons) are traveling along different paths, in an attempt to reach a destination. Their power decays quickly over distance, according to a power-law, with an empirical free-space limit of 2 m. But in this case, votons from different directions meet coherently, joining forces and managing to maintain strength and collectively carry sufficiently many vitbits.

Unfortunately, the vitbits find their target. There is no outage, just Bob’s outrage. He became a victim of his vir-ility.

Inspired by current events, let us focus on the “successful” transmission of a virus from one host to another. In the case of a coronavirus, such transmission happens when the infected person (the source) emits a “signal” by breathing, coughing, sneezing, or speaking, and another person (the destination) gets infected when the respiratory droplets land in the mouth or nose or are inhaled. Fundamentally, the process looks very similar to that of a conventional RF wireless transmission. There are notions of signal strength, directional transmission, path loss, and shadowing in both, resulting in a probability of “successful” reception. In both cases, distances play a critical role – there is the well-known 2 m separation in the case of coronaviral transmission that (presumably) causes an outage in the transmission with high probability; similarly, in the wireless case, there is sharp decay of reception probabilities as a function of distance due to path loss.

Given the critical role of the geometry of host (transceiver) locations, it appears that known models and analytical tools from stochastic geometry could be beneficially applied to learn more about the spread of an epidemic. In contrast to the careful modeling of channels and transmitter and receiver characteristics in wireless communications, the models for the spread of infectious diseases are usually based on the basic reproduction number R0, which is a population-wide parameter that reflects distances between individual agents very indirectly only. Since it is important to understand the effect of physical (Euclidean) distancing (often misleadingly referred to as “social distancing”) and masking (i.e., shadowing), it seems that incorporating distances explicitly in the models would increase their predictive power. Such models would account for the strong dependence of infection probabilities on distance, akin to the path loss function in wireless communications, as well as directionality, akin to beamforming. Time of proximity or exposure can be incorporated if mobility is included.

This line of thought raises some vital questions: How can expertise in wireless transmission be applied to viral transmission? What can stochastic geometry contribute to the understanding of how infectious diseases are spreading (spatial epidemics)? Is there a wireless analogy to “herd immunity”, perhaps related to percolation-theoretic analyses of how broadcasting over multiple hops in a wireless network leads to a giant component of nodes receiving an information packet?

I believe it is quite rewarding to think about these questions, both from a theoretical point of view and in terms of their real-world impact.

In the literature, the probability that the signal-to-interference ratio (SIR) at a given location and time exceeds a certain value is often referred to as the coverage probability. Is this sensible terminology, consistent with the way cellular operators define “coverage”? All publicly accessible cellular service coverage maps are static, i.e., their view of “coverage” is purely based on location and not on time. This seems natural since a rapidly changing coverage map, say at the level of seconds, would not be of much use to the user, apart from the fact that it would be very hard to collect the information at such time scales.

In contrast, the event SIR(x,t)>θ depends not only on the location x but also on the time t. A location may be “covered” at SIR level θ at one moment but “uncovered” just a little bit (one coherence time) later. Accordingly, a “coverage” map based on this criterion would have to be updated several times per second to accurately reflect this notion of coverage. Moreover, it would have to have a very high spatial resolution due to small-scale fading – one location may be “covered” at time t while another, half a meter away, may be “uncovered” at the same time t. Lastly, there is no notion of reliability. For each x and t, SIR(x,t)>θ either happens or not. It seems natural, though, to include reliability in a coverage definition; for example, by declaring that a location is covered if an SIR of θ is achieved with probability 95%, or 95 times out of 100 transmissions.

Hence there are three disadvantages of using SIR(x,t)>θ as the criterion for coverage:

The event depends on time (at the level of the coherence time)

The event depends on space at a very small scale (at the granularity of the coherence length of the small-scale fading)

The event does not allow for a reliability threshold to define coverage.

How can we define “coverage” without these shortcomings? First, we interpret coverage as a purely spatial term, consistent with the coverage maps we find on the web; it should not include a temporal component, at least not in the short-term – hopefully cellular coverage keeps improving over the years, but it should not vary randomly many times per second. Put differently, coverage should only depend on the network geometry (locations of base stations relative to the position x) and shadowing, but not on the rapid signal strength fluctuations due to small-scale fading. The solution to eliminate the temporal component is fairly straightforward – we just need to average over the temporal randomness, i.e., the small-scale fading. Such averaging eliminates the other two shortcomings as well. For a base station point process Φ, we define the conditional SIR distribution at location x as

Here, the probability is taken over the small-scale fading, which eliminates the dependence on time (assuming temporal ergodicity of the fading process, which means that the ensemble average here could be replaced by a time average over a suitable long period). If shadowing is present, it can be incorporated in Φ. Then, introducing a reliability threshold ν, we declare

The reliability threshold ν appears naturally in this definition. The probability that P(x)>ν is the meta distribution of the SIR, since it is the distribution of the conditional SIR distribution given Φ. For stationary and ergodic point processes Φ, it does not depend on x and gives the area fraction that is covered at SIR threshold θ and reliability threshold ν. The figure below shows a coverage map where the colors indicate the reliability threshold at which locations are covered, from dark blue (ν close to 0) to bright yellow (ν close to 1). So if P(SIR>θ) is not the coverage probability, what is it? It is simply the complementary cumulative distribution (ccdf) of the SIR, often interpreted as the success probability of a transmission.

When visualizing distributions with infinite support, we face the challenge that they can only be shown partially. Usually we try to judiciously choose the interval of our plot so that the interesting part is revealed, and it is understood that outside that interval the function is essentially zero (for a density) or essentially zero or one (for a cumulative distribution). However, there are two disadvantages to that approach: First, if two distributions are not shown on the same interval, it is hard to compare them. Second, interesting asymptotic behavior in the tails are masked. In many cases, using a linear scale suffers from a third shortcoming: Interesting features may occur on a significantly different scale. Which would require choosing a large interval, but that, in turn, may mask the behavior in some parts.

When plotting distribution of signal-to-interference ratios (SIRs), the standard approach is to use a dB scale. The complementary cumulative distribution (ccdf) is usually interpreted as the success probability of a transmission (in an interference-limited setting). While the dB scale allows for visualization the cccdf over a larger range, it has its own shortcomings: First, it turns the one-sided infinite support [0,∞) into the two-sided infinite support (-∞,∞), which can make selecting a suitable interval harder. Second, it distorts the ccdf, which prevents the viewer from obtaining insight into asymptotic behaviors.

So how can we resolve these issues? It turns out that there is a straightforward solution. It is based on a homeomorphic mapping of [0,∞) to the unit interval [0,1]. This mapping is given by the function T(x)=x/(1+x), and the resulting scale is called MH (Möbius homeomorphic) scale. For comparison, the dB scale has the mapping 10 log_{10}(x). In the figures below, the dummy variable is θ, and we have θ dB=10^{θ/10}, and θ MH=θ/(1-θ). In the important high-reliability regime θ→0, θ MH ∼ θ, i.e., there is no distortion.
The three figures show 6 ccdfs on a linear, dB, and the MH scale. Clearly, the linear scale plots mask the information about the tail. Some curves go to zero (too) quickly, while the behavior of the yellow one past 100 is completely hidden. The dB scale displays the transitional regime more prominently, but all curves have an inverted S shape, which reduces the discriminative power of these plots. In particular, for small θ, they all become flat. For example, the blue (Pareto) and the cyan (Lévy) curve look similar in the dB scale, but with some shift. The MH scale, however, reveals that the asymptotic behavior on both ends is, in fact, quite different. Also, the red (another Pareto distribution) and green (gamma) curve look fairly similar in the dB scale, while the MH scale emphasizes the difference between the two. Generally, the MH plots enhance the differences because the slopes at 0 and at 1 can assume any value, while the slopes in the dB plots always approach 0 – assuming the range is chosen wide enough.

In summary, the MH scale has the following advantages:

There is a single finite interval that reveals the complete distribution. There is never a question of what interval to choose, and nothing remains hidden.

The asymptotic behaviors are clearly visible. In comparison, on the dB scale, the behavior near 0 and towards infinity is always obscured.

In the case of SIRs, the MH scale has the additional interpretation as visualizing the distribution of the signal fractionS/(S+I) (SF) on a linear scale: If F(θ) is the ccdf of the SIR S/I, then F(T(θ)) is the ccdf of the SF.

The MH mapping and its application to SIRs and signal fractions was first introduced in the invited paper M. Haenggi, “SIR Analysis via Signal Fractions”, IEEE Communications Letters, vol. 24, pp. 1358-1362, July 2020.

Reading the literature, we often encounter sums of the form

for a point process Φ. This is an instance of what may be called “over-indexing”, since the subscript k is clearly not needed. It makes the notation unnecessarily cumbersome, but there is nothing mathematically wrong with it. If it is tacitly assumed that using a dummy variable of the form x_{k} somehow defines k, problems arise. For instance, what exactly is meant by this formula?

To make this concrete, let us focus on this simple form:

What is the result? It is not 14 but undefined, since k is undefined. This problem occurs fairly frequently in the form

where h(k) is used to denotes a fading random variable indexed by some (undefined) k and f is a path loss function. What is meant is usually

Alternatively,

Here the fading random variables are indexed by the points, i.e., they can be interpreted as marks associated with the points. This is the form I personally prefer, for it does not require a particular way of indexing the points and it is more compact.

The concept of the typical point is important in point process theory. In the wireless context, it is often concretized to the typical user, typical receiver, typical vehicle, etc. The typical point is an abstraction in the sense that it is not a point that is selected in any deterministic fashion from the point process, neither is it an arbitrary point. Also, speaking of “a typical point” is misleading, since it suggests that there exist a number of such typical points in the process (or perhaps even in a realization thereof), and we can choose one of them to “obtain a typical point”. This does not work – there is nothing “typical” about a point selected in a deterministic fashion, let alone an arbitrary point. That said, if the total number of points is finite, we can, loosely speaking, argue that the typical point is obtained by choosing one of the points uniformly at random. Even this is not trivial since picking a point in a single realization of the point process does generally not produce the desired result, for example of the point process is not ergodic. Many realizations need to be considered, but then the question arises how exactly to pick a point uniformly from many realizations. If the number of points is infinite, any attempt of picking a point uniformly at random is bound to fail because there is no way to select a point uniformly from infinitely many, in much the same way that we cannot select an integer uniformly at random.

So how do we arrive at the typical point? If the point process is stationary and ergodic, we can generate the statistics of the typical point by averaging those of each individual point in a single realization, observed in an increasingly large observation window. This leads to the interpretation of the typical point as a kind of “average point”. Similarly, we may find that the “typical American male” weighs 89.76593 kg, which does not imply that there exists an individual with that exact weight but means that if we weighed every(male)body or a large representative sample, we would obtain that average weight. For general (stationary) point processes, we obtain the typical point by conditioning on a point to exist at a certain location, usually the origin o. Upon averaging over the point process (while holding on to this point at the origin), that point becomes the typical point. The distribution of this conditioned point process is the Palm distribution. In the case of the Poisson process, conditioning on a point at o is the same as adding a point at o. This equivalence is called Slivnyak’s theorem.

In the non-stationary case, the typical point at location x may have different statistical properties than the typical point at another location y. As a result, there exist Palm measures (or distributions) for each location in the support of the point process. In this case, the formal definition of the Palm measure is given by the Radon-Nikodym derivative (with respect to the intensity measure) of the Campbell measure. In the Poisson case, Slivnyak’s theorem still applies.

What’s the point of stochastic geometry? In this blog, I am sharing random and deterministic thoughts about the use of stochastic geometry, mainly in wireless network modeling and analysis.