Interstellar Travel and Galactic Colonization: Insights from Percolation Theory and the Yule Process

Abstract

In this paper, percolation theory is employed to place tentative bounds on the probability p of interstellar travel and the emergence of a civilization (or panspermia) that colonizes the entire Galaxy. The ensuing ramifications with regard to the Fermi paradox are also explored. In particular, it is suggested that the correlation function of inhabited exoplanets can be used to observationally constrain p in the near future. It is shown, by using a mathematical evolution model known as the Yule process, that the probability distribution for civilizations with a given number of colonized worlds is likely to exhibit a power-law tail. Some of the dynamical aspects of this issue, including the question of timescales and generalizing percolation theory, were also studied. The limitations of these models, and other avenues for future inquiry, are also outlined. Key Words: Complex life—Extraterrestrial life—Panspermia—Life detection—SETI. Astrobiology 16, 418–426.

1. Introduction

The Drake equation (Drake, 1961; Shklovskii and Sagan, 1966) has been used to estimate the number of intelligent extraterrestrial civilizations capable of communicating with one another. Yet the (current) absence of such extraterrestrial contact led to the coinage of the Fermi paradox. The Fermi paradox was extensively studied by Hart (1975), and there have been several attempts to reconcile this apparent disparity in the potentially large number of intelligent civilizations and the lack of any communications with them. A summary of these aspects can be found in the works of Brin (1983), Webb (2002), Chyba and Hand (2005), Javaux and Dehant (2010), Ćirković (2009), and Wright et al. (2014); and some noteworthy attempts to resolve (or better understand) this paradox include careful study of the temporal aspects (Newman and Sagan, 1981; Fogg, 1987; Haqq-Misra and Baum, 2009; Forgan et al., 2013; Barlow, 2013), catastrophic events (Clarke, 1981; Annis, 1999; Horvath and Galante, 2012), and critiques of the Drake equation itself (Tipler, 1980, 1981; Ćirković, 2004).

In recent times, techniques from other areas of mathematics and physics have been brought to bear upon this problem. An earlier pioneering endeavor in this regard is the Newman and Sagan (1981) study in which a suitable population dynamics model was designed and implemented to examine the idea of spacefaring civilizations; evolutionary models along these same lines were developed by Maccone (2011) and Glade et al. (2012).

Other avenues that have been explored include cellular automata (Vukotić and Ćirković, 2012), phase transitions (Annis, 1999; Ćirković and Vukotić, 2008), and percolation models (Landis, 1998; Hair and Hedman, 2013; Cartin, 2014, 2015; Lin and Loeb, 2015). In the present study, we address issues that pertain to the Fermi paradox, the distribution of spacefaring civilizations, and other associated matters. To do so, we invoke results from discrete mathematics and explore their implications.

At this stage, we wish to set forth some of the generic limitations of our approach. There are several crucial sociological factors that we ignore here. For instance, civilizations can exhibit mutual coexistence and regulate their growth and colonization processes, and so on. Moreover, their expansion history is likely to be complicated with later waves of colonization that may overtake the previous ones. Some of these objections can be found in the works of Hart (1975), Brin (1983), Ćirković (2009), Wright et al. (2014) and references therein, although this list (of factors) is by no means exhaustive. Our analysis also ignores secular dynamics in our galaxy, and we refer the reader to the work of Lin and Loeb (2015), where a brief account of these effects is presented.

In this work, we describe the basics of percolation theory in Section 2 and explore some of its consequences for the Fermi paradox. In Section 3, we address the issue of the distribution of spacefaring civilizations by employing the Yule process. A discussion of the dynamical aspects of these models is presented in Section 4. Lastly, we summarize our findings in Section 5.

2. Percolation Theory: Results and Implications

We begin with a brief heuristic description of percolation theory before presenting some of the results and their consequences.

2.1. An intuitive picture

We primarily follow the approach outlined by Newman (2005). Assume that we consider a square lattice that is composed of (square) tiles. We then select each tile at random and assign it a color that represents a probability p. After this process is complete, we expect a fraction p to be colored, and 1 − p to be uncolored. We then ask: What structures have formed, and how can they be characterized?

Ultimately, this amounts to making assertions regarding the nature of the “clusters” (groups of colored tiles) that were formed. More precisely, what is of interest is the distribution of the “size” of these clusters. The number of adjacent squares that are colored measures the “size” of the cluster, which is denoted by s; for instance, if none of the tiles are colored, we have clusters of size zero everywhere. We will introduce the probability distribution P(s) for a given tile to belong in the cluster of size s, and offer some general comments on its features in Section 2.3.

2.2. Some results from percolation theory

Percolation theory is a vast field, and it is beyond the scope of this paper to provide a comprehensive overview of the results; such summaries can be found in the works of Stauffer (1979), Essam (1980), and Isichenko (1992).

In a general sense, percolation theory entails the study of N-dimensional lattices, and there are two broad classes involved: bond percolation and site percolation. In bond percolation, the lattice is modeled as being composed of vertices (sites) and edges (bonds). It is assumed that the probability of a bond that exists between two sites is p, while the probability of its absence is 1 − p. Thus, it follows immediately that if p is small, only small clusters will form due to the unavailability of edges that link different vertices. As p increases, the size of the clusters is also expected to increase.

From a naive perspective, it would be expected that the transition to larger clusters is fairly continuous, but the final result that arises from a rigorous treatment is quite surprising—the existence of phase transitions. In particular, it can be seen that there exists a critical probability threshold p _c such that

• for p < p _c, small and isolated clusters are scattered throughout the lattice.

• for p > p _c, a giant cluster emerges that spans the entire lattice.

In common with the theory of phase transitions, critical exponents for several quantities, including the percolation probability, mean cluster size, and so on, can be determined (Stauffer, 1979; Albert and Barabási, 2002). In Table 1, the bond percolation thresholds for different types of 3-D lattices are presented. Note that a similar study, pertaining to site percolation, was first undertaken (in the astrobiological context) by Landis (1998) and further explored by Hair and Hedman (2013) and Cartin (2014, 2015).

Table 1.

Bond Percolation Thresholds for Various Three-Dimensional Lattices

Model	Bond percolation threshold (p_c)	Reference
Simple cubic	0.24881182	Wang et al. (2013)
Body-centered cubic	0.1802875	Lorenz and Ziff (1998)
Face-centered cubic	0.1201635	Lorenz and Ziff (1998)
Hexagonal close packed	0.1201640	Lorenz et al. (2000)
Octagonal stack	0.1752	van der Marck (1997)

The table illustrates the values of the bond percolation threshold (p _c) for different types of 3-D lattices, and the corresponding references are also included.

At this stage, we also introduce other auxiliary results of considerable importance. Suppose a given “occupied” site A is considered, and another site B at a distance r from A is considered as well. The correlation function C(r) is defined to be the probability that site B belongs to the same finite cluster as site A. A general result from percolation theory is that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \bf C } ( { \bf r } ) \,= \, \exp \left( { - \frac { R } { \xi } } \right) \tag { 1 } \end{align*} \end{document}

where ξ is the correlation length and R = |r|/l, with l denoting the characteristic lattice spacing. It is well known that ξ = 0 when p = 0 and that it diverges when p = p _c (Albert and Barabási, 2002; Christensen, 2002). Moreover, in the vicinity of the critical region, it is also known that ξ exhibits a scaling behavior, viz ξ ∝ |p − p _c|^−ν when p → p _c; it is thus easy to verify that ξ → ∞ for p = p _c, provided that ν > 0. In general, ν cannot be determined for an arbitrary percolation model, but it is known that ν = 1 in 1-D and ν = 1/2 for infinite dimensions (Stauffer, 1979; Albert and Barabási, 2002; Christensen, 2002). Similarly, in the vicinity of the critical percolation threshold, the average cluster size <s> also exhibits a critical behavior, that is, <s> ∼ |p − p _c|^−γ where γ = 1 for both 1-D and infinite dimensions (Albert and Barabási, 2002; Christensen, 2002). Other quantities also exhibit a critical behavior, but we abstain from discussing them here as they do not play a central role in our subsequent analysis; detailed summaries can be found in the works of Stauffer (1979) and Essam (1980).

2.3. Cluster sizes and their associated probability distributions

As stated earlier, the chief interest here is to determine the probability distribution of a given site that belongs to a cluster of size s and is denoted by P(s). In a study by Albert and Barabási (2002), some exact results are presented for this distribution on a Cayley tree (a special type of lattice) that demonstrate the emergence of a power-law behavior. However, it is more illuminating to consider the arguments provided by Newman (2005), which are fairly general. It is important to emphasize that the generality of these results is predicated on the assumption that the cluster sizes are “large.”

The tool employed herein is dimensional analysis. Since P(s) is dimensionless, it must be constructed of dimensionless quantities. We have already seen that the “size” s is one of them; note that this size is an area on a 2-D lattice. Another chief parameter is the average “size,” which we denote by <s>. A third is the “unit” size, which we shall denote by s ₀—this could refer, for instance, to the size of the unit tile on a 2-D lattice, that is, the basic building block (or element) of the model. Thus, we end up with three dimensionless expressions s/<s>, s/s ₀, and s ₀/<s>, of which only two are independent.

Thus, our probability distribution has the form \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}P \left( s \right) = Cf \left( { \frac { s } { { { s_0 } } } , { \frac { { s_0 } } { \langle s \rangle } } } \right) \tag { 2 } \end{align*} \end{document}

where C is a suitable normalization constant.

Next, in accordance with Newman (2005), the “unit” size is scaled by a factor of λ, which thereby moves to s ₀/λ as the fundamental unit instead. It is assumed that the overall structure of the lattice, namely, the cluster sizes and numbers, stays approximately the same. Quite evidently, this is only likely to be true for the very large clusters, that is, the large s limit, where the increase/decrease in “size” is not particularly significant. Thus, in the limit of large s, it is argued that the probability distribution thus stays the same and takes the form \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}P \left( s \right) = Bf \left( { \frac { s } { { {{ { s_0 } } \mathord { \left/ { \vphantom { { { s_0 } } \lambda }} \right. } \lambda } } } , { \frac { { { { s_0 } } \mathord { \left/ { \vphantom { { { s_0 } } \lambda } } \right. } \lambda } }{ \langle s \rangle } } } \right) \tag { 3 } \end{align*} \end{document}

where B is a new normalization constant, which is taken to exhibit an implicit dependence on λ. Now, suppose that we undertake the transformations s → λs and <s> → λ<s> in (2); this amounts to stating that we have rescaled the cluster size, and the average cluster size, by λ. Next, we take the limit <s> → ∞ in (3) and (2), but only after applying this rescaling in the latter expression. We arrive at the relation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}P \left( { \lambda s } \right) = \frac { C } { B } P \left( s \right) = F \left( \lambda \right) P \left( s \right) \tag { 4 } \end{align*} \end{document}

and the second equality follows from the implicit dependence of B on λ. From the above expression, note that F(λ) = P(λ)/P(1). Now, differentiate both sides of the expression with respect to λ and set λ = 1 afterward. This leaves us with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}s { \frac { dP \left( s \right) } { ds } } = { \frac { P {^ \prime } \left( 1 \right) } { P \left( 1 \right) } } P \left( s \right) \tag { 5 } \end{align*} \end{document}

which can be integrated to yield \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}P \left( s \right) = P \left( 1 \right) {s^{ - \beta }} \tag{6}\end{align*} \end{document}

where β = −P(1)/P′(1). This clearly demonstrates the power-law behavior of P(s). However, it is important to recognize that this behavior arises only when s is large and in the limit <s> → ∞, namely, when the average cluster size diverges. A more exact treatment can be formulated by using the machinery of renormalization group theory.

The probability distribution P(s) discussed above is proportional to the number of clusters of size s per lattice site (Albert and Barabási, 2002; Christensen, 2002), which is denoted by n_s .

2.4. Implications and restrictions of percolation theory

Previously, we presented a compendium of mathematical results. It is necessary to interpret these results in an astrobiological context.

First, we suppose that our 3-D lattice comprises the inhabited planets, which constitute the sites. We interpret a “bond” between two planets to be the occurrence of interstellar travel between them that thereby leads to contact. Thus, the bond percolation model becomes a stand-in for modeling the propagation of extraterrestrial contact via interstellar travel. At this juncture, we record two immediate limitations that spring up. First, we have arranged the planets on a regular 3-D lattice, while a realistic distribution of planets in the Galaxy is likely to be inhomogeneous and anisotropic.

Second, we have ignored the effects of velocity shear present in the Galaxy (Binney and Tremaine, 1987); we do not expect this factor to be important provided that the propagation speed (of interstellar travel) is greater than the shearing velocity (Sellwood and Binney, 2002; Lin and Loeb, 2015). If we consider the opposite limit, we do not expect percolation theory in its present form to yield meaningful results; this is because the shear would ensure that the stars, and their planets, migrate faster than the time taken for the colonizers (civilizations/panspermia) to catch up with them. In such an event, where the migration effects and shear are dominant, a continuum model along the lines of an advection-diffusion system is warranted. Lin and Loeb (2015) showed that a subdiffusive behavior is exhibited, as the “bubbles” of these colonizers slowly continue to grow and fill out the Galaxy; in particular, it was shown that the size of the bubble R scales with time t as R ∝ t ^1/2 as opposed to the usual R ∝ t. Lingam (2016) adopted a different formalism by modeling panspermia through nonlocal self-replication and extinction.

As stated above, interstellar travel can be modeled as a bond percolation problem. Thus, it is now possible to invoke one of the key results discussed earlier, namely, the existence of a bond percolation threshold. From Table 1, it can be seen that the percolation thresholds for most lattices, barring the simple cubic, fall within the 0.1–0.2 range. Recalling that the bond percolation probability p corresponds to the probability of interstellar travel, we postulate that

• If p < 0.1, it is quite likely that any randomly selected (inhabited) planet will either belong to a small cluster or remain in isolation. In other words, the chances of extraterrestrial contact are quite low, as such a civilization would have a high chance of being isolated (or in a small cluster).

• If p > 0.2, a randomly chosen node (planet) will almost definitely belong to the giant cluster, as the percolation threshold has been exceeded. Hence, this makes it very likely that such a planet will have witnessed extraterrestrial contact, as it is part of a large and interconnected system of interstellar travel and/or communication.

Considering this from a different perspective, we suppose that a given civilization with spacefaring abilities sets out, and we assume that the percolation probability exceeds the threshold. This implies that the civilization can cover (percolate through) virtually all the Galaxy, over a suitable timescale, ensuring that it comes in contact with other intelligent civilizations. The colonization timescale is absolutely essential when making statements about the Fermi paradox, and we analyze this further in Section 4.1. With some minor modifications, this formalism can also be adapted so as to model panspermia (Lin and Loeb, 2015) or deduce the likelihood of civilizations that embark on extensive colonization; by “panspermia” we make reference to the process of transporting organisms (or organic material) across space (Crick and Orgel, 1973; Hoyle and Wickramasinghe, 1978; Wesson, 2010).

Thus, we are led to the hypothesis that percolation theory, in our simple model, places certain bounds on the probability of interstellar travel and communication, thereby placing bounds on the Fermi paradox itself (Landis, 1998). It is important to recognize that these bounds must be viewed as indicators as opposed to rigorous mathematical results. For instance, the current absence of extraterrestrial contact does not indicate p < 0.1, which in turn would indicate that interstellar travel is a low-probability process. Nevertheless, it is also important to recognize that these results are not inconsistent with such a scenario and may lend it some credibility.

Now, we consider the question of colonization by civilizations. We can ask the question: What is the distribution of civilizations with a specified number of colonies? This question bears some similarities with identifying the form of the cluster size distribution, which was undertaken in Section 2.3. These clusters can be envisioned as regions that are interconnected by bonds (interstellar travel), which thereby constitute the domain of a given civilization. A somewhat similar argument for panspermia is also feasible, assuming that each panspermia “species” is akin to that of a spacefaring (and colonizing) civilization. In either scenario, from the results of Section 2.3, we can conjecture that a power-law tail is to be expected; a more detailed treatment of this is deferred to Section 3.

However, the existence of a power-law tail in the number function (per lattice site), described in Section 2.3, is not yet detectable with our current level of technology. To do so, we would need a large enough sample of unambiguous alien markers (either biological or artificial) demarcating a given civilization. Although several promising techniques for detecting alien civilizations have been proposed [see Wright et al. (2014) for a recent summary of this area], none have yet borne fruit. In recent times, sophisticated data analysis techniques have been developed in the context of sparse data (Yao et al., 2005; Zou et al., 2006), but it is still likely that one would need to detect O (10) to O (10²) civilizations before the hypothesis of a power-law tail can be observationally validated. Nevertheless, the power-law tail serves as a useful tool in underscoring the following point: if we detect a given alien civilization, it makes sense to look “nearby” given the tendency of civilizations to form clusters (as seen from our preceding discussion). Moreover, in principle, we note that the detection of a sufficiently large number of civilizations would also make it possible to compute quantities such as the clustering coefficient (Albert and Barabási, 2002) and compare them against theoretical predictions.

It would seem, based on the preceding paragraph, that observational tests of this model may not yet be feasible. Fortunately, the presence of a well-defined correlation function enables us to constrain our model. It was argued in Section 2.2 that percolation models exhibit a correlation length as seen from (1). The correlation function could potentially be computed from observations (through biosignatures or some other process) in the near future. If we find that C → 0, this suggests that ξ → 0, and hence p → 0 also follows; the converse chain of reasoning is also equally valid. Hence, if we fail to detect any correlations, it would suggest that alien civilizations are incapable of percolation.

This conclusion is in exact agreement with the studies of Lin and Loeb (2015) and Lingam (2016), where the absence of panspermia (zero percolation) led to a vanishing correlation function. On the other hand, we can suppose that ξ → ∞, implying that a giant cluster emerges. In such a scenario, we find that C → 1 and can be viewed as maximal seeding/colonization at p = p _c. The result fully concurs with the analysis of Lin and Loeb (2015) and is also consistent with the continuum model of Lingam (2016).

If we find that 0 < C < 1, we can determine the value of ξ which enables us to gain a qualitative idea of the magnitude of p, and thereby constrain it. In the case of a simple 1-D model, it is known that the relation ξ = −[ln(p/p _c)]⁻¹ holds true (Christensen, 2002), which implies that knowledge of ξ can translate to knowledge of p and vice versa. Of course, the situation is more complicated for more realistic models, but it is still quite plausible that the correlation function (and the correlation length) can play an important role in observationally constraining the magnitude of p. When computing the correlation functions observationally, it is also possible that we may observe other spatial structures not predicted by percolation theory, and the existence of such patterns could also rule out, or expose the limitations of, a percolation model. Examples of such structures include “ring” or “bubble” patterns, which will be manifest as oscillatory behavior (Lingam, 2016).

As discussed in Section 2.2, the average cluster size <s> also exhibits a scaling behavior when p is close to the percolation threshold. A sufficiently advanced civilization, which can detect a large number of civilizations, could also use <s> as a means of constraining the value of p and/or p _c. However, in the case of our own civilization, we expect that P(s) and <s> are not likely to serve as useful constraints in the near future. On the other hand, for the reasons outlined above, the correlation function C may serve as a potentially robust tool for experimentally testing percolation models; this is also supported by the analysis undertaken by Lin and Loeb (2015).

One of the chief limitations of percolation theory, or its counterpart in graph theory—the Erdős-Renyí model—must be documented here. These models rely solely on analyses of the topology of the graph and do not take into account dynamical behavior. Yet it is clear that any process of interstellar travel (or panspermia) must involve a dynamical element. Some further comments in this regard can be found in Section 4.2. In the next section, we indicate how this difficulty may be partly bypassed through the adoption of an implicitly dynamical model.

A second major limitation of our discussion thus far is the dependence on lattice-based models. It is evident that the Milky Way is not a lattice, and a more realistic model is called for. There exists such a class of models in the literature, collectively referred to as continuum percolation theory (Meester and Roy, 1996). The analog of p _c for the lattice percolation models is given by φ_c, the critical “volume” fraction, in the continuum limit. It has been shown that a variety of continuum percolation models exhibit a phase-transition behavior, and we refer the reader to the works of Balberg (1985) and Meester and Roy (1996) for a discussion of the same. As it has been shown that lattice and continuum percolation models belong to the same universality class, it is reasonable to suppose that the qualitative features remain identical, although the specifics are likely to be altered. The critical volume fractions for some continuum percolation models are summarized in Table 2, and it is apparent that the tabulated properties are quite analogous to those described in Table 1. Collectively, the preceding (heuristic) arguments suggest that lattice percolation theories still yield fairly useful and accurate statements, albeit on a semiquantitative level. A more accurate treatment would necessitate the use of continuum percolation models, which are more realistic and detailed than their lattice counterparts.

Table 2.

Critical Percolation Thresholds for Various Continuum Percolation Models

Model	Critical volume fraction (φ_c)	Reference
Spheres	0.2895	Rintoul and Torquato (1997)
Spheroids with aspect ratio of 1/2	0.2629	Garboczi et al. (1995)
Spheroids with aspect ratio of 2	0.2618	Garboczi et al. (1995)
Randomly oriented cubes	0.2168	Baker et al. (2002)

The table illustrates the values of the critical volume fraction (φ_c) for differing continuum percolation models, and the corresponding references are also included.

Before concluding this section, we observe that our results are mostly consistent with previous studies of percolation theory in the context of Fermi's paradox; see for example the pioneering early work of Landis (1998), which was extended by Hair and Hedman (2013) and Cartin (2014, 2015).

3. The Yule Process and Its Consequences

In this section, we outline the basics of a model originally proposed by Yule (1925) as a mathematical model of evolution. After outlining the mathematical basics, we proceed to analyze the consequences of this model.

3.1. A description of the Yule process

Our treatment mirrors the approach adopted by Newman (2005). The Yule process shares a very close similarity with the field of scale-free networks (Barabási and Albert, 1999), and it generalizes some of the results obtained in the above model.

The system under consideration is assumed to comprise n objects, each of which is characterized by an intrinsic property. The latter is quantified via k, which denotes the number of constituent members for a given object, and we seek the probability distribution P(k,n), which signifies the fraction of objects with k members when there exist n objects in all. For instance, if the object were a city, then k could represent the number of people in that city. We start with an initial value of k ₀ and calibrate each time step such that m new members (e.g., people) are introduced while the total number of objects (e.g., cities) is incremented by unity. Each object acquires a new member depending on its current number of members—this is just a manifestation of the preferential attachment principle evoked in scale-free networks (Barabási and Albert, 1999; Albert and Barabási, 2002) or the Matthew effect (Merton, 1968). In other words, the probability that an object acquires a new member is proportional to k + C where k is the current number of members. Note that C is a corrective factor to account for the case k ₀ = 0, as such an object would never be able to “accrete” any further members.

A master equation is set up, which is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \left( { n + 1 } \right) P \left( { k , n + 1 } \right) & = m \left( { { \frac { k - 1 + C } { { k_0 } + C + m }} } \right) P \left( { k - 1 , n } \right) \\ & - m \left( { { \frac { k + C } { { k_0 } + C + m } } } \right) P \left( { k , n} \right) \\ & + { \rm { } } nP \left( { k , n } \right) \qquad \qquad { \rm for } \ k > k_0 \tag { 7 } \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \left( { n + 1 } \right) P \left( { { k_0 } , n + 1 } \right) = &- m \left( { { \frac { { k_0 } + C } { { k_0 } + C + m } } } \right) P \left( { { k_0 } , n } \right) \\ & + { \rm { } } nP \left( { { k_0 } , n } \right) + 1 \quad { \rm for } \ k = k_0 \tag { 8 } \end{align*} \end{document}

and further details of the algebra can be found in the work of Newman (2005) and references therein. The case with k < k ₀ does not occur, since we start with k = k ₀ with subsequent increments. Over a sufficiently large number of time steps, we expect n → ∞ as well, and we introduce the definition P(k) = lim_n→∞ P(k,n). Upon solving for the stationary state solutions, we arrive at \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}P \left( { { k_0 } } \right) = { \frac { { k_0 } + C + m } { \left( { m + 1 } \right) \left( { { k_0 } + C } \right)+ m } } \tag { 9 } \end{align*} \end{document}

while the expression for P(k) is now given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}P \left( k \right) = { \frac { B \left( { k + C , \gamma } \right) } { B \left( { { k_0 } + C , \gamma } \right) }} P \left( { { k_0 } } \right) \tag { 10 } \end{align*} \end{document}

as demonstrated by Newman (2005), where B(a,b) represents the beta function and γ is defined as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\gamma = 2 + \frac { { { k_0 } + C } } { m } \tag { 11 } \end{align*} \end{document}

As the beta function possesses a power-law tail, one finds that P(k) ∝ k ^−γ. For the special case with k ₀ = 0 and C = m, the probability distribution of Barabási and Albert (1999) is recovered.

3.2. Implications and limitations of the Yule process

In the preceding subsection, we gave an example where the “objects” were cities and the constituent “members” were people for the sake of clarity. It is thus important to recognize that the Yule model leads to the famous Zipf's law, which has been studied extensively in the literature [see, e.g., Gabaix (1999) and Newman (2005) and references therein]. Hence, it is likely that the same formalisms developed to study Zipf's law can be applied, as in the recent work of Lin and Loeb (2016), in which cosmological techniques were employed for modeling the spread of cities (or, alternatively, alien civilizations). Now, we can consider a different system wherein the “objects” are comprised of spacefaring, expansionist civilizations and the “members” constitute the colonized worlds, and repeat the same analysis.

We are led to the conclusion that even a dynamical model such as the Yule process leads to a power-law behavior, in the steady state limit, for the fraction of civilizations that feature a specified number of colonized worlds. If we consider another system wherein each “object” was a particular panspermia “species” and each “member” was a world seeded by that species, the same results follow upon applying the Yule process. This suggests that the probability distribution of panspermia species that have seeded a fixed number of worlds exhibits a power-law tail. In both these cases, it is reasonable to assume that the initial number of colonized/seeded worlds is zero, which is expressed as k ₀ = 0. Thus, we obtain γ = 2 + C/m, implying that a knowledge of C and m will suffice to determine the power-law tail exponent. Normally, we expect m to be fairly large, indicating that γ≈2 may occur.

It is also crucial to recognize the inherent limitations of the Yule process. New objects and members are continually added and/or created, but we do not consider their extinction. This is clearly unphysical, all the more so in the case of civilizations that are likely to be beset by a host of instabilities, some of which were discussed by Brin (1983) and Ćirković (2009). Generalizations of the Yule process such as the Simon model (Simon, 1955) can bypass some of the limitations of the former model.

4. A Discussion of the Dynamical Aspects

We cover some of the dynamical aspects of our models, including a discussion of the timescales involved and potential generalizations and alternatives for the models described previously.

4.1. The timescale for colonization and its implications

In Section 2.4, we covered some of the limitations of percolation theory, including the case where the colonizing species moves at low propagation speeds. We also indicated the key issue with the standard percolation theory approach, namely, the absence of dynamics. In other words, even if we suppose that a fairly high value of p is prevalent, the process may be too slow to satisfactorily resolve Fermi's paradox.

To gain some toy estimates of the timescales involved, we suppose that the Milky Way is modeled as a cylinder with R∼20 kpc and H∼1 kpc, and we suppose that there exist N∼10¹⁰ planets (an order of magnitude less than the number of stars present) in the Galaxy that are capable of being terraformed. Assuming a homogeneous distribution of these planets—a clearly simplistic approximation—the average distance <r> can be estimated as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}N \times \left( { \frac { { 4 \pi } } { 3 } { { \left\langle r \right\rangle } ^3 } } \right) = \pi { R^2 } H \tag{ 12 } \end{align*} \end{document}

which yields <r> ∼ 0.003 kpc ≈ 10¹⁷ m. The above expression arises from the simple geometric picture of “filling” a cylinder with small spheres. Thus, we expect the spacing between two habitable planets to be <d> = 2<r>. If we consider a civilization at roughly our technological level, we know that spacecraft analogous to Pioneer 10 and 11 would travel at a speed of 10⁴ m/s. This suggests that an average timescale for interstellar travel would be approximately τ ∼ 6 × 10⁵ yr. Now, we suppose that a civilization would begin to move outward and terraform a world in a timescale comparable to, or smaller than, the travel timescale τ. Note that we already assumed that N was the number of terraformable planets, which implies that the conditions on the planet would not be too alien or hostile to the colonizing species. Thus, such a timescale for terraforming might be fairly reasonable.

After the first world has been terraformed and settled, there exist two such worlds populated by this species, and they can set out subsequently to colonize other worlds. Hence, we see that this process constitutes a geometric series. All the terraformable planets are colonized in approximately 2^k = N iterations, and solving for k we arrive at k∼30. We would expect the entire Galaxy to be colonized in T = (2τ) × k∼3.5 × 10⁷ yr; note that the factor of 2 is present to account for the terraforming if its timescale is equal to the travel timescale. It is clear from (12) that <r> ∝ N ^−1/3, which constitutes a fairly weak dependence. Hence, even if we are somewhat incorrect in our estimate of N, our result is still reasonably robust, provided that we expect the geometric series of 1, 2, 4, … colonized worlds to continue indefinitely. We are led to conclude that, with enough supply of raw materials, it is feasible to envision the Galaxy being populated by a sufficiently expansionist civilization over a timescale far lesser than the age of the Galaxy. More sophisticated analyses of the total colonization time exist, ranging from 10⁶ to 10¹⁰ years (Jones, 1976; Newman and Sagan, 1981; Ćirković, 2009; Forgan et al., 2013; Nicholson and Forgan, 2013), and a recent overview of this subject suggests that T≲10⁸ years (Wright et al., 2014). Thus, we see that our simple model is not too far off the mark.

A chief limitation (albeit an implicit one) briefly alluded to at the end of Section 3.2 must be reiterated: all the models described assume that intelligent civilizations do not undergo extinction, either through natural causes or self-destruction. Hence, we will continue to operate with this implicit assumption throughout, especially since our value of T ∼ 35 Myr is not overly long by cosmological standards.

4.2. Additional remarks on dynamics

As we have emphasized, traditional percolation theory–based approaches do not incorporate dynamics. However, there exists an important class of problems for which such a treatment is crucial—epidemics. It can be readily seen that modeling the spread of an epidemic is quite similar to colonization/panspermia. This analogy can be understood as follows: the pathogen in an epidemic is equivalent to the colonizing species (either a civilization or panspermia), and the pathogen/colonizer “infects” others, thereby spreading further. Epidemics have been modeled based on a percolation theory approach (Hethcote, 2000; Dorogovtsev et al., 2008), and they take into account the fact that this “growth” cannot last forever; the food supply for the pathogen is exhausted, and the epidemic dies out.

Nevertheless, even in such a scenario, the emergence of critical phenomena has been demonstrated (Dorogovtsev et al., 2008). However, we wish to note that the field of temporal networks is a fairly nascent one (Holme and Saramäki, 2012), and extensive results akin to those from “classical” percolation theory are not yet available. However, the seminal work of Grassberger (1983) on modeling epidemics using a dynamical variant of percolation theory deserves a mention. It was shown that the epidemic can be mapped to a bond percolation problem, and the percolation threshold and critical exponents were computed and shown to be in good agreement with the results from orthodox percolation theory (Stauffer, 1979). It is important to recognize that we cannot take it to be conclusive that the percolation thresholds described in Table 1 remain near-identical even when modeling colonization via dynamical percolation. However, we suggest that the chief findings of Grassberger (1983) do not appear to directly contradict the assumption of a certain degree of (qualitative) universality. Moreover, we can envision a time-dependent percolation process as composed of different “snapshots” taken at different times, and we conjecture that some of the key properties, such as the percolation threshold, may exist for some length of time. Temporal evolution, which is widely modeled as a stochastic process, has been typically studied via directed percolation models, which also exhibit the kind of phase transitions and critical phenomena associated with normal percolation models (Hinrichsen, 2000).

At this stage, we note, with regard to the Yule process as mentioned in Section 3, that the time steps are calibrated such that one “new” object is introduced at each iteration. Hence, this introduces, quite naturally, a timescale into the picture, and this timescale quantifies the period over which a new civilization is birthed. It is also useful to draw parallels with the scale-free network paradigm developed by Barabási and Albert (1999), as this can, in some respects, be viewed as a simpler (and more transparent) case of the Yule process.

We are primarily interested in P(k), which denotes the probability distribution for a randomly chosen node to have a degree k; the “degree” here refers to the number of connections that it shares with other nodes on the network. In this model, each time step introduces one node and m edges; it is quite similar to the Yule process where one object and m members were introduced. After the adoption of a continuum model (Albert and Barabási, 2002), it is found that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}P \left( k \right) = { \frac { 2 { m^2 } t } { { m_0 } + t } } { k^ { - 3 } } \tag { 13 } \end{align*} \end{document}

where m ₀ is the initial number of nodes present and t represents the time evolved. For t → ∞, the power-law scaling P(k) ∝ k ⁻³ is recovered, but it is interesting to observe that P(k) ∝ tk ⁻³ at early times. Scale-free networks thereby illustrate the dynamical aspect of the probability distribution and strengthen the idea that a power-law behavior is reached asymptotically.

5. Conclusion

It has been proposed that a colonizing spacefaring civilization can populate the entire Galaxy in a span of time that is far less than the age of the Galaxy. Such a result would have important consequences for the Fermi paradox, as it would ensure that other intelligent civilizations (spacefaring or otherwise) would come into contact with such a civilization, which thereby suggests that the current (and apparent) absence of such contact would not be the case.

It is then reasonable to ask whether we can place bounds on the probability of such a spacefaring civilization equipped with interstellar travel. Through a suitable reformulation of this question, we modeled this question with the use of percolation theory. Exceeding a certain percolation threshold would likely lead to a galactic colonizing civilization as described in this paper. Thus, having taken into account the quantitative calculations in the literature on percolation theory, we suggest that the probability of interstellar flight may need to exceed the (bond) percolation threshold for a Galaxy-wide colonizer to be feasible. We also tackled the prospect of observationally constraining percolation-based models. By employing universal results from percolation theory, we suggest that the correlation function could serve as a robust measure, as it clearly differentiates between the cases of zero percolation (no interstellar colonization/panspermia) and finite percolation. We have shown that such a conclusion is generally in agreement with similar results presented by Lin and Loeb (2015) and Lingam (2016).

Moreover, we also expect any colonizer to exhibit a “sphere of influence,” that is, neighboring worlds it has colonized. To quantify the distribution of such “spheres of influence,” we analyzed the number of clusters of a given size (per lattice site) using percolation theory, and we suggest that a power-law tail behavior is expected. As percolation theory does not incorporate a dynamical element, we resorted to the use of the Yule process, which served as a mathematical model for characterizing the evolution of multiple spacefaring civilizations. We have shown that the distribution of civilizations with a given number of colonized worlds exhibits a power-law tail. Along the way, we have also elucidated some of the limitations of our model(s) and emphasized that the bounds and constraints recovered constitute guidelines rather than exact rules.

We touched upon some of the dynamical aspects in Section 4, highlighting some additional restrictions of our reasoning along the way. We also presented a simple argument to suggest that the galactic colonization should occur on a fairly fast timescale in comparison to the age of the Galaxy. This suggests that Fermi's paradox cannot be resolved by claiming that the process of colonization is slow, which agrees with the recent analysis undertaken by Wright et al. (2014). We have also indicated how percolation theory can be studied on a dynamical footing and used the scale-free network of Barabási and Albert (1999) to demonstrate the explicit time-dependence for some of the probability distributions studied previously.

Although our discussion has been primarily centered on the propagation of intelligent life, we reiterate that many of the same techniques are also equally applicable to panspermia. Two recent studies (Lin and Loeb, 2015; Lingam, 2016) suggest that the distinction between panspermia (nonzero percolation) and independent, spontaneous emergence of life (zero percolation) can be made via the spatial autocorrelation function. Furthermore, these authors also hypothesized the potential existence of critical behavior in the case of panspermia. We find that these two features are consistent with the analysis undertaken in this paper.

It is also possible that the formation of bubbles during cosmological phase transitions (Turner et al., 1992), which entails percolation, may also be worthy of further investigation by way of the paradigms highlighted in this paper. Finally, we wish to emphasize that the power-law tails described herein can also arise from a variety of other processes that have not been considered here, some of which may prove to be more relevant in certain astrophysical contexts. Examples of these processes include random walks (Newman, 2005), self-organized criticality (Bak et al., 1987), and the maximum entropy principle (Peterson et al., 2013).

Footnotes

Acknowledgments

The author thanks Abraham Loeb for his insightful commentary and for pointing out important issues and references. The helpful feedback received from the anonymous referees, which helped improve the quality of the manuscript, is also gratefully acknowledged. The author was supported by the NSF Grant No. AGS-1338944 and the DOE Grant No. DE-AC02-09CH-11466.

Author Disclosure Statement

No competing financial interests exist for this work.

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