Jekyll2019-09-24T15:07:15+00:00/feed.xmlKepler Lounge
The math journal of Aidan Rocke
Derivation of the isoperimetric inequality from the ideal gas equation, part I2019-09-23T00:00:00+00:002019-09-23T00:00:00+00:00/nonlinear/elasticity/2019/09/23/isoperimetry<h2 id="introduction">Introduction:</h2>
<p>Why is it that whenever balloons are inflated they converge towards the shape of a sphere regardless of their initial geometry?</p>
<p>On one level this may be a purely geometrical problem due to thermodynamic constraints on a body with finite surface area.
This suggests the necessity of solving a global optimisation problem. However, the sequence of deformations undergone may be
facilitated by the elastic material the balloons are made of.</p>
<p>In this article I consider the contribution of the latter by analysing the problem in two dimensions and demonstrate that a minimal
surface may be entirely due to local mechanical instabilities.</p>
<h2 id="the-role-of-material-properties">The role of material properties:</h2>
<p>Let’s consider an object that is only allowed to extend in one dimension. If you were to elongate such an object it would assume
a roughly cylindrical shape.</p>
<p>It follows that we must pay careful attention to the material properties of the balloon.</p>
<h2 id="reasonable-assumptions">Reasonable assumptions:</h2>
<p>A two-dimensional balloon <script type="math/tex">\mathcal{B} \in \mathbb{R}^2</script> is essentially an elastic loop that initially has perimeter of length:</p>
<p>\begin{equation}
\lvert \partial \mathcal{B}(t=0) \rvert = l_0
\end{equation}</p>
<p>Furthermore, we may make the following reasonable assumptions:</p>
<ol>
<li>
<p>The balloon contains an astronomical number of gas particles that collectively satsify the ideal gas equation.</p>
</li>
<li>
<p>The balloon is surrounded by a heat bath.</p>
</li>
<li>
<p>The balloon itself is made of elastic filaments that form a Hamiltonian circuit.</p>
</li>
</ol>
<p>Furthermore, we may assume that the mechanical behaviour of the balloon is largely driven by energy-minimisation processes that I shall
detail in the next couple sections.</p>
<h2 id="isobaric-inflation-as-a-consequence-of-energy-minimisation">Isobaric inflation as a consequence of energy minimisation:</h2>
<p>If we consider the force required to elongate the elastic boundary of the balloon we may define an associated potential energy function:</p>
<p>\begin{equation}
U((\lvert \partial \mathcal{B}(t) \rvert - l_0)^2) \geq 0
\end{equation}</p>
<p>such that:</p>
<p>\begin{equation}
U(\cdot) = 0 \iff \lvert \partial \mathcal{B}(t) \rvert = l_0
\end{equation}</p>
<p>Now, if we consider that physical systems tend to minimise potential energy we may infer that the balloon would tend to increase in volume
without increasing <script type="math/tex">\lvert \partial \mathcal{B}(t) \rvert</script>, the length of its perimeter.</p>
<p>In the case of inflation, after accumulating a pressure difference with respect to its environment the evolution of <script type="math/tex">\partial \mathcal{B}(t)</script>
would be guided by an approximately isobaric process provided that <script type="math/tex">\lvert \partial \mathcal{B}(t) \rvert \leq l_0</script>:</p>
<p>\begin{equation}
PV = nRT
\end{equation}</p>
<p>\begin{equation}
\frac{\Delta V}{V} = \frac{\Delta T}{T}
\end{equation}</p>
<p>We can go further with this type of reasoning. Not only does the elastic membrane constrain the type of thermodynamic processes that is likely to guide
inflation; it also constrains the mechanism for modifying the geometry of the balloon.</p>
<h2 id="local-deformations-of-elastic-filaments-lead-to-minimal-surfaces">Local deformations of elastic filaments lead to minimal surfaces:</h2>
<p>If we assume that the balloon constrains an ideal gas that my be modelled as an astronomical number of Newtonian particles, it’s reasonable to suppose
that equal pressure is applied to equal areas. Now, if this is the case we may consider pressure-driven deformations of <script type="math/tex">\partial \mathcal{B}</script> that
exploit a local mechanism that is operational everywhere on the boundary. What might such a mechanism look like?</p>
<p>Under a coarse-grained approximation, the boundary <script type="math/tex">\partial B</script> consists of a large chain of cylindrical elastic rods. If each individual rod is much larger
than the characteristic length where bending occurs any amount of bending will guarantee tensile stress.</p>
<p>It follows that the elastic membrane <script type="math/tex">\partial B</script> will try, as much as possible, to increase the enclosed volume while minimising the elongation globally.
This global minimisation happens by minimising the bending angle locally. No global coordination is required.</p>
<p>Another way of understanding this process is that deformations of the elastic membrane are mainly driven by local mechanical instabilities that lead to
a global minimisation of potential energies.</p>
<h2 id="a-polygonal-approximation-to-two-dimensional-elastic-boundaries">A polygonal approximation to two-dimensional elastic boundaries:</h2>
<p>One approach to modelling the activity of elastic boundaries is to approximate them as polygons with <script type="math/tex">N</script> sides of equal length where <script type="math/tex">N</script> is large.
Given that the sum of the interior angles <script type="math/tex">\theta_i \in (0,2\pi)</script> must add up to <script type="math/tex">(N-2)\cdot \pi</script> we may define the potential energy:</p>
<p>\begin{equation}
U = \frac{1}{2} \sum_{i=1}^N (\theta_i - \pi \cdot \big(\frac{N-2}{N}\big))^2
\end{equation}</p>
<p>\begin{equation}
\sum_{i=1}^N \theta_i = \pi \cdot (N-2)
\end{equation}</p>
<p>where:</p>
<p>\begin{equation}
\frac{\partial U}{\partial \theta_i} = \theta_i - \pi \cdot \big(\frac{N-2}{N}\big)
\end{equation}</p>
<p>\begin{equation}
\Delta \theta_i \propto \frac{\partial U}{\partial \theta_i}
\end{equation}</p>
<p>and we find that if we choose the local update with <script type="math/tex">\lambda \in (0,1)</script>:</p>
<p>\begin{equation}
\begin{split}
\theta_i^{t+1} & = \theta_i^{t} - \Delta \theta_i \\
& = \theta_i^{t} - \lambda \frac{\partial U}{\partial \theta_i} \\
& = (1-\lambda) \cdot \theta_i^t + \lambda \cdot \pi \cdot \big(\frac{N-2}{N}\big)
\end{split}
\end{equation}</p>
<p>and we can show that <script type="math/tex">\lim\limits_{t \to \infty} \theta_i^t = \big(\frac{N-2}{N}\big)</script> very quickly since:</p>
<p>\begin{equation}
x_{n+1} = (1-\lambda) \cdot x_n + \lambda \cdot \alpha \implies x_{n+1} - \alpha = (1-\lambda) \cdot (x_n - \alpha)
\end{equation}</p>
<p>\begin{equation}
\frac{(x_{n+1}-\alpha)^2}{(x_n - \alpha)^2} = (1-\lambda)^2
\end{equation}</p>
<p>so if we define:</p>
<p>\begin{equation}
\epsilon_{n+1}^2 = (x_{n+1}-\alpha)^2
\end{equation}</p>
<p>\begin{equation}
\epsilon_{n}^2 = (x_{n}-\alpha)^2
\end{equation}</p>
<p>we find that:</p>
<p>\begin{equation}
\lim_{n \to \infty} \epsilon_{n+1}^2 = \epsilon_1^2 \cdot \prod_{n=1}^\infty \frac{\epsilon_{n+1}^2}{\epsilon_{n}^2} = \lim_{n \to \infty} \epsilon_1^2 \cdot (1-\lambda)^{2n} = 0
\end{equation}</p>
<p>so we have exponentially fast convergence to a spherical geometry.</p>
<h2 id="discussion">Discussion:</h2>
<p>In this article I propose the existence of a local mechanical instability present everywhere in a closed elastic membrane with aspherical geometry. Surprisingly, the net action of this instability leads to exponentially fast convergence to the global minimum. But, this analysis may be refined.</p>
<p>The above analysis is entirely based on phenomenological studies of rubber bands by alternately dropping and manipulating rubber bands on a table. This led me to a useful
phenomenological model which may help simulate kinematics but doesn’t approximate the forces involved i.e. dynamics.</p>Aidan RockeIntroduction:Are 20 amino acids necessary and sufficient?2019-09-13T00:00:00+00:002019-09-13T00:00:00+00:00/optimisation/2019/09/13/amino-I<h2 id="motivation">Motivation:</h2>
<p>There is a persistent belief among evolutionary biologists that chance played a significant role in biological evolution. This would imply that life and
its building blocks could have been significantly different. In particular, some protein scientists suspect that evolution could have used as few as 10
amino acids to build life rather than the original 20. We may arrive at such a conclusion by considering the minimum number of letters required to fold
a protein [1].</p>
<p>However, there are a couple limitations with such analyses. First they ignore second-order, third-order and fourth-order constraints on the structure of amino
acids which quickly lead to a combinatorial explosion in the number of constraints facing any minimal set of amino acids. Second, they ignore the
time and space constraints on the search problem of finding the last universal common ancestor containing the set of 20 amino acids that we know.</p>
<p>By taking these two points into consideration we find that there was actually very little room for chance, leading us to a biological equivalent of the
anthropic principle.</p>
<h2 id="the-argument-for-sufficiency">The argument for sufficiency:</h2>
<p>There isn’t much to be said about sufficiency except that the diversity of multicellular life forms are an existence proof that 20 amino acids are
effectively sufficient. Establishing necessity is a lot harder.</p>
<h2 id="the-compositional-structure-of-life-forms">The compositional structure of life forms:</h2>
<p>It’s worth noting that biological systems have the following compositional structure:</p>
<p>\begin{equation}
\text{amino acids}(\mathcal{A}) \rightarrow \text{proteins}(\mathcal{P}) \rightarrow \text{cells}(\mathcal{C}) \rightarrow \text{eukarya}(\mathcal{E})
\end{equation}</p>
<p>where each level of abstraction is consistent with but not reducible to the levels of abstraction below it. These may also be represented as a
hierarchy of nonlinear transformations:</p>
<p>\begin{equation}
\mathcal{A} \overset{f_1}{\rightarrow} \mathcal{P} \overset{f_2}{\rightarrow} \mathcal{C} \overset{f_3}{\rightarrow} \mathcal{E}
\end{equation}</p>
<p>\begin{equation}
E = f_3 \circ \mathcal{C}
\end{equation}</p>
<p>\begin{equation}
E = f_3 \circ f_2 \circ \mathcal{P}
\end{equation}</p>
<p>\begin{equation}
E = f_3 \circ f_2 \circ f_1 \circ \mathcal{A}
\end{equation}</p>
<p>where the nonlinearity of the <script type="math/tex">f_i</script> implies that eukaryotes are more sensitive to variations in their basic set of amino acids than variations in cell type.
In other words, if <script type="math/tex">f</script> and <script type="math/tex">g</script> are nonlinear mappings then <script type="math/tex">f \circ g</script> is most probably even more nonlinear in its behaviour.</p>
<p>We may proceed in this manner with a stability analysis of functions on trees [2]. However, given that the functional form of the <script type="math/tex">f_i</script> isn’t exactly known a combinatorial analysis may be a better approach. Specifically, we may try to infer the number of independent constraints on <script type="math/tex">\mathcal{A}</script> given the number of independent constraints on <script type="math/tex">\mathcal{E}</script>.</p>
<h2 id="complexity-and-robustness">Complexity and Robustness:</h2>
<p>We may make the reasonable assumption that common life forms persist because they are robust. From an algorithmic perspective one way for a pattern to be robust
is if it is maximally informative. In other words, given a biological pattern <script type="math/tex">L</script> the Kolmogorov Complexity of <script type="math/tex">L</script> is given by:</p>
<p>\begin{equation}
3 \leq K(L) \approx \lvert L \rvert
\end{equation}</p>
<p>where <script type="math/tex">\lvert L \rvert</script> is the length of <script type="math/tex">L</script> and <script type="math/tex">3</script> is a reasonable lower-bound.</p>
<p>In order to apply this method to our analysis of compositional structures we must choose a representative from each set: <script type="math/tex">E \in \mathcal{E},C \in \mathcal{C},P \in \mathcal{P}</script>.</p>
<p>Using the principle that mundane life forms must be robust and the fact that biological systems have a compositional structure we define <script type="math/tex">E,C,P</script>
as follows:</p>
<p>\begin{equation}
E := \text{most common eukaryote}
\end{equation}</p>
<p>\begin{equation}
C := \text{most common cell type in } E
\end{equation}</p>
<p>\begin{equation}
P := \text{set of proteins found in } C
\end{equation}</p>
<p>\begin{equation}
\mathcal{A} := \text{set of 20 amino acids}
\end{equation}</p>
<p>Now, let’s suppose that the number of independent constraints on the reproduction of <script type="math/tex">\hat{E}</script> is approximately given by:</p>
<p>\begin{equation}
K(E) = \text{Kolmogorov Complexity of genome of } E
\end{equation}</p>
<p>\begin{equation}
K(E) \geq 3
\end{equation}</p>
<p>If each independent constraint on <script type="math/tex">E</script> may be expressed in terms of <script type="math/tex">C</script>, each independent constraint on <script type="math/tex">C</script> may be expressed
in terms of <script type="math/tex">P</script>…the number of constraints on <script type="math/tex">\mathcal{A}</script> must be on the order of:</p>
<p>\begin{equation}
\mathcal{L} = K(A)^{K(P)^{K(C)^{K(E)}}} \geq e^{e^{e^{e}}} > 10^{1000}
\end{equation}</p>
<p>assuming that:</p>
<p>\begin{equation}
\min(K(\mathcal{A}),K(P),K(C),K(E)) \geq 3
\end{equation}</p>
<p>At present I can’t fully explain the origin of the iterated exponential but I think it arises naturally in complex systems with emergent behaviour. The basic idea is that you can think of unfolding higher-level abstractions into more fundamental lower-level abstractions in higher-dimensional spaces using a sequence of multipartite graphs.</p>
<p>Edges that define interactions in one space become points in another.</p>
<h2 id="the-number-of-independent-constraints-on-the-fundamental-set-of-amino-acids">The number of independent constraints on the fundamental set of amino acids:</h2>
<p>It follows that finding a fundamental set of amino acids that simultaneously satify all <script type="math/tex">\mathcal{L}</script> constraints is a multi-objective discrete optimisation
problem in a search space that is at least as large as:</p>
<p>\begin{equation}
2^{\mathcal{L}} > 2^{10^{1000}}
\end{equation}</p>
<p>since for each independent constraint there are at least two options. The probability that this set of amino acids was discovered by chance is probably small…
but exactly how small?</p>
<h2 id="time-and-space-constraints-on-biological-evolution">Time and Space constraints on biological evolution:</h2>
<p>I’d like to make a compelling argument that the time-bounded search space for the 20 amino acids we know is much smaller than <script type="math/tex">2^\mathcal{L}</script>.</p>
<p>Consider that the smallest organism weighs on the order of one picogram. If there are less than <script type="math/tex">10^{20}</script> kilograms of living organism at any moment on any Earth-like planet, bacteria reproduce every 20 mins and the universe is less than <script type="math/tex">10^{10}</script> years old then the effective search space for a universal common ancestor containing
all 20 amino acids is less than:</p>
<p>\begin{equation}
S = \text{time} \times \text{space} \leq 10^{20} \cdot 10^{10} \cdot 10^{15} \cdot 365 \cdot 24 \cdot 3 \leq 10^{36}
\end{equation}</p>
<p>so the probability that the 20 amino acids were discovered by chance has to be much smaller than:</p>
<p>\begin{equation}
\frac{S}{2^\mathcal{L}} < \frac{1}{10^{1000}}
\end{equation}</p>
<p>which suggests that either a highly effective search procedure was responsible for their discovery or that fundamental laws of physics impose regularities on the search
space for amino acids and thus considerably simplify the optimisation problem. Both of these ideas lead us to the conclusion that there must be a biological equivalent
to the anthropic principle in cosmology.</p>
<h2 id="discussion">Discussion:</h2>
<p>The above calculations started as back-of-the-envelope calculations and I think they can be refined. One approach would be to think of this as a problem in programming language design. What if we allowed programming languages for dynamic control problems to evolve in such a way that higher-level languages emerge? Would we find that the
more primitive instruction set is more sensitive to random variation?</p>
<p>Alternatively, it’s a good idea to take a closer look at the problem from the perspective of protein folding. How many amino acids are required to cover the most
fundamental types of protein interactions?</p>
<h2 id="references">References:</h2>
<ol>
<li>Ke Fan and Wei Wang. What is the minimum number of letters required to fold a protein? Journal of Molecular Biology.</li>
<li>Roozbeh Farhoodi, Khashayar Filom, Ilenna Simone Jones, Konrad Paul Kording. On functions computed on trees. Arxiv. 2019.</li>
<li>Yi Liu and Stephen Freeland. On the evolution of the standard amino-acid alphabet. Genome Biology. 2006.</li>
<li>H. James Cleaves. The origin of the biologically coded amino acids. Journal of Theoretical Biology. 2009.</li>
<li>Matthias Granold et al. Modern diversification of the amino acid repertoire driven by oxygen. PNAS. 2017.</li>
</ol>Aidan RockeMotivation:Predicting doubling times during brain organoid development2019-09-06T00:00:00+00:002019-09-06T00:00:00+00:00/brain/organoids/2019/09/06/organoids-I<center><img src="https://raw.githubusercontent.com/Kepler-Lounge/Kepler-Lounge.github.io/master/_images/spherical_organoid.jpg?token=AH6QLODM3JMBDEAGYUQJWOS5PNX7I" width="75%" height="75%" align="middle" /></center>
<center>A spherical brain organoid grown in Berkeley [3]</center>
<h2 id="motivation">Motivation:</h2>
<p>Let’s suppose we have a lab which uses brain organoids to investigate human brain development. By seeding appropriate extracellular matrices(ECMs)
with thousands of human pluripotent stem cells(hPSCs) we may grow non-vascularized brain organoids. These tend to develop into spheroids for reasons that I try to
explain below.</p>
<p>Now, if the brain organoid’s spherical surface applies diffusion constraints on the transport of oxygen and nutrients to all cells in the interior, we may ask
how much time is required for the volume of a spherical brain organoid with radius <script type="math/tex">r</script> to double. The value of this analysis is that if we can estimate the
expected doubling time with reasonable confidence, we may predict the time to maturation.</p>
<p>Furthermore, I propose that predicting doubling times during brain organoid development as a fundamental challenge that could advance first principles
approaches to understanding organoid development.</p>
<p><strong>Caveat:</strong> Spherical Brain Organoids aren’t directly comparable to the human brain but they may be likened to the hydrogen atom for human brain development.</p>
<h2 id="assumptions">Assumptions:</h2>
<p>In order to proceed with our analysis a number of assumptions are necessary. The following are considered sufficient:</p>
<ol>
<li>
<p>An insignificant fraction of cells(< 5%) die before the spherical brain organoid has attained maximal volume, implying that the spherical organoid hasn’t grown too large.</p>
</li>
<li>
<p>During development, the distribution of each cell type converges to an equilibrium distribution where the distribution of each cell type(neurons,glia, oligodendrocytes)
is unimodal and tightly concentrated around its mean. Furthermore, we assume that the equilibrium distribution is isotropic i.e. spatially homogeneous.</p>
</li>
<li>
<p>Spherical symmetry is maintained via efficient mechanisms for cell signalling that coordinate the entire resource allocation process.</p>
</li>
<li>
<p>The packing density of cells is invariant to slight perturbations of the spherical geometry and therefore if the brain organoid’s geometry is denoted by <script type="math/tex">\mathcal{B}</script>:</p>
<p>\begin{equation}
\text{Mass}(\mathcal{B}) \approx k_1 \cdot \text{Vol}(\mathcal{B}) \approx k_2 \cdot N
\end{equation}</p>
<p>where <script type="math/tex">N</script> is the total number of cells and <script type="math/tex">k_1</script> and <script type="math/tex">k_2</script> are constants.</p>
</li>
<li>
<p>Half the organoid volume is exposed to air and the other half is embedded in ECM. Symmetry of this sort is necessary for our analytical arguments to be plausible.</p>
</li>
</ol>
<h2 id="a-rational-account-for-the-spherical-shape-of-brain-organoids">A rational account for the spherical shape of brain organoids:</h2>
<p>The isoperimetric inequality states that given a compact Euclidean manifold <script type="math/tex">\mathcal{B} \in \mathbb{R}^3</script> with fixed boundary <script type="math/tex">\text{Vol}(\partial \mathcal{B})</script>, then <script type="math/tex">\text{Vol}(\mathcal{B})</script> satisfies the following inequality:</p>
<p>\begin{equation}
\text{Vol}(\mathcal{B}) \leq \frac{1}{6 \sqrt{\pi}} \cdot \text{Vol}(\partial \mathcal{B})^{3/2}
\end{equation}</p>
<p>where we have equality if and only if <script type="math/tex">M</script> is a sphere.</p>
<p>Given the uniqueness of the sphere it’s reasonable to suppose that this shape isn’t an accident and that it’s probably advantageous to the brain organoid.
Here I posit two possible advantages in terms of energy loss and cell signalling.</p>
<ol>
<li>
<p>Minimisation of energy loss:</p>
<p>If heat is mainly lost by means of conduction via the boundary of the brain organoid then it would be advantageous to the brain organoid if this surface was
minimal.</p>
</li>
<li>
<p>Efficient cell signalling:</p>
<p>If we assume that the cells in an embryoid body communicate by means of some complex network and that the packing density of cells is isotropic then it’s sufficient
to minimise the average euclidean distance between cells. This minimisation process yields the sphere.</p>
</li>
</ol>
<p>At this point a mathematical biologist might remark that brain organoids aren’t vascularized and therefore resource allocation must be diffusion-constrained. Surely a flat
disk-like morphology would be more appropriate? The error in this argument is that it fails to consider that resource allocation is at the service of coordinating the
developmental process. Whatever is ideal for cell signalling shall constrain how resource allocation operates.</p>
<h2 id="the-expected-number-of-doubling-episodes-during-brain-organoid-development">The expected number of doubling episodes during brain organoid development:</h2>
<p>Before trying to estimate doubling times it might be instructive to analyse a related question. If <script type="math/tex">M_{\mathcal{B}}</script> is the mass of a spherical brain organoid,
<script type="math/tex">\rho_{\text{brain}}</script> is the average density of a human brain and the vast majority of cell divisions are symmetric:</p>
<p>\begin{equation}
M_{\mathcal{B}} \approx N_0 \cdot \overline{m_c} \cdot 2^D
\end{equation}</p>
<p>\begin{equation}
M_{\mathcal{B}} \approx \frac{4}{3} \pi r^3 \cdot \rho_{\text{brain}}
\end{equation}</p>
<p>\begin{equation}
\rho_{\text{brain}} \approx \frac{1400 g}{1260 \text{cm}^3} \approx \frac{1.1 \cdot 10^{-3} g}{1 \text{mm}^3}
\end{equation}</p>
<p>where <script type="math/tex">\overline{m_c}</script> is the average mass of a mature cell, <script type="math/tex">D</script> is the average number of cell divisions and <script type="math/tex">N_0</script> is the number of cells seeded per
embryoid body.</p>
<p>By equating (2) and (3) we find that:</p>
<p>\begin{equation}
D(N_0,r) = \frac{1}{\ln 2} \cdot \ln \big(\frac{4 r^3 \cdot \rho_{\text{brain}}}{N_0 \cdot \overline{m_c}}\big)
\end{equation}</p>
<p>Now, if we make the reasonable assumption that the mass of a eukaryotic cell is bounded between one nanogram and a thousand nanograms we may infer that [2]:</p>
<p>\begin{equation}
\overline{m_c} \approx 10^2 \text{ng} = 5 \cdot 10^{-7} \text{grams}
\end{equation}</p>
<p>so we have:</p>
<p>\begin{equation}
D(N_0,r) \approx \ln(4r^3 \cdot \rho_{\text{brain}}) - \ln (N_0) + 7\ln(10)
\end{equation}</p>
<p>and if we use the bounds from [1]:</p>
<p>\begin{equation}
5000 \leq N_0 \leq 10000
\end{equation}</p>
<p>\begin{equation}
1.5 \text{mm} \leq r \leq 2.5 \text{mm}
\end{equation}</p>
<p>we find that:</p>
<p>\begin{equation}
5.00 \leq D(N_0,r) \leq 5.83
\end{equation}</p>
<h2 id="estimating-the-doubling-times-during-brain-organoid-development">Estimating the doubling times during brain organoid development:</h2>
<p><strong>Disclaimer:</strong> In the analysis that follows we don’t make any assumptions on the proportion of cell divisions that are symmetric. This makes it more
robust than the previous analysis on the expected number of doubling episodes during brain organoid development.</p>
<p>Given the formula for the volume of a sphere, if <script type="math/tex">V_n</script> denotes the volume of a spheroid with radius <script type="math/tex">r_n \leq r_{\text{max}}</script> where <script type="math/tex">r_{\text{max}} = 2.5 \text{mm}</script> we have:</p>
<p>\begin{equation}
V_{n+1} = 2 \cdot V_n \implies r_{n+1} = 2^{\frac{1}{3}} \cdot r_n
\end{equation}</p>
<p>and given that brain organoids aren’t vascularized they must be diffusion-constrained. In this scenario, it’s reasonable to assume that:</p>
<p>\begin{equation}
\text{growth rate} \sim \text{metabolic rate} \sim \frac{\text{vol}(\partial \mathcal{B})}{\text{vol}(\mathcal{B})} \approx \frac{4 \pi r^2}{\frac{4}{3} \pi r^3} = \frac{3}{r}
\end{equation}</p>
<p>where we made the implicit assumption that during the elapsed time for doubling we have an approximate equality of the following averages:</p>
<p>\begin{equation}
\langle \text{growth rate of cell population} \rangle \approx \langle \text{growth rate of organoid volume} \rangle
\end{equation}</p>
<p>Now, given (13) if we denote the growth rate by <script type="math/tex">g_r</script> we have:</p>
<p>\begin{equation}
\frac{3k}{2^{\frac{1}{3}} \cdot r_n} \leq g_r \leq \frac{3k}{r_n}
\end{equation}</p>
<p>where <script type="math/tex">k</script> is an unknown constant.</p>
<p>It follows that if the volume of the brain organoid is currently <script type="math/tex">V_n</script> the expected doubling time <script type="math/tex">T_{n}</script> must be approximately:</p>
<p>\begin{equation}
T_n \cdot g_r = V_{n+1}
\end{equation}</p>
<p>\begin{equation}
V_{n+1} = 2 \cdot V_n = \frac{8}{3} \pi r_n^3
\end{equation}</p>
<p>using (15) we find that the doubling time must be in the interval:</p>
<p>\begin{equation}
\frac{8}{9k} \pi r_n^4 \leq T_n \leq \frac{8 \cdot 2^{\frac{1}{3}}}{9k} \pi r_n ^4
\end{equation}</p>
<p>and if our uncertainty over <script type="math/tex">T_n</script> is expressed as a uniform distribution on this interval the expected doubling time is given by:</p>
<p>\begin{equation}
\mathbb{E}[T_n] = \frac{4 \pi r_n^4}{9k} \cdot (1+2^{\frac{1}{3}})
\end{equation}</p>
<h2 id="discussion">Discussion:</h2>
<p>I must clarify that this theoretical analysis represents just the first attempt at a first-principles approach to predicting the time
required for a brain organoid to double its volume. The main objective of this analysis was to advance concepts that are useful for
analysing the development of brain organoids. This includes the metabolic activity of cells, their packing density, mechanisms for cell
signalling and equilibrium distributions over cell types at the terminal phase of development.</p>
<p>To validate my formulas that predict the expected waiting time for a spherical brain organoid to double its volume we may use tools
from data analysis. Specifically, we may use a combination of computer vision and non-linear regression to infer a functional relationship
between the doubling time and potentially relevant variables.</p>
<p>If the fourth derivative of the interpolated curve resulting from such an analysis is a positive constant then my theoretical analysis is
broadly correct.</p>
<p><strong>Acknowledgements:</strong> I would like to thank <a href="https://bradly-alicea.weebly.com">Bradly Alicea</a> for constructive feedback on this theoretical analysis.</p>
<h2 id="references">References:</h2>
<ol>
<li>Yakoub AM, Sadek M. Development and Characterization of Human Cerebral Organoids: An Optimized Protocol. 2018.</li>
<li>Haifei Zhang. Cell. http://soft-matter.seas.harvard.edu/index.php/Cell. 2009.</li>
<li>Modeling a neurodevelopmental disorder with human brain organoids: a new way to study conditions such as epilepsy and autism. https://neuroscience.berkeley.edu/modeling-neurodevelopmental-disorder-human-brain-organoids-new-way-study-conditions-epilepsy-autism/. 17/09/2018.</li>
</ol>Aidan RockeA spherical brain organoid grown in Berkeley [3]How neuroscientists can help address climate change2019-09-01T00:00:00+00:002019-09-01T00:00:00+00:00/neuroscience/2019/09/01/neuro4climate<center><img src="https://raw.githubusercontent.com/Kepler-Lounge/Kepler-Lounge.github.io/master/_images/flights.png?token=AH6QLOF7DV633JQD5LEK53K5SNMFO" width="75%" height="75%" align="middle" /></center>
<center>Total passengers carried by planes has grown by a factor of eight in the last 40 years(source: Data Bank)</center>
<h2 id="introduction">Introduction:</h2>
<p>The following analysis arose from a simple question. Why is it that the MathOverflow, an Internet forum for mathematicians, thrives and the analogous forum for neuroscientists doesn’t? What is the nature of the problem and if so how should it be addressed?</p>
<p>But, let’s start with an easier question. Why are all the neuroscientists on Twitter?</p>
<h2 id="why-are-all-the-neuroscientists-on-twitter">Why are all the neuroscientists on Twitter?:</h2>
<p>When Twitter was created with a constraint of 140 words per tweet I doubt that the Twitter product team expected their platform to be heavily used by scientists. You can’t render mathjax/latex and Twitter isn’t ideal for expressing subtleties but it has many desirable features for networking:</p>
<ol>
<li>Information exchange is efficient. A tweet represents <script type="math/tex">\sim 10^{-5}</script> kg CO2.</li>
<li>The message-length constraint incents quasi-synchronous exchanges.</li>
<li>All the scientists are already there for other reasons: sports, politics…etc.</li>
</ol>
<p>On any given day scientists on Twitter will share their preprints, explain how they achieved their results, and effectively conduct Q&A sessions on their research. Some scientists even joke that it has become the place for peer-review. I think it’s fair to say that Twitter has brought great value to the scientific community by allowing frictionless communication between scientists across the globe; scientists who probably wouldn’t communicate with each other unless they met at a conference. Crucially, relative to science conferences Twitter has a relatively small carbon footprint.</p>
<p>It seems almost like Twitter should be a public good except that it isn’t and this got me thinking about a public neuroscience forum for neuroscientists, a bit like the <a href="https://mathoverflow.net/">MathOverflow</a> for mathematicians.</p>
<h2 id="the-psychology-and-neuroscience-stackexchange-and-its-limits">The Psychology and Neuroscience stackexchange and its limits:</h2>
<p>What drew me to the Psychology and Neuroscience stack-exchange was that it had several functionalities that weren’t available on Twitter:</p>
<ol>
<li>You can easily find whether an identical/related question was asked.</li>
<li>Latex is available for mathematical formulas.</li>
<li>Shared tags for easy discoverability of posts.</li>
</ol>
<p>But, as I started making regular use of the forum I noticed sexist and racist behaviour at all user-reputation levels on the forum including the moderator-level:</p>
<ol>
<li><a href="https://psychology.stackexchange.com/questions/8277/how-to-interpret-a-bbc-news-article-on-the-effect-of-race-on-intelligence/8286#8286">How to interpret a BBC news article on the effect of race on intelligence?</a></li>
<li><a href="https://psychology.stackexchange.com/questions/20652/iq-gap-by-race-truth-or-myth/20664#20664">IQ gap by race, truth or myth?</a></li>
<li><a href="https://psychology.stackexchange.com/questions/10701/is-the-logic-of-herrnsteins-syllogism-sound-and-are-its-premises-true/17269#17269">Is the logic of “Herrnstein’s syllogism” sound, and are its premises true?</a></li>
</ol>
<p>On the balance, the current moderators nurtured an environment where racist and sexist views can coexist with research-level neuroscience questions thus offering them legitimacy. These questions are also terribly outdated, hailing back to a time when intelligence tests were used to justify colonial mentalities i.e. right-to-rule.</p>
<p>Having said this, I am not here to incite outrage and would stop short of labelling them as racist/sexist. We should be skeptical of the desire to punish as it often prevents us from seeing the bigger picture.</p>
<p>There are a couple problems with stack exchange forums which make the problem of sexism and racism difficult to tackle:</p>
<ol>
<li>Users can create one or more accounts under a pseudonym.</li>
<li>A moderator of a stack-exchange forum may be one of several accounts controlled by a single user.</li>
</ol>
<p>This allows the possibility of sockpuppeting at the moderator level, a fault that is exploitable on forums where users might want to share racist/sexist viewpoints. For the above reasons, I am not sure the neuroscience and psychology stack exchange is salvageable in its current form.</p>
<p>Finally, I’d like to address the view that scientists should be free to do research of a sexist/racist nature. First, if you had to list the twenty most important problems in neuroscience you would be hard-pressed to find any that require torturing data to ‘discover’ differences in intelligence between people of different gender or race. Second, there have always been ethical limits on scientific inquiry.</p>
<p>In a globalized world, scientists are free to pursue their scientific interests provided that it benefits a multicultural and inclusive society.</p>
<h2 id="the-sociology-of-neuroscience">The sociology of neuroscience:</h2>
<p>Besides sexist and racist behaviour, the Psychology and Neuroscience stack-exchange faces rather unique challenges unlike the MathOverflow, the Physics stack-exchange or the Theoretical Computer Science stack-exchange.</p>
<p>In neuroscience unlike math, physics or theoretical computer science the fundamental concepts are still in development. This is partly due to the complexity of the brain, possibly the most complex object in the universe, and partly due to a relative lack of data. This greater degree of uncertainty in neuroscience encourages much greater specialisation. In fact, it’s fair to say that neuroscience has many tribes that don’t share a common language.</p>
<p>This has a couple consequences:</p>
<ol>
<li>The probability that a research-level question will be answered on a forum lacking a critical-mass of researchers with diverse research backgrounds is small.</li>
<li>A researcher or masters student is more likely to address another specialist directly via email.</li>
</ol>
<p>For these reasons, the ratio of research-level questions to lower-tier questions is going to be biased towards lower-level questions and the Psychology and Neuroscience stack-exchange is unlikely to be dominated by research-level queries.</p>
<p>In contrast, neuroscience conferences are much better places for exchanging scientific information and building trust. This is partly because communication isn’t bandwidth-limited, face-to-face communication builds trust and a conference effectively directs the collective intelligence of scientists via synchronous communication. These are the types of forums that really matter to neuroscientists.</p>
<p>No wonder neuroscientists fly to hundreds of conferences per year. But, at what cost?</p>
<h2 id="the-carbon-footprint-of-neuroscience-conferences">The carbon footprint of neuroscience conferences:</h2>
<p>Plane flights emit on the order of ~,1 kg of CO2 per person per km. This explains why at the level of universities, the carbon footprint of academic conferences may represent up to a third of that universities’ carbon budget. At the level of the individual scientist the situation is much worse, easily representing more than 40% of their carbon
footprint.</p>
<p>Most scientists I have met care about the environment and wouldn’t deviate significantly from the European average of ~10 tonnes of CO2 per person per year. However, if we take into account that planes emit ~,1 kg of CO2 per person per km a scientist can easily add six tonnes of CO2 to their carbon footprint. They simply have to get on three return flights from San Francisco to Berlin which represents about 60,000 km of plane flight in total.</p>
<p>Does this mean that scientists should travel less? Probably. But, this doesn’t mean scientists should attend fewer conferences.</p>
<p>In the same way that Twitter has allowed scientists to communicate seamlessly across the Atlantic, I believe virtual reality may replace most brick-and-mortar conferences in the next five years.</p>
<h2 id="are-virtual-science-conferences-possible">Are virtual science conferences possible?</h2>
<p>Regarding the feasibility of virtual science conferences, we are not talking about science fiction. Virtual Reality is a technology which is already on the market in the form of the Oculus Rift S and the HTC Vive.</p>
<p>More than a technology, VR realises the vision of philosophers and mathematicians dating back a thousand years who believed that the world we perceive is a construction of the mind:</p>
<blockquote>
<p>Nothing of what is visible, apart from light and color, can be perceived by pure sensation, but only by discernment, inference, and recognition, in addition to sensation. -Alhazen</p>
</blockquote>
<p>From this perspective, robust VR requires progress on multi-sensory integration theories in order to understand how our senses can be tricked. The challenge is to find the right priors over multi-sensory data streams. On this front, we must recognise the important contributions made by behavioural and perceptual neuroscientists to VR
research and development.</p>
<p>If VR technologists can solve multi-party face-to-face interaction in the same way that Twitter has solved global public messaging, this would remove the need for almost all brick-and-mortar conferences. In the process, neuroscientists will make a historic contribution to addressing climate change.</p>
<h2 id="discussion">Discussion:</h2>
<p>While I think that behavioural and perceptual neuroscientists will play a crucial role in realising the vision of VR and consequently make most flights unnecessary I believe that the broader community of neuroscientists must do better in communicating their role to help address climate change. This will have the effect of unifying neuroscientists around neuroscience-driven solutions for climate change.</p>
<p>I also think this is only the beginning. There are other substantial ways neuroscientists can help address climate change.</p>
<h2 id="references">References:</h2>
<ol>
<li>WorldBank Data Bank. Air transport, passengers carried. https://data.worldbank.org/indicator/IS.AIR.PSGR. 01/09/2019.</li>
<li>Amanda Thompson. Scientific Racism: The Justification of Slavery and Segregated Education in America. 2003.</li>
<li>Cesare V. Parise , Marc O. Ernst. Noise, multisensory integration, and previous response in perceptual disambiguation. PLOS Biology. 2017.</li>
<li>Fast Company. How Much Energy Does a Tweet Consume? https://www.fastcompany.com/1620676/how-much-energy-does-tweet-consume. 19/04/2010. 01/09/2019.</li>
</ol>Aidan RockeTotal passengers carried by planes has grown by a factor of eight in the last 40 years(source: Data Bank)A comparison of credit-assignment models in mathematics and biology2019-08-27T00:00:00+00:002019-08-27T00:00:00+00:00/complex/networks/2019/08/27/authorship-I<h2 id="introduction">Introduction:</h2>
<p>In the world of science, scientists are rewarded for the quality of their publications. But, sometimes they are also rewarded for the relative ordering
of author names-what we may call the first-author model. This incents different kinds of citation behaviour and these distinct credit-assignment models
probably lead to different citation networks.</p>
<p>Among mathematicians and physicists who adhere to the alphabetical ordering of author names, this incents scientists to find brilliant collaborators.
On the other hand, if relative author ordering matters as is the case with biologists we might expect scientists to prioritise finding brilliant collaborators
and first-authorship, probably not in equal measure.</p>
<p>To a first-order approximation, we may understand the difference between these two types of credit-assignment systems by comparing the number of alphabetical
orderings with the number of first-author orderings as a function of <script type="math/tex">N</script>, the number of co-authors.</p>
<h2 id="alphabetical-order">Alphabetical order:</h2>
<p>Traditionally, in math and physics a group of <script type="math/tex">N</script> researchers that co-author a paper use alphabetical orderings by default so we have:</p>
<p>\begin{equation}
\forall N \in \mathbb{N}, A(N)=1
\end{equation}</p>
<p>where <script type="math/tex">A(\cdot)</script> stands for the number of alphabetical orders as a function of <script type="math/tex">N</script>. Although some information may be lost by adhering to alphabetical ordering one of its
advantages is that it reduces the risk of internal friction within the group of authors.</p>
<h2 id="an-upper-bound-on-author-orderings">An upper-bound on author orderings:</h2>
<p>In a world as complex as ours, each author might have their own metric so <script type="math/tex">3^{N \choose 2}</script> orderings are possible where each author is a node in a fully-connected
graph and it’s assumed that there are three possible labels <script type="math/tex">% <![CDATA[
\{<,>,=\} %]]></script> for each edge in the graph.</p>
<p>However, most of these orderings aren’t linear orders. In order to have a linear order all authors must organise to use a single metric. How does this consensus emerge?
Politics? Meritocracy? I have no idea. In any case, if <script type="math/tex">F(\cdot)</script> is the number of first-author orderings it’s reasonable to believe that:</p>
<p>\begin{equation}
\forall N \in \mathbb{N}, F(N) \ll 3^{N \choose 2}
\end{equation}</p>
<p>where <script type="math/tex">3^{N \choose 2}</script> represents a maximally diverse number of orderings.</p>
<h2 id="first-author-orderings">First-author orderings:</h2>
<p>If no ties between authors are possible then <script type="math/tex">F(\cdot)</script> is simply the number of hamiltonian paths in the fully-connected graph with <script type="math/tex">N</script> nodes so we have:</p>
<p>\begin{equation}
F(N) \geq N!
\end{equation}</p>
<p>But, if we allow ties then for each of the <script type="math/tex">N-1</script> edges in a hamiltonian path there are two options, <script type="math/tex">% <![CDATA[
\{<,=\} %]]></script>. So in general we have:</p>
<p>\begin{equation}
F(N) = 2^{N-1} \cdot N!
\end{equation}</p>
<p>In a group of <script type="math/tex">N</script> co-authors we might deduce that the fraction of orderings where a particular author comes first is given by:</p>
<p>\begin{equation}
\frac{F(N-1)}{F(N)} = \frac{1}{2N}
\end{equation}</p>
<p>so there’s a risk that the degree of selfish behaviour might increase as the number of co-authors increases because you might need
to do more work to convince the other co-authors that you contributed more than them.</p>
<p>Furthermore, we may intuit that <script type="math/tex">2^{N} \cdot N! \ll 3^{N \choose 2}</script> but we can make this comparison precise using:</p>
<p>\begin{equation}
\forall e \leq A \leq B, \frac{A}{B} \leq \frac{\ln A}{\ln B}
\end{equation}</p>
<p>Using the above inequality we find that:</p>
<p>\begin{equation}
\frac{2^{N-1} \cdot N!}{3^{N \choose 2}} \sim \frac{2^{N-1} (\frac{N}{e})^{N}}{3^{\frac{N^2}{2}}} \leq \frac{2N \ln N}{\frac{N^2}{2} \ln 3} < \frac{4 \ln N}{N}
\end{equation}</p>
<p>so the extent to which first-author orderings can capture a diversity of views vanishes faster than <script type="math/tex">\frac{4 \ln N}{N}</script>. From this analysis I can infer that the first-author
model is more suitable for small numbers of authors.</p>
<h2 id="discussion">Discussion:</h2>
<p>At this point I must acknowledge that this constitutes the beginning of a mathematical analysis which must be refined. How can we model the outcome of sequential self-centered
behaviour under both paradigms?</p>
<p>What kind of citation dynamics do these different credit-assignment models encourage? What if authors regularly co-author papers together? These are questions to be addressed
in a future article.</p>Aidan RockeIntroduction:A simple proof of Euler’s product formula2019-08-21T00:00:00+00:002019-08-21T00:00:00+00:00/number/theory/2019/08/21/euler_1<h2 id="introduction">Introduction:</h2>
<p>The Euler product formula states that if <script type="math/tex">\zeta(s)</script> is the Riemann zeta function and <script type="math/tex">p</script> is prime:</p>
<p>\begin{equation}
\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p} \frac{1}{1-p^{-s}}
\end{equation}</p>
<p>holds for all <script type="math/tex">s \in \mathbb{C}</script> such that <script type="math/tex">\zeta(s)</script> is absolutely convergent.</p>
<h2 id="proof">Proof:</h2>
<p>Every positive integer <script type="math/tex">n \in \mathbb{N^*}</script> has a unique prime factorization:</p>
<p>\begin{equation}
\forall n \in \mathbb{N^*} \exists c_p \in \mathbb{N}, n = \prod_p p^{c_p}
\end{equation}</p>
<p>where <script type="math/tex">% <![CDATA[
\sum c_p < \infty %]]></script>.</p>
<p>Furthermore, we note that:</p>
<p>\begin{equation}
\prod_p \frac{1}{1-p^{-s}} = \prod_p \big(\sum_{c_p = 0}^\infty p^{-c_p s} \big)
\end{equation}</p>
<p>due to elementary properties of geometric series.</p>
<p>In the formal expansion (2) we note that each term has a unique prime factorization and that every possible prime factorization occurs once. It follows that
if <script type="math/tex">\sum n^{-s}</script> converges absolutely we may rearrange the sum however we wish and so:</p>
<p>\begin{equation}
\zeta(s) = \prod_{p} \frac{1}{1-p^{-s}}
\end{equation}</p>
<p>provided that the hypotheses on <script type="math/tex">s</script> are satisfied.</p>Aidan RockeIntroduction:Almost all simple graphs are small world networks2019-07-31T00:00:00+00:002019-07-31T00:00:00+00:00/graphs/2019/07/31/small-worlds<h2 id="introduction">Introduction:</h2>
<p>Two days ago, while thinking about brain networks, it occurred to me that almost all simple graphs are small world networks in the sense that if <script type="math/tex">G_N</script> is a simple graph with <script type="math/tex">N</script> nodes sampled from the Erdös-Rényi random graph distribution with probability half then when <script type="math/tex">N</script> is large:</p>
<p>\begin{equation}
\mathbb{E}[d(v_i,v_j)] \leq \log_2 N
\end{equation}</p>
<p>My strategy for proving this was to show that when <script type="math/tex">N</script> is large, <script type="math/tex">\forall v_i \in G_N</script> there exists a chain of distinct nodes of length <script type="math/tex">\log_2 N</script> originating from <script type="math/tex">v_i</script> almost surely. This implies that:</p>
<p>\begin{equation}
\forall v_i, v_j \in G_N, d(v_i,v_j) \leq \log_2 N
\end{equation}</p>
<p>almost surely when <script type="math/tex">N</script> is large.</p>
<p>Now, by using the above method of proof I managed to show that almost all simple graphs are <em>very small</em> in the sense that:</p>
<p>\begin{equation}
\mathbb{E}[d(v_i,v_j)] \leq \log_2\log_2 N
\end{equation}</p>
<p>when <script type="math/tex">N</script> tends to infinity. We can actually do even better.</p>
<p>Using my proof that <a href="https://keplerlounge.com/math/2019/07/02/connected-graphs.html">almost all simple graphs are connected</a>, I can show that almost all simple graphs have diameter 2. However, I think there is more value in going through my original proof which in my opinion provides greater insight into the problem.</p>
<h2 id="degrees-of-separation-and-the-neighborhood-of-a-node">Degrees of separation and the neighborhood of a node:</h2>
<p>We may think of degrees of separation as a sequence of ‘hops’ between the neighborhoods of distinct nodes <script type="math/tex">v_i</script>. Given a node <script type="math/tex">v_i</script> we may define <script type="math/tex">\mathcal{N}(v_i)</script>
as follows:</p>
<p>\begin{equation}
\mathcal{N}(v_i) = \{v_j \in G_N: \overline{v_i v_j} \in G_N \}
\end{equation}</p>
<p>where <script type="math/tex">G_N = (V,E)</script> is a graph with <script type="math/tex">N</script> nodes.</p>
<p>Now, given the E-R model we can say that <script type="math/tex">v_i \neq v_j</script> implies:</p>
<p>\begin{equation}
P(v_k \notin \mathcal{N}(v_i) \land v_k \notin \mathcal{N}(v_j)) = P(v_k \notin \mathcal{N}(v_i)) \cdot P(v_k \notin \mathcal{N}(v_j)) = \frac{1}{4}
\end{equation}</p>
<p>and by induction:</p>
<p>\begin{equation}
P(v_k \notin \mathcal{N}(v_{1}) \land … \land v_k \notin \mathcal{N}(v_{n})) = \frac{1}{2^n}
\end{equation}</p>
<p>It follows that if there is a chain of distinct nodes <script type="math/tex">\overline{v_1 ... v_n}</script> we can say that:</p>
<p>\begin{equation}
P(d(v_1,v_k) \leq n) = 1- \frac{1}{2^n}
\end{equation}</p>
<h2 id="almost-all-simple-graphs-are-very-small-world-networks">Almost all simple graphs are very small world networks:</h2>
<h3 id="a-chain-of-distinct-nodes-v_i_i1log_2log_2-n-exists-almost-surely">A chain of distinct nodes <script type="math/tex">\{v_i\}_{i=1}^{\log_2\log_2 N}</script> exists almost surely:</h3>
<p>The probability that there exists a chain of nodes of length <script type="math/tex">\log_2\log_2 N</script>:</p>
<p>\begin{equation}
\overline{v_1 … v_{\log_2\log_2 N}}
\end{equation}</p>
<p>such that <script type="math/tex">v_i = v_j \iff i=j</script> is given by:</p>
<p>\begin{equation}
P(\overline{v_1 … v_{\log_2\log_2 N}} \in G_N) = \prod_{k=1}^{\log_2\log_2 N} \big(1-\frac{1}{2^{N-k}} \big) \geq \big(1- \frac{\log_2 N}{2^N}\big)^{\log_2\log_2 N}
\end{equation}</p>
<p>and we note that:</p>
<p>\begin{equation}
\lim\limits_{N \to \infty} \big(1- \frac{\log_2 N}{2^N}\big)^{\log_2\log_2 N} = 1
\end{equation}</p>
<p>this guarantees the existence of a chain of distinct nodes of length <script type="math/tex">\log_2 N</script> originating from any <script type="math/tex">v_i \in G_N</script> almost surely.</p>
<h3 id="given-that-overlinev_1--v_log_2log_2-n-exists-almost-surely-we-may-deduce-that-forall-i-in-1log_2log_2-n-dv_iv_k-leq-log_2log_2-n-almost-surely">Given that <script type="math/tex">\overline{v_1 ... v_{\log_2\log_2 N}}</script> exists almost surely we may deduce that <script type="math/tex">\forall i \in [1,\log_2\log_2 N], d(v_i,v_k) \leq \log_2\log_2 N</script> almost surely:</h3>
<p>If <script type="math/tex">\overline{v_1 ... v_{\log_2\log_2 N}}</script> exists we have:</p>
<p>\begin{equation}
\forall \{v_i\}_{i=1}^n, v_k \in G_N, P(d(v_1,v_k) \leq \log_2\log_2 N) = 1 - \frac{1}{2^{\log_2\log_2 N}}
\end{equation}</p>
<p>and so we have:</p>
<p>\begin{equation}
\lim\limits_{N \to \infty} \forall \{v_i\}_{i=1}^n, v_k \in G_N, P(d(v_1,v_k) \leq \log_2\log_2 N) = 1
\end{equation}</p>
<h2 id="discussion">Discussion:</h2>
<p>I must say that initially I found the above result quite surprising and I think it partially explains why small world networks frequently occur in nature.
Granted, in natural settings the graph is typically embedded in some kind of Euclidean space so in addition to the degrees of separation we must consider
the Euclidean distance. But, I suspect that in real-world networks with small world effects the Euclidean distance plays a marginal role.</p>
<p>In particular, I believe that wherever small-world networks prevail the Euclidean distance is dominated by ergodic dynamics between nodes. There is probably
some kind of stochastic communication process between the nodes.</p>Aidan RockeIntroduction:Fractional Cartesian Products2019-07-08T00:00:00+00:002019-07-08T00:00:00+00:00/set/theory/2019/07/08/fractional_cartesian<h2 id="introduction">Introduction:</h2>
<p>Recently, I wondered whether we could define hypercubes with non-integer dimension. It occurred to me that this would
require a generalisation of the usual Cartesian Product to fractional dimensions.</p>
<p>A few Google searches indicated that previous work [1], [2] has been done on this subject by Ron C. Blei. However,
I usually try to develop my own ideas first as this sometimes allows me to develop a perspective that is particularly
insightful. For this problem I decided to start by considering hypercube volumes.</p>
<h2 id="hypercube-volumes">Hypercube volumes:</h2>
<p>If the volume of a regular hypercube with integer dimension is given by:</p>
<p>\begin{equation}
\forall n \in \mathbb{N}, \text{Vol}([-1,1]^n) = 2^n
\end{equation}</p>
<p>then I think we may define the volume of hypercubes with non-integer dimension as follows:</p>
<p>\begin{equation}
\forall x \in \mathbb{R}_+ \setminus \mathbb{N}, \text{Vol}([-1,1]^x) = 2^x
\end{equation}</p>
<p>but the challenge is how should we define <script type="math/tex">[-1,1]^x</script> analytically so that this hypercube reduces to the
usual hypercube when <script type="math/tex">x \in \mathbb{N}</script>. I think this requires a suitable representation of the Cartesian Product.</p>
<p>One idea that occurred to me was to represent Cartesian Products as multipartite graphs.</p>
<h2 id="references">References:</h2>
<ol>
<li>Ron C Blei. Fractional cartesian products of sets. 1979.</li>
<li>Ron Blei, Fuchang Gao. Combinatorial dimension in fractional Cartesian products. 2005.</li>
</ol>Aidan RockeIntroduction:The number of ways to partition a graph2019-07-05T00:00:00+00:002019-07-05T00:00:00+00:00/graph/theory/2019/07/05/graph-partition<h2 id="introduction">Introduction:</h2>
<p>Let’s suppose we have a graph with <script type="math/tex">N</script> vertices. How many ways can these vertices be wired to each other assuming that these vertices are distinct and each vertex
<script type="math/tex">v_i</script> may be connected to at most <script type="math/tex">N-1</script> distinct vertices? Alternately, let’s consider the set <script type="math/tex">G_N</script> of simple graphs with <script type="math/tex">N</script> vertices. This set may correspond
to the set of potential social networks among a community of <script type="math/tex">N</script> individuals.</p>
<p>What is the cardinality of <script type="math/tex">G_N</script>? We may show that:</p>
<p>\begin{equation}
\lvert G_N \rvert = \sum_{k=0}^{N \choose 2} { {N \choose 2} \choose k} = 2^{N \choose 2}
\end{equation}</p>
<p>and we may note that <script type="math/tex">\lvert G_N \rvert</script> very quickly becomes astronomical:</p>
<p>\begin{equation}
\forall N > 50, \lvert G_N \rvert > 10^{368}
\end{equation}</p>
<p>which is many times greater than the number of atoms in the universe.</p>
<h2 id="a-few-observations">A few observations:</h2>
<h3 id="lvert-g_n-rvert-grows-more-than-exponentially-fast"><script type="math/tex">\lvert G_N \rvert</script> grows more than exponentially fast:</h3>
<p>It’s worth noting that <script type="math/tex">\lvert G_N \rvert</script> grows more than exponentially fast as a function of <script type="math/tex">N</script> since:</p>
<p>\begin{equation}
\frac{\lvert G_{N+1} \rvert}{\lvert G_{N} \rvert} = 2^N
\end{equation}</p>
<p>so we have:</p>
<p>\begin{equation}
\lvert G_{N+1} \rvert = 2^N \cdot \lvert G_{N} \rvert
\end{equation}</p>
<p>and this means that whenever we add a vertex <script type="math/tex">\widehat{v_{N+1}}</script> to a network with <script type="math/tex">N</script> vertices the number of
possible networks grows by a factor of <script type="math/tex">2^N</script>. The reason for this is that when a new vertex <script type="math/tex">\widehat{v_{N+1}}</script>
is added to a graph with <script type="math/tex">N</script> vertices there are <script type="math/tex">N</script> possible new edges between <script type="math/tex">\widehat{v_{N+1}}</script> and the
existing set of vertices.</p>
<p>Another way to think about (3) is that given a graph with <script type="math/tex">N</script> vertices an additional vertex <script type="math/tex">\widehat{v_{N+1}}</script>
adds <script type="math/tex">N</script> bits of information. Between any two vertices we have either a connection or we don’t so:</p>
<p>\begin{equation}
\log_2(\lvert G_{N+1} \rvert) - \log_2(\lvert G_{N} \rvert) = N
\end{equation}</p>
<h3 id="probabilistic-analysis">Probabilistic analysis:</h3>
<p>Now, let’s consider the probability of a connection between a random pair of vertices <script type="math/tex">(v_i,v_j)</script> in a graph <script type="math/tex">\Gamma_N</script>
sampled uniformly from <script type="math/tex">G_N</script>:</p>
<p>\begin{equation}
P(\overline{v_iv_j} \in \Gamma_N) = \frac{ {N \choose 2} }{2^{N \choose 2}}
\end{equation}</p>
<p>and this probability goes down exponentially quickly since:</p>
<p>\begin{equation}
\frac{P(\overline{v_iv_j} \in \Gamma_{N+1})}{P(\overline{v_lv_k} \in \Gamma_N)} = \frac{(N+1) \cdot 2^{-N}}{N-1} \approx 2^{-N}
\end{equation}</p>
<p>Within the context of social networks, if we suppose that a connection between any pair of individuals occurs with probability half
then the probability of a connection between a randomly chosen pair of individuals drops off to zero exponentially fast as the size
of the network, i.e. number of individuals, grows.</p>
<p>We can make one more relatively simple observation that is also useful. Given the symmetry of binomial coefficients, the probability
that a randomly chosen graph <script type="math/tex">\Gamma_N \sim G_N</script> has more than half the maximum number of edges, <script type="math/tex">{N \choose 2}</script>, is also <script type="math/tex">\frac{1}{2}</script>.
This doesn’t contradict the last observation since the number of possible edges grows quadratically <script type="math/tex">\sim \frac{N^2}{2}</script> while the
number of possible vertices grows linearly <script type="math/tex">\sim N</script>.</p>
<h2 id="discussion">Discussion:</h2>
<p>This analysis actually preceded my last article on <a href="https://keplerlounge.com/math/2019/07/02/connected-graphs.html">simple graphs that are connected</a>
and due to the rapid growth of <script type="math/tex">\lvert G_N \rvert</script> we may ask what does a typical graph look like.</p>
<p>We know that almost all simple graphs are connected but generally speaking what are the properties of almost all simple graphs?</p>Aidan RockeIntroduction:Almost all simple graphs are connected2019-07-02T00:00:00+00:002019-07-02T00:00:00+00:00/math/2019/07/02/connected-graphs<h2 id="introduction">Introduction:</h2>
<p>Recently, I wondered whether given the set of graphs with <script type="math/tex">N</script> distinguishable vertices, <script type="math/tex">G_N</script>, whether most of these graphs might be connected. This set may correspond to the state space of a biological network whose connectivity varies over time such as a brain. The relevance of connectivity here is that it guarantees a path
between different fundamental nodes.</p>
<p>Initially, I thought we might need to derive the asymptotic formula for the number of connected graphs with <script type="math/tex">N</script> vertices. It turns out that there’s a much simpler approach
using the Erdős–Rényi random graph model. I realised this after discussing a related question with <a href="https://mathoverflow.net/questions/334936/asymptotic-formula-for-the-number-of-connected-graphs">mathematicians on the MathOverflow</a>.</p>
<h2 id="demonstration">Demonstration:</h2>
<p><a href="https://www.math.u-psud.fr/~fouquet/">Olivier Fouquet</a> and lambda made very helpful remarks regarding the connection with random graphs. In particular, I would like to point out lambda’s remark that:</p>
<blockquote>
<p>…the Erdős–Rényi random graph model with edge probability 1/2 gives the
uniform distribution on labelled graphs</p>
</blockquote>
<p>Using this insight we may proceed as follows:</p>
<p>Let’s first note that the Erdős–Rényi random graph model with edge probability 1/2 gives the uniform distribution on labelled graphs since for each pair of vertices they are either joined by an edge or not. It follows that given a graph with <script type="math/tex">N</script> vertices the probability that any finite subset of <script type="math/tex">k</script> vertices, <script type="math/tex">V \subset \{v_i\}_{i=1}^N</script> and <script type="math/tex">\lvert V \rvert=k</script>, are joined to a common vertex <script type="math/tex">v_l \notin V</script> is given by:</p>
<p>\begin{equation}
1 - {N \choose k}\big(1-\frac{1}{2^k} \big)^{N-k}
\end{equation}</p>
<p>Now, we would like to show that:</p>
<p>\begin{equation}
\lim\limits_{N \to \infty}{N \choose k}\big(1-\frac{1}{2^k} \big)^{N-k}=0
\end{equation}</p>
<p>Let’s first note that:</p>
<p>\begin{equation}
{N \choose k}=\frac{N!}{k!(N-k)!} \leq N^k
\end{equation}</p>
<p>\begin{equation}
\big(1-\frac{1}{2^k} \big)^{N-k} \propto \big(1-\frac{1}{2^k} \big)^N \sim e^{-\frac{N}{2^k}}
\end{equation}</p>
<p>and taking logarithms we find that for fixed <script type="math/tex">k \in \mathbb{N}</script>:</p>
<p>\begin{equation}
\lim_{N \to \infty} \frac{\ln N}{N} < \frac{1}{k2^k}
\end{equation}</p>
<p>so we may conclude that a simple graph is connected with probability 1.</p>
<p>As a corollary we may deduce that for large <script type="math/tex">N</script> the number of connected graphs <script type="math/tex">K_N</script> is given by:</p>
<p>\begin{equation}
\lvert K_N \rvert \sim 2^{N \choose 2}
\end{equation}</p>
<h2 id="discussion">Discussion:</h2>
<p>I’m quite satisfied with this demonstration using probabilistic arguments as it’s a lot simpler than the approach
proposed by [1] and [2]. However, I must say that [1] and [2] contain interesting insights and methods that I haven’t
seen before. For this reason both of these publications are on my reading list.</p>
<h1 id="references">References:</h1>
<ol>
<li>E. Bender, E. Canfield & B. McKay. The Asymptotic Number of labeled Connected Graphs with a Given Number of I/ertices and Edges. 1990.</li>
<li>Example II.15 in Flajolet and Sedgewick, Analytic Combinatorics. 2009.</li>
</ol>Aidan RockeIntroduction: