An analysis of silk density in spider webs

Abstract

This study explores the structural complexity of spider webs through information-theoretic and harmonicity-based frameworks to quantify spatial patterns in silk density across different web regions and reveal the underlying resource allocation strategies. Currently, there is no normalized approach for describing web structure and complexity, particularly for sheet webs, and this methodology allows for non-destructive scanning and quantification of web characteristics. By analysing the entropy of silk density distributions, a single scalar that captures the heterogeneity of material investment across the entire web, we observed that the entropy values follow a normal distribution with a mean of 1.24 $\pm$ 0.22 bits when using 10 quantization levels. In the second part of the paper, by measuring the harmonicity of the silk density, we reveal that the silk density at a given point can be inferred from its neighbours, with an average harmonicity value of 0.0039 $\pm$ 0.0017 (fraction of total points in point cloud data). The harmonic behaviour is notable for its maximum principle, suggesting that the strongest parts of the web appear at the boundaries, aligning with existing knowledge of spider web construction. These findings provide a new technique for quantifying web-building strategies and offer new insights into spider behaviour and evolution.

1. Introduction

Spiders are exceptional biological engineers, using self-manufactured silk to design and build webs many times larger than their bodies [1]. As a material, spider silk is stronger and more elastic than almost any other known natural or man-made material, while simultaneously being biodegradable and biocompatible [2–5]. Spiders use silk for a variety of purposes, including prey capture and wrapping, encasing egg sacks, during courtship and copulation, and perhaps most visibly: building webs [6]. A spider’s web serves as a tool for prey capture, an extension of the senses and a retreat from predators [6–10]. Based on the most up-to-date phylogenetic analysis, orb webs are the most ancient type of prey capture web, evolving more than 165 Ma [11–14]. Whether the web evolved once or multiple times is still under debate, but regardless of the origin, many extant families abandoned orb webs later in their evolution, adopting cursorial lifestyles or a variety of other web structures, including sheet webs, cob-webs and funnel webs [6,15].

The linyphioids group (Linyphiidae and Pimoidae) constitute about 10% of all spider species worldwide, with more than 5000 named species [16]. Despite linyphioid diversity and abundance, web architecture and mechanics are poorly studied. Linyphiids generally build clearly suspended horizontal sheet webs that are sparsely attached to the surrounding substrate; however, variations in this design are prevalent in most genera [17]. Sheet web structures are generally described as an unordered meshwork of silk fibres varying in thickness and strength, with some threads bearing viscid droplets [7,18]. However, building a web is an energetically costly process, depleting glucoproteins and significantly increasing the metabolic rate [19,20]. It is expected that over evolutionary time, selection would favour economical webs that maximize function with the least amount of silk [21,22]. However, quantifying the effectiveness and economy of webs is a difficult process with no universally accepted metric, although mathematical modelling has precedence for cob and orb web species [23–26].

Another important aspect in spider web complexity is the spatial correlations in silk density [26]. By examining how the density of silk in one region of the web relates to the density in neighbouring regions, we can infer patterns and predict the distribution of silk throughout the web. Understanding how silk densities vary across the three-dimensional surface of a sheet web can potentially provide further insight into spider web building behaviour and identify areas of the web used for different purposes, for example, prey capture compared with predator retreat [6,17].

In this paper, we develop a method to quantify sheet web complexity by measuring Shannon entropy and harmonicity for the sheet webs of three species of linyphiid common to Northern California, Neriene digna, Neriene litigiosa and Microlinyphia dana. Using these metrics, we are able to discern new features of sheet webs and deepen our understanding of spider web-building behaviour.

The remainder of the paper is organized as follows. In §§2 and 3, we provide background information on the mathematical principles and experimental methodology, respectively. Section 4 discusses entropy calculations and the corresponding statistical analyses. Section 5 discusses the harmonicity of silk density and the analysis of local spatial correlations across the webs. Finally, §6 concludes the paper and suggests directions for future research.

2. What are entropy and harmonicity?

2.1. Mathematical framework for entropy

2.1.1. Motivation for entropy

Quantities that capture macroscopic properties of a system are valuable since they enable meaningful analysis without requiring a detailed account of the system’s microscopic structure. For example, temperature measures the thermal energy of a substance without reference to the motion of individual atoms or molecules. Silk density in a spider web is directly proportional to the spider’s material investment and energetic cost of construction, but there is currently no standard metric for quantifying this information aside from simple summary figures such as total silk mass [1,27].

By quantizing local silk‐mass measurements into a discrete distribution and computing their Shannon entropy, we obtain a single scalar that captures the heterogeneity of that investment across the entire web. This macroscopic value allows us to quantify global organizational features of the web while abstracting away the intricate and often irregular details of individual thread placement and allows for comparison between webs of different sizes and structures. Crucially, our entropy‐based measure is also insensitive to the absolute scale or the particular choice of segmentation, which facilitates robust comparisons across webs of different sizes, shapes and species.

This methodology shows that in all scanned webs, 98–99% of the space is empty and has zero density value, leaving us with an apparent sparsity of data to analyse. However, this is due to the implicit nature of spider webs, which are by definition minute amounts of silk occupying otherwise empty space. By removing the zero density values from our analysis, we are focusing our analysis and observations on only the biological structure, rather than excluding or failing to describe any part of the web itself.

2.1.2. Mathematical framework for entropy computation

Generally, Shannon entropy measures the amount of information needed to describe the spread of data across all possible values [28]. By examining the silk density distribution across a web, we can assign an entropy value to each web. A low entropy value would indicate that the silk density across the web is primarily concentrated at one value, while a high entropy value would indicate a web with a full range of density values spread uniformly at random. The lowest possible entropy value is zero bit, and the highest is ${\log }_2(10)\approx 3.3219$ bits when quantizing with $10$ density levels.

We develop a mathematical framework for computing the entropy of silk density distributions. The silk density distribution of a web can be determined by subdividing the three-dimensional space where the web is constructed by a cubic grid and then quantizing the silk density measurements into discrete levels. It should be noted that this framework demonstrates a convergence property: as we refine the subdivisions of the spatial domain, the resulting entropy becomes asymptotically independent of how the region is subdivided.

Following Shannon’s definition of entropy, we compute entropy of the density of a web as follows: divide the whole box with measurements given by height, width and depth into $N=N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}\times N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}\times N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}$ smaller boxes of size ${\displaystyle \frac{\mathtt{\text{height}}}{N_{\mathtt{\text{height}}}}\times \frac{\mathtt{\text{width}}}{N_{\mathtt{\text{width}}}}\times \frac{\mathtt{\text{depth}}}{N_{\mathtt{\text{depth}}}}}$. For $0\le i<N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}},0\le j<N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}},0\le k<N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}$, let $n_{i,j,k}$ be the number of points in the corresponding small box $B_{i,j,k}$. Then the local density of a box is

$${\displaystyle {\displaystyle {\rho }_{i,j,k}=\frac{n_{i,j,k}}{\mathrm{\#}{B}_{i,j,k}}=\frac{n_{i,j,k}}{\frac{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}{N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}\frac{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}{N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}}\frac{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}{N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}}}.}}$$ (2.1)

We group these densities into $L$ different levels (bins), and compute the entropy of the corresponding distribution. To do this, we first compute ${\rho }_{\max }$, the greatest density among all boxes. Then we subdivide the range from $0$ to ${\rho }_{\max }$ into $L$ equal intervals. Each density can then be categorized into its corresponding interval. A box $B_{i,j,k}$ is assigned to level 0 if and only if $n_{i,j,k}=0$, i.e. it contains no point.

Definition 2.1 (Entropy of a web’s silk density). Define the entropy of a web $W$’s silk density as

$${\displaystyle {\displaystyle {\displaystyle H(W;N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}},N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}},N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}},L)=-\sum _{1\le \ell \le L}{p}_{\ell }{\log }_2{p}_{\ell }}}}$$

where

$$p_{\ell }=\frac{\#\{(i,j,k):\mathrm{level}({\rho }_{i,j,k})=\ell \}}{N}$$

are the probability densities for each level $\ell \in \{1,2,\dots ,L\}$. Observe that empty boxes (level 0) do not enter into the entropy sum since they are not part of the web.

Note that our definition of entropy depends on the choice of the subdivisions $N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}},N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}},N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}$ and the number of levels $L$. However, we remark that in theory, the entropy converges as we take finer subdivisions.

Theorem 2.2 (Convergence of silk density entropy with respect to subdivision). Assuming there exists a measurable underlying field $\rho$ of silk density over the box, as the subdivisions become finer (i.e., $N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}},N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}},N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}\to \mathrm{\infty }$), the entropy $H(W;N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}},N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}},N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}},L)$ converges to a finite limit $H(W;L)\in {ℝ}_{\ge 0}.$

Proof. Since Shannon’s entropy is continuous with respect to the variational distance ($L^1$ distance), it is enough to show that the empirical distribution of silk density levels converges in $L^1$ to the true distribution of the density levels as the number of subdivisions $N=N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}\times N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}\times N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}$ goes to infinity. The average silk density ${\rho }_{i,j,k}$ of the $ijk$-th subdivided box $B_{i,j,k}$ can be written as an integral ${\rho }_{i,j,k}=\frac{1}{\delta B_{i,j,k}}{\int }_{B_{i,j,k}}\rho (\mathbf{x})\mathrm{d}\mathbf{x}$ (here $\delta B_{i,j,k}$ is a volume of $B_{i,j,k}$), and Lebesgue’s differentiation theorem implies that ${\rho }_{i,j,k}$ converges to $\rho (\mathbf{x})$ as $N\to \infty$. Since silk density functions (both empirical and true) are measurable, one can use Lebesgue’s differentiation theorem to show that ${\rho }_{i,j,k}$ converges to $\rho (\mathbf{x})$ as $\delta B_{i,j,k}\to 0$, i.e. $N\to \infty$. The level function is also measurable, hence the indicator function for each level of empirical density distribution converges to the indicator function of the true density distribution by dominated convergence theorem. By summing over all levels, we can conclude the $L^1$ convergence.∎

Figure 1 shows the convergence of entropy as the number of subdivisions grows, which empirically verifies the claim of theorem 2.2.

Figure 1.

Convergence of entropy with respect to number of subdivisions along each dimension for three different webs. The spider that constructed web 1 and web 3 is a randomly selected juvenile female M. dana The spider that constructed web 2 is a randomly selected male M. dana.

2.1.3. Choice of the number of quantization levels

For the following experiments, we choose $L=10$ for the number of levels. To justify our choice of density levels, we both provide a theoretical bound and show that, in practice, entropy values increase linearly if we use $L=100$ levels instead, and therefore, increasing the number of quantization levels from 10 to 100 does not yield additional information (figure 2). The theorem below demonstrates that raising the number of levels from $L$ to $mL$ increases the entropy by at most ${\log }_{2}m$. As a result, we find $L=10$ sufficiently fine to capture all relevant heterogeneity while remaining computationally efficient.

Figure 2.

Scatter of $H_{100}$ versus $H_{10}$ with fitted regression line ${\hat{H}}_{100}=1.4538H_{10}+2.4725$. The model explains approximately 95.9% of the variance ($R^2=0.9589$), with a slope highly significant ($t=32.76$, $p<{10}^{-32}$) and a residual s.e. of 0.09195 bits ($n=48$).

Theorem 2.3. Fix $N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}},N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}},N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}$ and let $H(W;L)=H(W,N_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}},N_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}},N_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}},L)$. For an integer $m\ge 1$, we have

$$H(W;mL)\le H(W;L)+{\log }_{2}m$$

Proof. Let $n_{\ell }$ (resp. $n_{\ell }^{\prime }$) be the number of samples (local densities) of level $\ell$ with maximum level $L$ (resp. $mL$). Then we have $n_{\ell }=n_{m\ell -(m-1)}^{\prime }+\cdots +n_{m\ell }^{\prime }$ (the bin corresponds to the level $\ell$ is partitioned into $m$-bins with new levels $m\ell -(m-1),m\ell -(m-2),\dots ,m\ell$). By Jensen’s inequality (applied to $f(x)=-x{\log }_{2}x$), we have

$$\begin{array}{rl}& -\frac{n_{\ell }}{mn}{\log }_2\frac{n_{\ell }}{mn}\ge \frac{1}{m}\sum _{0\le j<m}-\frac{n_{m\ell -j}^{\prime }}{n}{\log }_2\frac{n_{m\ell -j}^{\prime }}{n}\iff -\sum _{0\le j<m}\frac{n_{m\ell -j}^{\prime }}{n}{\log }_2\frac{n_{m\ell -j}^{\prime }}{n}\le -\frac{n_{\ell }}{n}{\log }_2\frac{n_{\ell }}{n}+\frac{n_{\ell }}{n}{\log }_{2}m.\end{array}$$

Then we get

$$\begin{array}{rl}H(W;mL)& =\sum _{1\le \ell \le mL}-\frac{n_{\ell }^{\prime }}{n}{\log }_2\frac{n_{\ell }^{\prime }}{n}=\sum _{1\le \ell \le L}\sum _{0\le j<m}-\frac{n_{m\ell -j}^{\prime }}{n}{\log }_2\frac{n_{m\ell -j}^{\prime }}{n}\\ & \le \sum _{1\le \ell \le L}[-\frac{n_{\ell }}{n}{\log }_2\frac{n_{\ell }}{n}+\frac{n_{\ell }}{n}{\log }_{2}m]=H(W;L)+{\log }_{2}m.\end{array}$$

∎

Figure 2 is a scatterplot showing the correlation between the distribution of entropy with $L=10$ and $L=100$ levels, which shows that the distributions are highly positively correlated.

2.1.4. Visualization of low- and high-entropy webs

Figures 3 and 4 present heat maps of the silk density distributions for two samples: one with a relatively low entropy value for the silk density distribution and the other with a relatively high entropy value.

Figure 3.

The figure on the left shows the silk density distribution for a web sample exhibiting a relatively low entropy value (0.5589 bits), calculated by using 10 quantization levels and excluding the empty level. The figure on the right shows the silk density distribution for a web sample exhibiting a relatively high entropy value (1.3300 bits), calculated by using 10 quantization levels and excluding the empty level.

Figure 4.

The figure on the left shows one representative frame of the web scan from the low entropy web in figure 3 (left). The figure on the right shows one representative frame of the web scan from the high entropy web in figure 3 (right).

In the low‐entropy web (figure 3, figure 4 left), silk density remains uniformly low across almost the entire structure. When subdividing the region 100 subdivisions on the length and width dimensions and 75 subdivision on the height dimension, the density level distribution is

$$[746096,3520,271,78,22,6,3,2,1,1],$$

where the first entry represents the number of blocks with 0 density, the next entry represents the number of blocks with the lowest positive density level, and so on. This distribution indicates that most of the space is empty, with the lowest density level remaining dominant.

By contrast, the high‐entropy web (figure 3, right) shows greater heterogeneity. When subdividing the region 100 subdivisions on the length and width dimensions and 75 subdivision on the height dimension, the density level distribution is

$$[737043,8924,2630,951,298,111,27,11,3,2].$$

This distribution indicates that most of the space is still empty, with the lowest density level remaining relatively dominant; however, the spider employed a substantially wider range of density levels throughout the structure.

It should be noted that, in theory, if the majority of the web has high silk density, the entropy would still remain low due to consistency. However, our observations indicate that as the spider invests more energy in adding silk to the web, the additional silk density is distributed in a relatively random manner. This randomness prevents the density levels from concentrating on specific values, thereby increasing the entropy as the spider enhances the web’s density. Additionally, while high-density silk on the boundary provides structural integrity, the variability in silk density elsewhere reflects the spider’s adaptive energy investment [27]. For example, varying silk densities can optimize the web for prey capture, as areas with lower density allow prey to escape more easily [1].

2.2. Mathematical framework for harmonicity

2.2.1. Motivation for harmonicity

If you know your neighbours, do you know yourself? An important aspect in spider web complexity is the spatial correlations in silk density. By examining how the density of silk in one region of the web relates to the density in neighbouring regions, we can infer patterns and interpolate the distribution of silk throughout the web. In particular, we want to infer a silk density of a subdivided box $B_{i,j,k}$ based on densities of its neighbouring subdivided boxes.

There are some nice properties that a discrete harmonic function (which will be defined in the following section) satisfy. For example, the value of a harmonic function at a point is determined as the average of the values of its neighbours. In particular, harmonic functions obey the maximum principle, which states that the highest values of the function appear at its boundary, an observed but typically unquantified trait of sheet webs [1,17].

2.2.2. Mathematical framework for harmonicity computation

Let $f$ be a function on a lattice ${\mathbf{Z}}^3$, the set of integer triples. We call $f$ harmonic if the value of $f$ at a point is equal to the average of $f$ over neighbouring points:

Definition 2.4 (Harmonic functions). A function $f:{\mathbf{Z}}^3\to \mathbf{R}$ is harmonic if $f(\mathbf{x})$ is equal to

$$\stackrel{\mathrm{~}}{f}(\mathbf{x}):=\frac{1}{6}\sum _{\mathbf{y}:\Vert \mathbf{y}-\mathbf{x}{\Vert }_1=1}f(\mathbf{y})$$

for any $\mathbf{x}\in {\mathbf{Z}}^3$.

Equivalently, $f$ is harmonic if and only if the discrete Laplacian ${\displaystyle \mathrm{\Delta }f}$ vanishes identically [29,30]:

Definition 2.5 (Laplacian and harmonicity). For a function $f:{\mathbf{Z}}^3\to \mathbf{R}$, the Laplacian of $f$ is a function ${\displaystyle \mathrm{\Delta }f:{\mathbf{Z}}^3\to \mathbf{R}}$ defined as

$${\displaystyle {\displaystyle \mathrm{\Delta }f(\mathbf{x})=\sum _{\mathbf{y}:\Vert \mathbf{y}-\mathbf{x}{\Vert }_1=1}(f(\mathbf{x})-f(\mathbf{y}))\mathrm{\forall }\mathbf{x}\in {\mathbf{Z}}^3.}}$$

Harmonicity of $f$ is defined as an $L^2$-norm of ${\displaystyle \mathrm{\Delta }f}$.

As we mentioned above, we are interested in the spatial correlations in silk density. We want to infer the silk density of a box based on densities of its neighbouring boxes. The simplest way to do this is by taking the average of neighbour densities, and this would work perfectly if and only if the silk density function $\rho$ is harmonic.

By our earlier discussion, ${\displaystyle \mathrm{\Delta }\rho }$ will be identically zero if and only if $\stackrel{\mathrm{~}}{\rho }=\rho$. However, there is no reason for $\rho$ to be a harmonic function a priori. Instead, we can use a generalized version of averages, and we consider power means as better alternatives. More precisely, we consider

$${\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{(\alpha )}:={\left(\frac{1}{6}\sum _{\Vert (i^{\prime },j^{\prime },k^{\prime })-(i,j,k){\Vert }_1=1}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\right)}^{1/\alpha }$$

and see if we get a better inference result for some $\alpha \ge 0$ other than $\alpha =1$, which corresponds to the usual mean. The $\alpha$-power mean would give a zero error if and only if ${\rho }^{\alpha }$ is harmonic. This lets us to define $\alpha$-harmonicity of silk density:

Definition 2.6 (α-harmonicity of silk density). Define the $\alpha$-harmonicity of a web $W$’s silk density $\rho$ as ${\displaystyle \Vert \mathrm{\Delta }({\rho }^{\alpha }){\Vert }_2}$, the harmonicity of ${\rho }^{\alpha }$.

We simply call 1-harmonicity as harmonicity. Like entropy, $\alpha$-harmonicity depends on the choice of subdivisions $N$ and the number of levels $L$. Under mild hypothesis, we can show that there always exists best $\alpha ={\alpha }_{\ast }\in [0,\infty )$ that minimizes the $\alpha$-harmonicity:

Theorem 2.7. (Existence of the best $\alpha$) If a web $W$ with silk density satisfies the inequality

$$\begin{array}{lcr}& \sum _{(i,j,k)}\mathrm{sign}({\rho }_{i,j,k}-{\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{\infty })\cdot {(\frac{1}{6}\sum _{(i^{\prime },j^{\prime },k^{\prime })}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\log {\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }-\log \left(\frac{1}{6}\sum _{(i^{\prime },j^{\prime },k^{\prime })}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\right))}^2>0,& \end{array}$$ (2.2)

there exists finite $\alpha ={\alpha }_{\ast }\ge 0$ where $\alpha$-harmonicity of silk density $\rho$ is minimized.

Before we give a proof, we explain the condition (2.2). We found that the error function eventually increases for all of our webs, and this is a typical case in the following sense. The term after sign inside the summation in (2.2) is non-negative by Jensen’s inequality. Also, $\mathrm{sign}({\rho }_{i,j,k}-{\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{\infty })=-1$ when ${\rho }_{i,j,k}<{\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{\infty }$, i.e. the silk density attains local maximum at $(i,j,k)$. In this case, all the neighbouring indices $(i^{\prime },j^{\prime },k^{\prime })$ with $\Vert (i,j,k)-(i^{\prime },j^{\prime },k^{\prime }){\Vert }_1=1$ satisfy ${\rho }_{i^{\prime },j^{\prime },k^{\prime }}\ge {\stackrel{\mathrm{~}}{\rho }}_{i^{\prime },j^{\prime },k^{\prime }}^{\infty }$ and the corresponding signs are plus. In other words, most of the signs in the summation are $+1$, and this explains why the condition (2.2) is met.

Proof. First, the $\alpha$-power average converges to the max function as $p\to \infty$:

$$\underset{\alpha \to \infty }{\lim }({\stackrel{\mathrm{~}}{\rho }}^{(\alpha )}{)}_{i,j,k}=({\stackrel{\mathrm{~}}{\rho }}^{(\infty )}{)}_{i,j,k}:=\underset{\Vert (i^{\prime },j^{\prime },k^{\prime })-(i,j,k){\Vert }_1=1}{\max }{\rho }_{i^{\prime },j^{\prime },k^{\prime }}.$$

The logarithmic derivative of ${\stackrel{\mathrm{~}}{\rho }}^{(\alpha )}$ is

$${\displaystyle {\displaystyle {\displaystyle \begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}\alpha }\log {\tilde{\rho }}^{(\alpha )}& =\frac{\mathrm{d}}{\mathrm{d}\alpha }\frac{1}{\alpha }\log \left(\frac{1}{6}\sum _{{\Vert \left(i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}\right)-(i,j,k)\Vert }_1=1}{\rho }_{i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}}^{\alpha }\right)\\ & =-\frac{1}{{\alpha }^2}\log \left(\frac{1}{6}\sum _{{\Vert \left(i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}\right)-(i,j,k)\Vert }_1=1}{\rho }_{i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}}^{\alpha }\right)+\frac{1}{\alpha }\frac{\sum _{{\Vert \left(i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}\right)-(i,j,k)\Vert }_1=1}{\rho }_{i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}}^{\alpha }\log {\rho }_{i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}}}{\sum _{{\Vert \left(i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}\right)-(i,j,k)\Vert }_1=1}{\rho }_{i^{\mathrm{\prime }},j^{\mathrm{\prime }},k^{\mathrm{\prime }}}^{\alpha }}\end{array}}}}$$

and by Jensen’s inequality applied to the convex function $f(x)=x\log x$ with $x={\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }$, we have

$$\begin{array}{c}\frac{1}{6}\sum _{(i^{\prime },j^{\prime },k^{\prime })}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\log {\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\ge \left(\frac{1}{6}\sum _{(i^{\prime },j^{\prime },k^{\prime })}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\right)\log \left(\frac{1}{6}\sum _{(i^{\prime },j^{\prime },k^{\prime })}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\right).\end{array}$$

Hence the function $\alpha \mapsto \log {\stackrel{\mathrm{~}}{\rho }}^{(\alpha )}$ is increasing, as is $\alpha \mapsto {\stackrel{\mathrm{~}}{\rho }}^{(\alpha )}$.

Since $\alpha \mapsto {\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{(\alpha )}$ is increasing, $\alpha \mapsto |{\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{(\alpha )}-{\rho }_{i,j,k}{|}^2$ has two possible behaviours: increases for large $\alpha$ when ${\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{\infty }<{\rho }_{i,j,k}$, or decreases for large $\alpha$ when ${\stackrel{\mathrm{~}}{\rho }}_{i,j,k}\le {\rho }_{i,j,k}$. The derivative of $L^2$-error $\Vert {\stackrel{\mathrm{~}}{\rho }}^{(\alpha )}-\rho {\Vert }_2$ is

$$\frac{\text{d}}{\text{d}\alpha }\Vert {\stackrel{\mathrm{~}}{\rho }}^{(\alpha )}-\rho {\Vert }_2^2=\frac{1}{{\alpha }^2}\sum _{(i,j,k)}\mathrm{sign}({\rho }_{i,j,k}-{\stackrel{\mathrm{~}}{\rho }}_{i,j,k}^{\infty })\cdot {(\frac{1}{6}\sum _{(i^{\prime },j^{\prime },k^{\prime })}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\log {\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }-\log \left(\frac{1}{6}\sum _{(i^{\prime },j^{\prime },k^{\prime })}{\rho }_{i^{\prime },j^{\prime },k^{\prime }}^{\alpha }\right))}^2$$

and the conclusion follows.∎

3. Methodology

3.1. Previous methodologies

In [23], scanned images were stacked to form a three-dimensional model of the web to understand the process of web construction. This process has been used to determine the purpose and importance of different stages in web construction, discern discrete stages of web construction and estimate the mechanical properties of silk under stress in orb and tangle web species [31]. Instead of capturing and stacking individual frames sequentially as in [23] which takes 2 hours, we generate the entire scan as continuous video in 30 seconds and reconstruct the three-dimensional volume in post-processing—drastically reducing acquisition time for each web scan.

Namazi used the fractal dimension and Shannon entropy to quantify the complexity and information content of orb weaving spiders to correlate cognitive function with web structure by interpreting the brightness level of digitized images [32]. However, web complexity was assessed using two-dimensional projections of inherently three-dimensional structures, and important volumetric features such as out-of-plane geometries may be lost.

3.2. Specimen collection and web construction

Local linyphiid species were used as test species for developing this new methodology and were hand-collected from the UC Berkeley campus (figure 5). Not all specimens built webs during the allotted time, and only fully completed webs were included in the analysis. Collected spiders were kept in small containers for one to two weeks without food prior to introduction to experimental space to encourage web development [33]. The total number of spiders collected are 23, 5 and 13 for N. digna, N. litigiosa and M. dana, respectively. Specimens were placed inside 10.16 × 10.16 × 10.16 cm (4″ × 4″ × 4″) acrylic boxes with clear sides, a black base and no lid (figure 6). Vaseline was coated on the inside top 2.54 cm (1″) of each side to prevent specimens from escaping. Box size was selected based the sizes of the species (all individuals measured less than 8 mm in length excluding legs) and scanning capabilities of the experimental set-up. Each spider was given at least two weeks in ambient conditions to complete a web, construction duration was not recorded in detail. Linyphiids construct webs over the course several days, completing full construction by seven days, and web completion was defined as the point at which no additional silk was added to the web after several consecutive days [33].

Figure 5.

A linyphiid web on UC Berkeley campus (unidentified species). The web was dusted with cornstarch prior to photographing (quarter included for scale).

Figure 6.

A representative image of a completed web in the experimental chamber. The web was dusted with cornstarch after scanning but prior to photographing for better visualization of the structure.

3.3. Web scanning

Completed webs were placed into an enclosed experimental scanning apparatus blocking all outside light. Scanning was conducted by a laser (Quarton Inc. VLM-520-56) and camera (Canon EOS R5C mirrorless camera using a Canon RF 100 mm F 2.8 Macro lens) mounted on a track at a fixed distance (0.67 metre). A small motor (Habow Technic Power-Functions XL-Motor-Set 8882) was used to advance the laser and camera simultaneously to scan the length of the experimental chamber from front to back, recording at 120 frames s⁻¹ for 20−30 s. An infrared laser distance sensor (DFROBOT DFRduin Uno v. 3.0[R3]) monitored the location of the camera relative to its start position to provide z-axis information for three-dimensional reconstruction.

3.4. Data processing and extraction

Individual video frames representing two-dimensional slices of the web were extracted for cleaning and analysis using a custom Python script [34]. For each video, a set cropped region was identified in each frame that excluded the box edges and background, and accounted for any minute changes in elevation across the length of the box. Red and blue colour values were reset to the minimum value and the image converted to greyscale. Background noise was removed using a minimum threshold filter of 0.55 max RGB value. Three-dimensional point cloud data (PCD) representations of each web were created using the filtered data. Examples of the three-dimensional PCD plots are shown in figure 7. All data are original scans collected as described above.

Figure 7.

Visual representations of the silk density plots for four sheet webs from the three experimental species. These visualizations are derived from the PCD representation of the spider webs. The density in each region is computed based on the fraction of points within that region, relative to the total number of points. Purple colours represent less dense (or more sparse) areas of silk, while red areas indicate higher silk density.

3.5. Quantification of silk density

We approximate the silk density in a region by counting the number of coordinate points in a subdivided box $B_{i,j,k}$ (§2.1.2) of the corresponding PCD document, divided by volume of $B_{i,j,k}$. The historical method to determine silk density would be to physically cut each web over a partition and then measure the silk mass for each subdivision [35]. However, this approach poses practical challenges and destroys the web in the process. Using PCD representations allows us to gather accurate data while maintaining the web for further scanning or experimentation.

3.6. Statistical analysis

The data for the entropy and harmonicity values for each web were analysed in R 4.3 (https://www.r-project.org/) using the dplyr, ggplot2, lme4 and car packages: H (entropy), global harmonicity and ${\alpha }_{\ast }$. Data exploration and analysis followed Zuur et al. [36]. Outliers were removed based on interquartile range (IQR) quartiles and the data were tested for normality using Shapiro–Wilk normality test ($p$-value = $0.441$ for the entropy and $p$-value = $0.421$ for harmonicity and ${\alpha }_{\ast }$) and Levene’s test for homogeneity of variance (for entropy: ${\displaystyle F_{(2,38)}=0.364,p=0.697}$; harmonicity: ${\displaystyle F_{(2,41)}=0.253,p=0.778}$). Heteroscedasticity was tested using Breusch–Pagan test (${\displaystyle \mathrm{B}\mathrm{P}=9.095,\mathrm{d}f=6,p=0.168}$). We fit independent general linear mixed models to describe each response variable (H, global harmonicity and ${\alpha }_{\ast }$) and assess the influence of species, genus, sex and age, as well as the random effects of the date. The influence of each factor was modelled both independently and additively to test all possible combinations; statistical significance of the effect of the factors (species, genus, sex, age and date) was not detected in any of the models. Comparisons between models were made using Akaike information criterion (AIC), where the best fit model was identified based on the lowest AIC values. To assess multicollinearity, mixed models with multiple variables were tested using the variance inflation factor (VIF), and all ${\mathrm{GVIF}}^{1/(2\cdot df)}$ values were less than 1.40 (species: $1.07$; genus: $1.10$; sex: $1.37$; age: $1.40$). Residual autocorrelation was checked with the Durbin–Watson test (${\displaystyle \mathrm{D}\mathrm{W}=2.43,p=0.168}$).

4. Entropy of the silk density results and discussion

In this section, we address the question: is the silk density of a web consistent across its structure? Our analysis reveals that the silk density exhibits a high degree of consistency, as indicated by the low entropy of the density level distribution—the average of the entropy H across the samples consistently converges to approximately 1.24 $\pm$ 0.22 bits (excluding zero-density-level) with respect to the number of samples for the spider species analysed in this study, when using $10$ quantization levels (table 1). Based on our convergence analysis (figure 1), which shows that the entropy reaches convergence once the number of subdivisions along the longest axes exceeds approximately 80, we therefore adopt 100 subdivisions (1 mm cell edge length) along length and width (and 75 along height) as a conservative yet efficient choice. To reduce sensitivity to the choice of subdivision size, we apply a moving average approach to the subdivision parameters. Specifically, the subdivisions are varied from 100 to 120 along the length and width dimensions, and from 75 to 100 along the height dimension. No significant effects of species ($F(2,38)=1.897,p=0.164$), genus ($F(\mathrm{1,39})=1.757,p=0.193$), date ($F(\mathrm{22,18})=1.111,p=0.415$), sex ($F(2,38)=0.756,p=0.476$) and age ($F(2,38)=2.600,p=0.087$) were identified. When the additive influence of species, date, sex and age was modelled, no significant factors were identified (best fit model—$F(6,36)=1.906,p=0.1083$). Further details on the specific sex and life stage for each specimen are included in the Dryad Repository [37]. One might wonder whether using more quantization levels than 10 could capture additional information about the silk density distribution. However, we computed entropy values up to 100 quantization levels, and the resulting distributions yielded a shape similar to that obtained with 10 levels. Increasing the number of density levels does not significantly alter the results under the current experimental conditions. The entropy value ($H$, excluding zero-density-level) is 73% lower than the theoretical maximum of ${\log }_{2}10\approx 3.32$, reflecting an uneven distribution of density levels.

Table 1.

Calculated entropy values by species, presented as mean $\pm$ s.d. [minimum value–maximum value]. For the computation of the entropy $H$ , we remove empty space (level 0). Excluding level 0 allows us to compare the density levels only where silk is present, and more accurately reflects the organization of the sheet web.

species	number of individuals	$H$ (in bits) and range of values
N. digna	23	1.18 $\pm$ 0.23 [0.40–1.65]
N. litigiosa	5	1.33 $\pm$ 0.23 [1.26–1.65]
M. dana	13	1.31 $\pm$ 0.19 [0.55–1.88]

The majority of the silk density falls within the lowest quantization level, suggesting an energy-saving behaviour by the spiders. This low entropy may reflect a bias toward minimal silk usage, possibly prioritising efficiency of resource use [19,20,38]. The production of silk and the construction of a web are energy-intensive processes and are directly influenced by the physiological state of the spider and a number of ecological factors [6]. Among linyphiid species, silk density can range from quite sparse, as seen in Laminacauda ansoni to very dense in Pocobletus versicolor, where distinguishing individual thread lines is difficult [17]. Our results indicate that the sheet web spiders tested may optimize the amount of silk needed for effective prey capture while conserving energy; however, we cannot make decisive conclusions due to the limitations of the current study. The low silk density may be a factor of starvation, but this is unlikely, as starved spiders typically invest larger amounts of capture silk compared with sated individuals [6].

Alternatively, a higher entropy value would imply a more uniform use of different density levels, which could indicate that the spider may prioritize different factors of web performance across neighbouring web sections related to varying functional purposes. For example, a dense region necessary for structural support adjacent to a sparse region of sticky silk for prey capture. This was not observed in the species used in this paper; however, the restrictions of the study may contribute to the observed consistency, as all specimens were kept in identical controlled conditions and had a relatively low sample size across all species, sexes and life stages. Integrating greater numbers, species and habitat diversity into future studies will either support low entropy as a universal feature of sheet webs or reject this conclusion and illuminate environmental and behavioural traits linked with differing entropy levels.

5. Harmonicity results and discussion

In this section, we address the question: can we infer silk density of a web by neighbours? Using the mathematical framework described in §2.2, we calculated the global harmonicity of silk density (${\displaystyle \Vert \mathrm{\Delta }{\Vert }_2}$). No significant effects of species ($F(2,38)=0.892,p=0.419$), genus ($F(\mathrm{1,39})=1.757,p=0.193$), date ($F(\mathrm{22,18})=1.772,p=0.111$), sex ($F(2,38)=0.6976,p=0.504$) and age ($F(2,38)=0.035,p=0.965$) were identified. When the additive influence of species, date, sex and age was modelled, no significant factors were identified (best fit model—$F(6,34)=0.576,p=0.746$). For all webs, the average harmonicity value was 0.0039 $\pm$ 0.0017 (fraction of total points in PCD, table 2). The harmonic behaviour displayed in the silk density indicates that the silk is lain at similar densities across the surface of the sheet web, even when the specific shape of the web varies.

Table 2.

Harmonicity and ${\alpha }_{\ast }$ presented as mean $\pm$ s.d. [minimum value–maximum value] for web scans separated by species. We use 20 subdivisions on the length and width dimensions and 15 subdivision on the height dimension. No significant differences were found between the species studied.

species	number of individuals	$\Vert \mathrm{\Delta }{\Vert }_2$ and range of values	${\alpha }_{\ast }$ and range of values
N. digna	23	0.00360 $\pm$ 0.0017 [0.0007–0.0099]	0.951 $\pm$ 0.061 [0.81–1.02]
N. litigiosa	5	0.00496 $\pm$ 0.0016 [0.0024–0.0067]	0.942 $\pm$ 0.036 [0.92–0.97]
M. dana	13	0.00398 $\pm$ 0.0018 [0.0018–0.0076]	0.938 $\pm$ 0.053 [0.85–1.00]

Also, we measure the $\alpha$-harmonicity of each web and found the best $\alpha ={\alpha }_{\ast }$ that minimizes the error, as mentioned at the end of §2.2 (see also theorem 2.7). We compute the harmonicity of ${\rho }^{\alpha }$ by varying the $\alpha$ value in increments of 0.01 and find the $\alpha ={\alpha }_{\ast }$ that minimizes ${\displaystyle \Vert \mathrm{\Delta }({\rho }^{\alpha }){\Vert }_2}$.

We found that the average ${\alpha }_{\ast }$ is identically consistent to (${\displaystyle \Vert \mathrm{\Delta }{\Vert }_2}$) over species, genus, sex and age, at $0.95$ $\pm$ 0.06 (table 2). In other words, $(\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{k}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}{)}^{0.95}$ is approximately harmonic. ${\alpha }_{\ast }$ for the three linyphiid species.

6. Conclusions

In three species of linyphiid spiders, completed sheet webs exhibit traits consistent with organization, optimization and harmonicity, despite appearing random to the human eye. While the specifics of the organization and design are not yet known, the low entropy values provide evidence that the ‘unordered meshwork’ is rather a deliberate network of silk fibre [7]. The limitations of the current study restrict our ability to draw broad biological and ecological conclusions, as only three, closely related species of linyphiid were tested in a controlled space. Future studies integrating a wide diversity of species across varying environments will allow us to investigate how web structure relates to lifestyle, ecological conditions and evolutionary history.

Our non-destructive approach allows us to draw quantifiable data from web structures. Moving forward, we can use these imaging and quantification methods to investigate web design in a wide variety of spider species across different environmental conditions and stressors. While only completed webs were scanned in this study, this non-destructive approach will allow us to obtain three-dimensional scans of webs across every stage of construction, providing further insight into the step-by-step process used to create a viable model of the sheet web. The three-dimensional web scans can be analysed in more detail to discern additional physical and geometric properties, and may be used to inform computational models of sheet webs.

Most importantly, combining microscopic and macroscopic analysis of three-dimensional web scans based on computed entropy and harmonicity values can bridge small-scale structural patterns with large-scale functional design that might reveal how spiders optimize their webs. From a microscopic perspective, metrics such as clustering coefficients, degree distributions and modularity can be computed to analyse the graph structure of sheet webs. Viewing sheet webs as complex networks can provide deeper insights towards the specifics of thread organization and how this contributes towards the functionality of the web as both a shelter and a prey trap [39]. From a macroscopic perspective, efforts are already underway to derive equations that can accurately model the surfaces of the webs we have already scanned. With these equations, we will investigate whether sheet webs exhibit characteristics of minimal surfaces. We can also simulate these webs with such equations to test the viability of using webs as a model for applications in electronics and material science. With a sufficiently large dataset of spider web scans, machine learning models can be trained to generate web-like structures with similar properties.

Understanding the physical properties of sheet webs may also provide inspiration for engineering and architectural designs that minimize material usage while maintaining structural function. The ability to directly measure, experiment and manipulate webs during construction will open new doors into the study of spider behaviour and ecology, and allow us to quantify what makes an effective web. Identifying the unseen patterns of nature is a powerful tool, and the non-destructive scanning technique and quantification of entropy and harmonicity described here have the potential to illuminate many undiscovered nuances in the structure and design of spider webs.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

All data can be accessed here [37].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

F.L.: conceptualization, data curation, formal analysis, investigation, methodology, resources, software, supervision, validation, visualization, writing—original draft, writing—review and editing; K.M.N.: data curation, formal analysis, investigation, methodology, resources, software, validation, visualization, writing—original draft, writing—review and editing; S.L: conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization, writing—original draft, writing—review and editing; J.J.: data curation, formal analysis, investigation, methodology, resources, software, validation, visualization, writing—original draft, writing—review and editing; G.Y.: data curation, formal analysis, investigation, methodology, resources, software, validation, visualization, writing—original draft; P.C.: data curation, investigation, software; S.C.L.: data curation, investigation, software; N.S.: conceptualization, data curation, formal analysis, investigation, methodology, project administration, resources, supervision, validation, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

No funding has been received for this article.