Linking as voting: how the Condorcet jury theorem in political science is relevant to webometrics

Abstract

A webmaster’s decision to link to a webpage can be interpreted as a “vote” for that webpage. But how far does the parallel between linking and voting extend? In this paper, we prove several “linking theorems” showing that link-based ranking tracks importance on the web in the limit as the number of webpages grows, given independence and minimal linking competence. The theorems are similar in spirit to the voting, or jury, theorem famously attributed to the 18th century mathematician Nicolas de Condorcet. We argue that the linking theorems provide a fundamental epistemological justification for link-based ranking on the web, analogous to the justification that Condorcet’s theorems bestow on majority voting as a basic democratic procedure. The analogy extends to the practical limitations facing both kinds of result, in particular due to limited voting/linking independence. However, we argue, referring to the theoretical developments inspired by the jury theorem, that some of the pessimism expressed in the webometrics literature regarding the possibility of a “theory of linking” may be unjustified. The present study connects the two academic disciplines of webometrics in information science and epistemic democracy in political science by showing how they share a common structure. As such, it opens up new possibilities for theoretical cross-fertilization and interdisciplinary transference of concepts and results. In particular, we show how the relatively young field of webometrics can benefit from the extensive and sophisticated literature on the Condorcet jury theorem.

Appendix: Proofs

All of the following proofs assume that the distribution of importance values is given by some density $\rho \left( {pi} \right):\left[ {0,1} \right] \mapsto \left[ {0,\infty } \right]$, such that:

$$ P\left( {pi \in \left[ {a,b} \right]} \right)\text{ := }\mathop \int \limits_{a}^{b} \rho \left( {pi} \right)dpi. $$

Proof of Theorem 1

It is assumed in the basic model that linking probability is determined solely by the importance of the target page. Hence, the web ecology is such that the probability to link to the ith page from any of the other $n - 1$ pages of the web-graph is a simple function of the importance of the ith page.

$$ \forall j \ne i\left[ {P\left( {j \to {\text{i}}} \right) = {\mathcal{F}}\left( {pi_{i} } \right)} \right], $$

for ${\mathcal{F}}\left( {pi_{i} } \right): \left[ {0,1} \right] \mapsto \left[ {0,1} \right].$ The precise form, and parameters, of ${\mathcal{F}}\left( {pi_{i} } \right)$ do not matter for the proof.

The assignment of up to $n - 1$ links to the ith page, each with the same (independent) probability for assignment, satisfies the conditions of a Bernoulli trial. Thus the probability distribution over the number of links to the ith page is the binomial distribution with mean—the expected number of in-links to the ith page $\overline{N}_{i}$—given by

$$ \overline{N}_{i} = \left( {n - 1} \right){\mathcal{F}}\left( {pi_{i} } \right). $$

The In-Degree of the ith webpage ID _i is herein defined as the ratio of incoming links to maximum possible number of incoming links where each page may link to another only once and no page can link to itself:

$$ ID_{i } : = \frac{{N_{i} }}{n - 1}. $$

As the assignment of up to $n - 1$ links to the ith page, each with the same probability for assignment, satisfies the conditions of a Bernoulli trial, so the law of large numbers applies. Thus, with probability 1, $\frac{{N_{i} }}{n - 1} \to \frac{{\overline{N}_{i} }}{n - 1}$ as $n \to \infty$. As $ID_{i } = \frac{{N_{i} }}{n - 1}$ and ${\mathcal{F}}\left( {pi_{i} } \right) = \frac{{\overline{N}_{i} }}{n - 1}$, so we have that, with probability 1, $ID_{i } \to {\mathcal{F}}\left( {pi_{i} } \right)$ as $n \to \infty$.

As the above holds for each webpage in the web-graph, so the probability is 1 that page In-Degree converges to a function of page importance as graph size goes to infinity: which is to say that the probability is 1 that In-Degree and page importance go toward perfectly correlation as the number of pages to the web-graph goes to infinity. That is, the probability is 1 that $R^{2} \to 1$ with ${\mathcal{F}}\left( {pi_{i} } \right)$ as the regression function as $n \to \infty$.□

Proof of Theorem 2

As shown in Fortunato et al. (2008), the average PageRank $PR_{i}$ of a node $i$ with $N_{i}$ in-links is

$$ PR_{i} = \frac{q}{n} + \frac{1 - q}{n}\frac{{N_{i} }}{{\overline{N}_{i} }} $$

where $q$ is the probability for the random surfer to jump to a new random page rather than follow a link. This equation holds given that the probability of a source page to link to a target page is independent of the number of links from, or to, the source page itself, which is easily checked to hold given the assumption of the basic model that linking probability is determined solely by the importance of the target page. Substituting $(n - 1) ID_{i}$ for $N_{i}$, we get

$$ PR_{i} = \frac{q}{n} + \frac{1 - q}{n}\frac{{\left( {n - 1} \right) ID_{i} }}{{\overline{{\left( {n - 1} \right) ID}}_{i} }} = \frac{q}{n} + \frac{1 - q}{n}\frac{{\left( {n - 1} \right) ID_{i} }}{{\left( {n - 1} \right)\overline{{ID_{i} }}_{i} }} = \frac{q}{n} + \frac{1 - q}{n}\frac{{ ID_{i} }}{{\overline{{ID_{i} }} }} $$

As $n$ increases, this is ever better approximated by:

$$ PR_{i} = \frac{1 - q}{n}\frac{{ ID_{i} }}{{\overline{{ID_{i} }} }} = c ID_{i} $$

for a constant $c$. Thus $PR_{i}$ is linearly proportional to $ID_{i}$ in the limit of large $n$. This explains why their $R^{2}$ values converge in larger web-graphs and so explains why PageRank goes toward perfect correlation with page importance as web-graph size increases in web ecologies where In-Degree does the same and the linking probability is independent of the number of links to, or from, the source page.□

Proof of Theorem 3

By the assumptions of the extended model, the linking probability depends on the importance of the source page and the importance of the target page. Hence, the probability of page $j$ linking to page $i$ is a function ${\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right): \left[ {0,1} \right] \times \left[ {0,1} \right] \mapsto \left[ {0,1} \right]$ of the importance of both these pages:

$$ \forall j \ne i\left[ {P\left( {j \to {\text{i}}} \right) = {\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)} \right], $$

The precise form, and parameters, of ${\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)$ do not matter for the proof.

The (independent) assignment of up to $n - 1$ links to the ith page, each with varying probability for assignment, satisfies the conditions of a Poisson trial. The mean of the Poisson distribution—the expected number of in-links to the ith page $\overline{N}_{i}$—is given by:

$$ \overline{N}_{i} = \mathop \sum \limits_{j = 1}^{n - 1} {\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right). $$

As the independent assignment of up to $n - 1$ links to the ith page, each with varying probability for assignment, satisfies the conditions of a Poisson trial, so the law of large numbers applies. Thus, with probability 1, $\frac{{N_{i} }}{n - 1} \to \frac{{\overline{N}_{i} }}{n - 1}$ as $n \to \infty$. As $ID_{i } = \frac{{N_{i} }}{n - 1}$ and $\overline{N}_{i} = \mathop \sum \limits_{j = 1}^{n - 1} {\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)$, so we have that, with probability 1, $ID_{i } \to \frac{{\mathop \sum \nolimits_{j = 1}^{n - 1} {\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)}}{n - 1}$ as $n \to \infty$.

In words, we have that the probability is 1 that the In-Degree of the ith page converges to the average value of ${\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right) {\text{for a given }}pi_{i}$ as the number of pages goes to infinity. But in that same limit, again by the law of large numbers, the probability is 1 that the average value of ${\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right) {\text{for a given }}pi_{i}$ converges to the expected value of that function relative to how the $pi_{j}$ are distributed. The expected value for ${\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)$ relative to how the $pi_{j}$ are distributed, for a given value of $pi_{i}$, is given by:

$$ \overline{{{\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)}} = \mathop \int \limits_{0}^{1} {\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)\rho \left( {pi_{j} } \right)dpi_{j} = k\left( {pi_{i} } \right), $$

where $\rho \left( {pi_{j} } \right)$ is the distribution of page importance assumed in the model and $k\left( {pi_{i} } \right)$ is whatever function that results from carrying out the integral.

Thus as both

$$ ID_{i} \to \frac{{\mathop \sum \nolimits_{j = 1}^{n - 1} {\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)}}{n - 1} $$

and

$$ \frac{{\mathop \sum \nolimits_{j = 1}^{n - 1} {\mathcal{F}}\left( {pi_{i} ,pi_{j} } \right)}}{n - 1} \to k\left( {pi_{i} } \right) $$

as $n \to \infty$ with probability 1, so it trivially follows that

$$ ID_{i} \to k\left( {pi_{i} } \right) $$

as $n \to \infty$ with probability 1.As the above holds for each webpage in the web-graph, so the probability is 1 that page In-Degree converges to a function of page importance as graph size goes to infinity: which is to say that the probability is 1 that In-Degree and page importance go toward perfectly correlation as the number of pages to the web-graph goes to infinity. That is, the probability is 1 that $R^{2} \to 1$ with $k\left( {pi_{i} } \right)$ as the regression function as $n \to \infty$.□