Claudiu Musat

Julien Velcin

Stefan Trausan-Matu

Marian-Andrei Rizoiu

Improving Topic Evaluation Using Conceptual Knowledge

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Introduction

Topic models have recently retained a lot of attention in dealing with textual corpora. In brief, topics are multinomial distributions over words or key phrases which aim at capturing the meaning of huge volume of textual data in an unsupervised way. These mathematical models based on probabilistic Bayesian networks have been designed to address various issues, such as: multi-topics allocation [START_REF] Blei | [END_REF], super and sub-topic hierarchies [Blei et al., 2004], temporal evolution of topics [START_REF] Wang | Topics over time: a non-Markov continuoustime model of topical trends[END_REF], etc. Plenty of applications can take great benefits from topic models, including information retrieval, database summarization or ontology learning.

However, the comparison of the proposed models remains rather difficult, especially when considering models built on different theoretical basis. A lot of effort is currently put into evaluating topic models. Recent work has proved that using only numerical measures cannot alone solve this issue; human judgment can be of great benefit in the task of topic evaluation [Chang et al., 2009a]. Very recent work [Newman et al., 2010] uses external resources, especially the Web, as an alternative evaluation measure but deems working with ontologies unfeasible. To our knowledge, no previous work has successfully used ontologies to evaluate the meaning of topic models.

In this paper, our main contribution is to automate the topic model evaluation using an external concept hierarchy -here, WordNet [Miller, 1995] To tackle this problem, we propose the notion of conceptual relevance. The idea behind it is to find the most related concepts to each topic given the concept hierarchy and to evaluate the topic based on these concepts and the strength of the (topic, concept) relations. This semantic approach is very different from that of Newman [2010] which is largely based on statistics.

We prove the correlation between our semantic measure and the human judgment given by 37 external judges. The experiments were made on two different datasets: the first Suall [START_REF] Wang | Topics over time: a non-Markov continuoustime model of topical trends[END_REF]] is a general dataset on American history and the second is a specific dataset containing exclusively economic articles. Although the evaluation correlation shows a similar pattern for both corpora, some quantitative differences exist. Relevant evaluation differences have emerged when confronting the human evaluators with different types of topic word mixes that need to be separated. The influence of these and other external factors on the overall system accuracy is also discussed.

Topic models and their previously used evaluation methods are outlined further in the introduction, the proposed system is detailed in section 2, the experiments and their results are presented in section 3 and our conclusions and future work follow in section 4.

Topic Modeling

Topic models are graphical hierarchical Bayesian networks used to extract the latent meaning of textual datasets. Latent Dirichlet Allocation (LDA) is the prototypic model [START_REF] Blei | [END_REF] following the work of Hofmann [1999]. The main idea of probabilistic models lies in the assumption that the observed texts are derived from a generative model. In such a model, there are unseen latent variables (the topics) from which words and documents are generated. The latent variables are represented as random variables over the set of words or n-grams. Thereby these models attempt to estimate

Improving Topic Evaluation Using Conceptual Knowledge

Claudiu Cristian Musat 1 , Julien Velcin 2 , Stefan Trausan-Matu 1 and Marian-Andrei Rizoiu 2

1 Computer Science Department "Politehnica" University of Bucharest, Romania 2 ERIC Laboratoire Université Lumière, Lyon 2, France {Claudiu.Musat, Trausan}@cs.pub.ro , {Julien.Velcin, Marian-Andrei.Rizoiu}@univ-lyon2.fr the probability distributions of the latent variables by using maximum likelihood or Bayesian inference. On top of that basic approach, other generative models have been developed in order to address topics extraction in complex cases, such as for correlated topics [Blei and Lafferty, 2007], ngram handling [START_REF] Wang | [END_REF], social networks [Chang et al., 2009b], opinion mining [Mei et al., 2007b],etc.

The English language lexical database WordNet [Miller, 1995] has a long tradition of being used in text classification tasks [START_REF] Scott | Sam Scott and Stan Matwin. Text Classification using WordNet Hypernyms[END_REF]]. Also, the idea to mix topic models and ontologies is not new. LDAWN, latent Dirichlet allocation with WordNet [Boyd-Graber et al., 2007] is a version of LDA that uses the word sense as a hidden variable and becomes a system for word sense disambiguation.

Topic Evaluation

Topic models have been proven to be accurate both quantitatively and qualitatively. Quantitatively, it has been shown [START_REF] Wallach | [END_REF], using the perplexity measure, that they possess a high generalization ability on unseen data. This method allows for the estimation of the log-likelihood either on a fraction of the train set (said ``held-out'' data) in a cross-validation way, or on new documents.

Qualitatively, a sample of topics is usually exhibited in order to convince the reader of their usefulness. Each exhibited topic ݖ is a short list of the first terms ,ݓ from a decreasing probability perspective, associated (sometimes) with their probability values ሺݓȁݖሻ, and most of the times with a name given manually by the authors. Here is, for instance, an extract of a topic description illustrating the output of the LDA algorithm [START_REF] Blei | [END_REF]: ``Arts'' (new, film, show, music, movie, play, musical, best, actor, first, york, opera, theater, actress, love).

Furthermore, it has been shown [Chang et al., 2009] that human judgment does not coincide with the common automatic evaluation measures. We believe this to be the greatest shortfall of the above measures. This finding has prompted other researchers to look for novel evaluation systems such as that proposed in [Newman et al., 2010]. The latter relies on external resources for the evaluation task and that postulates topic coherence to be at the core of the idea of evaluation. Results obtained by employing Wikipedia or Google were satisfying, while those extracted with WordNet were deemed patchy at best.

The evaluation task is conceptually similar to that of labeling. A "good" label implies an understandable underlying meaning. [Mei et al., 2007a] suggested multiple solutions which fell short of being able to generalize concepts through a hypernymy relation, which is usually appropriate, while [Andrzejewski et al., 2009] uses domain knowledge through "must link" or "cannot link" relations in a semi automated model. The latter introduces an important concept -that some words can wrongfully be inserted into a topic. This we believe leads to our conceptual disagreement with Newman's [2010] conclusion on the use of WordNet in topic evaluation. Although some topical words can be outliers, the quality of the topic can remain elevated and the topic itself humanly comprehensible.

Proposed System

The system we proposed is designed to rank topics according to their relevance to the user, which we defined as a function of their cohesion -how many of the topic words are related and in what degree -and specificity -how general their common hypernym is.

Given a text collection T and V the employed vocabulary we aim to evaluate various probability distribution sets over T with respect to knowledge regarding related concepts. We will use z to denote a discrete probability distribution function ሼሺݓȁሻሽ ௪א over T, which we will further refer to as a topic. Each topic is a member of Ĭ ൌ ሼ ଵ ǡ ǥ ǡ ୩ ሽ a finite set of k topics extracted using one of the known algorithms given T, with Ĭ ؿ ܼ , where ܼ represents the set of all possible topics over T. Furthermore, Ĭ א ܲሺܼሻ, the space of all possible probability distribution sets over T.

Concept Representation

As previously said, prior knowledge about related concepts is considered. Let ܱ ൌ ሺǡ जሻǡ be the pair of the set of relevant concepts , and a set of relevant relations over , further referred to as ज. We address the particular case where there exists a relation ݎ א ज, according to which all concepts in a subset ߜሺࣝሻ ؿ form a tree ࣝ ൌ ሺߜሺࣝሻǡ ݎሻ in which the ߜሺࣝሻ elements form the tree nodes and ݎ is the relation between them. ࣝ is of course a member of all possible trees over ߜሺࣝሻ, given all possible relations ज, a space which we will refer to as ડ. The ݎ relation can be exemplified as a either hypernymy or hyponymy relations between concepts such as those present in WordNet.

Employed Distances

Within ࣝ, we define a branch as a path between two concepts ܿ and ܿ that are either directly or indirectly related, as the pairs ሺ ଵ ǡ ሻ or ሺ ଷ ǡ ሻ in Fig. 1. In the case of a hypernymy or hyponymy relation, the fact that two concepts are related also implies that one is an ancestor (or a generalization) of the other. To determine the distance between two concepts with respect to ࣝ, ݀൫ܿ ǡ ܿ ൯, we use a slightly modified version of the ancestral path distance with the result being infinity if one is not a direct ancestor of the other. Let ߜሺܿ ǡ ܿ ሻ be the set containing all the nodes within the branch that connects ܿ and ܿ .

We further assume that there exists a subset of words in the vocabulary ࣝ ؿ where, for each word in ࣝ there exists at least one concept in ࣝ that is a sense of the given word. Let ߜሺݓሻ ؿ ߜሺࣝሻ be the set of all senses of the word ݓ within ࣝ. We define the distance between a word and a concept, ǡݓሺݐݏ݅݀ ܿሻ, as the smallest number of transitions between any concept in ߜሺݓሻ and the target concept ܿ: ݀ሺݓǡ ܿሻ ൌ ୡ ౭ אఋሺ௪ሻ ൫݀ሺ ୵ ǡ ܿሻ൯ א ࣝǡ ݓ א ࣝ Let ܿ ሺࣝሻ א ߜሺࣝሻ be the root of the ࣝ tree. For instance, in Fig. 1 the distance between ݓ ଵ and the subtree root ܿ ሺࣝሻ is 2, because the distances between its senses and ܿ ሺࣝሻ are . It is note-

worthy that ࣝ ୵ǡୡ is a generalization of ࣝ ௪ . ࣝ ୵ భ ǡୡ భ is in Fig 1. a reunion of the ሺܿ ଵ ǡ ܿ ଷ ሻ and ሺܿ ଷ ǡ ܿ ሻ arcs.
Starting from a topic's most important words, we construct a substructure of the concept representation -a topical subtree that comprises all the concepts related to the selected topic words. The idea of using the topic's most important words is conceptually similar to the method employed in [Chang et al., 2009a] where those words alone are sufficient to lead to a satisfying assessment of a topic's quality. For a given topic model Ĭ ൌ ሼ ଵ ǡ ǥ ǡ ୩ ሽ we define a term importance function ߬ǣ Ĭ ƍ ՜ Թ ା Ǣ ୧ ǡ ݓ ߬ሺ ୧ ǡ ݓሻǡ  ୧ א Ĭǡ ݓ א ƍ. Thus, for each ୧ א Ĭ we extract a set of relevant words ȇሺ ୧ ሻ as the words that have ߬ሺ ୧ ǡ ݓሻ ߙ, with ߙ א Թ ା previously determined.

Let a topic's relevant concepts ߜሺ ୧ ሻ within ߜሺࣝሻ be the reunion of the ࣝ senses of the topic's relevant words:

ߜሺ ୧ ሻ ൌ ڂ ߜሺሻ ǡ ௪אȇሺ ሻ
 ୧ א Ĭ Definition 4. A topical subtree within ࣝ is the reunion of all the ࣝ subtrees of all the topic's relevant words:

ࣝ ൌ ሺߜሺ ୧ ǡ ࣝሻǡ ݎሻǢ ߜሺ ୧ ǡ ࣝሻ ൌ ራ ራ ߜሺܿ ௪ ǡ ܿ ሺࣝሻሻ ೢ אఋሺ୵ሻ ǡ ௪אȇሺ ሻ  ୧ א Ĭ In Fig 1.
ࣝ is outlined with a bold line (dotted or not). Definition 5. A topical subtree of a concept ܿ within ࣝ is the reunion of all the ࣝ ୵ǡୡ subtrees of all topical words:

ࣝ ǡୡ ൌ ሺߜሺ ୧ ǡ ࣝ ሻǡ ݎሻǢ ߜሺ ୧ ǡ ࣝ ሻ ൌ ራ ራ ߜሺܿ ௪ ǡ ܿሻ ೢ אఋሺ୵ሻ ǡ ௪אȇሺ ሻ  ୧ א Ĭǡ

Concept Metrics

We aim to identify the topical subtrees that include at least one sense for as many of the topic's words as possible while at the same time having a root concept as specific as possible. We are thus trapped in the age old tradeoff between coverage and specificity.

We define a subtrees' coverage ǣݒܿ

ߜሺ ୧ ǡ ࣝሻ ൈ Ĭ ՜ Գ ା Ǣ ܿ ܿݒሺܿǡ ୧ ሻ ൌ ௗሼځ ఋሺ ሻǡఋሺ ǡࣝ ሻሽ ௗሼఋሺ ሻሽ
. In order to determine a concept's specificity we rely on two additional featuresits height ݄ǣ ߜሺࣝሻ ՜ Գ ା Ǣ ܿ ݄ሺܿሻ ൌ ݀ሺܿǡ ܿ ሺࣝሻሻ, and depth ߩ with respect to the given topic:

ߩǣ ߜሺ ୧ ǡ ࣝሻǡ Ĭ ՜ Գ ା Ǣ ሺܿǡ ୧ ሻ ߩሺܿǡ ୧ ሻ ൌ ൏ ݂ ௪אఋሺ ሻ ൫݀ሺݓǡ ܿሻȁܿ א ߜሺ ୧ ሻ൯ǡ
 ୧ א Ĭǡ ܿ א ߜሺ ୧ ǡ ࣝሻǡ where ൏ ݂ is a member of a given family of functions such as the minimum, maximum or average that obtain a single scalar value for an input vec-

tor: ൏ ݂ ǣ Թ ା ୩ ՜ Թ ା Ǣ ݔ ଵ Ǥ Ǥ ݔ ൏ ݂ ൭ ݔ ଵ Ǥ Ǥ ݔ ൱ ൌ ݕ א Թ ା .
There are more than one possible definitions for the specificity, one of which is a weighted average of the concept's height and depth: ݁ܿǣݏ ࣝ ൈ Ĭ ՜ Թ; ܿǡ ୧ ܿ݁ݏሺܿǡ ୧ ሻ ൌ ߱ ڄ ݄ሺܿሻ ߱ ఘ ڄ ߩሺܿǡ ୧ ሻ, with the weights set a priori.

Topic Evaluation

We define the evaluation of the whole topic model as an aggregate evaluation function: for each distribution set ȣǡ ߟǣ ܲሺܼሻ ՜ Թ ା Ǣ ȣ ߟሺȣሻ ൌ൏ ݂ ൫ߟሺሻ൯ǡ  א ȣ.

The evaluation of the model as a whole depends on the individual evaluation of each of its topics ߟǣ ȣ ՜ Թ ା ǡ ୧ ߟሺ ୧ ሻ ൌ ൏ ݂ ሺ߶ሺࣝ ሻǡ  ୧ א ȣ, in which ࣝ is the topical subtree of ୧ given ࣝ. Each node from ࣝ is assigned a positive fitness value based on its relevance to the given topic ୧ , ߶ǣ ࣝ ൈ ȣ ՜ Թ ା . We express this relevance as a weighted average of its coverage and specificity, with ߱ ௩ and ߱ ௦ set a priori, ܿ ߶ሺܿሻ ൌ ߱ ௩ ȉ ܿݒሺǡ ୧ ሻ ߱ ௦ ȉ ܿ݁ݏሺǡ ୧ ሻ The higher the coverage of the concept with the highest fitness value, the higher the topic's cohesion given that concept. Furthermore, more specific concepts are attached to more specific topics. The total relevance of an individual topic is thus a function of its cohesion and specificity.

Experiments

Three types of experiments were devised, in order to capture the correlation between the human verdict and the calculated topic relevance. The LDA [START_REF] Blei | [END_REF] model built into the Mallet suite [START_REF] Mccallum | [END_REF] was used to generate the topics and all automatically determined relevancies were calculated using the framework above while human evaluations similar to those employed by [Chang et al., 2009a] were gathered using a binary question answering system. Evaluators were asked to extract the unrelated words from a group containing one topic and an additional spurious word. One or more unrelated words were chosen for each group.

In the first experiment, the analyzed topics were separated into relevant or not from an algorithmic point of view and the aim is to see whether an improvement of the spurious word detection is visible from one category to the other. The second experiment shows how the improvement gains and the confidence in the experiment vary when modifying k (the number of topics in the model). These variations are shown for three different metrics -the chance to hit the spurious word, the same limited to the evaluator's first choice word and the total number of chosen words. The third experiment regarded the correlation between evaluator agreement, accuracy and ontological topic relevance.

Two corpora were used to find whether results would differ greatly if the focus is changed from a general purpose corpus such as the Suall [START_REF] Wang | Topics over time: a non-Markov continuoustime model of topical trends[END_REF] to a more specific one, in our case an economic corpus. We built the second corpus from publicly available Associated Press articles published in the Yahoo! Finance section. A total of 23986 news broadcasts which had originally appeared between July and October 2010 were gathered.

Spurious Word Types

As previously said, human evaluation of a topic depends on the chance that the evaluator correctly detects a spurious word that is mixed with the topic's words. The choice of the spurious word is not obvious as it influences the outcome of the experiment. While [Chang et al. 2009] use a random word from those relevant to the other topics but not relevant to the current one, we believe that a discussion is necessary. Within each model, we compute all the inter-topic Kullback -Leibler divergences and for each topic we select a word from both the closest and the farthest remaining topics which shall serve as spurious words. The aim is to detect differences in evaluator agreement and evaluation performance depending on the spurious word choice. For instance, one of the topics obtained from the AP corpus was {drug, treatment, company, patient, cancer}. The word chosen from the closest KL neighbor was hospital while the choice from the farthest topic was pound.

A total of 37 evaluators were each given 40 groups of six words in a randomized order containing the five most important words for a topic (the ones carrying the highest probability in the LDA model) and a spurious word. In the example above, one examiner will be asked to choose a spurious word from the {cancer, drug, pound, treatment, com-pany, patient} group while {company, patient, cancer, hospital, drug, treatment} will be shown to another person.

The questions were balanced to have an equal number of topics evaluated for the two corpora, for each topic number ݇ א ሼ͵Ͳǡ ͷͲǡ ͳͲͲǡ ʹͲͲǡ ͵ͲͲሽ and for each of the two spurious word types.

Spurious Word Detection Variation

In the first experimental setup the analyzed topics were separated into relevant and irrelevant from an algorithmic point of view. The first lot contained the top ten topics for each k above and each corpus, ranked by conceptual relevance while the second lot contained the bottom ten. The aim was to see whether an improvement of the spurious word detection is visible from one category to the other. The setup was duplicated for the close or far spurious word poison type.

For each examiner's answers we computed the average ratio of topics where the spurious word was detected in the two situations -top and bottom topics -݄ଓݐ ା തതതതതത and ݄ଓݐ ି തതതതതത . The difference between the two, as a percentage of ݄ଓݐ ି തതതതതത is shown as the gain, in the last column of Table 1 The influence of the spurious word choice on the outcome of the experiment is noteworthy. On average, as shown in Table 1., the detection rate was 92.7% higher when the spurious words had been taken from far topics than in the close topic scenario.

The standard deviations for each topic type ߪሺ݄݅ݐ ା ሻ and ߪሺ݄݅ݐ ି ሻ are also shown and the evolution is mixed, for reasons which will be discussed in section 3.4.

Another metric is whether the spurious word was detected from the first word in the evaluator's answer rather than the others, ݂݄ଓݐ തതതതതത . For instance, in the second example presented earlier, an evaluator answered (company, hospital), which means that, although the spurious word has been detected, the initial bias was towards company. The gains obtained when passing from the low quality topics to high ranking ones are even more pronounced in this case.

This implies that if the evaluators were forced to only give one answer they would have detected the inserted words in even twice as many cases for the good topics than for the lower quality ones. A third metric is the number of chosen words to answer one question. A higher number signals a lower quality topic where the evaluator is unable to choose an outlying word with enough confidence. On average for each corpus and for each spurious word type 23 out of the 37 evaluators gave at least an answer containing a minimum of two words. Results show that the number of chosen words ݓ ഥ and the standard deviation are always smaller for good topics in all cases shown. The better the quality of the topic is, the lesser the need to also insert one of the real topic words alongside with the poison.

Metric and Topic Number Correlation

The second experiment setup shows the improvement gains presented in the previous experiment divided according to the number of LDA topics. Results obtained using the two spurious word types are presented separately. The wide differences between the results obtained show the need for a finer granularity. Although the ratios are computed based on the same top-bottom topics dichotomy, we observe that, for these two corpora, with ݇ ൌ ሼ͵ͲǡͷͲሽ the algorithmically best topics perform poorer than their counterparts while for the upper k values the scales turn pronouncedly. As we will show in the following experiment, this is due to the fact that the relevance difference between the better and the poorer quality topics varies with k. We underline the fact that the presented ratios move in tandem when varying k, regardless of the type of poisoning.

Evaluator Agreement

In the third experimental framework we divided the question instances by their spurious word type and again by the number of classes of the model the contained topic came from. For each topic i we bundled all the examiners' answers and we computed their average ݄ଓݐ ା ప തതതതതത or ݄ଓݐ ି ప തതതതതത and standard deviation ߪሺ݄݅ݐ ି ሻ or ߪሺ݄݅ݐ ା ሻ respectively.

We then computed the average of The difference ݐ݄߲݅ ା െ ݐ݄߲݅ ି shows how much easier the evaluators were able to find the spurious words relative to the whole experiment average, given a fixed k value. Also ߲ߪ ା െ ߲ߪ ି shows the variation of the degree of uncertainty in the previous improvement, above the experiment average. Results are shown in Table 5.

The goal is to find the correlations between the accuracy increases, uncertainty decreases and relevance variations for different k values. We computed the Pearson correlation between the ݐ݄߲݅( ା െ ݐ݄߲݅ ି ) and ݈݁ݎ߲( ା െ ݈݁ݎ߲ ି ) and also between (߲ߪ ା െ ߲ߪ ି ) and ݈݁ݎ߲( ା െ ݈݁ݎ߲ ି ) for the two spurious word cases. A value close to 1 or -1 implies strong positive or negative correlation while values close to 0 show a lack of linear correlation. Results prove our assumption that the chance of the spurious word being detected is directly correlated with the quality of the topic, with values of 0.557 for the distant spurious words and 0.704 for the second case. Moreover the inverse correlation of -0.979 is extremely strong for the distant spurious word uncertainty variation. The better the topics, compared to the average value, the better the evaluator's answers and the greater their agreement.

݇ ݐ݄߲݅ ି ߲ߪ ି ݈݁ݎ߲ ି ݐ݄߲݅ ା ߲ߪ ା ݈݁ݎ߲ ା ݐ݄߲݅ ା െ ݐ݄߲݅ ି ߲ߪ െ ߲ߪ െ ݈݁ݎ߲ ା െ ݈݁ݎ߲ ି Close Spurious Word 30

Conclusions and Future Work

We have successfully proven that there is a strong correlation between the ontological evaluation of topics and the way humans interpret them. We have outlined the important impact corpus choice or question formation have on that correlation. By shifting to a conceptual perspective we believe that we will be better equipped to rank topics and their respective models in a manner congruent with user needs. Topic labeling and corpus summarization are just two of the applications that could benefit from the model.

A natural follow up of the evaluation task is the attempt to improve the given models. We have already developed a system based on a similar WordNet topical subtree framework to detect conceptual outliers within the topic relevant words, which should be functional in the near future. moreover we will analyze the quantitative impact of automatically labeling topics from a conceptual standpoint rather than a statistical one. Another inciting application of the framework is to create conceptual neighborhoods within a topic model and detect the importance of the context created by the other topics when labeling or evaluating a single one.

  ݀൫ݓ ଵǡଵ ǡ ܿ ሺࣝሻ൯ ൌ ͵ and ݀൫ݓ ଵǡଶ ǡ ܿ ሺࣝሻ൯ ൌ ʹ, where ݓ ଵǡଵ and ݓ ଵǡଶ are ݓ ଵ 's senses.

Figure 1

 1 Figure 1. Topical Subtrees 2.3 Concept and Topical Subtrees Definition 1. A word's subtree within ࣝ is the reunion of all the branches that relate concepts within ߜሺݓሻ to ܿ ሺࣝሻ, ࣝ ௪ ൌ ሺߜሺݓǡ ࣝሻǡ ݎሻǢ ߜሺݓǡ ࣝሻ ൌ ڂ ߜ൫ܿ ௪ ǡ ܿ ሺࣝሻ൯ ೢ אఋሺ௪ሻ Ǥ In Fig. 1 ݓ ଵ 's subtree within ࣝ is marked with a dotted line. Definition 2. The subtree of an arbitrary concept ܿ in ࣝ ࣝ ൌ ሺߜሺࣝ ሻǡ ݎሻ is the subtree of ࣝ whose root concept is ܿǡ ߜሺࣝ ሻ ؿ ߜሺࣝሻǡ ߜሺࣝ ሻ ൌ ሼܿ א ߜሺࣝሻȁܿ

  which in the case of ࣝ ǡୡ మ is shown in the Fig 1. ellipse.

Table 1 .

 1 . Spurious Word Detection Ratios

	Data	Type	݄ଓݐ ା തതതതതത	ߪሺ݄݅ݐ ା ሻ ݄ଓݐ ି തതതതതത ߪሺ݄݅ݐ ି ሻ +(%)
	AP	Close	0.37	0.29	0.27	0.23	39.33
		Far	0.69	0.29	0.65	0.23	6.93
	Suall	Close	0.51	0.23	0.3	0.24	66.76
		Far	0.75	0.24	0.59	0.33	28.55

Table 2 .

 2 Spurious Word Hit from the First Chosen Word

	Data	Type	݂݄ଓݐ ା തതതതതതത ߪሺ݂݄݅ݐ ା ሻ ݂݄ଓݐ ି തതതതതതത ߪሺ݂݄݅ݐ ି ሻ +(%)
	AP	Close	0.27	0.26 0.21	0.2	32.08
		Far	0.6	0.32 0.53	0.24	13.92
	Suall Close	0.48	0.23 0.23	0.18	105.4
		Far	0.71	0.23 0.44	0.32	60.35

Table 4 .

 4 Metric and Topic Number Correlation

		ሺ݄ଓݐ ା തതതതതത െ ݄ଓݐ ି തതതതതത ሻȀ	ሺ݂݄ଓݐ ା തതതതതതത െ		ሺݓ ି തതതത െ ݓ ା തതതതሻȀ
		݄ଓݐ ି തതതതതത (%)		݂݄ଓݐ ି തതതതതതത ሻȀ݂݄ଓݐ ି തതതതതതത (%)	ݓ ା തതതത (%)
	k	Close	Far	Close	Far	Close Far
	30	-9.55	-2.52 -29.17	-10.26	27.3 11.91
	50	-37.09 -20.27	-49.02 -28.68	45.9 41.16
	100	35.29	32.8	83.75	111.52	0.17 10.25
	200	80.06	42 183.33	139.79 23.19 35.24
	300	100	65.42	160	665.71	14.8	24.9

Table 5 .

 5 Evaluator Agreement

			0.08	0.04	0.22	-0.02	0.05	-0.21	-0.1	0.01	-0.43
		50	0.02	0.06	0.06	-0.16	0	-0.03	-0.18	-0.06	-0.08
		100	-0.05	-0.06	0	-0.05	0.03	0.08	-0.01	0.09	0.08
		200	0	0.01	-0.11	0.23	-0.04	0.1	0.23	-0.05	0.21
		300	-0.06	-0.05	-0.17	0	-0.04	0.06	0.06	0.01	0.22
	Correlation								0.704	-0.016
	Distant	30	-0.11	-0.01	0.22	-0.18	0.11	-0.21	-0.07	0.12	-0.43
	Spurious	50	-0.02	0.04	0.06	-0.08	0.04	-0.03	-0.05	0	-0.08
	Word	100	0.02	-0.01	0	0.13	-0.03	0.08	0.11	-0.03	0.08
		200	0.04	0.01	-0.11	0.02	-0.04	0.1	-0.02	-0.05	0.21
		300	0.07	-0.04	-0.17	0.1	-0.08	0.06	0.03	-0.04	0.22
	Correlation								0.557	-0.979

Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence

Acknowledgments

This work is supported by the European Union Grant PO-SDRU/6/1.5/S/19 7713


Miguel A Rodríguez-Gironés

Guadalupe Corcobado

Jordi Moya-Laraño

Silk elasticity as a potential constraint on spider body size

Keywords: biomechanics, dispersal, minor ampullate silk, radiation, sexual size dimorphism (SSD)

Silk is known for its strength and extensibility and has played a key role in the radiation of spiders. Individual spiders use different glands to produce silk types with unique sets of proteins. Most research has studied the properties of major ampullate and capture spiral silks and their ecological implications, while little is known about minor ampullate silk, the type used by those spider species studied to date for bridging displacements. A biomechanical model parameterised with available data shows that the minimum radius of silk filaments required for efficient bridging grows with the square root of the spider's body mass, faster than the radius of minor ampullate silk filaments actually produced by spiders. Because the morphology of spiders adapted to walking along or under silk threads is ill suited for moving on a solid surface, for these species there is a negative relationship between body mass and displacement ability. As it stands, the model suggests that spiders that use silk for their displacements are prevented from attaining a large body size if they must track their resources in space. In particular, silk elasticity would favour sexual size dimorphism because males that must use bridging lines to search for females cannot grow large.

Silk has played a key role in the radiation of spiders [START_REF] Blackledge | Reconstructing web evolution and spider diversification in the molecular era[END_REF][START_REF] Vollrath | The role of behavior in the evolution of spiders, silks, and webs[END_REF] following the colonisation of land by their ancestors in the early Devonian [START_REF] Selden | A spider and other arachnids from the Devonian of New-York, and reinterpretations of Devonian Araneae[END_REF][START_REF] Shear | A Devonian spinneret -Early evidence of spiders and silk use[END_REF]. The number of silk-producing glands and silk types that individual spiders produce has increased during the evolution of the group, and orb-weaving spiders spin seven different kinds of silk, including five fibrous silks that contain unique sets of proteins [START_REF] Foelix | Biology of Spiders[END_REF]. Regardless of the selective pressures leading to the initial production of silk by spiders, silk was quickly used to produce hunting devices, ranging from isolated threads to sophisticated webs [START_REF] Vollrath | The role of behavior in the evolution of spiders, silks, and webs[END_REF] that have fostered the coevolutionary arms race between insects and spiders [e.g. [START_REF] Blackledge | Are three-dimensional spider webs defensive adaptations?[END_REF]] -to the point that it has been suggested that the exponential diversification of insects may have resulted from selection favouring insects escaping from spider predation [START_REF] Penney | Does the fossil record of spiders track that of their principal prey, the insects?[END_REF]. It is possibly for this reason that most research on spider silk, be it on the mechanical properties of silk or their ecological implications, has focused on the major ampullate and spiral silks used in web building [START_REF] Denny | Physical properties of spiders silk and their role in design of orb-webs[END_REF][START_REF] Gosline | The mechanical design of spider silks: From fibroin sequence to mechanical function[END_REF][START_REF] Kitagawa | Mechanical properties of dragline and capture thread for the spider Nephila clavata[END_REF][START_REF] Köhler | Thread biomechanics in the 2 orb-weaving spiders Araneus diadematus (Araneae, Araneidae) and Uloborus walckenaerius (Araneae, Uloboridae)[END_REF] and very little is known about other silk types, such as the minor ampullate silk that those spider species that have been studied use for bridging displacements [START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF]. In this paper we study the constraints that the physical properties of minor ampullate silk could set on the displacement ability of spiders.

The morphological traits allowing efficient locomotion in spiders that spend most of their life hanging upside-down from their webs are very different from those favoured in spiders that must support their weights on their legs while walking on solid surfaces [START_REF] Moya-Laraño | Morphological evolution of spiders predicted by pendulum mechanics[END_REF]. As a result, spiders adapted to moving along silk fibres are clumsy when moving on hard surfaces. But these spiders have turned their weakness into strength, developing an efficient displacement mechanism: bridging. To use this displacement method, a spider releases a bridging line until it adheres to some substrate, spanning a gap in the vegetation. The spider then tightens the line [START_REF] Moya-Laraño | Morphological evolution of spiders predicted by pendulum mechanics[END_REF][START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF]) and crawls across the silk threads. Although bridging has received relatively little attention in the literature, in the only species that have been studied in detail (Linyphia triangularis and Araneus diadematus) the bridging line is composed of two minor ampullate silk filaments [START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF]. Bridging frees spiders at once from the need to support their own weight [START_REF] Moya-Laraño | Morphological evolution of spiders predicted by pendulum mechanics[END_REF] and the constraint to follow the complex, fractal-like three-dimensional structure of vegetation for their displacements. Bridging is widespread in the Orbicularia clade. It is such an efficient displacement mechanism that it has been secondarily adopted by vegetationdwelling sit-and-wait predatory groups like the Thomisidae -a family belonging to the Dyonicha group which lost the web-building ability when it diverged within the RTA clade [START_REF] Coddington | Systematics and evolution of spiders (Araneae)[END_REF].

Although bridging is made possible by the remarkable mechanical properties of silk, even the finest materials have their limits. Large spiders are seldom seen bridging in the field, and [START_REF] Morse | Experimental and observational studies of patch choice at different scales by the crab spider Misumena Vatia[END_REF] have suggested that the elasticity of silk fibres prevents large crab spiders from bridging. Bridging lines sag under the weight of crossing spiders, and bridging will only be successful if the sag is less than the clearance under the line -otherwise the bridging spider will be dragged into the soil or vegetation. To evaluate the possibility that the elasticity of minor ampullate silk makes bridging inefficient for large spiders, we develop a biomechanical model to calculate the relationship between the cross section of the fibre, its elastic properties, the mass of the spider, m, and the sag of the line, k (hereafter defined as the maximal vertical displacement of the bridging spider divided by the horizontal distance between the attachment points of the line), as the spider crosses it. For a given body mass, the relative sag of the line, k, depends on the stress, extensibility and cross section of the silk fibre. Because increasing the stress on the fibre eventually leads to its breaking, heavier spiders must release thicker silk filaments to prevent excessive sagging. Although in general the thickness of the silk fibre released by a spider depends on the number and type of silk glands engaged in the production of silk [START_REF] Foelix | Biology of Spiders[END_REF], bridging lines are composed of two filaments, produced by the left and right minor ampullate glands [START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF], limiting the control that spiders have over the thickness of their bridging lines essentially to the ability of increasing the diameter of these two fibre silks. Thus, the radius of the minor ampullate silk filaments produced by a spider may determine its bridging ability.

The model

A spider of mass m bridges along a fibre composed of two silk filaments, each of them with radius r 0 and length L 0 when the fibre suffers no stress. The force exerted on the spider by the gravitational pull of the earth is directed downwards and has magnitude m • g, where g = 9.8 m•s -2 is the acceleration of gravity. We will neglect the gravitational pull acting directly on the silk fibre itself -the mass of the bridging fibre is a tiny fraction of the mass of the spider.

Let ϕ max be the true breaking strain (log[stretched length/rest length of fibre] at the breaking point, [START_REF] Roesler | Mechanical behaviour of engineering materials[END_REF]) of the minor ampullate silk fibre used for bridging and τ max To calculate the relationship between the radius at rest of the individual silk filaments, r 0 , the relative sag, k, and safety coefficient, β, we note that both the maximal vertical displacement of the spider and maximal stress on the fibre are achieved when the spider is at the midpoint of the fibre. By definition of relative sag, when the spider is midway through crossing its vertical displacement y satisfies the equation |y| = k•d, and hence its true breaking stress (force/ instantaneous cross section at the breaking point, [START_REF] Roesler | Mechanical behaviour of engineering materials[END_REF]). The breaking strain is an index of the extensibility of the fibre: larger values of the breaking strain indicate that fibres suffer greater elongation before breaking. The breaking stress, on the other hand, denotes how much force (per unit area) we must apply before the fibre breaks. The safety coefficient under which the fibre operates, β, is defined as the true breaking stress of the fibre divided by the maximum stress to which the fibre is actually exposed [START_REF] Osaki | Spider silk as mechanical lifeline[END_REF][START_REF] Osaki | Safety coefficient of the mechanical lifeline of spiders[END_REF]. Large safety coefficients indicate that the maximum stress (force per unit area) applied on the fibre is much lower than the stress required to break the fibre. The safety coefficient cannot be less than one: by definition, fibres break when the stress applied equals the breaking stress. Values only slightly larger than one indicate that the maximum stress applied on the fibre is close to the breaking stress.

, 1 • 4 • 2 cos 2 + = k k α
where d is the distance between the endpoints of the bridging line (greater than L 0 if the spider has tightened the line) and α is the angle between the vertical axis and the segment connecting the spider with an endpoint (Fig. 1). The vertical component of the forces exerted by the silk fibre on the spider (forces F in figure 1) must balance the gravitational pull. Therefore:

, 1 • 4 • • • • 8 1 • 4 • • 4 •cos • 2 • 2 2 2 + = + = = k k T r k k F F g m π α (1)
where r is the instantaneous radius of the silk filaments and T the true stress to which they are subject. Note that, in going from forces to stress we have multiplied the numerator by the cross section of the two minor ampullate silk filaments that constitute the bridging fibre (i.e.,

F = 2•π•r 2 •T). From the definition of the safety coefficient, β•T = τ max . • • • 8 1 • 4 • • • max 2 τ π β k k g m r + =
. Substituting in equation 1 and rearranging,

(2) Assuming constant volume during extension [START_REF] Vollrath | The effect of spinning conditions on the mechanics of a spider's dragline silk[END_REF],

. • • • • 8 1 • 4 • • • 0 max 2 0 L L k k g m r τ π β + = (3)
The stress-strain diagram gives us the relationship between elongation and stress, and hence between L and β. For simplicity, we will assume a linear relationship between true stress and true strain, so that ( )

. log max max 0 τ ϕ T L L ⋅ =
[Note that, due to the strain hardening of minor ampulate silk -i.e., its stress-strain curve is jshaped [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF], this assumption underestimates the value of r 0 , 1 max β τ = T : spiders will have to produce bridging lines slightly thicker than the model predictsalthough the approximation is fairly good when β ≤ 2.] With this final assumption, and noting that because L in equation 3 represents the maximum length of the fibre, so that we have

. • 1 •exp • • • 8 1 • 4 • • • max max 2 0         + = ϕ β τ π β k k g m r (4) Comparison of the value of r 0 π 739 . 0 9 • 10 • 93 . 2 m
computed from equation 4 with the radius of the minor ampullate silk filaments produced by a spider of mass m tells us whether the spider can bridge with the specified values of the safety coefficient and relative sag. We have only been able to obtain the radii of minor ampullate silk filaments produced by adult Argiope trifasciata [START_REF] Hayashi | Molecular and mechanical characterization of aciniform silk: Uniformity of iterated sequence modules in a novel member of the spider silk fibroin gene family[END_REF] and A. argentata [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF] females of unreported body mass. We know, however, that the average radius (in metres) of single major ampullate silk filaments produced by Araneus diadematus, ranging from 0.00036 g first-instar hatchlings to gravid 1.15 g females, increases with the mass of the spider, m (in Kg) according to r ≈ [START_REF] Ortlepp | The scaling of safety factor in spider draglines[END_REF]. Assuming that this relationship is similar in magnitude within and among species, we can write the radius of minor ampullate silk filaments, r min

, • 10 • 93 . 2 • 739 . 0 9 min π ξ m r - = , as (5)
where ξ is the ratio between the radii of minor and major ampullate silk filaments. Equating r min from equation 5 and r 0 from equation 4 we obtain the bridging threshold, or the maximum body size for which bridging is possible for a specified set of parameter values, m max :

. •exp 1 • 4 • • • • • 10 • 34 . 2 261 . 0 1 max 2 max 2 8 max                 - + = - β ϕ β τ ξ k g k m (6)

Choice of parameter values

Although the mechanical properties of minor ampullate silk fibres have been little studied, the breaking stress τ max and breaking strain ϕ max of minor ampullate silk have been measured for a few species. For Argiope trifasciata, τ max = 752 MPa and ϕ max = 0.442 [START_REF] Hayashi | Molecular and mechanical characterization of aciniform silk: Uniformity of iterated sequence modules in a novel member of the spider silk fibroin gene family[END_REF], and for A. argentata τ max = 923 MPa and ϕ max The average diameter of single major ampullate silk filaments was 3.24 and 3.5 µm for A. trifasciata and A. argentata, respectively, while the average diameter of single minor ampullate silk filaments was 0.69 and 1.1 µm for the same species [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF][START_REF] Hayashi | Molecular and mechanical characterization of aciniform silk: Uniformity of iterated sequence modules in a novel member of the spider silk fibroin gene family[END_REF]. According to the relationship found by Ortlepp and Gosline [START_REF] Ortlepp | The scaling of safety factor in spider draglines[END_REF] for Araneus diadematus, the major ampullate silk diameters would correspond to spiders of 350 and 450 mg, very reasonable for A. trifasciata and A. argentata adult females. In these two species, the ratio of minor over major ampullate silk filament diameter is ξ = 0.69/3.24 = 0.21 and ξ = 1.1/3.5 = 0.31. Inspection of scanning electron microscope photographs of bridging fibres produced by A. diadematus and Linyphia triangularis [START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF], however, suggests values closer to ξ = 0.5. It seems likely that the value of ξ changes with the size of spiders, and it is possible that the value of ξ differs between spiders producing silk fibres for bridging [START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF] and spiders from which silk is actively extracted by experimenters [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF][START_REF] Hayashi | Molecular and mechanical characterization of aciniform silk: Uniformity of iterated sequence modules in a novel member of the spider silk fibroin gene family[END_REF]. To obtain a conservative estimate of the constraints that silk imposes on bridging spiders, we calculate the maximum body size for which bridging is possible (equation 6) for ξ = 0.5 and 0.667. = 0.330 [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF]. These are the values we use for our calculations. [START_REF] Osaki | Spider silk as mechanical lifeline[END_REF][START_REF] Osaki | Safety coefficient of the mechanical lifeline of spiders[END_REF] reports that the safety coefficient for dragline silk used as lifeline (silk used for hanging while dropping from relatively high places) is β ≥ 2. Our calculations are based on β ≤ 2. Note that by choosing relatively small values of β and relatively large values of ξ, as compared with those measured in the lab, we ensure that we have conservative results: if silk elasticity limits spider's bridging, the real constraints will most likely be stronger than those we find.

Results

Figure 2 shows the square-root relationship between the mass of a spider, m, and the minimum radius of single minor ampullate silk filaments, r 0 , that bridging A. trifasciata and A. argentata individuals require (eq. 4) for different values of the safety coefficient, β = 1.0, 1.5 and 2.0, if the relative sag cannot exceed the value k = 0.3. In a log-log plot, the safety coefficient and elastic properties of silk affect the intercept, but not the slope, of r 0 . The intercept of r 0 increases with the safety coefficient and the breaking strain of the silk, and decreases with its breaking stress. For comparison, the dotted and dashed lines in figure 2 represent the expected radii of single minor ampullate silk filaments, r min (equation 5), calculated with ξ = 0.5 (dashed lines) or ξ = 0.67 (dotted lines). Because r 0 increases with m faster than r min , beyond a certain body mass the Figure 3. Bridging threshold for A. argentata and A. trifasciata. Relationship between bridging threshold, mmax, and relative sag, k, when the silk fibre has the mechanical properties characteristic of A. argentata (solid lines) and A. trifasciata (dashed lines) minor ampullate silk and the safety coefficient is β = 1.5 (solid circles) or β = 2 (empty circles). Because ξ has a multiplicative effect, it changes the scale, but not the shape, of the relationship. As a consequence, the same graph can be used to plot the bridging threshold when the ratio between major and minor ampullate silk filament radii is ξ = 0.5 (left scale) or ξ = 0.667 (right scale). silk filaments produced by spiders are thinner than the filaments they would need for bridging with the desired parameters (r min < r 0 ) and therefore bridging is not possible without increasing the relative sag or decreasing the safety coefficient. The body mass beyond which bridging is not feasible with the desired specifications is the bridging threshold, m max (equation 6).

The bridging threshold, m max , depends on the safety coefficient, β, the maximum relative sag allowed, k, and the physical properties of minor ampullate silk, τ max and ϕ max . Figure 3 plots the bridging threshold of A. argentata and A. trifasciata as a function of the relative sag, k, for β = 1.5 and 2. Note (equation 6) that the coefficient ξ used to estimate the radii of the minor ampullate silk filaments produced by spiders (equation 5) has a multiplicative effect on the bridging threshold and therefore acts as a scale parameter.

Due to the uncertainty in the estimate of τ max , ϕ max and ξ, we have used equation 6 to calculate the relationship between the bridging threshold, m max , and these parameters when the safety coefficient is β = 2 and the maximum relative sag allowed is k = 0.3 (Fig. 4). If spiders produce minor ampullate silk filaments substantially thinner than major ampullate silk filaments (ξ ≤ 0.6), bridging should be inefficient for spiders over 200 mg for reasonable values of τ max and ϕ max . When ξ = 0.7, the bridging threshold should lie around 250 mg if the values of τ max and ϕ max measured for A. trifasciata and A. argentata are representative of the elastic properties of minor ampullate silk in other species, but the bridging threshold could increase substantially in species with stiffer (lower ϕ max ) or stronger (higher τ max ) minor ampullate silk. Finally, if minor ampullate silk filaments are about as thick as major ampullate silk filaments (ξ ≥ 0.8), it seems unlikely that bridging is inefficient even for the larger spiders.

Discussion

Simple mechanical considerations allow us to calculate the minimal radius of the silk filaments that the spider must use for bridging once we know the mechanical properties of silk, the relative sag that the line can accept without hitting the ground and the safety coefficient under which the line can operate. This radius increases with the square root of the spider's body mass and eventually becomes greater than the radius of the minor ampullate silk filament that spiders produce. When the model is parameterised with available data, it predicts that the elasticity of minor ampullate silk severely limits the bridging abilities of heavy spiders.

There is considerable across-species variability in the mechanical properties of major ampullate silk [START_REF] Swanson | Spider dragline silk: Correlated and mosaic evolution in highperformance biological materials[END_REF] and it could be argued that heavy spiders are selected to produce minor ampullate silk stronger and stiffer than smaller spiders. Indeed, the minor ampullate silk of A. argentata is stronger and stiffer than the silk of A. trifasciata [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF][START_REF] Hayashi | Molecular and mechanical characterization of aciniform silk: Uniformity of iterated sequence modules in a novel member of the spider silk fibroin gene family[END_REF]. Nevertheless, the model has been parameterised with data from relatively large spiders, A. argentata and A. trifasciata, and predicts that neither of these species is an efficient bridger: an A. argentata of 400 mg producing minor ampullate silk filaments with a radius of 0.82 µm (to be compared with the 0.55 µm measured by [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF]) and operating under a safety coefficient of β = 1.5 would suffer a vertical displacement as large as the horizontal distance it tries to span through bridging (k = 1, figure 3) -the relative sag would be much larger for an adult A. trifasciata female -which would mean an overall travel distance 40% longer than in a normal bridging event. Therefore, even if heavier spiders produce stronger and stiffer minor ampullate silk than small spiders, they do not manage to compensate the effect of increasing body mass.

The relationship between spider size and the thickness of the minor ampullate filaments it produces, controlled by the parameter ξ of equation 5, plays an important role in determining bridging thresholds (equation 6, Figs. 2 and4). If increasing ξ from 0.5 to 0.667 leads to a nine-fold increase in bridging threshold, reducing ξ from 0.5 to 0.4 (respectively 0.31) leads to a five-fold (respectively 39-fold) decrease in bridging threshold. The bridging thresholds we have calculated assume that minor ampullate silk filaments are at least half as thick as major ampullate silk filaments (ξ = 0.5 and 0.667), which is much thicker than the values actually measured (ξ = 0.21 and 0.31; [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF][START_REF] Hayashi | Molecular and mechanical characterization of aciniform silk: Uniformity of iterated sequence modules in a novel member of the spider silk fibroin gene family[END_REF]. We have chosen such conservative estimates because filament diameter has been measured forcibly silking spiders [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF][START_REF] Hayashi | Molecular and mechanical characterization of aciniform silk: Uniformity of iterated sequence modules in a novel member of the spider silk fibroin gene family[END_REF], and it is therefore possible that the diameter of silk fibres produced in natural conditions is actually thicker.

Heavy spiders are less efficient bridgers than light ones. However, this does not mean that heavy spiders cannot bridge. Gravid giant orbweaving females such as some species in the genera Argiope, Araneus, Nephila or Caerostris, for example, must use bridging during the initial stages of web building [START_REF] Burnet | The Silken Web. A Natural History of Australian Spiders[END_REF][START_REF] Levi | The Diadematus group of the orb-weaver genus Araneus north of Mexico (Araneae: Araneidae)[END_REF]. But they will only be able to do so when the clearance under their lines is large compared to the horizontal distance they must span (Fig. 3). Thus, according to the model, parameterised with conservative versions of available data, although spiders living in the canopy will be able to use bridging as a displacement mechanism, spiders moving through shrubs and annual plants, where horizontal distances can often be longer than the height of lines, will be unable to use bridging if their body mass exceeds 100 or 200 mg. The predictions of the model are supported by laboratory trials where 204 individual spiders of 13 species, in the families Araneidae, Tetragnathidae, Linyphiidae, Theridiidae and Thomisidae, were stimulated to bridge in order to escape from simulated high predation risk. There was a strong negative correlation between body mass and bridging propensity. No spider over 200 mg (N = 46) used bridging to emigrate from simulated risky patches, only one individual between 100 and 200 mg (N = 7) used bridging, while 81% of the individuals under 100 mg (N = 151) bridged away from risky patches -large individuals mainly dropped to the ground to leave the risky patch (Corcobado et al., unpublished manuscript).

In low vegetation, therefore, species adapted to moving along silk lines may have little mobility once they cross the 100 mg body mass limit and may only be able to grow as large as they can continue growing without further displacements. Crab spiders over 200 mg, for instance, cannot move to patches some 10 m away once their patch has wilted [START_REF] Morse | Predator upon a flower. Life history and fitness in a crab spider[END_REF]. Since predator-prey body size ratios largely determine "who eats whom" in food webs [START_REF] Brose | Consumer-resource body-size relationships in natural food webs[END_REF][START_REF] Woodward | Body size in ecological networks[END_REF], this displacement constraint prevents crab spiders from reaching sizes sufficient to exploit resources that would, otherwise, be available to them, such as large carpenter bees and moths. Notice, however, that this constraint only applies to lifestyles that must track their resources in space. Strategies consisting in finding as a juvenile a productive site where the individual remains through its life will be unaffected, and can continue growing as long as prey are abundant at their site.

The constraints imposed by minor ampullate silk elasticity on the displacement of large spiders are not limited to individuals looking for food: they apply to spiders searching for any other resources, such as mates. With few exceptions [START_REF] Aisenberg | Daring females, devoted males, and reversed sexual size dimorphism in the sand-dwelling spider Allocosa brasiliensis (Araneae, Lycosidae)[END_REF], males are the searching sex in spiders. After their last moult, males abandon their web or nest and disperse looking for receptive females [START_REF] Foelix | Biology of Spiders[END_REF]. When females are too large for efficient bridging, males must be smaller than females or forego the possibility of using bridging for mate searching. Because, in many habitats, bridging could be a more efficient displacement mechanism than walking, the mechanical properties of silk may contribute to the evolution of sexual size dimorphism in spiders (Corcobado et al. unpublished manuscript).

Bridging plays an important role in the life of spiders adapted to moving along silk fibres (i.e. crab spiders within the RTA clade and Orbiculariae spiders). Because their morphology is unsuitable for walking along solid surfaces [START_REF] Moya-Laraño | Morphological evolution of spiders predicted by pendulum mechanics[END_REF], they are seldom seen walking on the ground and depend on bridging for their mid-range displacements. As we have seen, the elastic properties of minor ampulalte silk may make bridging an inefficient displacement mechanism for large spiders. The extent to which large body size constraints bridging ability, however, will depend on the precise values of the silk elasticity parameters and on the diameter of the filaments produced by spiders of different sizes. While we have used published data to parameterise the model, and have chosen rather conservative parameter values rather than the published ones, the scarcity of data prevents us from reaching final conclusions. Nevertheless, we hope that the results of our model prompt groups studying the mechanical properties of silk to devote more effort to the analysis of minor ampullate silk. It is also important to confirm that all (or at least most) spider species rely on minor ampullate silk for their bridging lines, as this result has been obtained from only two species [START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF]. It is important to understand how the physical properties of minor ampullate silk, including the thickness of the filaments produced, change with the size and ecology of species -and the extent to which aciniform silk filaments attached to minor silk filaments, presumably to increase its drag [START_REF] Peters | On the structural and glandular origin of the bridging lines used by spiders for moving to distant places[END_REF], affect the mechanical properties of bridging lines. This knowledge is essential for a detailed understanding of the relationship between size and mobility in bridging spiders. Even more interesting, if confirmed, is the fact that spiders rely on minor ampullate silk fibres for bridging. Major ampullate silk fibres are thicker, stronger and stiffer: the breaking stress of major ampullate silk for A. argentata is 1495 MPa, and its breaking strain 0.205 [START_REF] Blackledge | Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775)[END_REF]. If A. argentata used two major ampullate silk filaments for bridging, an adult female could bridge with a relative sag of less than k = 0.1 and a safety coefficient of β = 2. Why, then, should spiders use minor ampullate silk filaments for bridging? Minor ampullate silk may form stronger attachments to the substrate, or it may be easier to initiate in the absence of any tension pulling the silk from the spider [START_REF] Eberhard | How spiders initiate airborne lines[END_REF].

Role of the funding source

This work was supported by Ministerio de Ciencia e Innovación/FEDER (projects CGL2007-63223/BOS to MARG and CGL2007-60520/BOS to JML, and pre-doctoral grant FPI BES-2005-9234 to GC) and Junta de Andalucía/FEDER (project RNM-340). The funding sources were not involved in the study design; in the collection, analysis and interpretation of data; in the writing of the report; or in the decision to submit the paper for publication.

Figure 1 .

 1 Figure 1. Force diagram. Schematic representation of the forces acting on the spider when it is midway between the starting and finishing points of the bridging line, when the vertical displacement (|y|) and forces along the silk lines (F) attain their maximum value. If d represents the distance between the starting and finishing points of the bridging line, the relative sag is k = |y|/d.

Figure 2 .

 2 Figure 2. Minimal filament radius required for bridging. The solid lines represent the relationship between the mass of a spider, m, and the minimum radius of single minor ampullate silk filaments that spiders need for bridging, r0 (equation 4), when the relative sag cannot exceed the value k = 0.3. The values of r0 have been calculated using the mechanical properties of A. trifasciata (top panel) and A. argentata (bottom panel) minor ampullate silk. Each line represents a different safety coefficient (from top to bottom, β = 1.0, 1.5 and 2.0). The dotted and dashed lines represent the expected radii of single minor ampullate silk filaments, rmin (equation 5), calculated with ξ = 0.5 (dashed lines) or ξ = 0.67 (dotted lines).

Figure 4 .

 4 Figure 4. Contour plots of the bridging threshold. Panels represent, for different values of ξ, the relationship between the bridging threshold, mmax, and the elastic properties of minor ampullate silk, τmax and ϕmax, in a wide region that probably includes the true values for most species. The properties of minor ampullate silk in A. trifasciata (star) and A. argentata (circle) are indicated for comparison. Within each panel, the bottom line represents the set of points where the bridging threshold is mmax = ∆mmax. As we move upwards within each panel, lines represent the set of points where the bridging threshold is mmax = 2•∆mmax, mmax = 3•∆mmax, and so on. The values of ξ and ∆mmax are given in the lower-left corner of each panel. Bridging thresholds have been calculated from equation 6, with β = 2 and k = 0.3.

Acknowledgements

We thank D. Vinković, T. Blackledge and three anonymous reviewers for their helpful comments on previous versions of this ms. This work was supported by Ministerio de Ciencia e Innovación/FEDER (projects CGL2007-63223/BOS to MARG and CGL2007-60520/BOS to JML, and pre-doctoral grant FPI BES-2005-9234 to GC) and Junta de Andalucía/FEDER (project RNM-340).

Author's contributions

MAR-G, GC and JM-L conceived and designed the study. MAR-G developed the model and wrote the manuscript. GC compiled the information about the mechanical properties of spider silks used to parameterise the model. All authors read, made significant comments and approved the final manuscript.

Conflicts of interests

The authors declare that no conflict of interest may have affected their work.