Student(s):
Paul Fornia, Margaret Noble, Tiana Stastny
Dates of Involvement:
Summer 2009 - Present
Faculty Advisor(s):
Jim Curry, Anne Dougherty
Graduate Mentor:
Spider Webs - Do They Exhibit a Natural Network?
Background Throughout the world, there are approximately 40,000 known species of spiders. However, only some spider breeds construct webs to ensnare prey. The type of web that a spider constructs depends on the family in which the spider is a member. The various types of spider webs include orb webs, sheet webs, tent webs, funnel webs, cobwebs, and purse webs. The most well known type of web is the orb web, depicted in Figure 1.
Orb webs are composed of both sticky strands which compose the spiral of the web, and non-sticky strands which compose the radial threads. Spiders catch prey by using specialized body parts to detect the location of the prey on the web. The two types of strands each have different modes of elasticity. The mode of elasticity affects how vibrations travel through the web and how spiders detect prey.
Spider Webs as Networks
In recent years, the study of social networks has greatly expanded. In response to these studies, our research project asks the question whether networks can occur in nature, or are purely a result of a social setting. To begin, we are studying the orb webs of spiders. The orb web indeed has the structure of a network, but it is unknown whether the orb web behaves as a network. To model an orb web as a network, we must begin by defining which parts of the web are nodes and which parts are edges. In a typical social network, a person or group could define a node, while the edges represent the relationships between these groups. We also must determine the entity that is traveling across the network. In a social network, an example of this entity could be a certain opinion or a disease. As vibrations on a spider web alert the spider to both prey and potential mates, we will attempt to formulate a network that models how vibrations travel through a web. When choosing what to define as a node, it is important to reference another type of network that has properties similar to the orb web. Many network studies have been done on traffic on urban streets, and the nodes in this case are defined as streets, and the edges are defined as intersections of streets. In the case of our spider web, we will define nodes as the connecting threads between each intersection, and the intersections will be defined as our edges. This model is logical, as a node in network is defined as an individual point, and the edges of the network define how these nodes are connected. In the case of the spider web, the connection of the individual strands, or nodes, is the actual intersection of the strands of silk. This may seem to be a confusing model at first, so we have developed a network that resembles the basic shape of a simplified orb web for clarification, as seen in Figure 2.
Figure 2: A simplified spider web network with the nodes (silk strands)
labeled from 1-35 (left) and its corresponding network graph (right). A line
connects the nodes in the graph if the silk strands of the web are touching.
This definition for the nodes and edges of the orb web also makes sense, as prey will not always land exactly on the intersections of the strands of silk. To simplify our model, we will assume that our prey is just a single point that lands on a single node.
When modeling a spider web as a network, we need to create an adjacency matrix to show how different nodes are related to one another. As we are modeling how vibrations travel across a web, each node is going to have a different relationship with every other node that is connected to it. As the radial threads have a greater mode of elasticity than the spiral threads, the radial threads will carry the most vibrations, while the sticky spiral threads will carry fewer vibrations. Because of this fact, if an insect lands on a radial node, the vibrations traveling outward from that node are not going to disperse evenly. More of the waves are going to travel through the radial threads that are connected to the original radial node. Therefore, the adjacency matrix is going to have different weights depending on the dispersion of the vibrations from the point of origin.
Next Steps: Weighted Networks
In Analysis of Weighted Networks, Newman discusses a method for studying weighted networks. Many networks in our world are more complex than unweighted networks, having entries other than either a zero or one in the adjacency matrix. In fact, Newman speculates that although weighted networks have not been deeply analyzed due to their seemingly complex nature, a wealth of information could be found from investigating these networks. Newman's method involves mapping a weighted network to an unweighted multigraph. He then proposes that one can study the weighted network by applying the standard techniques used for studying unweighted graphs to the unweighted multigraph.
In an effort to determine the most accurate and useful weighted clustering measure for a spider web network, we explore two different equations. We begin by looking at the individual weighted clustering coefficient presented by Barrat:
The Barrat equation provides a clustering coefficient between the values 0 and 1. It takes into account both topological information as well as the weight distribution of the network. However, this equation only considers the weights of edges adjacent to node k and overlooks the weights of edges between the neighbors of node k.
A different approach to the calculation of the weighted clustering coefficient is provided by Zhang and Horvath:
The Zhang and Horvath equation also provides a clustering coefficient between 0 and 1. This equation depends only on the network weights, and accounts both for weights between a node k and its adjacent nodes, as well as the weights of edges between the neighbors of k.
We will continue our research by using Newman’s technique for weighted networks, as well as determining the optimal definitions of measurements, such as the clustering coefficient, for an orb web. Our hope is to construct an accurate model of an orb web and determine if it behaves as a network.
About Paul Fornia, Margaret Noble, Tiana Stastny:
Paul Fornia is a sophomore in Applied Mathematics and Economics. He is also working towards an emphasis in Finance. While he has no definite plans for after school, possibilities include graduate school for mathematics or economics, teaching, or possibly working for an engineering firm or financial institute.
Margaret Noble is a senior in Applied Mathematics with an emphasis in Biology. She is working towards her B.S., and though she doesn’t know exactly what she wants to do in the future, teaching has always appealed to her.
Tiana Stastny is currently a senior in Applied Mathematics and is working towards a BS degree with a minor in electrical engineering. She hopes to one day apply her knowledge to the medical field as a biomedical engineer or perhaps a physician. Tiana loves to play her flute, which often helps her gain new insight into her studies.
References:
[1] F.G. Barth, A Spider's World Senses and Behavior, Springer, New York, 2001.
[2] A. Barrat, M. Barthelemy, R. Pastor-Satorras, and A. Vespignani, The architecture of weighted networks, PNAS, 101 (2004), 3747-3752.
[3] S. Doak, How Spiders Catch Their Prey, http://www.helium.com/items/1060744-how-spiders-catch-their-prey (June 5, 2009).
[4] M. Newman, Analysis of weighted networks, Physical Review, 70 (5) (2004), 1-9.
[5] M. Quinn, Southeastern Social Cobweb Spider, Oct. 8, 2007, http://www.texasento.net/Anelosimus.htm (June 6, 2009).
[6] T.H. Savory, The Spider's Web, Frederick Warne & Co. Ltd., London, 1952.
