Peanuts Cookbook Ingredient Network
We have reached a fascinating point in the history of science, where once again distant fields are shedding their differences and beginning to merge. An elegant yet simple example of this is found in graph theory, which is continually finding applications in describing a wide variety of natural and artificial systems ranging from the networks of protein interactions within a single cell, to food chains, to the World Wide Web.
This is a graph of the ingredients found in the ``Peanuts Cook Book'' (Scholastic Book Services, New York, Publication date unknown), which was on my parents' bookshelf when I was a child, and from which I learned to make "Lucy's Lemon Squares", a delicious citrous treat. The book contains simple recipes that even a child can prepare and that is illustrated by the late Charles M. Schulz. It contains a total of 17 recipes made from 40 ingredients (the index actually lists 21 recipes, but some of these are missing from my copy, as it is rather tattered). The above graph shows the network of ingredients used in the various recipes: Each node is an ingredient, and the edges (clearly not directed) indicate the ingredients that are used together in a recipe.

The network of ingredients is not connected within the context of the cookbook. In the upper right-hand region, one can find three nodes that are disconnected from the main body of the graph. These nodes represent ice cream, chocolate sauce, and cola, the ingredients for "Everybody's Chocolate Soda" (p.38). Because these ingredients are separate from the main body of the graph, the average distance between nodes becomes infinite. With the addition of other recipes to the cookbook, however, these nodes might become connected. For example, a bridge may be drawn between chocolate sauce and pecans if a recipe for a chocolate sundae is included in a future edition of the cookbook. In the large connected part of the graph, the maximum distance between any two nodes is 2, and the average distance between nodes is 1.76.

The short path length between arbitrary nodes is partially due to the presence of four highly connected nodes in the center of the graph. These nodes are salt, sugar, water, and corn syrup, with respective degrees of 27, 22, 19 and 17. Upon further inspection, attention may be drawn to a tight cluster of eight nodes near the top center of the graph. These nodes: pumpkin, baking powder, flour, cinnamon, nutmeg, ginger, raisins, and pecans, are ingredients that are key to ``Great Pumpkin Cookies'' (p.72). In fact, of these eight ingredients, only baking powder, flour, and cinnamon are found in other recipes in the book. This cluster of ingredients illustrates an interesting aspect of analyzing cooking through networks: spices tend to be well-connected, even though they may be seldom used. In a larger network of this sort, one might expect clusters of spices to form, indicating those spices that are commonly used together in a dish.

The connectivity of the graph is shown in the above figure, where the squares indicate the number of nodes with a given degree, and the solid line is present just to guide the eye along the general shape of the distribution. This plot seems to be bimodal, with fully a quarter of the ingrediants displaying a degree of 10, and a second smaller peak at a degree of 4. The average degree of the network is 8.0.