The study of trophic relationships is central to ecology, and ecologists have a long history of describing and analyzing food webs, networks of who eats whom in ecological communities. Explicit food-web research dates back to work by Elton in the 1920’s, with further development in the 1940’s and 50’s by Lindeman, Odum, MacArthur, and others. The increasingly sophisticated empiricism, analysis, and modeling of the last quarter century were kick-started by mathematical analyses of community stability published by May in 1973. In this tutorial I will briefly discuss the history of food-web concepts and research, and then will explore current key topics, with an emphasis on the structure, dynamics, and stability of complex trophic networks, and how such research fits within the broader arena of network theory.

Suggested Reading:
The Network Structure of Food Webs, Jennifer Dunne
Food-web Structure and network theory: The role of connectance and size, Dunne, Williams, Martinez; PNAS October 1, 2002

The mutualistic interactions between plants and the animals that pollinate them or disperse their fruits have molded the organization of Earths’s biodiversity. These interactions can form complex networks involving dozens and even hundreds of species. Recently, it has been shown that mutualistic networks are very heterogeneous, nested and build upon weak and asymmetric links among species. These network patterns have far-reaching consequences for species persistence and coevolution, and thus can be regarded as the architecture of biodiversity. Past evolutionary history conveyed in the phylogenies of both plants and animals contribute to shaping these coevolutionary networks. Because pylogenetically similar species tend to play similar roles in the network, extinction events trigger non-random coextinction cascades, strongly reducing taxonomic diversity.

REFERENCES
Bascompte, J., P. Jordano, and J.M. Olesen. (2006). Asymmetric coevolutionary networks facilitate biodiversity maintenance. Science, 312: 431-433.
Bascompte, J., P. Jordano, C.J. Melián and J.M. Olesen. (2003). The nested assembly of plant-animal mutualistic networks. Proceedings of the National Academy of Sciences USA, 100: 9383-9387.
Jordano, P., J. Bascompte, and J.M. Olesen. (2003). Invariant properties in coevolutionary networks of plant-animal interactions. Ecology Letters, 6: 69-81.

We define the complex system underlying organizational culture by incorporating the social-psychological principles of balance and information (B-I) into models of influence (changes in attitudes as a function of interaction) and selection (changes in interaction). We identify information based influence as a potential anchor for actors' sentiments so that they are not overwhelmed by normative influence. In the model of selection, we identify the pursuit of information as an important counterbalance to the effect of homophily (interacting with others like oneself). While some forms of these models are emerging in the literature, we integrate social-psychological and information processing theories into a longitudinal framework. This allows us to generate different types of equilibria characteristic of complex and chaotic systems. We use the tools of dynamic systems to demonstrate that a basic model produces an explosion in actors' sentiments as actors continually influence each other through interactions unless restrictions are placed on actors' patterns of interaction or extent of influence on each other. We modify this model by governing the extent to which actors may influence one another in the absence of new information that has entered the system at any time. Thus our models require a longitudinal framework, and the behavior of the system cannot be predicted based on the state of the system at any given time period -- the system is non-Markovian. We then define a model of actors' interactions to complete the dynamic system. It is in this model that we incorporate counterbalancing feedback mechanisms that are critical to generating the full range of behaviors characteristic of a complex system. Specifically, we recognize that actors may both seek to interact with others who hold sentiments similar to their own and interact with others who are exposed to influences unlike their own. Through our representations and simulations of the basic systems of organizations we observe how actors construct organizational structures in response to existing internal structures and exogenous shocks. For example, we demonstrate the counterintuitive result that attempts to increase consensus may actually generate factions as previously neutral actors align with members of a given subgroup. We also demonstrate that an actor's centrality can be defined in terms of the longitudinal effects on the system of a shock to that actor.

Organizational Culture as a Complex System: balance and Information in Models of Influence and Selection. ,
Frank, K.A., & Fahrbach, K. (1999)
Special issue of Organization Science on Chaos and Complexity, Vol 10, No. 3, pp. 253-277.
Suggested Viewing

It is increasingly apparent that an ecological network perspective, which encompasses direct and indirect effects among interacting taxa, is critical for understanding, forecasting, and managing the impacts of species loss and invasion, habitat conversion, and climate change. At a basic research level, this suggests that we need to develop a more general framework for understanding ecological network robustness at whole-system and component levels, in terms of both external perturbations and internal dynamics. An effective framework will have a scope that extends beyond the usual focus on contemporary systems. Examples from a biocomplexity project based in the Aleutian Islands and a “paleofoodweb” project demonstrate how research at the interface of ecology and network theory can be fruitfully extended across archaeological and paleontological time scales, deepening our understanding of different aspects of ecological robustness.
Suggested Reading:
Allometric scaling enhances stability in complex food webs, Brose, Williams, Martinez; Ecology Letters 2006
Simple rules yield complex food webs, Williams, Martinez; Nature 9 March, 2000

This tutorial introduces some network-appropriate tools available in R, a free and widely used environment for statistical computing and graphics.  We illustrate these tools using current research studying the effect of various network characteristics on the prevalence of a particular disease.  The networks on which the disease spread is simulated are themselves simulated from individual-level data, using tools available in R.  The probabilistic models that give rise to these simulations, which are called exponential-family random graph models (ERGMs), are also introduced.

Individuals within a population are not equal; they differ in their exposure and susceptibility to parasites. These heterogeneities in infection status create "super-spreaders": hosts that have a disproportionate contribution to parasite transmission and persistence. Network models offer a useful method for investigating the per capita contribution of individuals. Empirical wildlife data, such as capture-mark-recapture allow us to approximate a contact network. Here, I detail a time-series of data on known individuals within a rodent population in conjunction with their parasitic status. I work through an example of how we can produce a contact network from these data. I discuss questions such as how can we define a disease contact and what are the biases introduced in the observation process.

All organisms interact with their environment, and in doing so shape it, modifying resource availability. Termed niche construction, this process has been studied primarily at the ecological level with an emphasis on the consequences of construction across generations. We focus on the behavioural process of construction within a single generation, identifying the role a robustness mechanism—conflict management—has in promoting interactions that build social resource networks or social niches. Using 'knockout' experiments on a large, captive group of pigtailed macaques (Macaca nemestrina), we show that a policing function, performed infrequently by a small subset of individuals, significantly contributes to maintaining stable resource networks in the face of chronic perturbations that arise through conflict. When policing is absent, social niches destabilize, with group members building smaller, less diverse, and less integrated grooming, play, proximity and contact-sitting networks. Instability is quantified in terms of reduced mean degree, increased clustering, reduced reach, and increased assortativity. Policing not only controls conflict, we find it significantly influences the structure of networks that constitute essential social resources in gregarious primate societies. The structure of such networks plays a critical role in infant survivorship, emergence and spread of cooperative behaviour, social learning and cultural traditions.

Suppose we wish to determine how individuals' characteristics predict the presence or absence of relationships in an observed social network.  If we assume that every potential relationship is independent of every other, then standard logistic regression will suffice.  However, one of the tenets of social network analysis is that social structure itself drives the formation of networks, meaning that relationships are not independent.  We might therefore turn to exponential-family random graph models (ERGMs).  This talk expands upon the discussion of ERGMs in the earlier tutorial, describing the special computational techniques that must be employed due to the mathematical intractability of the estimation problem along with some novel techniques for assessing model goodness of fit.  We illustrate these ideas using data on friendships among high school students.

In this talk, I introduce basic concepts and tools for the application of network theory to landscapes. Many concepts in spatial ecology hinge on the idea of spatial adjacency – who or what is in my neighborhood? Graph theory is a convenient way to express and communicate ideas about adjacency. It also allows one to compute meaningful quantities related to spatial pattern. Some properties of simple spatial graphs are discussed and their relationship to methods in spatial statistics are considered. Adding vertex and edge properties to graphs extends the problem domain to network modeling. In networks, we consider properties such as distance, cost and flow. Basic algorithms are introduced and connectivity measures such as the cut-set are introduced. Finally, I consider stochastic extensions for modeling random walks and relate these to some basic results in metapopulation theory.

The structure of spatial networks can give insight on the robustness of the ecological and evolutionary processes taking place on these networks.  I will introduce two case studies. First, the network of temporary ponds in Doñana National Park (Southern Spain) has a structure that makes the system robust to drought and thus provides a mechanism for amphibian persistence in stochastic environments. Second, the spatial mating network of an insect-pollinated tree (Prunus mahaleb) in Cazorla Natural Park (South Eastern Spain), determines how many different pollen donors contribute to the progeny of a receptor plant. This information has important implications for gene-flow in heterogeneous landscapes, and is determined by the interplay between the spatial distribution of trees and the shape of the pollination kernel. I will end up by discussing scenarios to address spatial networks of interaction networks.   

Graph theory is a body of mathematics, and associated computer algorithms, dealing with problems of connectivity, flow, and routing in networks ranging from social science, to traffic engineering, to computer networks such as the Internet. Graphs consist of nodes (here representing habitat patches or local populations) and edges or arcs, representing functional connections between nodes (here, via dispersal). Recently, graphs have become increasingly popular in conservation biology, particularly for applications couched in metapopulation theory. Here I illustrate graph models developed for habitat networks in a variety of systems, focusing on the conservation implications of network connectivity. In general, I find the graph model a remarkably robust framework for analysis and communication. I consider new efforts in three directions:  facilitating the construction of graph models from geospatial data, implementing graph-theoretic tools in user-friendly packages, and exploring new analytic methods for network optimization for conservation and ecological restoration. 

Connectivity is a fundamental property of landscapes and has important implications for ecological and evolutionary processes. Network theory provides a language and calculus for mapping process onto habitat patterns. The structure of network strong influences function. I discuss three concepts related to network structure: resilience (the capacity for recovery after disturbance), robustness (the capacity for recovery after alteration of the network itself) and resistance (the ability to avoid changes in function after disturbance). Using these concepts I consider approaches to mapping risk across landscapes. I show how connectivity influences resiliency and the relationship between connectivity-scale and robustness. The importance of movement bottlenecks in restricting dispersal and as potential hot spots for biological control or epidemic emergence are also considered.

Landscape ecologists and conservation scientists are increasingly using networks to measure the connectivity of landscapes. In this presentation I will describe methods that we have developed using commonly-used GIS software to: (a) generate landscape networks so that nodes and edges have ecological meaning; and (b) compute ecologically-relevant metrics that capture 1st and beyond 1st order configurational aspects of a landscape.

With a view to their possible application in the study of disease dynamics, we survey a range of network-based techniques and methodologies employed in the field  of Electrical and Computer Engineering, emphasizing the nature of the questions that are posed and the types of answers that may be obtained. These include queuing models, routing models, and dynamic programming techniques.  We pose one particular problem and suggest the insights that may be gained by viewing the question as one of finding the shortest path through a network.

Click here for Abstract
Recommended Reading: Low-rank Smoothing Splines on Complex Domains, (2007) Biometrics, 209-217.

Graphs and Networks are a popular tool to model interaction effects in the Sciences.  At the outset, graphs are simply relations on sets.  In applications, however, we often have weights on edges, and maybe even coordinates on the vertices.  That is to say, there is a whole lot of things which are called networks.  In the talk, I will summarize some important algorithms for graphs and networks.  It is important to realize that some properties of graphs are hard to compute (like isomorphism, traveling salesman, 1-factor), whereas there are other properties for which efficient algorithms are available (like network flow, connectivity, spanning tree).  Combinatorial optimization provides a means to solve some hard problems and I will report on that as well.

movie 1 Minimum cost spanning tree movie 2 TSP movie 3 Domino Portrait

The problem of searching a text for occurrences of a substring is familiar to anyone who has used a text editor.  When an algorithm needs to generate a large number of searches in a text under study, such as a genome, enormous improvements in efficiency can be obtained by representing the text with a network.  The talk will describe some representations that I have co-developed to accomplish this, as well as recent joint work with Asa Ben-Hur on inexact matching algorithms that make use of them.  It will also touch on some of my recent work on algorithms for constructing linear arrangements of sequence fragments, given a way to test which pairs of fragments overlap.