Exponential Random Graph Modeling for Complex Brain Networks

Exponential random graph models (ERGMs), also known as p* models, have been utilized extensively in the social science literature to study complex networks and how their global structure depends on underlying structural components. However, the literature on their use in biological networks (especially brain networks) has remained sparse. Descriptive models based on a specific feature of the graph (clustering coefficient, degree distribution, etc.) have dominated connectivity research in neuroscience. Corresponding generative models have been developed to reproduce one of these features. However, the complexity inherent in whole-brain network data necessitates the development and use of tools that allow the systematic exploration of several features simultaneously and how they interact to form the global network architecture. ERGMs provide a statistically principled approach to the assessment of how a set of interacting local brain network features gives rise to the global structure. We illustrate the utility of ERGMs for modeling, analyzing, and simulating complex whole-brain networks with network data from normal subjects. We also provide a foundation for the selection of important local features through the implementation and assessment of three selection approaches: a traditional p-value based backward selection approach, an information criterion approach (AIC), and a graphical goodness of fit (GOF) approach. The graphical GOF approach serves as the best method given the scientific interest in being able to capture and reproduce the structure of fitted brain networks

Characterizing Self-Developing Biological Neural Networks: A First Step Towards their Application To Computing Systems

Carbon nanotubes are often seen as the only alternative technology to silicon transistors. While they are the most likely short-term one, other longer-term alternatives should be studied as well. While contemplating biological neurons as an alternative component may seem preposterous at first sight, significant recent progress in CMOS-neuron interface suggests this direction may not be unrealistic; moreover, biological neurons are known to self-assemble into very large networks capable of complex information processing tasks, something that has yet to be achieved with other emerging technologies. The first step to designing computing systems on top of biological neurons is to build an abstract model of self-assembled biological neural networks, much like computer architects manipulate abstract models of transistors and circuits. In this article, we propose a first model of the structure of biological neural networks. We provide empirical evidence that this model matches the biological neural networks found in living organisms, and exhibits the small-world graph structure properties commonly found in many large and self-organized systems, including biological neural networks. More importantly, we extract the simple local rules and characteristics governing the growth of such networks, enabling the development of potentially large but realistic biological neural networks, as would be needed for complex information processing/computing tasks. Based on this model, future work will be targeted to understanding the evolution and learning properties of such networks, and how they can be used to build computing systems

Solution of the 2-star model of a network

The p-star model or exponential random graph is among the oldest and best-known of network models. Here we give an analytic solution for the particular case of the 2-star model, which is one of the most fundamental of exponential random graphs. We derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition.Comment: 5 pages, 3 figure

Community detection in network analysis: a survey

The existence of community structures in networks is not unusual, including in the domains of sociology, biology, and business, etc. The characteristic of the community structure is that nodes of the same community are highly similar while on the contrary, nodes across communities present low similarity. In academia, there is a surge in research efforts on community detection in network analysis, especially in developing statistically sound methodologies for exploring, modeling, and interpreting these kind of structures and relationships. This survey paper aims to provide a brief review of current applicable statistical methodologies and approaches in a comparative manner along with metrics for evaluating graph clustering results and application using R. At the end, we provide promising future research directions.Statistic

A cluster expansion approach to exponential random graph models

The exponential family of random graphs is among the most widely-studied network models. We show that any exponential random graph model may alternatively be viewed as a lattice gas model with a finite Banach space norm. The system may then be treated by cluster expansion methods from statistical mechanics. In particular, we derive a convergent power series expansion for the limiting free energy in the case of small parameters. Since the free energy is the generating function for the expectations of other random variables, this characterizes the structure and behavior of the limiting network in this parameter region.Comment: 15 pages, 1 figur

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference