Critical graph

Abstract: Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local network structures including exponential random graph models and motif counting. We show that none of the here-investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others.

Abstract: There is a large, popular, and growing literature on "scale-free" networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scale-free graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scale-free. This structural view can be related t...

Abstract: A large class of diagrams can be informally characterized as node-link diagrams. Typically nodes represent entities, and links represent relationships between them. The discipline of graph drawing is concerned with methods for drawing abstract versions of such diagrams. At the foundation of the discipline are a set of graph aesthetics (rules for graph layout) that, it is assumed, will produce graphs that can be clearly understood. Examples of aesthetics include minimizing edge crossings and minimizing the sum of the lengths of the edges. However, with a few notable exceptions, these aesthetics are taken as axiomatic, and have not been empirically tested. We argue that human pattern perception can tell us much that is relevant to the study of graph aesthetics including providing a more detailed understanding of aesthetics and suggesting new ones. In particular, we find the importance of good continuity (ie keeping multi-edge paths as straight as possible) has been neglected. We introduce a methodology for evaluating the cognitive cost of graph aesthetics and we apply it to the task of finding the shortest paths in spring layout graphs. The results suggest that after the length of the path the two most important factors are continuity and edge crossings, and we provide cognitive cost estimates for these parameters. Another important factor is the number of branches emanating from nodes on the path.