Communities and bottlenecks: Trees and treelike networks have high modularity
Physical Review E, 85, , 2012
Abstract: Much effort has gone into understanding the modular nature of complex networks. Communities, also known as clusters or modules, are typically considered to be densely interconnected groups of nodes that are only sparsely connected to other groups in the network. Discovering high quality communities is a difficult and important problem in a number of areas. The most popular approach is the objective function known as modularity, used both to discover communities and to measure their strength. To understand the modular structure of networks it is then crucial to know how such functions evaluate different topologies, what features they account for, and what implicit assumptions they may make. We show that trees and treelike networks can have unexpectedly and often arbitrarily high values of modularity. This is surprising since trees are maximally sparse connected graphs and are not typically considered to possess modular structure, yet the nonlocal null model used by modularity assigns low probabilities, and thus high significance, to the densities of these sparse tree communities. We further study the practical performance of popular methods on model trees and on a genealogical data set and find that the discovered communities also have very high modularity, often approaching its maximum value. Statistical tests reveal the communities in trees to be significant, in contrast with known results for partitions of sparse, random graphs.
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Bongard's work focuses on understanding the general nature of cognition, regardless of whether it is found in humans, animals or robots. This unique approach focuses on the role that morphology and evolution plays in cognition. Addressing these questions has taken him into the fields of biology, psychology, engineering and computer science.
Danforth is an applied mathematician interested in modeling a variety of physical, biological, and social phenomenon. He has applied principles of chaos theory to improve weather forecasts as a member of the Mathematics and Climate Research Network, and developed a real-time remote sensor of global happiness using messages from Twitter: the Hedonometer. Danforth co-runs the Computational Story Lab with Peter Dodds, and helps run UVM's reading group on complexity.
Laurent studies the interaction of structure and dynamics. His research involves network theory, statistical physics and nonlinear dynamics along with their applications in epidemiology, ecology, biology, and sociology. Recent projects include comparing complex networks of different nature, the coevolution of human behavior and infectious diseases, understanding the role of forest shape in determining stability of tropical forests, as well as the impact of echo chambers in political discussions.
Hines' work broadly focuses on finding ways to make electric energy more reliable, more affordable, with less environmental impact. Particular topics of interest include understanding the mechanisms by which small problems in the power grid become large blackouts, identifying and mitigating the stresses caused by large amounts of electric vehicle charging, and quantifying the impact of high penetrations of wind/solar on electricity systems.
Bagrow's interests include: Complex Networks (community detection, social modeling and human dynamics, statistical phenomena, graph similarity and isomorphism), Statistical Physics (non-equilibrium methods, phase transitions, percolation, interacting particle systems, spin glasses), and Optimization(glassy techniques such as simulated/quantum annealing, (non-gradient) minimization of noisy objective functions).