Signal Amplification in Highly Ordered Networks Is Driven by Geometry.

Here, we hypothesize that, in biological systems such as cell surface receptors that relay external signals, clustering leads to substantial improvements in signaling efficiency. Representing cooperative signaling networks as planar graphs and applying Euler's polyhedron formula, we can show that clustering may result in an up to a 200% boost in signaling amplitude dictated solely by the size and geometry of the network. This is a fundamental relationship that applies to all clustered systems regardless of its components. Nature has figured out a way to maximize the signaling amplitude in receptors that relay weak external signals. In addition, in cell-to-cell interactions, clustering both receptors and ligands may result in maximum efficiency and synchronization. The importance of clustering geometry in signaling efficiency goes beyond biological systems and can inform the design of amplifiers in nonbiological systems.

Transient amplifiers of selection and reducers of fixation for death-Birth updating on graphs.

The spatial structure of an evolving population affects the balance of natural selection versus genetic drift. Some structures amplify selection, increasing the role that fitness differences play in determining which mutations become fixed. Other structures suppress selection, reducing the effect of fitness differences and increasing the role of random chance. This phenomenon can be modeled by representing spatial structure as a graph, with individuals occupying vertices. Births and deaths occur stochastically, according to a specified update rule. We study death-Birth updating: An individual is chosen to die and then its neighbors compete to reproduce into the vacant spot. Previous numerical experiments suggested that amplifiers of selection for this process are either rare or nonexistent. We introduce a perturbative method for this problem for weak selection regime, meaning that mutations have small fitness effects. We show that fixation probability under weak selection can be calculated in terms of the coalescence times of random walks. This result leads naturally to a new definition of effective population size. Using this and other methods, we uncover the first known examples of transient amplifiers of selection (graphs that amplify selection for a particular range of fitness values) for the death-Birth process. We also exhibit new families of "reducers of fixation", which decrease the fixation probability of all mutations, whether beneficial or deleterious.

Evolutionary games on isothermal graphs.

Population structure affects the outcome of natural selection. These effects can be modeled using evolutionary games on graphs. Recently, conditions were derived for a trait to be favored under weak selection, on any weighted graph, in terms of coalescence times of random walks. Here we consider isothermal graphs, which have the same total edge weight at each node. The conditions for success on isothermal graphs take a simple form, in which the effects of graph structure are captured in the 'effective degree'-a measure of the effective number of neighbors per individual. For two update rules (death-Birth and birth-Death), cooperative behavior is favored on a large isothermal graph if the benefit-to-cost ratio exceeds the effective degree. For two other update rules (Birth-death and Death-birth), cooperation is never favored. We relate the effective degree of a graph to its spectral gap, thereby linking evolutionary dynamics to the theory of expander graphs. Surprisingly, we find graphs of infinite average degree that nonetheless provide strong support for cooperation.

Evolutionary dynamics on any population structure.

Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times of random walks. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small changes in population structure-graph surgery-affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.

Fundamental limitations of network reconstruction from temporal data.

Inferring properties of the interaction matrix that characterizes how nodes in a networked system directly interact with each other is a well-known network reconstruction problem. Despite a decade of extensive studies, network reconstruction remains an outstanding challenge. The fundamental limitations governing which properties of the interaction matrix (e.g. adjacency pattern, sign pattern or degree sequence) can be inferred from given temporal data of individual nodes remain unknown. Here, we rigorously derive the necessary conditions to reconstruct any property of the interaction matrix. Counterintuitively, we find that reconstructing any property of the interaction matrix is generically as difficult as reconstructing the interaction matrix itself, requiring equally informative temporal data. Revealing these fundamental limitations sheds light on the design of better network reconstruction algorithms that offer practical improvements over existing methods.