Admixed populations are formed by the merging of two or more ancestral populations, and the ancestry of each locus in an admixed genome derives from either source. Consider a simple pulse admixture model, where populations A and B merged t generation
s ago without subsequent gene flow. We derive the distribution of the proportion of an admixed chromosome that has A (or B) ancestry, as a function of the chromosome length L, t, and the initial contribution of the A source, m. We demonstrate that these results can be used for inference of the admixture parameters. For more complex admixture models, we derive an expression in Laplace space for the distribution of ancestry proportions that depends on having the distribution of the lengths of segments of each ancestry. We obtain explicit results for the special case of a two-wave admixture model, where population A contributed additional migrants in one of the generations between the present and the initial admixture event. Specifically, we derive formulas for the distribution of A and B segment lengths and numerical results for the distribution of ancestry proportions. We show that for recent admixture, data generated under a two-wave model can hardly be distinguished from that generated under a pulse model.
We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability proportional to r^{-a}) by means of an exact master equation for the process. We s
how that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as (ln L)^2 when a=d, and otherwise as L^x, with x=(d-a)/(d+1-a) for a<d, x=a-d for d<a<d+1, and x=1 for a>d+1. These values of x exceed the rigorous lower-bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of as.
We study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree. Each node is associated with, or attracted to its neighbor of maximal degree, as long as the degree is increasing. A node that
has no neighbors of higher degree is a peak, attracting all the nodes in its basin. Maximally random scale-free networks exhibit different behavior based on their degree distribution exponent $gamma$: for small $gamma$ (broad distribution) networks are dominated by a giant basin, whereas for large $gamma$ (narrow distribution) there are numerous basins, with peaks attracting mainly their nearest neighbors. We derive expressions for the first two moments of the number of basins. We also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale-free networks that resemble random nets. Finally, we generalize the problem to regular networks and lattices where all degrees are equal, and thus the attractiveness of a node must be determined by an assigned weight, rather than the degree. We derive the complete distribution of basins of attraction resulting from randomly assigned weights in one-dimensional chains.
We investigate the electrical current and flow (number of parallel paths) between two sets of n sources and n sinks in complex networks. We derive analytical formulas for the average current and flow as a function of n. We show that for small n, incr
easing n improves the total transport in the network, while for large n bottlenecks begin to form. For the case of flow, this leads to an optimal n* above which the transport is less efficient. For current, the typical decrease in the length of the connecting paths for large n compensates for the effect of the bottlenecks. We also derive an expression for the average flow as a function of n under the common limitation that transport takes place between specific pairs of sources and sinks.
We introduce a model for diffusion of two classes of particles ($A$ and $B$) with priority: where both species are present in the same site the motion of $A$s takes precedence over that of $B$s. This describes realistic situations in wireless and com
munication networks. In regular lattices the diffusion of the two species is normal but the $B$ particles are significantly slower, due to the presence of the $A$ particles. From the fraction of sites where the $B$ particles can move freely, which we compute analytically, we derive the diffusion coefficients of the two species. In heterogeneous networks the fraction of sites where $B$ is free decreases exponentially with the degree of the sites. This, coupled with accumulation of particles in high-degree nodes leads to trapping of the low priority particles in scale-free networks.
