Search Results for « Network Motifs »

ABSTRACT Natural populations display a variety of spatial arrangements, each potentially with a distinctive impact on genetic diversity and genetic differentiation among subpopulations. Although the spatial arrangement of populations can lead to intricate migration networks, theoretical developments have focused mainly on a small subset of such networks, emphasizing the island-migration and stepping-stone models. In this study, we investigate all small network motifs: the set of all possible migration networks among populations subdivided into at most four subpopulations. For each motif, we use coalescent theory to derive expectations for three quantities that describe genetic variation: nucleotide diversity, F ST , and half-time to equilibrium diversity. We describe the impact of network properties on these quantities, finding that motifs with a large mean node degree have the largest nucleotide diversity and the longest time to equilibrium, whereas motifs with small density have the largest F ST . In addition, we show that the motifs whose pattern of variation is most strongly influenced by loss of a connection or a subpopulation are those that can be split easily into several disconnected components. We illustrate our results using two example datasets—sky island birds of genus Brachypteryx and Indian tigers—identifying disturbance scenarios that produce the greatest reduction in genetic diversity; for tigers, we also compare the benefits of two assisted gene flow scenarios. Our results have consequences for understanding the effect of geography on genetic diversity and for designing strategies to alter population migration networks to maximize genetic variation in the context of conservation of endangered species.

Abstract Path homology is a powerful method for attaching algebraic invariants to digraphs. While there have been growing theoretical developments on the algebro-topological framework surrounding path homology, bona fide applications to the study of complex networks have remained stagnant. We address this gap by presenting an algorithm for path homology that combines efficient pruning and indexing techniques and using it to topologically analyze a variety of real-world complex temporal networks. A crucial step in our analysis is the complete characterization of path homologies of certain families of small digraphs that appear as subgraphs in these complex networks. These families include all digraphs, directed acyclic graphs, and undirected graphs up to certain numbers of vertices, as well as some specially constructed cases. Using information from this analysis, we identify small digraphs contributing to path homology in dimension two for three temporal networks in an aggregated representation and relate these digraphs to network behavior. We then investigate alternative temporal network representations and identify complementary subgraphs as well as behavior that is preserved across representations. We conclude that path homology provides insight into temporal network structure, and in turn, emergent structures in temporal networks provide us with new subgraphs having interesting path homology.