Ges in the sequence produced by the IDSS algorithm (Panel B, Fig 4). This simple graphic informs us about the relations between assemblages without the addition of the information regarding the composition of the types. The graph representation has an advantage over traditional centered-bar diagrams since it allows us to examine relations where assemblages may be shared in multiple valid solutions [89, 91]. The ability of graphs to reflect complex sets of relationships, however, can result in difficult interpretation of the results. The strength of seriation is that solutions are linear relations where the order reflects some combination of differences in time and space. However, if assemblages are found in more than one solution, additional analytic steps must be taken to reduce the results to something that can serve as a hypothesis about the structure of transmission and the relations between assemblages. As shown in Fig 6, we can proceed by accumulating valid solutions, and then pruning unnecessary edges. We begin at the top with three valid solutions output from the basic IDSS algorithm. Each meets the WP1066 price criteria for unimodality and the frequency differences are within the 0.1 tolerance limit set for continuity. In the middle of the figure, we show the results of agglomerating the graphs together, where an edge exists between two vertices if those vertices possess an edge in any of the three source graphs. The weights assigned to edges are proportional to the summed differences in type frequencies between pairs of assemblages. This aggregate graph allows us to construct the final solution. We follow the approach described by Lipo [89], starting with just the vertices, and iteratively adding edges from the summed graph starting with those which possess the lowest weight as measured by Euclidean distance between pairs of assemblages. This process produces a graph that includes the maximal set of vertices from the starting solutions but using the minimum number of edges that represent smallest distances between vertices and includes all equivalent values as options. The result is what we call the “minmax” graph.PLOS ONE | DOI:10.1371/FT011MedChemExpress FT011 journal.pone.0124942 April 29,14 /The IDSS Frequency Seriation AlgorithmFig 6. “Minmax” graph creation steps. In this example, we begin with the graph representations of three valid seriation solutions (1?) for a set of 9 assemblages (A-I). In the figure, the thickness of the edges reflects the summed differences in frequencies between each pair of assemblages. Each solution represents a valid and unique seriation. To combine the three seriations, we first sum the graphs to create a single aggregate solution that is composed of all nodes and edges from the individual graphs. Using the aggregate solution, we then reduce the graph by including the fewest edges that can be made between all vertices and starting with the edges that have the smallest weight, as calculated by the sum of the differences in frequencies. Edges that include new vertices are added sequentially until all of the connected vertices are included. Edges with equivalent weight values are retained as well. doi:10.1371/journal.pone.0124942.gPLOS ONE | DOI:10.1371/journal.pone.0124942 April 29,15 /The IDSS Frequency Seriation AlgorithmAs an example, Fig 7 represents a simulated case in which a set of assemblages that initially represent a single lineage with a single temporal order branches into two sub-populations, each having valid seriation.Ges in the sequence produced by the IDSS algorithm (Panel B, Fig 4). This simple graphic informs us about the relations between assemblages without the addition of the information regarding the composition of the types. The graph representation has an advantage over traditional centered-bar diagrams since it allows us to examine relations where assemblages may be shared in multiple valid solutions [89, 91]. The ability of graphs to reflect complex sets of relationships, however, can result in difficult interpretation of the results. The strength of seriation is that solutions are linear relations where the order reflects some combination of differences in time and space. However, if assemblages are found in more than one solution, additional analytic steps must be taken to reduce the results to something that can serve as a hypothesis about the structure of transmission and the relations between assemblages. As shown in Fig 6, we can proceed by accumulating valid solutions, and then pruning unnecessary edges. We begin at the top with three valid solutions output from the basic IDSS algorithm. Each meets the criteria for unimodality and the frequency differences are within the 0.1 tolerance limit set for continuity. In the middle of the figure, we show the results of agglomerating the graphs together, where an edge exists between two vertices if those vertices possess an edge in any of the three source graphs. The weights assigned to edges are proportional to the summed differences in type frequencies between pairs of assemblages. This aggregate graph allows us to construct the final solution. We follow the approach described by Lipo [89], starting with just the vertices, and iteratively adding edges from the summed graph starting with those which possess the lowest weight as measured by Euclidean distance between pairs of assemblages. This process produces a graph that includes the maximal set of vertices from the starting solutions but using the minimum number of edges that represent smallest distances between vertices and includes all equivalent values as options. The result is what we call the “minmax” graph.PLOS ONE | DOI:10.1371/journal.pone.0124942 April 29,14 /The IDSS Frequency Seriation AlgorithmFig 6. “Minmax” graph creation steps. In this example, we begin with the graph representations of three valid seriation solutions (1?) for a set of 9 assemblages (A-I). In the figure, the thickness of the edges reflects the summed differences in frequencies between each pair of assemblages. Each solution represents a valid and unique seriation. To combine the three seriations, we first sum the graphs to create a single aggregate solution that is composed of all nodes and edges from the individual graphs. Using the aggregate solution, we then reduce the graph by including the fewest edges that can be made between all vertices and starting with the edges that have the smallest weight, as calculated by the sum of the differences in frequencies. Edges that include new vertices are added sequentially until all of the connected vertices are included. Edges with equivalent weight values are retained as well. doi:10.1371/journal.pone.0124942.gPLOS ONE | DOI:10.1371/journal.pone.0124942 April 29,15 /The IDSS Frequency Seriation AlgorithmAs an example, Fig 7 represents a simulated case in which a set of assemblages that initially represent a single lineage with a single temporal order branches into two sub-populations, each having valid seriation.