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arxiv: 2605.26411 · v3 · pith:HLSTGI2Lnew · submitted 2026-05-26 · 🧬 q-bio.PE

Fixation location in structured populations

David A. Brewster , Gabor Lippner , Josef Tkadlec , Martin A. Nowak This is my paper

Pith reviewed 2026-06-30 11:21 UTC · model grok-4.3

classification 🧬 q-bio.PE
keywords fixation locationevolutionary graph theorystructured populationsbirth-death processneutral evolutionconstant selectiongraph structuremutant fixation
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The pith

The node hosting the last wild-type before mutant fixation follows a nonuniform probability distribution set by graph structure and selection strength.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In evolutionary dynamics on graphs the replacement of one type by another occurs at a specific node: the position of the final wild-type individual just before fixation. The paper establishes that the probability each node is this location varies strongly with the underlying graph and with the intensity of selection. The resulting distribution can be sharply different even for graphs that share the same fixation probability or fixation time. On many graphs certain nodes have exactly zero probability of ever being the fixation location. This spatial detail supplies information about the process that is invisible to the usual aggregate statistics.

Core claim

Conditional on fixation, the fixation location is the node occupied by the last wild-type individual immediately before mutant fixation. This location is described by a probability distribution over the nodes of the graph. The distribution is often highly nonuniform, depends strongly on the graph structure and the selection strength, and can differ sharply even when classical fixation statistics are similar. For many graphs some nodes can never be fixation locations. The result holds for neutral evolution, colonization, and constant selection on small graphs, cycles, tori, random graphs, and island populations.

What carries the argument

Conditional probability distribution of the fixation location, obtained from the transition probabilities of the Markov chain describing birth-death updating on the graph.

If this is right

  • The fixation-location distribution supplies information about evolutionary outcomes that is not contained in fixation probability or mean fixation time.
  • Certain nodes on many graphs are impossible sites for fixation, so the process cannot eliminate the wild-type last at those positions.
  • The nonuniformity changes with selection strength, becoming more or less pronounced as selection moves from neutral to strong.
  • The same qualitative features appear on cycles, tori, random graphs, and island models, indicating the phenomenon is not limited to special topologies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Network designers could select topologies that make the fixation location more predictable or that protect particular nodes from serving as the final replacement site.
  • In cultural or opinion dynamics the fixation location would correspond to the last individual holding an old trait, offering a spatial marker for tracking large-scale change.
  • Empirical tracking of the final wild-type position in spatially structured microbial or tumor populations could test the predicted distributions directly.
  • The same conditional-distribution calculation could be repeated for death-birth updating or frequency-dependent fitness to check whether nonuniformity persists.

Load-bearing premise

The evolutionary process on the graph is a Markovian birth-death updating rule under neutral evolution, colonization, or constant selection that permits the conditional distribution of the last wild-type node to be computed from the transition probabilities.

What would settle it

Simulations or exact calculations on a cycle graph under neutral drift showing that every node has exactly the same probability of being the fixation location, or a node with theoretically zero probability appearing as the fixation location in any realization.

Figures

Figures reproduced from arXiv: 2605.26411 by David A. Brewster, Gabor Lippner, Josef Tkadlec, Martin A. Nowak.

Figure 1
Figure 1. Figure 1: Fixation location probabilities for a directed graph on N = 12 vertices with r = 1.25. a, Locations are labelled i = 1, 2, . . . , 12. b, Initially, a single mutant occupies location i = 1 (green); wild types occupy the remaining locations (blue). c, Some locations cannot possibly be the fixation location (black with white Xs); locations i = 1, 2, 4, 7, 9, 11. d, Fixation location probabilities are illustr… view at source ↗
Figure 2
Figure 2. Figure 2: Fixation location probabilities for all 21 connected undirected graphs (circled, cream-colored vertices) on N = 5 vertices and all non-isomorphic initial mutant locations, for r → ∞. The initial mutant location is green. The locations with the highest fixation location probabilities are red. Impossible fixation locations are white. All other possible fixation locations are yellow. 2 [PITH_FULL_IMAGE:figur… view at source ↗
Figure 3
Figure 3. Figure 3: Directed cycle populations. Location i is connected to location i + 1 with a directed edge (location N is connected to location 1). Initially, a single mutant occupies location i = 1. The remaining population consists of wild types. a, Directed cycle with N = 10. b,c,d, Numerical computation of the fixation location probability p for r = 1, 2, 3, respectively. We show population sizes of N = 5, 10, 20, 30,… view at source ↗
Figure 4
Figure 4. Figure 4: Bidirectional (undirected) cycle populations. Location i is connected to location i + 1 with a directed edge and vice versa (location N is connected to location 1). Initially, a single mutant occupies location i = 1. The remaining population consists of wild types. a, Bidirectional cycle with N = 10. b,c,d, Numerical computation of the fixation location probability p for r = 1, 2, 3, respectively. We show … view at source ↗
Figure 5
Figure 5. Figure 5: Periodic square grid (torus). a, Periodic square grid on N = 112 = 121 vertices. b, Probability p that the fixation location is at a particular vertex when the initial mutant is located at the center (red star), under strong selection (r = 10). This is the result of simulations of 103 fixations. c, The average probability q that a vertex v that is distance d from initial mutant location is the fixation loc… view at source ↗
Figure 6
Figure 6. Figure 6: Undirected random regular graphs. All vertices have degree k. a, Instance of a random regular graph on N = 50 vertices with degree k = 10. b, Average distance from initial mutant location for r → ∞ (blue) versus a uniformly sampled node (orange), for 104 simulations, and k = 3, 5, 10 (panels from left to right). The plot has a logarithmic horizontal axis. The dotted lines are the lines of best fit with sha… view at source ↗
Figure 7
Figure 7. Figure 7: Island populations. There are two islands—island 1 (green) and island 2 (purple). Each island has a population size of n. Thus the total population size is N = 2n. Each island is a well-mixed population and there is migration between the islands. More concretely, m12 and m21 are migration probabilities. When an individual on island 1 reproduces, its offspring is placed on island 2 with probability m12 and … view at source ↗
read the original abstract

In stochastic evolutionary dynamics, the replacement of an existing genotype or cultural trait by a newly introduced mutant is typically characterized by the quantities of fixation probability and fixation time. But in a structured population, the disappearance of a lineage occurs at a specific place. For evolutionary dynamics on graphs, we define the fixation location as the node occupied by the last wild-type individual immediately before mutant fixation. Conditional on fixation, this location is described by a probability distribution over the nodes of the graph. We study the fixation location for neutral evolution, for the colonization process, and, more generally, for constant selection on small graphs, cycles, tori, random graphs, and island populations. We find that the distribution of the fixation location is often highly nonuniform, depends strongly on the graph structure and the selection strength, and can differ sharply even when classical fixation statistics are similar. For many graphs, some nodes can never be fixation locations. Our results identify fixation location as a fundamental aspect of evolutionary dynamics and suggest new ways to understand, monitor, and potentially mitigate extinction events in biological and social settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the fixation location as the node occupied by the last wild-type individual immediately before mutant fixation in birth-death processes on graphs. Conditional on fixation, it defines and computes the probability distribution of this location under neutral evolution, colonization, and constant selection. The authors examine this distribution on small graphs, cycles, tori, random graphs, and island models, reporting that it is frequently highly nonuniform, depends on graph structure and selection strength, can differ markedly even when fixation probability and time are similar, and that certain nodes are unreachable as fixation locations.

Significance. If the reported distributions hold, the work adds a spatially resolved observable to evolutionary graph theory that is not reducible to fixation probability or time. The systematic comparison across neutral, colonization, and selection regimes on standard graph families is a strength, as is the observation that some nodes are structurally excluded from being fixation locations. This could inform models of local extinction in biological and social networks.

minor comments (3)
  1. The manuscript should explicitly state the precise birth-death update rule (e.g., whether birth or death is selected first) and the normalization of fitness-dependent probabilities in the methods section, as these choices affect the transition matrix used to compute the conditional distribution.
  2. Figure captions and axis labels for the reported distributions should include the exact parameter values (selection strength, graph size) used in each panel to allow direct reproduction.
  3. A brief discussion of computational method (exact solution of the absorbing Markov chain versus Monte Carlo sampling) and any truncation or approximation employed for larger graphs would strengthen the results section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary and recommendation of minor revision. The assessment correctly identifies the core contribution: that the fixation location distribution is a distinct, spatially resolved observable that is not reducible to fixation probability or time and that varies with graph structure and selection regime. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces fixation location as a new observable (the node of the final wild-type individual before absorption) and derives its conditional distribution directly from the one-step transition matrix of the birth-death Markov chain on the graph. This computation is a standard consequence of solving the absorbing chain for the probability of each possible last-wild-type state; it requires no fitted parameters, no self-referential equations, and no load-bearing self-citations. The reported nonuniformity and impossible nodes are immediate algebraic outcomes of the adjacency matrix and fitness-dependent birth rates, with no reduction of the claimed results to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions of stochastic evolutionary dynamics on graphs; no free parameters or invented physical entities are indicated.

axioms (1)
  • domain assumption Evolutionary replacement on graphs follows a stochastic birth-death process determined by fitness and graph edges.
    This is the modeling framework invoked throughout the abstract for neutral, colonization, and constant-selection cases.

pith-pipeline@v0.9.1-grok · 5719 in / 1309 out tokens · 44530 ms · 2026-06-30T11:21:57.202408+00:00 · methodology

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