Here we describe the basic model and outline the mathematical methods. Further details are given in Supplementary Note 1.

Model

Population structure is captured by a graph G with N vertices and directed edges that are possibly weighted and include self-loops. The vertices represent individuals, the edges represent interactions (Fig. 1). The weight of an edge between vertices i and j is denoted by \(w_{ij}\), and captures the rate at which vertex i interacts with j. Alternatively, the rates can be captured by allowing multiple edges between vertices, in which case the structure is modeled by an unweighted multigraph. The degree of a vertex is the number of edges adjacent to it.

Individuals are of two types: residents with fitness 1 and (advantageous) mutants with fitness r > 1. The fitness of individual occupying vertex i is denoted by f i .

The population evolves according to birth–death updating. In each step, one individual is chosen for reproduction randomly and proportionally to its fitness, and then one of the adjacent edges is chosen randomly and proportionally to the edge weight. The selected individual produces a copy of itself (birth) and sends this copy along the selected edge to replace the individual at the other end of the edge (death). That is, individual i is selected for reproduction with probability \(f_i/\mathop {\sum}

olimits_j f_j\) and its adjacent edge \((i,i{\prime})\) is then selected with probability \(w_{ii{\prime}}/\mathop {\sum}

olimits_j w_{ij}\). The individual at vertex i′ then becomes the same type as individual at vertex i. Note that due to different degrees and weights, an edge between individuals i and \(i{\prime}\) can be used more frequently in direction “from i to \(i{\prime}\)” than in direction “from \(i{\prime}\) to \(i\)”.

The state of the process is given by vector \((x_1,x_2, \ldots ,x_N)\), where x i = 1 represents that the individual occupying vertex i is a mutant and x i = 0 for a resident. The process starts with a single mutant on one vertex and stops either when all the individuals become mutants (fixation) or residents (extinction) (Fig. 1a).

Initialization scheme

In the beginning, the position of the single mutant can be chosen either uniformly at random (uniform initialization, U) or proportionally to the temperature of the vertex (temperature initialization, T). The temperature t h of vertex h is defined as

$$t_h = \mathop {\sum}\limits_i \frac{{w_{ih}}}{{\mathop {\sum}\limits_j w_{ij}}}$$ (1)

and corresponds to how frequently each vertex is being replaced by reproduction events happening in the neighboring vertices.

Amplifiers

The probability of fixation for a single mutant with relative fitness r appearing on graph G according to initialization scheme U (or T) is denoted by \(\rho (G,r,U)\) (or \(\rho (G,r,T)\)).

Denoting by K N the complete graph on N vertices, we have

$$\rho \left( {K_N,r,U} \right) = \rho \left( {K_N,r,T} \right) = \frac{{1 - 1{\mathrm{/}}r}}{{1 - 1{\mathrm{/}}r^N}} \to 1 - 1{\mathrm{/}}r$$ (2)

as \(N \to \infty\). Graphs for which the fixation probability is greater than this for any r > 1 are called unif-amplifiers or temp-amplifiers based on the initialization scheme.

The most well-known unif-amplifiers are Star graphs S N (see Fig. 1d) for which \(\rho (S_N,r,U) \to 1 - 1/r^2 > 1 - 1/r\) as \(N \to \infty\). However, Star graphs are not temp-amplifiers, since \(\rho (S_N,r,T) \to 0\).

We are interested in the behavior for large population sizes. A sequence of graphs \(G_1,G_2, \ldots\) of increasing size is called a strong unif-amplifier if, in the limit, the fixation probability under uniform initialization tends to 1 for arbitrary, fixed r > 1 (that is, if for any r > 1, we have \(\mathop {{\rm{lim}}}

olimits_{i \to \infty } \rho (G_i,r,U) \to 1\)). Strong temp-amplifiers are defined analogously, requiring that \(\mathop {{\rm{lim}}}

olimits_{i \to \infty } \rho (G_i,r,T) \to 1\).

Fundamental questions

Despite the rich interest in amplifiers of natural selection, many basic questions have remained unanswered. The fundamental open questions are the following.

First, are there strong amplifiers for temperature initialization? More generally, are there population structures that function as strong amplifiers for both uniform and temperature initialization?

Second, similarly to the Star, does there exist a graph without self-loops and/or without weights that is an amplifier for temperature initialization, achieving fixation probability at least 1 − 1/r2 for large N?

Third, are there simple structures, such as graphs with bounded degree, that are strong amplifiers for uniform initialization?

Construction of strong amplifiers

In our positive result, we answer the first fundamental question by proving that almost every family of graphs of increasing population size can be turned into a strong amplifier (both strong unif-amplifier and strong temp-amplifier) by allowing self-loops and assigning weights to edges. Self-loops are natural. They indicate that the offspring can replace the parent25. The standard Moran process is given by a complete graph with self-loops.

In the proof47, we start by defining a subset of vertices called a hub and partitioning the remaining vertices into subsets called branches in such a way that each branch connects to the hub. The partitioning has the property that the hub is larger than each branch individually, but smaller than all of them combined. Such a partitioning is possible for all graphs that have diameter polynomially smaller than N (i.e., the distance between every pair of nodes is at most \(N^{1 - \varepsilon }\), where \(\varepsilon \,> \,0\) is fixed and independent of N).

Once we construct the partitioning with the required properties, we proceed by assigning weights in such a way that, intuitively, (i) in each branch, there is a sense of global flow directed toward the hub; and (ii) the hub is isothermal and evolves faster than the rest of the graph. The success of the construction then relies on the following two principles.

First, the weights create a sense of global flow. The weight assignment in the branches guarantees that every edge is used more frequently in the direction toward the hub than in the direction away from the hub. Moreover, by assigning suitable weights to the self-loops, we achieve that edges closer to the hub are used more frequently then edges further away from the hub. See Fig. 5a for a small numerical illustration. These two facts imply that a mutant arising in a branch will propagate toward the hub and repeatedly try to invade it.

Fig. 5 Details of steps to fixation. a Assigning different weights to edges and self-loops changes the frequency with which each edge is used in each direction. Thicker arrows indicate edges that are used more frequently. b Our weight assignment creates a global sense of flow in the branches, directed toward the hub. The hub itself is almost isothermal and evolves fast. c Three stages to fixation illustrated on a single branch and the connecting vertex in the hub. After fixating on the hub at the end of Stage 2 (hub becomes dark orange), mutants spread to all the branches and fixate on the whole graph Full size image

Second, there is an important asymmetry between mutants and residents on a well-mixed population. For large population size N, the fixation probability of a single mutant with fitness r > 1 invading a well-mixed population of N residents tends to the positive constant 1 − 1/r. On the other hand, the probability that a single resident takes over a large well-mixed population of advantageous mutants is ~r−N, i.e., exponentially small in N. The weight assignment within the hub makes the hub behave approximately like a well-mixed population. Therefore, once the mutants fixate in the hub, they are extremely likely to resist the upcoming invasion attempts of the residents.

With these two principles in mind, we can informally argue as follows. Since the hub occupies only a small portion of the graph, the first mutant most likely appears in some branch. We focus on that branch and the hub (see Fig. 5b) and prove that due to the biased flow toward the hub, the mutants spread all the way to the hub (see Fig. 5c, Stage 1). Once in the hub, the mutants have a constant chance to fixate there. If the first invading mutant lineage fails in the hub, another such lineage will be generated from the branch. Eventually, the mutants take over the hub (see Fig. 5c, Stage 2). From that point on, it is extremely unlikely that the residents could win the hub back. In order to fixate on the branch, the mutants have to proceed against the natural direction of the flow, which is, in absolute terms, fairly improbable. However, the alternative of residents taking over the hub is much more improbable. Thus, with high probability, the mutants will succeed in fixating on the branch (see Fig. 5c, Stage 3). Similarly, they fixate on all the other branches.

For details, see Supplementary Note 1, Section 5, and the references therein.

Necessary conditions for amplification

In our negative results, we answer the second and the third fundamental question by proving that both self-loops and weighted edges are essential for existence of strong amplifiers.

First, in order to address the second fundamental question, we consider temperature initialization on an arbitrary graph.

In order to prove that self-loops are necessary features for strong amplification, we consider a graph without self-loops (possibly with weighted edges). Then, given any possible starting position i of the mutant, we find that the probability, p i , that the mutant is replaced by one of its resident neighbors before it reproduces even once equals \(p_i = t_i/(t_i + r)\), where t i is the temperature of vertex i. Therefore, the fixation probability starting from a state with single mutant at vertex i is at most \(1 - t_i/(t_i + r)\). Taking all possible starting positions into account, we establish an upper bound \(1 - 1/(r + 1)\) on the fixation probability using Cauchy–Schwarz inequality. This implies that without self-loops, strong amplification under temperature initialization is not possible.

In order to prove that weighted edges are also necessary, we consider an unweighted graph (possibly with self-loops). We argue similarly and establish an upper bound \(1 - 1/(4r + 2)\) on the fixation probability. Thus, the second fundamental question is answered in negative.

Second, in order to address the third fundamental question, we consider uniform initialization on graphs with bounded degree. As above, we first consider a graph without self-loops (possibly with weighted edges). Given any such graph G, we single out a subset Vh of vertices with high temperature that we call hot vertices. Formally, Vh consists of vertices that are replaced by at least one of their neighbors with rate at least 1/c, where c is the constant that bounds the degree. We prove that the subset Vh is large (namely, \(\left| {V^h} \right| \ge N{\mathrm{/}}c\)) and that the fixation probability starting from a single hot vertex is small (namely, smaller than \(rc{\mathrm{/}}(1 + rc)\)). Accounting of all hot vertices, we establish that \(\rho (G,r,U) \le 1 - 1{\mathrm{/}}(c + cr^2)\). The case of unweighted graphs (that possibly have self-loops) follows similarly, by noticing that under bounded degree, all vertices are sufficiently hot. The bound we obtain is \(\rho (G,r,U) \le 1 - 1{\mathrm{/}}(1 + rc)\). Altogether, this answers the third fundamental question in negative. For details, see Supplementary Note 1, Section 4, and the references therein.

Data and code availability

The data sets generated and analyzed during the current study and the related computer code are available in the Figshare48 repository, https://doi.org/10.6084/m9.figshare.6323240.v1.