Natural Models for Evolution on Networks

Evolutionary dynamics have been traditionally studied in the context of homogeneous populations, mainly described my the Moran process. Recently, this approach has been generalized in \cite{LHN} by arranging individuals on the nodes of a network. Undirected networks seem to have a smoother behavior than directed ones, and thus it is more challenging to find suppressors/amplifiers of selection. In this paper we present the first class of undirected graphs which act as suppressors of selection, by achieving a fixation probability that is at most one half of that of the complete graph, as the number of vertices increases. Moreover, we provide some generic upper and lower bounds for the fixation probability of general undirected graphs. As our main contribution, we introduce the natural alternative of the model proposed in \cite{LHN}, where all individuals interact simultaneously and the result is a compromise between aggressive and non-aggressive individuals. That is, the behavior of the individuals in our new model and in the model of \cite{LHN} can be interpreted as an “aggregation” vs. an “all-or-nothing” strategy, respectively. We prove that our new model of mutual influences admits a \emph{potential function}, which guarantees the convergence of the system for any graph topology and any initial fitness vector of the individuals. Furthermore, we prove fast convergence to the stable state for the case of the complete graph, as well as we provide almost tight bounds on the limit fitness of the individuals. Apart from being important on its own, this new evolutionary model appears to be useful also in the abstract modeling of control mechanisms over invading populations in networks. We demonstrate this by introducing and analyzing two alternative control approaches, for which we bound the time needed to stabilize to the “healthy” state of the system.

💡 Research Summary

This paper investigates evolutionary dynamics on graphs by extending the classic Moran process in two distinct directions. The first extension follows the model introduced by Lieberman, Hauert, and Nowak (LHN), which we refer to as the “all‑or‑nothing” approach. In this discrete‑time process, at each step a single individual is chosen for reproduction with probability proportional to its fitness; its offspring replaces a randomly selected neighbor. The state of the system is a subset of vertices occupied by mutants, yielding a Markov chain with a state space of size 2ⁿ. The two absorbing states correspond to fixation (all vertices mutant) and extinction (no mutants). While the fixation probability on a complete graph is known analytically (ρ = (1‑1/r)/(1‑r⁻ⁿ)), computing it for arbitrary undirected graphs is intractable because of the exponential state space. The authors therefore derive two families of generic bounds that apply to any connected undirected graph. The first bound relates the fixation probability f(G) to the ratio of minimum and maximum vertex degrees (Δ_min/Δ_max) and the complete‑graph probability ρ, yielding (Δ_min/Δ_max)·ρ ≤ f(G) ≤ (Δ_max/Δ_min)·ρ. The second bound depends on local neighbourhood structure: for any vertex v, the sum of the reciprocals of the degrees of its neighbors provides a tighter upper bound. These results generalize earlier analyses that were limited to regular graphs, stars, and paths.

The second contribution of the paper is a new “aggregation” model, which can be viewed as a smooth counterpart of the LHN process. In this model all individuals interact simultaneously at each time step. The influence that a vertex u exerts on a neighbor v is proportional to u’s fitness f_u and to the normalized edge weight w_uv = 1/deg(u). The dynamics can be written compactly as x(t+1) = W·diag(f)·x(t), where x(t) is the vector of fitnesses, W is the row‑stochastic adjacency matrix, and diag(f) is a diagonal matrix of current fitnesses. The authors introduce a potential function Φ(x) = Σ_i (x_i – \bar{x})², where \bar{x} is the average fitness. They prove that Φ strictly decreases at every step unless the system is already at equilibrium, guaranteeing convergence to a unique stable state in which all vertices share the same fitness. For the complete graph K_n they obtain explicit convergence rates: the distance to equilibrium shrinks exponentially fast, with a mixing time of O(log n). Moreover, they derive almost tight bounds on the limiting fitness value as a function of the initial mutant proportion α and mutant fitness r, showing that the final fitness lies between (1‑α)+α·r·(1‑e^{-c·t/n}) and its symmetric counterpart.

A particularly notable result is the construction of the first known family of undirected graphs that act as suppressors of selection under the LHN model. The authors define the “clique‑wheel” graph G_n, consisting of an n‑vertex clique together with an n‑vertex cycle that is perfectly matched to the clique vertices. When a mutant is placed on a clique vertex, the probability of eventual fixation tends to zero as n grows; when placed on a cycle vertex, the fixation probability approaches the complete‑graph value 1‑1/r. Consequently, the overall fixation probability of G_n is bounded above by ½(1‑1/r), i.e., at most half of that of a complete graph, establishing G_n as a suppressor of selection. This resolves a long‑standing open problem: no undirected suppressor had been identified before.

Finally, the paper leverages the aggregation model to study two control strategies aimed at eradicating an invading population (e.g., a virus or cancer cells) on a complete graph. In the first “phased” strategy, a small fraction of vertices is periodically reset to the healthy fitness (≈1); after each reset the system is allowed to converge before the next phase. The authors prove that the number of phases required scales logarithmically with r‑1 and is independent of the population size n. In the second “continuous” strategy, a small fraction of vertices is adjusted at every time step. Here the total number of steps needed to reach the healthy state scales linearly with n and proportionally to r·ln(r‑1). Both strategies guarantee that the system reaches a state where all fitnesses are arbitrarily close to 1, demonstrating the practical utility of the aggregation model for designing intervention protocols.

In summary, the paper makes four major contributions: (1) it provides the first undirected suppressor of selection for the LHN process; (2) it establishes general upper and lower bounds on fixation probabilities for arbitrary undirected graphs; (3) it introduces an aggregation‑based evolutionary model equipped with a Lyapunov‑type potential function that ensures convergence and yields explicit convergence rates on complete graphs; and (4) it applies this model to devise and analyze two concrete control mechanisms for eliminating invasive mutants. These results bridge evolutionary game theory, stochastic processes on networks, and control of spreading phenomena, offering both theoretical insights and actionable tools for researchers in computational biology, network science, and related fields.