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.


Introduction
Evolutionary dynamics have been well studied (see [2,7,8,14,22,24,25]), mainly in the context of homogeneous populations, described by the Moran process [18,20].In addition, population dynamics have been extensively studied also from the perspective of the strategic interaction in evolutionary game theory, cf. for instance [10][11][12][13]23].One of the main targets of evolutionary game theory is evolutionary dynamics (see [11,26]).Such dynamics usually examine the propagation of intruders with a given fitness to a population, whose initial members (resident individuals) have a different fitness.In fact, "evolutionary stability" is the case where no dissident behaviour can invade and dominate the population.The evolutionary models and the dynamics we consider here belong to this framework.In addition, however, we consider structured populations (i.e. in the form of an undirected graph) and we study how the underlying graph structure affects the evolutionary dynamics.We study in this paper two kinds of evolutionary dynamics.Namely, the "all or nothing" case (where either the intruder overtakes the whole graph or die out) and the "aggregation" case (more similar in spirit to classical evolutionary game theory, where the intruder's fitness aggregates with the population fitness and generates eventually a homogeneous crowd with a new fitness).
In a recent article, Lieberman, Hauert, and Nowak proposed a generalization of the Moran process by arranging individuals on a connected network (i.e.graph) [16] (see also [21]).In this model, vertices correspond to individuals of the population and weighted edges represent the reproductive rates between the adjacent vertices.That is, the population structure is translated into a network (i.e.graph) structure.Furthermore, individuals (i.e.vertices) are partitioned into two types: aggressive and non-aggressive.The degree of (relative) aggressiveness of an individual is measured by its relative fitness; in particular, non-aggressive and aggressive individuals are assumed to have relative fitness 1 and r ≥ 1, respectively.This modeling approach initiates an ambitious direction of interdisciplinary research, which combines classical aspects of computer science (such as combinatorial structures and complex network topologies), probabilistic calculus (discrete Markov chains), and fundamental aspects of evolutionary game theory (such as evolutionary dynamics).
In the model of [16], one mutant (or invader ) with relative fitness r ≥ 1 is introduced into a given population of resident individuals, each of whom having relative fitness 1.For simplicity, a vertex of the graph that is occupied by a mutant will be referred to as black, while the rest of the vertices will be referred to as white.At each time step, an individual is chosen for reproduction with a probability proportional to its fitness, while its offspring replaces a randomly chosen neighboring individual in the population.Once u has been selected for reproduction, the probability that vertex u places its offspring into position v is given by the weight w uv of the directed arc uv .This process stops when either all vertices of the graph become black (resulting to a fixation of the graph) or they all become white (resulting to extinction of the mutants).Several similar models have been previously studied, describing for instance influence propagation in social networks (such as the decreasing cascade model [15,19]), dynamic monopolies [3], particle interactions (such as the voter model, the antivoter model, and the exclusion process [1,9,17]), etc.However, the dynamics emerging from these models do not consider different fitnesses for the individuals.
The fixation probability f G of a graph G = (V, E) is the probability that eventually fixation occurs, i.e. the probability that an initially introduced mutant, placed uniformly at random on a vertex of G, eventually spreads over the whole population V , replacing all resident individuals.One of the main characteristics in this model is that at every iteration of the process, a "battle" takes place between aggressive and nonaggressive individuals, while the process stabilizes only when one of the two teams of individuals takes over the whole population.This kind of behavior of the individuals can be interpreted as an all-or-nothing strategy.
Lieberman et al. [16] proved that the fixation probability for every symmetric directed graph (i.e. when w uv = w vu for every u, v) is equal to that of the complete graph (i.e. the homogeneous population of the Moran process), which tends to 1 − 1 r as the size n of the population grows.Moreover, exploiting vertices with zero in-degree or zero out-degree ("upstream" and "downstream" populations, respectively), they provided several examples of directed graphs with arbitrarily small and arbitrarily large fixation probability [16].Furthermore, the existence of directions on the arcs leads to examples where neither fixation nor extinction is possible (e.g. a graph with two sources).
In contrast, general undirected graphs (i.e. when uv ∈ E if and only if vu ∈ E for every u, v) appear to have a smoother behavior, as the above process eventually reaches fixation or extinction with probability 1.Furthermore, the coexistence of both directions at every edge in an undirected graph seems to make it more difficult to find suppressors or amplifiers of selection (i.e.graphs with smaller or greater fixation probability than the complete graph, respectively), or even to derive non-trivial upper and lower bound for the fixation probability on general undirected graphs.This is the main reason why only little progress has been done so far in this direction and why most of the recent work focuses mainly on the exact or numerical computation of the fixation probability for very special cases of undirected graphs, e.g. the star and the path [4][5][6].
Our contribution.In this paper we overcome this difficulty for undirected graphs and we provide the first class of undirected graphs that act as suppressors of selection in the model of [16], as the number of vertices increases.This is a very simple class of graphs (called clique-wheels), where each member G n has a clique of size n ≥ 3 and an induced cycle of the same size n with a perfect matching between them.We prove that, when the mutant is introduced to a clique vertex of G n , then the probability of fixation tends to zero as n grows.Furthermore, we prove that, when the mutant is introduced to a cycle vertex of G n , then the probability of fixation is at most 1 − 1 r as n grows (i.e. to the same value with the homogeneous population of the Moran process).Therefore, since the clique and the cycle have the same number n of vertices in G n , the fixation probability f Gn of G n is at most 1 2 (1 − 1 r ) as n increases, i.e.G n is a suppressor of selection.Furthermore, we provide for the model of [16] the first non-trivial upper and lower bounds for the fixation probability in general undirected graphs.In particular, we first provide a generic upper bound depending on the degrees of some local neighborhood.Second, we present another upper and lower bound, depending on the ratio between the minimum and the maximum degree of the vertices.
As our main contribution, we introduce in this paper the natural alternative of the all-or-nothing approach of [16], which can be interpreted as an aggregation strategy.In this aggregation model, all individuals interact simultaneously and the result is a compromise between the aggressive and non-aggressive individuals.Both these two alternative models for evolutionary dynamics coexist in several domains of interaction between individuals, e.g. in society (dictatorship vs. democracy, war vs. negotiation) and biology (natural selection vs. mutation of species).In particular, another motivation for our models comes from biological networks, in which the interacting individuals (vertices) correspond to cells of an organ and advantageous mutants correspond to viral cells or cancer.Regarding the proposed model of mutual influences, we first prove that it admits a potential function.This potential function guarantees that for any graph topology and any initial fitness vector, the system converges to a stable state, where all individuals have the same fitness.Furthermore, we analyze the telescopic behavior of this model for the complete graph.In particular, we prove fast convergence to the stable state, 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 enables also the abstract modeling of new control mechanisms over invading populations in networks.We demonstrate this by introducing and analyzing the behavior of two alternative control approaches.In both scenarios we periodically modify the fitness of a small fraction of individuals in the current population, which is arranged on a complete graph with n vertices.In the first scenario, we proceed in phases.Namely, after each modification, we let the system stabilize before we perform the next modification.In the second scenario, we modify the fitness of a small fraction of individuals at each step.In both alternatives, we stop performing these modifications of the population whenever the fitness of every individual becomes sufficiently close to 1 (which is considered to be the "healthy" state of the system).For the first scenario, we prove that the number of phases needed for the system to stabilize in the healthy state is logarithmic in r − 1 and independent of n.For the second scenario, we prove that the number of iterations needed for the system to stabilize in the healthy state is linear in n and proportional to r ln(r − 1).
Notation.In an undirected graph G = (V, E), the edge between vertices u ∈ V and v ∈ V is denoted by uv ∈ E, and in this case u and v are said to be adjacent in G.If the graph G is directed, we denote by uv the arc from u to v.For every vertex u ∈ V in an undirected graph G = (V, E), we denote by Organization of the paper.We discuss in Section 2 the two alternative models for evolutionary dynamics on graphs.In particular, we formally present in Section 2.1 the model of [16] and then we introduce in Section 2.2 our new model of mutual influences.In Section 3 we first provide generic upper and lower bounds of the fixation probability in the model of [16] for arbitrary undirected graphs.Then we present in Section 3.3 the first class of undirected graphs which act as suppressors of selection in the model of [16], as the number of vertices increases.In Section 4 we analyze our new evolutionary model of mutual influences.In particular, we first prove in Section 4.1 the convergence of the model by using a potential function, and then we analyze in Section 4.2 the telescopic behavior of this model for the case of a complete graph.In Section 5 we demonstrate the use of our new model in analyzing the behavior of two alternative invasion control mechanisms.Finally, we discuss the presented results and further research in Section 6.

All-or-nothing vs. aggregation
In this section we formally define the model of [16] for undirected graphs and we introduce our new model of mutual influences.Similarly to [16], we assume for every edge uv of an undirected graph that w uv = 1 deg u and w vu = 1 deg v , i.e. once a vertex u has been chosen for reproduction, it chooses one of its neighbors uniformly at random.

The model of Lieberman, Hauert, and Nowak (an all-or-nothing approach)
Let G = (V, E) be a connected undirected graph with n vertices.Then, the stochastic process defined in [16] can be described by a Markov chain with state space S = 2 V (i.e. the set of all subsets of V ) and transition probability matrix P , where for any two states S 1 , S 2 ⊆ V , Notice that in the above Markov chain there are two absorbing states, namely ∅ and V , which describe the cases where the vertices of G are all white or all black, respectively.Since G is connected, the above Markov chain will eventually reach one of these two absorbing states with probability 1.If we denote by h v the probability of absorption at state V , given that we start with a single mutant placed initially on vertex v, then by definition f G = v hv n .Generalizing this notation, let h S be the probability of absorption at V given that we start at state S ⊆ V , and let h = [h S ] S⊆V .Then, it follows that vector h is the unique solution of the linear system h = P • h with boundary conditions h ∅ = 0 and h V = 1.
However, observe that the state space S = 2 V of this Markov chain has size 2 n , i.e. the matrix P = [P S1,S2 ] in (1) has dimension 2 n × 2 n .This indicates that the problem of computing the fixation probability f G of a given graph G is hard, as also mentioned in [16].This is the main reason why, to the best of our knowledge, all known results so far regarding the computation of the fixation probability of undirected graphs are restricted to regular graphs, stars, and paths [4][5][6]16,21].In particular, for the case of regular graphs, the above Markov chain is equivalent to a birth-death process with n − 1 transient (non-absorbing) states, where the forward bias at every state (i.e. the ratio of the forward probability over the backward probability) is equal to r.In this case, the fixation probability is equal to cf. [21], chapter 8.It is worth mentioning that, even for the case of paths, there is no known exact or approximate formula for the fixation probability [6].

An evolutionary model of mutual influences (an aggregation approach)
The evolutionary model of [16] constitutes a sequential process, in every step of which only two individuals interact and the process eventually reaches one of two extreme states.However, in many evolutionary processes, all individuals may interact simultaneously at each time step, while some individuals have greater influence to the rest of the population than others.This observation leads naturally to the following model for evolution on graphs, which can be thought as a smooth version of the model presented in [16].Consider a population of size n and a portion α ∈ [0, 1] of newly introduced mutants with relative fitness r.The topology of the population is given in general by a directed graph G = (V, E) with |V | = n vertices, where the directed arcs of E describe the allowed interactions between the individuals.At each time step, every individual u ∈ V of the population influences every individual v ∈ V , for which uv ∈ E, while the degree of this influence is proportional to the fitness of u and to the weight w uv of the arc uv .Note that we can assume without loss of generality that the weights w uv on the arcs are normalized, i.e. for every fixed vertex u ∈ V it holds uv ∈E w uv = 1 .Although this model can be defined in general for directed graphs with arbitrary arc weights w uv , we will focus in the following to the case where G is an undirected graph (i.e.u i u j ∈ E if and only if u j u i ∈ E, for every i, j) and w uv = 1 deg(u) for all edges uv ∈ E. Formally, let V = {u 1 , u 2 , . . ., u n } be the set of vertices and r ui (k) be the fitness of the vertex u i ∈ V at iteration k ≥ 0. Let Σ(k) denote the sum of the fitnesses of all vertices at iteration k, i.e.Σ(k) = n i=1 r ui (k).Then the vector r(k + 1) with the fitnesses r ui (k + 1) of the vertices i.e. r(k In the latter equation, the elements of the square matrix P = [P ij ] n i,j=1 depend on the iteration k and they are given as follows: Note by ( 4) and ( 5) that after the first iteration, the fitness of every individual in our new evolutionary model of mutual influences equals the expected fitness of this individual in the model of [16] (cf.Section 2.1).However, this correlation of the two models is not maintained in the next iterations and the two models behave differently as the processes evolve.
In particular, in the case where G is the complete graph, i.e. deg(u i ) = n − 1 for every vertex u i , the matrix P becomes The system given by ( 4) and ( 5) can be defined for every initial fitness vector r(0).However, in the case where there is initially a portion α ∈ [0, 1] of newly introduced mutants with relative fitness r, the initial condition r(0) of the system in (3) is a vector with αn entries equal to r and with (1 − α)n entries equal to 1.
Observation 1 Note that the recursive equation ( 4) is a non-linear equation on the fitness values r uj (k) of the vertices at iteration k.
Since by ( 5) the sum of every row of the matrix P equals to one, the fitness r ui (k) of vertex u i after the (k + 1)-th iteration of the process is a convex combination of the fitnesses of the neighbors of u i after the k-th iteration.Therefore, in particular, the fitness of every vertex u i at every iteration k ≥ 0 lies between the smallest and the greatest initial fitness of the vertices, as the next observation states.
Observation 2 Let r min and r max be the smallest and the greatest initial fitness in r(0), respectively.Then r min ≤ r ui (k) ≤ r max for every u i ∈ V and every k ≥ 0. Degree of influence.Suppose that initially αn mutants (for some α ∈ [0, 1]) with relative fitness r ≥ 1 are introduced in graph G on a subset S ⊆ V of its vertices.Then, as we prove in Theorem 5, after a certain number of iterations the fitness vector r(k) converges to a vector [r S 0 , r S 0 , . . ., r S 0 ] T , for some value r S 0 .This limit fitness r S 0 depends in general on the initial relative fitness r of the mutants, on their initial number αn, as well as on their initial position on the vertices of S ⊆ V .The relative fitness r of the initially introduced mutants can be thought as having the "black" color, while the initial fitness of all the other vertices can be thought as having the "white" color.Then, the limit fitness r S 0 can be thought as the "degree of gray color" that all the vertices obtain after sufficiently many iterations, given that the mutants are initially placed at the vertices of S. In the case where the αn mutants are initially placed with uniform probability to the vertices of G, we can define the limit fitness r 0 of G as For a given initial value of r, the bigger is r 0 the stronger is the effect of natural selection in G.
Since r S 0 is a convex combination of r and 1, there exists a value f G,S (r) Then, the value f G,S (r) is the degree of influence of the graph G, given that the mutants are initially placed at the vertices of S. In the case where the mutants are initially placed with uniform probability at the vertices of G, we can define the degree of influence of G as Number of iterations to stability.For some graphs G, the fitness vector r(k) reaches exactly the limit fitness vector [r 0 , r 0 , . . ., r 0 ] T (for instance, the complete graph with two vertices and one mutant not only reaches this limit in exactly one iteration, but also the degree of influence is exactly the fixation probability of this simple graph).However, for other graphs G the fitness vector r(k) converges to [r 0 , r 0 , . . ., r 0 ] T (cf.Theorem 5 below), but it never becomes equal to it.In the first case, one can compute (exactly or approximately) the number of iterations needed to reach the limit fitness vector.In the second case, given an arbitrary ε > 0, one can compute the number of iterations needed to come ε-close to the limit fitness vector.

Analysis of the all-or-nothing model
In this section we present analytic results on the evolutionary model of [16], which is based on the sequential interaction among the individuals.In particular, we first present non-trivial upper and lower bounds for the fixation probability, depending on the degrees of vertices.Then we present the first class of undirected graphs that act as suppressors of selection in the model of [16], as the number of vertices increases.
Recall by the preamble of Section 2.2 that, similarly to [16], we assumed that w uv = 1 deg u and for every edge uv of an undirected graph G = (V, E).It is easy to see that this formulation is equivalent to assigning to every edge e = uv ∈ E the weight w e = w uv = w vu = 1, since also in this case, once a vertex u has been chosen for reproduction, it chooses one of its neighbors uniformly at random.A natural generalization of this weight assignment is to consider G as a complete graph, where every edge e in the clique is assigned a non-negative weight w e ≥ 0, and w e is not necessarily an integer.Note that, whenever w e = 0, it is as if the edge e is not present in G.Then, once a vertex u has been chosen for reproduction, u chooses any other vertex v with probability wuv x =u wux .Note that, if we do not impose any additional constraint on the weights, we can simulate multigraphs by just setting the weight of an edge to be equal to the multiplicity of this edge.Furthermore, we can construct graphs with arbitrary small fixation probability.For instance, consider an undirected star with n leaves, where one of the edges has weight an arbitrary small ε > 0 and all the other edges have weight 1.Then, the leaf that is incident to the edge with weight ε acts as a source in the graph as ε → 0. Thus, the only chance to reach fixation is when we initially place the mutant at the source, i.e. the fixation probability of this graph tends to 1 n+1 as ε → 0. Therefore, it seems that the difficulty to construct strong suppressors lies in the fact that unweighted undirected graphs can not simulate sources.For this reason, we consider in the remainder of this paper only unweighted undirected graphs.

A generic upper bound approach
In the next theorem we provide a generic upper bound of the fixation probability of undirected graphs, depending on the degrees of the vertices in some local neighborhood.
Proof.For the proof we construct a simple Markov chain M, in which the probability of reaching one of its absorbing states is at least the probability of fixation in the original Markov chain.Then, in order to provide an upper bound of the fixation probability in the original Markov chain, we provide an upper bound on the probability of reaching one of the absorbing states in M.
Let u be a vertex that maximizes the probability of fixation, that is, u ∈ arg max u∈V h u .Furthermore, assume that we end the process in favor of the black vertices when the corresponding Markov chain describing the model of [16] reaches three black vertices.To favor fixation even more, since u maximizes h u , we assume that, whenever we reach two black vertices and a backward step happens (i.e., a step that reduces the number of black vertices), then we backtrack to state u (even if vertex u was the one that became white).Finally, given that we start at vertex u and we increase the number of black vertices by one, we assume that we make black the neighbor v of u that maximizes the forward bias of the state {u, v}.Imposing these constraints (and eliminating self loops), we get a Markov chain M, shown in Figure 1, that dominates the original Markov chain.That is, the probability that M reaches the state of three black vertices, given that we start at u, is an upper bound of the fixation probability of G.For the Markov chain M, we have that where N (u) is the set of neighbors of u.Also, Let now hu (resp.huv ) denote the probability of reaching 3 blacks, starting from u (resp.starting from the state {u, v}) in M. We have that This completes the proof of the theorem.
Consider for instance a bipartite graph G = (U, V, E), where deg u = d 1 for every vertex u ∈ U and deg v = d 2 for every vertex v ∈ V .Then any edge of E has one vertex in U and one vertex in V .Using the above notation, consider now an arbitrary edge uv ∈ E, where u ∈ U and v ∈ V .Then Q u = d1 d2 and . The right side of ( 9) is maximized when d 1 < d 2 , and thus in this case Theorem .
In particular, for the star graph with n + 1 vertices, we have d 1 = 1 and But, as shown in [16], the fixation probability of the star is asymptotically equal to 1 − 1 r 2 , whereas the above bound gives f star ≤ , which is quite tight.

Upper and lower bounds depending on degrees
In the following theorem we provide upper and lower bounds of the fixation probability of undirected graphs, depending on the minimum and the maximum degree of the vertices.
Theorem 2 Let G = (V, E) be an undirected graph, where deg(v) deg(u) ≤ λ whenever uv ∈ E.Then, the fixation probability f G of G, when the fitness of the mutant is r, is upper (resp.lower) bounded by the fixation probability of the clique for mutant fitness r 1 = rλ (resp.for mutant fitness r 2 = r λ ).That is, Proof.For an arbitrary state S ⊆ V of the Markov Chain (that corresponds to the set of black vertices in that state), let ρ + (S) (resp.ρ − (S)) denote the probability that the number of black vertices increases (resp.decreases).In the case where G is the clique, the forward bias at state S is equal to ρ+(S) ρ−(S) = r, for every state S [16,21].
∈ S} be the set of edges with one vertex in S and one vertex in V \ S.Then, and Now, since by assumption By ( 11), (12), and (13) we get the following upper and lower bounds for the forward bias at state S.
Notice that the upper and lower bounds of ( 14) for the forward bias at state S are independent of S. Therefore, the process stochastically dominates a birth-death process with forward bias r λ , while it is stochastically dominated by a birth-death process with forward bias rλ (cf.equation ( 2)).This completes the proof of the theorem.

The undirected suppressor
In this section we provide the first class of undirected graphs (which we call clique-wheels) that act as suppressors of selection as the number of vertices increases.In particular, we prove that the fixation probability of the members of this class is at most 1 2 (1 − 1 r ), i.e. the half of the fixation probability of the complete graph, as n → ∞.An example of a clique-wheel graph G n is depicted in Figure 2(a).This graph consists of a clique of size n ≥ 3 and an induced cycle of the same size n with a perfect matching between them.We will refer in the following to the vertices of the inner clique as clique vertices and to the vertices of the outer cycle as ring vertices.Denote by h clique (resp.h ring ) the probability that all the vertices of G n become black, given that we start with one black clique vertex (resp.with one black ring vertex).We first provide in the next lemma an upper bound on h clique .
Lemma 1 For any r ∈ 1, 4  3 , Proof.Denote by S k the state, in which exactly k ≥ 0 clique vertices are black and all ring vertices are white.Note that S 0 is the empty state.Denote by F k the state where at least one ring vertex of G n and exactly k ≥ 0 clique vertices are black.Furthermore, for every state S, denote by h S the probability that, starting at the state S, we eventually reach the full state (i.e. the state where all vertices are black).In the following we denote for simplicity the probability h S k by h k , for every k = 0, 1, . . ., n.Note that h 0 = 0 and h 1 = h clique , since S 0 is the empty state and S 1 is the state with only one black clique vertex.In order to compute an upper bound of h 1 , we can set the value h n log 7 n and the values h F k for every k ≥ 1 to their trivial upper bound 1.That is, we assume that the state S n log 7 n , as well as all states F k , where k ≥ 1, are absorbing.After performing these relaxations (and eliminating self loops), we obtain a Markov chain, whose state graph is illustrated in Figure 2(b).
For any k = 1, . . ., n log 7 n − 1 in this Markov chain, where Notice now by ( 16) that since r ∈ 1, 4 3 by assumption.Furthermore, since 6 for sufficiently large n, it follows that for every k = 1, 2, . . ., n log 7 n − 1, Now, since α k + β k + γ k = 1, (15) implies by ( 17) and ( 18) that Thus, since h 0 = 0 and h k ≥ h k−1 for all k = 1, . . ., n log 7 n , it follows that for every k, Consequently, since h n log 7 n = 1 in the relaxed Markov chain, we have that Corollary 1 Starting with one black clique vertex, the probability that we make at least one ring vertex black, or that we eventually reach n log 7 n black clique vertices, is at most In the remainder of this section, we will also provide an upper bound on h ring , thus bounding the fixation probability f Gn of G n (cf.Theorem 4).Consider the Markov chain M that is depicted in Figure 3.Our analysis will use the following auxiliary lemma which concerns the expected time to absorption of this Markov chain.Lemma 2 Let p = q and p + q = 1.Then, as m tends to infinity, the expected number of steps needed for M to reach v m , given that we start at v 1 , satisfies Proof.For i = 0, 1, . . ., m, let µ i denote the expected number of steps needed to reach v m .Clearly, µ m = 0 and µ 0 = 1 + µ 1 .Furthermore, for i = 1, . . .m − 1, it follows that i.e.
which concludes the proof of the lemma for large m.
Denote in the following by M 1 the Markov chain of the stochastic process defined in [16] (see Section 2.1 for an overview), when the underlying graph is the clique-wheel G n , cf. Figure 2(a).The next definition will be useful for the discussion below.
Definition 1 (Ring steps) A transition of the Markov chain M 1 is called a ring step if it results in a change of the number of black vertices in the outer ring (i.e.ring vertices).
We now present some domination statements that simplify the Markov chain M 1 .More specifically, all these statements will increase the probability of reaching fixation when we start with one black ring vertex, such that we finally get an upper bound on h ring .D 1 : Let v be a vertex on the outer ring, and let v ′ be its (unique) neighbor in the clique.We will forbid transitions of the Markov chain M 1 , where a white colored v ′ tries to make white a black colored v.
D 2 : Fixation is forced when either of the following happens: The outer ring reaches log n black vertices.
A 2 : The number of ring steps in order to reach log n black ring vertices is more than Θ(log 2 n).A 3 : The clique reaches n black vertices.A 4 : A black clique vertex makes black a white vertex on the outer ring.
Let now M 2 be the modified Markov chain after these domination statements are imposed.Given these statements, we can prove the following.Proof.This follows by noticing that a second black arc can only be created either by a white colored vertex on the clique making white its black neighbor on the outer ring (this is impossible because of D 1 ), or by a black vertex on the clique making black its white neighbor on the outer ring (this will lead to absorption at The following definitions will be useful in the sequence. Definition 2 (Offspring) Given the history H t of the Markov chain up to time t, the current state S ⊆ V at time t (i.e. the set of black colored vertices in the graph), and a vertex v ∈ S, we will say that v is an offspring of u ∈ V if and only if there exists a transition path in H t that proves that v is black because of u.
Notice in Definition 2 that u is not necessarily black at time t.Definition 3 (Birth in the clique) We will say that a vertex v ′ is born in the clique if and only if its (unique) neighbor v in the outer ring is black and makes a transition to the clique.
Notice in Definition 3 that the color of v ′ is irrelevant before v ′ is born in the clique.We only need that the color of v is black.Furthermore, the above definition allows for a specific vertex to be born more than once (i.e. at different time steps).The proof of our main theorem can now be reduced to a collection of lemmas.Lemma 3 concerns the behavior of the ring.
Lemma 3 Let B 1 be the stochastic process describing the ring steps in Markov chain M 2 .Given that we do not have absorption at A 4 , then B 1 is a birth-death process with forward bias equal to r.Furthermore, given that we start with a single black vertex on the ring, the following hold: (1) The probability that the number of black vertices in the outer ring reaches log n before absorption at A 2 , A 3 or A 4 is at most (2) The probability that more than log 2 n ring steps are needed in order to reach log n black colored vertices in the ring, or to reach absorption in Proof.Recall that we do not allow transitions where the clique affects the number of black colored vertices in the outer ring (by the domination statements D 1 and A 4 ).Then, it can be easily seen that the forward bias of the birth-death process B 1 (i.e. the ratio of the forward probability over the backward probability) is = r, where W is the sum of the fitness of every vertex in the graph.Thus, part (1) of the lemma follows by equation ( 2) (for an overview of birth-death processes, see also [20,21]).
For part (2), let X denote the number of ring steps needed in order to reach log n black colored vertices in the ring, or to reach absorption in A 2 , A 3 or A 4 .Then X is stochastically dominated by the number of steps needed for Markov chain M (cf. Figure 3) to reach v m , with m = log n and p = r r+1 .Hence, by Lemma 2 and Markov's inequality, we get that Pr(X ≥ log 2 n) ≤ c 1 log n for some positive constant c = c(r).
The next lemma bounds the number of vertices that are born in the clique (see Definition 3).
Lemma 4 Given that we start with a single black vertex on the ring, the probability that we have more than log 7 n births in the clique is at most O 1 log n .
Proof.For the proof, we will ignore for the moment what happens in the clique and how the clique affects the ring, since these steps are either forbidden (by D 1 ) or lead to absorption (by A 4 ).
Let Y be the number of births in the clique (see Definition 3) that we observe between two ring steps.Notice that at any time before absorption, there will be exactly 2 white colored vertices in the outer ring that can perform a ring step (see Definition 1).Furthermore, if the number of black vertices in the ring is more than 2, then not all black vertices can affect the number of black vertices in the ring.We now restrict ourselves, to observe only ring-involved moves (forgetting about the clique), that is, transitions where only vertices of the ring that can cause a ring step or a birth in the clique are chosen.Given that M 2 (i.e. the modified Markov chain) has not been absorbed, the probability that a ring step happens next is where z is the number of black colored vertices in the outer ring.Similarly, the probability that a birth in the clique happens next is Consequently, the random variable Y + 1 is stochastically dominated by a geometric random variable with probability of success p = p step p step + p birth = 2r + 2 zr + 2r + 2 ≥ 1 log n where in the last inequality we used the observation that at any time before absorption, the number of black vertices in the ring is at most log n because of A 1 .But then, by Markov's inequality, we have that But by part (2) of Lemma 3, the probability that there are more than log 7 n births in the clique before the Markov chain is absorbed is by Boole's inequality at most which concludes the proof of the lemma.
The following lemma states that it is highly unlikely that the clique will affect the outer ring, or that the number of black vertices in the clique will reach n.
Lemma 5 Given that we start with a single black vertex on the ring, the probability of absorption at A 3 or A 4 is at most O 1 log n .Proof.For the purposes of the proof, we assign to each birth in the clique a distinct label.Notice by Lemma 4, we will use at most log 7 n labels with probability at least 1 − O 1 log n .If we end up using more than log 7 n labels (which happens with probability at most O 1 log n by Lemma 4), then we stop the process and assume that we have reached one of the absorbing states.Furthermore, whenever a black vertex v in the clique with label i replaces one of its neighbors with an offspring, then the label of v is inherited to its offspring.
In order for M 2 to reach absorption at A 3 , the clique must have n black vertices.Since each of these vertices has a label j ∈ [log 7 n], there exists at least one label i such that at least n log 7 n vertices have label i.Similarly, if M 2 reaches absorption at A 4 and v is the corresponding affected ring vertex, then there exists a label i, such that v has label i.We will call a label i winner if there are at least n log 7 n vertices in the clique that have label i, or the outer ring is affected by a clique vertex of label i.Clearly, if M 2 reaches absorption at A 3 of A 4 , there must be at least one winner.
Recall that, by Corollary 1, the probability that a single black vertex in the clique either reaches n log 7 n offsprings or affects the outer ring is at most 7 6n( 4 3r −1) + o 1 n .Consider now a particular label i.Then, if all the other black vertices of the graph that do not have label i (i.e.black ring vertices or black clique vertices with label j = i) had fitness 1, then the probability that i becomes a winner is by Corollary 1 at most 7 6n( 4 3r −1) + o 1 n .The fact that the other black vertices that do not have label i have fitness r can only reduce the probability that i becomes a winner.Therefore, considering all different labels i ∈ [log 7 n] and using Boole's inequality, we conclude that the probability to reach absorption at A 3 or A 4 is at most where the term O 1 log n in the left side corresponds to the probability that we have more than log 7 n labels.
Finally, the following theorem concerns the probability of absorption of M 2 .
Theorem 3 For n large, given that we start with a single black vertex on the ring, the probability that M 2 is absorbed at Proof.The bounds on the absorption at A 1 or A 2 follow from Lemma 3, while the bounds on absorption at A 3 or A 4 follow from Lemma 5.
Recall now that M 2 (the modified Markov chain) dominates M 1 (the original Markov chain).Furthermore, recall that the clique-wheel graph G n has n clique vertices and n ring vertices, and thus the fixation probability of G n is f Gn = h clique +hring 2 . Therefore, the next theorem is implied by Theorem 3 and Lemma 1.
Theorem 4 For the Markov chain M 1 , and any r ∈ 1, 4 3 , h ring ≤ (1+o(1)) 1 − 1 r .Therefore, as n → ∞, the fixation probability of the clique-wheel graph G n in Figure 2(a) is by Lemma 1 4 Analysis of the aggregation model In this section, we provide analytic results on the new evolutionary model of mutual influences.More specifically, in Section 4.1 we prove that this model admits a potential function for arbitrary undirected graphs and arbitrary initial fitness vector, which implies that the corresponding dynamic system converges to a stable state.Furthermore, in Section 4.2 we prove fast convergence of the dynamic system for the case of a complete graph, as well as we provide almost tight upper and lower bounds on the limit fitness, to which the system converges.

Potential and convergence in general undirected graphs
In the following theorem we prove convergence of the new model of mutual influences using a potential function.
Theorem 5 Let G = (V, E) be a connected undirected graph.Let r(0) be an initial fitness vector of G, and let r min and r max be the smallest and the greatest initial fitness in r(0), respectively.Then, in the model of mutual influences, the fitness vector r(k) converges to a vector [r 0 , r 0 , . . ., r 0 ] T as k → ∞, for some value Proof.Denote the vertices of G by V = {u 1 , u 2 , . . ., u n }.Let k ≥ 0. Then (5) implies that for any i = 1, 2, . . ., n, the element r ui (k + 1) of the vector r(k + 1) is and thus Therefore, by summing up the equations in (24) for every i = 1, 2, . . ., n it follows that ui∈V deg(ui) for every iteration k ≥ 0 of the process.Note by Observation 2 that Σ(k) = ui∈V r ui (k) ≤ nr max is a trivial upper bound for Σ(k).Therefore, (25) implies that Furthermore, note that r max • ui∈V 1 deg(ui) ≤ nr max is a trivial upper bound for φ(k).Therefore, since φ(k + 1) ≥ φ(k) for every k ≥ 0 by (25), it follows that φ(k) converges to some value φ 0 as k → ∞, where φ(0) ≤ φ 0 ≤ nr max .Consider now an arbitrary ε > 0 and let ε ′ = ε 2 n 3 rmax .Then, since φ(k) −→ k→∞ φ 0 , there exists k 0 ∈ N, such that |φ(k + 1) − φ(k)| < ε ′ for every k ≥ k 0 .Therefore, (26) implies that for every edge Thus, for every ε > 0, there exists k 0 ∈ N, such that |r uj (k) − r ui (k)| < ε for every k ≥ k 0 and for every edge u i u j ∈ E of G. Therefore, since G is assumed to be connected, all values r u (k), where u ∈ V , converge to the same value r 0 as k → ∞.Furthermore, since r u (k) ∈ [r min , r max ] by Observation 2, it follows that r 0 ∈ [r min , r max ] as well.This completes the proof of the theorem.

Analysis of the complete graph
The next theorem provides an almost tight analysis for the limit fitness value r 0 and the convergence time to this value, in the case of a complete graph (i.e. a homogeneous population).
The next corollary follows by Theorem 6.
Corollary 2 Let G = (V, E) the complete graph with n vertices.Suppose that initially exactly one mutant with relative fitness r ≥ 1 is placed in G and let r 0 be the limit fitness of G. Then 1 + r−1 n ≤ r 0 ≤ 1 + r 2 −1 2n .
Proof.Since we have initially one mutant, it follows that α = 1 n .Then, substituting this value of α in (29), we obtain the lower bound r 0 ≥ 1 + r−1 n .For the upper bound of r 0 , it follows by substituting α in (28) that

Invasion control mechanisms
As stated in the introduction of this paper, our new evolutionary model of mutual influences can be used to model control mechanisms over invading populations in networks.We demonstrate this by presenting two alternative scenarios in Sections 5.1 and 5.2.In both considered scenarios, we assume that αn individuals of relative fitness r (the rest being of fitness 1) are introduced in the complete graph with n vertices.Then, as the process evolves, we periodically choose (arbitrarily) a small fraction β ∈ [0, 1] of individuals in the current population and we reduce their current fitnesses to a value that is considered to correspond to the healthy state of the system (without loss of generality, this value in our setting is 1).In the remainder of this section, we call these modified individuals as "stabilizers", as they help the population resist to the invasion of the mutants.

Control of invasion in phases
In the first scenario of controlling the invasion of advantageous mutants in networks, we insert stabilizers to the population in phases, as follows.In each phase k ≥ 1, we let the process evolve until all fitnesses {r v | v ∈ V } become ε-relatively-close to their fixed point r < ε for every v ∈ V .Note by Theorem 5 that, at every phase, the fitness values always ε-approximate such a limit fitness r (k) 0 .After the end of each phase, we introduce βn stabilizers, where β ∈ [0, 1].That is, we replace βn vertices (arbitrarily chosen) by individuals of fitness 1, i.e. by resident individuals.Clearly, the more the number of phases, the closer the fixed point at the end of each phase will be to 1.In the following theorem we bound the number of phases needed until the system stabilizes, i.e. until the fitness of every vertex becomes sufficiently close to 1.
Theorem 7 Let G = (V, E) be the complete graph with n vertices.Let α ∈ [0, 1] be the portion of initially introduced mutants with relative fitness r ≥ 1 in G and let β ∈ [0, 1] be the portion of the stabilizers introduced at every phase.Let r (k) 0 be the limit fitness after phase k and let ε, δ > 0, be such that β 2 > √ ε and δ > 4 3 √ ε.Finally, let each phase k run until the fitnesses ε-approximate their fixed point r (k) 0 .Then, after phases, the relative fitness of every vertex u ∈ V is at most 1 + δ.
Proof.Consider the first phase, where initially there exist αn mutants with relative fitness r and (1 − α)n resident individuals with fitness 1 each.Then, since r ≥ 1, it follows by (28) for the fixed point r Suppose that we let each phase k ≥ 1 run until the fitnesses ε-approximate their fixed point r .Then, we obtain similarly to (45) that the fixed point r . Then the last inequality becomes and by induction we have Therefore, (45) implies that At the end of the kth phase, the relative fitness of each vertex is at most (1 + ε)r (k) 0 .Now, in order to compute at least how many phases are needed to reach a relative fitness (1 + ε)r This completes the proof of the theorem.

Observation 3
The bound in Theorem 8 of the number of iterations needed to achieve everywhere a sufficiently small relative fitness is independent of the portion α ∈ [0, 1] of initially placed mutants in the graph.Instead, it depends only on the initial relative fitness r of the mutants and on the portion β ∈ [0, 1] of the vertices, to which we introduce the stabilizers.

Concluding remarks
In this paper we alternative models for evolutionary dynamics on graphs.In particular, we first considered the evolutionary model proposed in [16], where vertices of the graph correspond to individuals of the population.We provided in this model generic upper and lower bounds of the fixation probability on a general graph G and we presented the first class of undirected graphs (called clique-wheels) that act as suppressors of selection.Specifically, we proved that the fixation probability of the clique-wheel graphs is at most one half of the fixation probability of the complete graph (i.e. the homogeneous population) as the number of vertices increases.An interesting open question in this model is whether there exist functions f 1 (r) and f 2 (r) (independent of the size of the input graph), such that the fixation probability of every undirected graph G lies between f 1 (r) and f 2 (r).Another line of future research is to investigate the behavior of the model of [16] in the case where there are more than two types of individuals (aggressive vs. non-aggressive) in the graph.
As our main contribution, we introduced in this paper a new evolutionary model based on mutual influences between individuals.In contrast to the model presented in [16], in this new model all individuals interact simultaneously and the result is a compromise between aggressive and non-aggressive individuals.In other words, the behavior of the individuals in our new model and in the model of [16] can be interpreted as an "aggregation" vs. an "all-or-nothing" strategy, respectively.We prove that our new evolutionary model admits a potential function, which guarantees the convergence of the system for any graph topology and any initial fitnesses on the vertices of the underlying graph.Furthermore, we provide almost tight bounds on the limit fitness for the case of a complete graph, as well as a bound on the number of steps needed to approximate the stable state.Finally, our new model appears to be useful also in the abstract modeling of new control mechanisms over invading populations in networks.As an example, we demonstrated its usefulness by analyzing the behavior of two alternative control approaches.Many interesting open questions lie ahead in our new model.For instance, what is the speed of convergence and what is the limit fitness in arbitrary undirected graphs?What happens if many types of individuals simultaneously interact at every iteration?

Figure 2 :
Figure 2: (a) The clique-wheel graph G n and (b) the state graph of a relaxed Markov chain for computing an upper bound of h 1 = h clique .

k
This completes the proof of the lemma, since 4 3r > 1.The next corollary follows by the proof of Lemma 1.

Proposition 1
If M 2  has not yet reached an absorbing state, then the outer ring consists of a single arc of consecutive black vertices, or has no black vertex at all.
that, at the start of the process, (1 − α)n vertices have fitness 1 and αn vertices have fitness r.Similarly, before the kth phase starts, βn vertices have fitness 1 and (1 − β)n vertices have fitness at most (1 + ε)r (k−1) 0