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Let us first recall the result in the indecomposable case (see Athreya and Ney ). Consider a super-critical branching process with [MATH] and suppose that the vector valued random variable [MATH] satisfies a technical condition called |
Kesten-Stigum “ [MATH] ” condition (see Lyons et al. and Olofsson ), which is always satisfied in our case, since the probability distribution of the offsprings has finite support. It is natural to define the normalized random vector [MATH] This normalized random vector has a limit when [MATH] , that is, there exists a... |
[EQUATION] where [MATH] is the normalized right eigenvector corresponding to the malthusian parameter [MATH] [MATH] ) and [EQUATION] |
where [MATH] is the normalized left eigenvector corresponding to the malthusian parameter [MATH] [MATH] ). An important step in the proof of Kesten-Stigum theorem is the Kurtz |
convergence of classes theorem: [EQUATION] Combining this with the Perron-Frobenius theorem (see Meyer ) one obtains [EQUATION] where the convergence in the first limit is in probability. The approach adopted in Cuesta et al. |
relates only to the second equality involving the limit of mean values of equation ( 17 ). By explicitly considering the microscopic model as a multivariate branching process the equality of the two limits in equation ( 17 ) is guaranteed. This result may be useful for computational simulations of the model, since one ... |
The meaning of the Kesten-Stigum theorem is that the total size of the population divided by [MATH] , converges to a random vector, but the relative proportions of the various “classes” approach fixed limits. Since we are assuming that the process is indecomposable the normalized right eigenvector [MATH] is positive an... |
[MATH] It is called the asymptotic distribution of classes of the multitype branching process. In order to extend these results to the case where the branching process is decomposable one should employ the Frobenius normal form of the mean matrix [MATH] , which is reducible in this case (see Gantmatcher |
). Kesten and Stigum show that it is possible, by rearranging the rows and columns, to rewrite the mean matrix in a block upper triangular form in such a way that the diagonal blocks are irreducible square matrices associated to components of the decomposable branching process. By a component of a decomposable branchin... |
[MATH] ordered according to which [MATH] if there is a sequence of directed edges leading from some [MATH] to some [MATH] Given two components [MATH] and [MATH] define the sub-matrix |
[EQUATION] Then, for each [MATH] , the square sub-matrix [MATH] is the irreducible mean matrix of the sub-process [EQUATION] Now the order of the components [MATH] allows us to rearrange the rows and columns of |
[MATH] in such a way that [EQUATION] Therefore, the sub-process [MATH] “receives input” from the sub-process [MATH] , with [MATH] , throughout the sub-matrix [MATH] Note that if the sub-matrices [MATH] are all zero then the branching process splits as a sum of independent indecomposable branching processes. |
Example A.4 The matrices ( ) and ( ) of the simple and the general phenotypic models, respectively, already are in the normal form: |
(i) In the simple phenotypic model we have that [MATH] , with one-dimensional diagonal sub-matrices [MATH] , with one-dimensional sub-matrices [MATH] and one-dimensional sub-matrices |
[MATH] if [MATH] (ii) In the general phenotypic model we have [MATH] , with the first diagonal sub-matrix [MATH] (or [MATH] for the first variation of the model), the second diagonal sub-matrix [MATH] with [MATH] and [MATH] |
Now observe that if [MATH] with [MATH] then for [MATH] , the sub-process [MATH] for all [MATH] That is, the branching process behaves as if the sub-processes [MATH] |
for all [MATH] did not exist. Since each non-zero diagonal sub-matrix [MATH] is irreducible, it has a largest positive eigenvalue [MATH] and then we may define the |
effective malthusian parameter of the sequence of sub-processes [MATH] to be [EQUATION] The simplest case is when all [MATH] are simple eigenvalues of their respective sub-matrices [MATH] – they are distinct amongst each other – this is exactly the case for matrices ( ) and ). |
In Kesten and Stigum the result about the asymptotic behaviour of irreducible super-critical branching process is generalized to the reducible case. The main theorem applied to the case where all [MATH] are simple eigenvalues of their respective sub-matrices [MATH] states that if the effective malthusian |
[MATH] and the “ [MATH] ” condition holds then for the normalized random vector [MATH] there exists a scalar random variable [MATH] such that, with probability one, |
[EQUATION] where [MATH] is the normalized right eigenvector corresponding to the effective malthusian parameter [MATH] and [EQUATION] |
where [MATH] is the left eigenvector corresponding to the effective malthusian parameter [MATH] Moreover, Kurtz’s convergence of classes theorem ( 16 ) still holds. But one should note that the normalized right and left eigenvectors are not positive anymore. In fact, [MATH] may have negative entries, but only those ass... |
[MATH] , and so is a probability distribution. Critical Behavior and Regime Transition The critical state separates the super-critical and the sub-critical regimes where the branching process has two distinct behaviors in time and thus characterizes the existence of regime transition with genuine critical behavior. In ... |
Although in a critical branching process [MATH] , almost surely, when [MATH] , one still may obtain a meaningful asymptotic law by conditioning on non-extinction. See Mullikin |
and Joffe and Spitzer for the indecomposable case and Foster and Ney for certain decomposable cases. In the indecomposable critical case [MATH] grows at a linear rate proportional to [MATH] (see Harris |
or Athreya and Ney ), and so one should consider the normalized random vector [MATH] If the second moments are finite and the branching process is non-singular, there is a scalar random variable [MATH] such that |
[EQUATION] where [MATH] is the normalized right eigenvector corresponding to the malthusian parameter [MATH] and with convergence only in distribution , which is weaker than the almost surely convergence in the super-critical case. |
# Source: arxiv 1202.0678 # Title: Influence of Topological Features on Spatially-Structured Evolutionary Algorithms Dynamics # Sections: all # Downloaded: 2026-03-03T01:56:54.442732+00:00 |
Influence of Topological Features on Spatially-Structured Evolutionary Algorithms Dynamics Abstract In the last decades, complex networks theory significantly influenced other disciplines on the modeling of both static and dynamic aspects of systems observed in nature. This work aims to investigate the effects of netwo... |
Index Terms: Evolutionary Computation, Complex Networks, Spatially-Structured Evolutionary Algorithms Introduction Spatially-Structured Evolutionary Algorithms (ssEAs) are defined as evolutionary algorithms (EAs) where the mating between individuals is based on a graph (or network, in this work we will consider these t... |
The first studies on ssEAs focused on the effects of the shape of the neighbourhood, commonly a regular lattice, on selection pressure Sarma and Jong, 1996 . SsEAs have been reported as being useful in maintaining diversity with multimodal and epistatic problems Alba and Troya, 2000 considering neighbourhood shape on 2... |
This paper presents a basic ssEA with the purpose to investigate the relationship between graph structure and diversity maintenance, without any explicit mechanism (e.g. fitness sharing). Given the advancements in the last decade in the field of complex networks, this study aim to find a link between complex networks d... |
A particular attention has been paid to Small-World network models, proposed by Watts and Strogatz Watts and Strogatz, 1998 . This model has been chosen because of the possibility to tune an important graph feature such as the average path length (APL) changing the value of the rewiring factor [MATH] |
The rest of the paper is organized as follows. In section II we provide a brief description of ssEAs then explaining accurately the particular algorithm we use in this paper. A description about complex networks theory and network dynamics follows in Section III . We present the experimentation on ONEMAX problem in Sec... |
II Multi-modal Functions Optimization with ssEAs Multi-modal functions have multiple optimal solutions, which may be local or global optima. In the case where more than a single global optimum exists, there might be the necessity to find all the optima and not only a single one at the end of the algorithm execution. Us... |
The most common ssEA models in literature are the Island Model and Cellular EAs. The Island Model consists of multiple populations running in parallel which exchange solutions (migration) with a specific strategy. |
A cellular EA (cEA) structures the population by the means of “local” small neighbourhoods, maintaining a population whose individuals are spatially distributed in cells. A cellular Genetic Algorithm (cGA) is a genetic algorithm whose selection, recombination and mutation are performed within the neighbourhood of each ... |
This kind of EA tries to preserve the diversity by restricting the mating (and the consequent exchange of genetic material) on “physical” distance between individuals. Commonly cGAs are defined on a 1D lattice or square lattice (see Alba and Dorronsoro, 2008 ) but, as stated before, cGAs based on other graphs topologie... |
As we explained before, in this work we use the term Spatially-Structured EAs (ssEAs) for all the EAs where individuals’ interactions are bound to a directed or undirected graph. |
II-A ssEA Algorithm We implemented a ssEA where each individual is associated to a node of an undirected graph (interaction graph). The neighbourhood [MATH] of the individual [MATH] is defined with all the nodes connected to it with an edge. Hence the size of the neighbourhood, and so the mating pool, is the same of th... |
The selection mechanism is random within the neighbourhood of individual [MATH] , that is each individual [MATH] connected with [MATH] has the same uniform probability ( [MATH] ) to be selected, thus having: |
[EQUATION] with [MATH] the degree of node [MATH] The selected individual is mutated with a uniform bitwise mutation method: each bit of the genotype of length [MATH] is flipped with probability [MATH] . Then a “replace if better” strategy is applied: if the fitness of the mutated solution is higher or equal than the on... |
[EQUATION] with [MATH] the number of individuals with a fitness better or equal in the neighborhood, [MATH] the degree of node [MATH] and [MATH] the fitness value of individual [MATH] . The first part of Eq. is the probability to select an individual in the neighbourhood with better fitness without mutating any bits of... |
We can see that in this algorithm the downgrade probability, i.e. the probability that an individual makes its fitness value worst after the selection/mutation, is zero and so once an optimal solution is found it will start to spread among the population assuming that the graph is undirected and connected. |
In this work we classify the ways a new optimal solution can appear in a node as ‘cloning’ and ‘mutation’. II-A Cloning The ‘cloning’ dynamic, as the name suggests, is when an individual get copied remaining unvaried. In fact, an optimal solution [MATH] can be selected by an individual [MATH] with probability shown in ... |
II-A Mutation The ‘mutation’ dynamic describes the achievement of an optimal solution starting from a not-optimal one. Given the mutation mechanism of ssEA algorithm, we can always obtain an optimal solution with a probability [MATH] with [MATH] the Hamming distance from the optimal string. The latter formula represent... |
Algorithm 1 A simple ssEA with Random Selection Mutation 1: [MATH] 2: for each individual [MATH] in [MATH] do 3: [MATH] 4: end for |
5: [MATH] 6: while not termination criteria do 7: for each individual [MATH] in [MATH] do 8: [MATH] 9: [MATH] 10: [MATH] 11: if [MATH] |
then 12: [MATH] 13: end if 14: end for 15: [MATH] 16: end while III Complex Topologies and ssEAs As the structure of the interactions (e.g. neighbourhood shape) between individuals has showed to play a key role on the spreading of solutions and selection pressure Giacobini et al., 2005b Tomassini, 2005 , in this work w... |
In recent years, thanks to the advances in computer science and to the progressive digitalization of huge amount of data it was possible to analyze the topology of many real-world systems from different scientific fields ranging from biology, social sciences, economics to electronics, physics and computer science. At t... |
In this section the CN literature is reviewed to introduce network models and analytical characterizations that will be exploited in the rest of the paper. Here we define the synthetic networks model that we use as a substrate for ssEAs with special attention on the ones that favor or slow down spreading and diffusion ... |
Since the beginning of the last century real networks have been represented as random graphs with random connections between the nodes and thus with node degrees (the number of connections for a node) distributed homogenously, i.e. with only few fluctuations around the mean value. In the last decade the analysis of big... |
Power-law functions have the property of maintaining the same functional form at different scales and hence defined as scale-free functions. Thus, networks showing a degree distribution fitted by a power-law function are defined as Scale-Free (SF) networks. The presence of hubs has profound implications on the structur... |
A common distance measure on graphs is the so called average path length (APL), also defined as characteristic path length. It is a measure of distance in the network and it represents the mean distance between two arbitrary nodes in a graph. Where the distance between two nodes [MATH] and [MATH] is defined as the mini... |
[EQUATION] Another quantity strictly related with spreading processes on networks is the clustering coefficient (CC). It represents the probability that two nodes sharing a neighbor have also a link connecting them. |
In Table we present the topological features of the networks we use in experimental part (Sections IV and VI ). III-A Selected network models |
To get a first insight on the effects of topology on algorithms’ dynamics we perform our analysis comparing the following topologies: random networks, scale-free networks and small-worlds networks. Here it follows a detailed introduction for each of the used topologies, with references and creation algorithm. |
Random Networks The simplest example of random network is represented by the so-called Erdös and Rényi (ER) random graphs Erdös and Rényi, 1959 presented in [MATH] by the hungarian mathematicians Paul Erdös and Alfréd Rényi. ER graphs are based on the assumption that nodes connect randomly between them and all the node... |
To create an ER graph it is possible to follow the original algorithm presented in Erdös and Rényi, 1959 1. To create a graph with [MATH] nodes start with a set of [MATH] disconnected nodes. |
2. For each couple of nodes [MATH] connect them with probability [MATH] The resulting graph will be composed by [MATH] nodes and [MATH] links. |
Scale-Free networks In SF networks, differently from ER graphs, nodes do not have the same connection probability. Specifically, nodes with a high connectivity have a higher probability to be selected as end of a link respect with other nodes. In the last years several models have been proposed in the literature to gen... |
1. For each node [MATH] choose the desired final degree [MATH] as a random number from a power-law probability distribution of the form [MATH] |
2. For each node [MATH] stubs (the ending part of a link) are attached to it. 3. The process follows choosing randomly two stubs and connecting them to create a link, avoiding self loops and multiple links. This point is repeated until all the stubs have been connected. |
Once the process ends, the resulting network will be characterized by the same starting degree distribution [MATH] . Another important feature of SF networks is that, due to the random selection and the presence of highly connected nodes, the distances between nodes in the network are even smaller than an ER graph and ... |
Small-World Networks We follow in our study using as a substrate a class of networks that can interpolate, via a tunable parameter, from a fully regular 1D lattice (a ‘ring’), characterized by both a high CC and APL, to a completely random network with small distances between nodes and a vanishing CC. These networks ar... |
1. The process starts with a 1D lattice with [MATH] nodes. 2. For each link [MATH] , with probability [MATH] one end of the link is rewired to another node selected randomly avoiding loops and duplications. |
With [MATH] no rewiring is performed, the 1D lattice is preserved and characterized by [MATH] and [MATH] . With [MATH] all the links are rewired and a fully random topology (similar to an ER network) is achieved with [MATH] for large [MATH] and [MATH] |
Boccaletti et al., 2006 IV ONEMAX Analysis In this section we investigate the behavior of the ssEA algorithm presented in Sec. II-A on a common pseudo-boolean function optimisation problem, OneMax . The OneMax fitness function is very simple: |
[EQUATION] with [MATH] the length of the boolean string. The simplicity of this problem makes it an optimal choice as the first step on the investigation of the effect of diverse graph topologies applied to evolutionary algorithms. Our algorithm will be tested with various graph topologies with the aim of analyze conve... |
All the results presented has been obtained after 100 independent simulations for each configuration, the stopping criteria is the reach of 5000 generations. We measure the generation where the fitness converges (FCT), i.e. all the individuals have the same fitness, and the first hitting time (FHT). Let [MATH] be the n... |
[EQUATION] Note that, being the OneMax problem a unimodal problem, the FCT coincides with Takeover Time measure. In this analysis we included a random, a Scale-free network and Small-Worlds graph with [MATH] |
In Figure we show how each proposed configuration behaves on the problem with [MATH] . In Table III we give instead the average values of FHT and FCT with standard deviations with [MATH] |
As we studied the convergence speed to the global optimum we measured the genotypic entropy Rosca, 1995 of the population [MATH] as follows: |
[EQUATION] where [MATH] is the fraction of solutions with a given distance (in this case Hamming distance) from the origin (the 0-bit string). This metric, derived from statistical thermodynamics, permits to measure the quantity of different solutions (states) inside a population. |
In Figure we show the average genotypic entropy of selected graph topologies on OneMax problem. Obviously, inside the population there is a convergence towards to the most ordered configuration, i.e. the optimal string. The logarithmic scale we used allows to observe accurately how the population reaches the ‘most-orde... |
In general, Small-World graphs exhibit a dynamic clearly different from random and Scale-Free graphs as well as panmictic algorithms (which can be represented by a fully-connected graph), both for FHT/FCT and genotypic entropy. This difference can be explained by the differences in topological features, mainly due to t... |
IV-A Spreading Analysis As Payne and Eppstein, 2009a observed, Average Path Length may be not the only topological feature to influence algorithm dynamics. For this reason, we analyze in this section the spread of the optimal solution with respect to the graph distance from the node where the optimum appeared. The use ... |
The spreading of the optimal solution is more evident in Figure 5c , where the same plot is performed on a more regular topology (a Small-World network with [MATH] ) and where a kind of linear relationship between distance from node [MATH] and interval of optimal solution appearing is evident in the first distance valu... |
For the algorithm with Random Graph the average FCT for the problem [MATH] is 856.4 ( [MATH] ), sensibly lower than the value shown in Table III |
Differently, using Small-Worlds networks with mutation disabled the FCT becomes drastically larger as for graphs with such as large diameter (see Section III ) optimal solutions can appear independently in other part of the graph. Instead, without ‘mutation’ dynamic the convergence can be achieved only copying the firs... |
NMAX Experimentations This section introduces a combinatorial optimisation problem created by a composition of TwoMax functions Hoyweghen et al., 2002 Goldberg and Pelikan, 2000 , a bimodal equivalent of OneMax . This pseudo-Boolean function, called Nmax , has been created by concatenating [MATH] |
TwoMax strings of [MATH] bit each leading to a [MATH] global optima. The fitness function is defined as follows: [EQUATION] with [MATH] the i-th substring of length [MATH] inside the global problem string. The fitness of twomax problem is: |
[EQUATION] Given that a twomax problem has two distinct optima (the 0-string and the 1-string), with the concatenation of [MATH] strings we obtain a problem with [MATH] distinct global optima. In this paper we use [MATH] |
As in the previous section, we used the algorithm described in Section II-A with the parameters given in Table II As for OneMax problem, we simulate 100 runs of the algorithm described in section II-A with different typologies of graphs: random graph, Small-World with [MATH] , and Scale-Free. A panmictic version of the... |
In Table IV we show the average results (all the differences are significant, we omitted the standard deviations for sake of clarity). We observe similar FHT and FCT values both on Nmax and OneMax results shown in Tab. III |
Moreover, Figure illustrates the evolution of the ratio of optimal individuals (i.e. the value of [MATH] means that all the individuals have an optimal fitness). Because of the replacing strategy we can see that when the algorithm finds more than an optimum, during the generations this number tends to decrease. In fact... |
Observing the figures and the number of global optima found, panmictic, Random graphs as well as scale-free networks behave similarly generally leading to a single optimal solution. In general we can see how larger problems (i.e. with increasing values of [MATH] ) tend to have lower diversity, in fact with smaller prob... |
Similarly to the genotypic diversity described in the previous sections, the phenotypic diversity is defined as: [EQUATION] where [MATH] is the fraction of solutions with a given fitness value. |
As we can see in Figure and in nmax problem genotypic and phenotypic entropy highlight the multimodality of the problem, in fact when the population converges the phenotypic entropy goes to zero (all the individuals have the same fitness) but the genotypic entropy is greater than zero when there are optima with differe... |
V-A Small-World topologies In this section we investigate the dynamics of the proposed algorithm against the change of the rewiring factor [MATH] on the Watts-Strogatz model (see Section III ). |
We considered the same [MATH] different values as in Sec. IV . Table shows the numerical results on nmax problem with [MATH] and on a subset of the [MATH] values. As expected, we can see that as [MATH] grows the algorithm tends to perform similarly to a random network ssEA (see Section III ). A logarithmic plot in Figu... |
In Figure 10 is shown the average number of distinct optima over all simulations on Small-World networks for each value of rewiring factor [MATH] . It is interesting to note that all the three problem sizes lead to the same behavior with respect to the rewiring factor [MATH] , showing a kind of inflection point between... |
In order to compare the three Nmax problems we introduce a relative FHT measure ( [MATH] ), computed dividing the FHT value of the simulation [MATH] by the minimum FHT achieved by all the runs on every topology with Nmax with the same size ( [MATH] ): |
[EQUATION] This makes the FHTs obtained on different problem sizes comparable. Figure 11 shows all the runs performed on Nmax problems on Small-World networks with all the [MATH] values and with the random graph. This figure clearly illustrates the ssEA dynamics: in general we can observe a sort of linear relationship ... |
VI Weighted Networks In the previous section we analysed how each different graph topology leads to a specific algorithm’s behavior, considering both exploration speed (the number of epochs needed to find an optimal solution, i.e. FHT) and diversity (the number of distinct optimal solutions found), given an optimizatio... |
In this section we transform the underlying graph to a weighted graph in order to better point out the existing trade-off between exploring speed and diversity, a trade-off recalling the well-known trade-off between exploration and exploitation usually related to evolutionary search (see Eiben and Schippers, 1998 for a... |
[EQUATION] At the first generation, we have all the weights set to one, i.e. having a uniform selection probability. Then, at each generation we update the weights depending on the fitness value of the extreme nodes with the following formula: |
[EQUATION] with [MATH] and [MATH] the fitness values of nodes [MATH] and [MATH] , and [MATH] a real-valued parameter. This update formula has been proposed to allow two different behaviors: with [MATH] the probability to select a neighbour individual increases at each generation (and otherwise for individual with lower... |
VI-A Results with Weighted Networks In this section we describe the results of the simulations performed with a set of different [MATH] values on Small-World, Random and Scale-Free topologies. For sake of brevity, only the results with Nmax problem with [MATH] are presented. |
We explored the behavior of selected network models with integer values of [MATH] between the interval [MATH] with the addition of [MATH] values. |
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