Local Optima Networks of NK Landscapes with Neutrality
aa r X i v : . [ c s . A I] J u l Local Optima Networksof NK Landscapes with Neutrality
S´ebastien Verel, Gabriela Ochoa, Marco Tomassini ∗ Abstract
In previous work, we have introduced a network-based model that abstractsmany details of the underlying landscape and compresses the landscape informa-tion into a weighted, oriented graph which we call the local optima network . Thevertices of this graph are the local optima of the given fitness landscape, whilethe arcs are transition probabilities between local optima basins. Here, we extendthis formalism to neutral fitness landscapes, which are common in difficult com-binatorial search spaces. By using two known neutral variants of the NK family(i.e. NK p and NK q ) in which the amount of neutrality can be tuned by a param-eter, we show that our new definitions of the optima networks and the associatedbasins are consistent with the previous definitions for the non-neutral case. More-over, our empirical study and statistical analysis show that the features of neutrallandscapes interpolate smoothly between landscapes with maximum neutrality andnon-neutral ones. We found some unknown structural differences between the twostudied families of neutral landscapes. But overall, the network features studiedconfirmed that neutrality, in landscapes with percolating neutral networks, mayenhance heuristic search. Our current methodology requires the exhaustive enu-meration of the underlying search space. Therefore, sampling techniques should ∗ S´ebastien Verel is with INRIA Lille - Nord Europe, and University of Nice Sophia-Antipolis / CNRS,Nice, France. E-mail: [email protected] . Gabriela Ochoa is with The Automated Scheduling, Op-timisation and Planning (ASAP) Group, School of Computer Science, University of Nottingham, Notting-ham, UK. E-mail: [email protected] . Marco Tomassini is with the Information Systems Department,HEC, University of Lausanne, Switzerland. E-mail:
[email protected] e developed before this analysis can have practical implications. We argue, how-ever, that the proposed model offers a new perspective into the problem difficultyof combinatorial optimization problems and may inspire the design of more effec-tive search heuristics. Studying the distribution of local optima in a search space is of utmost importancefor understanding the search difficulty of the corresponding landscape. This under-standing may eventually be exploited when designing efficient search algorithms. Forexample, it has been observed in many combinatorial landscapes that local optima arenot randomly distributed, rather they tend to be clustered in a ”central massif” (or ”bigvalley” if we are minimizing). This globally convex landscape structure has been ob-served in the
N K family of landscapes [1, 2], and in many combinatorial optimizationproblems, such as the traveling salesman problem [3], graph bipartitioning [4], andflowshop scheduling [5]. Algorithms that exploit this global structure have, in conse-quence, been proposed [3, 5].Combinatorial landscapes can be seen as a graph whose vertices are the possibleconfigurations. If two configurations can be transformed into each other by a suitableoperator move, then we can trace an edge between them. The resulting graph, with anindication of the fitness at each vertex, is a representation of the given problem fitnesslandscape. A useful simplification of the graphs for the energy landscapes of atomicclusters was introduced in [6, 7]. The idea consists of taking as vertices of the graphnot all the possible configurations, but only those that correspond to energy minima.For atomic clusters these are well-known, at least for relatively small assemblages.Two minima are considered connected, and thus an edge is traced between them, ifthe energy barrier separating them is sufficiently low. In this case there is a transitionstate, meaning that the system can jump from one minimum to the other by thermalfluctuations going through a saddle point in the energy hyper-surface. The values ofthese activation energies are mostly known experimentally or can be determined bysimulation. In this way, a network can be built which is called the ”inherent structure”2r ”inherent network” in [6].In [8, 9, 10], we proposed a network characterization of combinatorial fitness land-scapes by adapting the notion of inherent networks described above. We used the well-known family of
N K landscapes as an example. In our case, the inherent network wasthe graph where the vertices are all the local maxima, obtained exhaustively by runninga best-improvement (steepest-ascent) local search algorithm from every configurationof the search space. The edges accounted for the notion of adjacency between basins.In our work we call this graph the local optima network or since it also represents theinteraction between the landscape’s basin the basin adjacency network . We proposedtwo alternative definitions of edges. In the first definition [8], two maxima i and j wereconnected (with an undirected edge without weight), if there exists at least one pairof directly connected solutions s i and s j , one in each basin of attraction ( b i and b j )(Fig. 1, top). The second, more accurate definition, associated weights to the edgesthat account for the transition probabilities between the basins of attraction of the localoptima (Fig. 1, bottom). More details on the relevant algorithms and formal definitionsare given in section 3. This characterization of landscapes as networks has brought newinsights into the global structure of the landscapes studied, particularly into the distri-bution of their local optima. Therefore, the application of these techniques to morerealistic and complex landscapes, is a research direction worth exploring.The fitness landscape metaphor [11] has been a standard tool for visualizing bio-logical evolution and speciation. It has also been useful for studying the dynamics ofevolutionary and heuristic search algorithms applied to optimization and design prob-lems. Traditionally, fitness landscapes are often depicted as ‘rugged’ surfaces withmany local ‘peaks’ of different heights flanked by ‘valleys’ of different depth [1, 2].This view is now acknowledged to be only part of the story. In both natural and ar-tificial systems a picture is emerging of populations engaged not in hill-climbing, butrather drifting along connected networks of genotypes of equal (or quasi equal) fitness,with sporadic jumps between these so called neutral networks. The importance of selective neutrality as a significant factor in evolution was stressed by Kimura [12] inthe context of evolutionary theory, and by Eigen et al. [13] in the context of molec-ular biology. Interest in selective neutrality was re-gained in the 90s by the identifi-3 j arbitrary configurationlocal maximum ! ij ij arbitrary configurationlocal maximum ! ij ! ji Figure 1: A diagram of the local optima or basin adjacency networks. The darknodes correspond to the local optima in the landscape, whereas the edges represent thenotion of adjacency among basins. Dashed lines separate the basins. Two alternativedefinitions of edges are sketched as undirected (top plot) and directed weighted arcs(bottom).cation of neutral networks in models for bio-polymer sequence to structure mappings[14, 15, 16, 17, 18, 19, 20]. It has also been observed that the huge dimensionality ofbiologically interesting fitness landscapes, considering the redundancy in the genotype-fitness map, brings naturally the existence of neutral and nearly neutral networks [21].In this context, the metaphor of ‘holey adaptive landscapes’ has been put forward as analternative to the conventionally view of rugged adaptive landscapes, to model macro-evolution and speciation in nature [21, 22, 23]. The relevance and benefits of neutralityfor the robustness and evolvability in living systems has been recently discussed in[24].There is growing evidence that such large-scale neutrality is also present in artificiallandscapes. Not only in combinatorial fitness landscapes such as randomly generatedSAT instances [25], cellular automata rules [26] and many others, but also in complexreal-world design and engineering applications such as evolutionary robotics [27, 28],evolvable hardware [29, 30, 31], genetic programming [32, 33, 34, 35] and grammatical4volution [36].Not only the structure of interesting natural and artificial landscapes, as discussedabove, is different from the conventional view of rugged landscapes; the evidence alsosuggests that the dynamics of evolutionary (or more generally search) processes onfitness landscapes with neutrality are qualitatively very different from the dynamics onrugged landscapes [17, 29, 37, 38, 39, 40, 41, 42]. As a consequence, techniques foreffective evolutionary search on landscapes with neutrality may be quite different frommore traditional approaches to evolutionary search [40, 43].In this paper, we apply our previous network definitions and analysis of combi-natorial search spaces to landscapes with selective neutrality. In particular, it is ourintention to investigate whether our graph-based approach is still adequate when neu-trality is present. This is apparently simple but, in reality, requires a careful redefinitionof the concept of a basin of attraction. The new notions will be presented in the nextsection. We also study how neutrality affects the landscape graph structure and statis-tics, and discuss the implications for the dynamic of heuristic search on these land-scapes. Following our previous work on N K landscapes [8, 9, 10], we selected twoextensions of the
N K family as example landscapes with synthetic neutrality, namely:the
N K p (‘probabilistic’ N K ) [39], and
N K q (‘quantized’ N K ) [44] families. The
N K p landscape introduces neutrality by setting a certain proportion p of the entries ina genotypes fitness tables to 0; whilst the N K q landscape does so by transforming thegenotype fitness entries from real numbers to integer values (in the range [0, q)). Theselandscapes posses two statistical features: fitness correlation and selective neutrality,which are relevant to combinatorial optimization.The paper begins by describing in more detail the neutral families of landscapesunder study (section 2). Thereafter, section 3 includes the relevant definitions andalgorithms used. The empirical network analysis of our selected neutral landscape in-stances is presented next (section 4), followed by a summary and discussion (section 5)and our conclusions and ideas for future work (section 6).5 N K landscapes with neutrality
The
N K family of landscapes [2] is a problem-independent model for constructingmultimodal landscapes that can gradually be tuned from smooth to rugged. In themodel, N refers to the number of (binary) genes in the genotype (i.e. the string length)and K to the number of genes that influence a particular gene (the epistatic interac-tions). By increasing the value of K from 0 to N − , N K landscapes can be tunedfrom smooth to rugged.The fitness function of a
N K -landscape f NK : { , } N → [0 , is defined onbinary strings with N bits. An ‘atom’ with fixed epistasis level is represented by afitness component f i : { , } K +1 → [0 , associated to each bit i . Its value dependson the allele at bit i and also on the alleles at the K other epistatic positions. ( K mustfall between and N − ). The fitness f NK ( s ) of s ∈ { , } N is the average of thevalues of the N fitness components f i : f NK ( s ) = 1 N N X i =1 f i ( s i , s i , . . . , s i K ) where { i , . . . , i K } ⊂ { , . . . , i − , i + 1 , . . . , N } . Several ways have been proposedto choose the K other bits from N bits in the bit string. Two possibilities are mainlyused: adjacent and random neighborhoods. With an adjacent neighborhood, the K bits nearest to the bit i are chosen (the genotype is taken to have periodic boundaries).With a random neighborhood, the K bits are chosen randomly on the bit string. Eachfitness component f i is specified by extension, i.e. a number y is i ,s i ,...,s iK from [0 , is associated with each element ( s i , s i , . . . , s i K ) from { , } K +1 . Those numbers areuniformly distributed in the range [0 , .The two variants of N K landscapes are representative of the way to obtain neu-trality in additive fitness landscapes. Indeed, for the two families, the fitness value of asolution is computed as a sum. Modifying a term in the sum would alter the probabilityto get the same fitness value.The
N K p landscapes have been introduced by Barnett [39]. In this variant, oneterm of the sum is null with probability p . Formally, the fitness components are mod-ified and tuned by the parameter p ∈ [0 , which controls the neutrality of the land-6cape. The fitness component y is i ,s i ,...,s iK is null with probability p , i.e. P ( y is i ,s i ,...,s iK =0) = p . The probability that two neighboring solutions have the same fitness value in-creases with the parameter p .The N K q landscapes have been introduced by Newman et al [44]. For these land-scapes, the terms of the sum are integer numbers between and q − . Thus, whensome terms are modified, it is possible to get the same sum. Formally, as for N K p landscapes, the fitness components are defined with a parameter q which tunes the neu-trality. Parameter q is an integer number above or equal to . Each y is i ,s i ,...,s iK is oneof the fractions kq where k is an integer number randomly chosen in [0 , q − .Neutrality is maximal when q is equal to , and decreases when q increases. Thisfamily of landscapes was shown to model the properties of neutral evolution of molec-ular species [44]. We include the relevant definitions and algorithms to obtain the local optima network inlandscapes with neutrality. For completeness, we also include some relevant definitionsthat apply to non-neutral landscapes [9, 10].
Fitness landscape:
A landscape is a triplet ( S, V, f ) where S is a set of admissible solutions i.e. asearch space, V : S −→ | S | , a neighborhood structure, is a function that assigns toevery s ∈ S a set of neighbors V ( s ) , and f : S −→ R is a fitness function that can bepictured as the height of the corresponding solutions.In our study, the search space is composed of binary strings of length N , thereforeits size is N . The neighborhood is defined by the minimum possible move on a binarysearch space, that is, the 1-move or bit-flip operation. In consequence, for any givenstring s of length N , the neighborhood size is | V ( s ) | = N . Neutral neighbor:
A neutral neighbor of s is a neighbor configuration x with thesame fitness f ( s ) . V n ( s ) = { x ∈ V ( s ) | f ( x ) = f ( s ) } Neutral network:
A neutral network, denoted as
N N , is a connected sub-graphwhose vertices are configurations with the same fitness value. Two vertices in a
N N are connected if they are neutral neighbors.With the bit-flip mutation operator, for all solutions x and y , if x ∈ V ( y ) then y ∈ V ( x ) . So in this case, the neutral networks are the equivalent classes of therelation R ( x, y ) iff ( x ∈ V ( y ) and f ( x ) = f ( y ) ) .We denote the neutral network of a configuration s by N N ( s ) . In this section, we define the notion of a basin of attraction for landscapes with neu-trality. The analogous notion for non-neutral landscapes has been given in [10].First let us define the standard notion of a local optimum, and its extension forlandscapes with neutral networks.
Local optimum:
A local optimum, which is taken to be a maximum here, is a so-lution s ∗ such that ∀ s ∈ V ( s ) , f ( s ) ≤ f ( s ∗ ) .Notice that the inequality is not strict, in order to allow the treatment of the neutrallandscape case. Local optimum neutral network (LONN):
A neutral network is a local optimumif all the configurations of the neutral network are local optima. Our definition of neutrality is strict. It also possible to define a concept of quasi-neutrality [26] but wedo not use it in this work.
8o extract the basins of attraction of the local optima neutral networks, the ”Stochas-tic Hill Climbing” algorithm is used. In this algorithm (illustrated below) one neigh-bour solution with maximum fitness is randomly chosen, and solutions with equal orimproved fitness are accepted.
Algorithm 1
Stochastic Hill ClimbingChoose initial solution s ∈ S repeat randomly choose s ′ from { z ∈ V ( s ) | f ( z ) = max { f ( x ) | x ∈ V ( s ) }} if f ( s ) ≤ f ( s ′ ) then s ← s ′ end ifuntil s is in a LONNLet us denote by h , the stochastic operator which associates to each solution s ,the solution obtained after applying the Stochastic Hill Climbing algorithm for a suffi-ciently large number of iterations to converge to a solution in a LONN.The size of the landscape is finite, so we can denote by N N , N N , N N . . . , N N n ,the local optima neutral networks. These LONNs are the vertices of the local optimanetwork in the neutral case. So, in this scenario, we have an inherent network whosenodes are themselves networks.Now, we introduce the concept of basin of attraction to define the edges and weightsof our inherent network. Note that for each solution s , there is a probability that h ( s ) ∈ N N i . We denote p i ( s ) the probability P ( h ( s ) ∈ N N i ) . We have for each solution s ∈ S , P ni =1 p i ( s ) = 1 .In non-neutral fitness landscapes where the size of each neutral network is , foreach solution s , there exists only one neutral network (in fact one solution) N N i suchthat p i ( s ) = 1 . In this case, the basin of attraction of a local optimum neutral network i is the set b i = { s ∈ S | p i ( s ) = 1 } which exactly correspond to our previous definitionin [10]. We cannot use this definition in neutral fitness landscapes, but we can extendit in the following way: 9 asin of attraction: The basin of attraction of the local optimum neutral network i is the set b i = { s ∈ S | p i ( s ) > } . This definition is consistent with our previousdefinition [8, 9] for the non-neutral case.The size of each basin of attraction can now be defined as follows: Size of a basin of attraction:
The size of the basin of attraction of a local optimumneutral network i is P s ∈S p i ( s ) .We are ready now to define the landscape’s local optima network. Local optima network:
The local optima network G = ( N, E ) is the graph wherethe nodes are the local optima N N and there is an edge between nodes
N N i and N N j when there are two solutions s i ∈ b i and s j ∈ b j such that s i ∈ V ( s j ) . Edge weight:
We first reproduce the definition of edge weights for the non-neutral landscape [9]:For each solutions s and s ′ , let p ( s → s ′ ) denote the probability that s ′ is a neighborof s , i.e. s ′ ∈ V ( s ) . The probability that a configuration s ∈ S has a neighbor in abasin b j , is therefore: p ( s → b j ) = X s ′ ∈ b j p ( s → s ′ ) The total probability of going from basin b i to basin b j is the average over all s ∈ b i ofthe transition probabilities to solutions s ′ ∈ b j : p ( b i → b j ) = 1 ♯b i X s ∈ b i p ( s → b j ) Figure 2 illustrates the complete network of a small non-neutral
N K landscape( N = 6 , K = 2 ). The circles represent the local optima basins (with diameters indi-cating the size of basins), and the weighted edges the transition probabilities as definedabove.For landscapes with neutrality, we have defined the probability p i ( s ) that a solution s belongs to a basin i . So, we can modify the previous definitions to consider neutral10andscapes: p ( s → b j ) = X s ′ ∈ b j p ( s → s ′ ) p j ( s ′ ) and in the same way : p ( b i → b j ) = 1 ♯b i X s ∈ b i p i ( s ) p ( s → b j ) where ♯b i is the size of the basin b i .In the non-neutral case, we have p k ( s ) = 1 for all the configurations in the basin b k . Therefore, the definition of weights for the non-neutral case is consistent with theprevious definition. Now, we are in a position to define the weighted local optimanetwork: fit=0.7046 fit=0.7133fit=0.7657 Figure 2: Visualization of the weighted local optima network of a small
N K landscape( N = 6 , K = 2 ). The nodes correspond to the local optima basins (with the diameterindicating the size of basins, and the label ”fit”, the fitness of the local optima). Theedges depict the transition probabilities between basins as defined in the text. Weighted local optima network:
The weighted local optima network G w =( N, E ) is the graph where the nodes are the local optima neutral networks, and thereis an edge e ij ∈ E with the weight w ij = p ( b i → b j ) between two nodes i and j if p ( b i → b j ) > .According to our definition of edge weights, w ij = p ( b i → b j ) may be differentthan w ji = p ( b j → b i ) . Thus, two weights are needed in general, and we have anoriented transition graph. 11 Analysis of the local optima networks
In order to minimize the influence of the random creation of landscapes, we considered30 different and independent landscapes for each parameter combinations: N , K and q or p . The measures reported, are the average of these 30 landscapes. We conductedour empirical study for N = 18, which is the largest possible value of N that allows theexhaustive extraction of inherent networks. The remaining set of parameters exploredare: K ∈ { , , , , , , , , } , for N K q landscapes q ∈ { , , } , and for N K p landscapes, p ∈ { . , . , . } . This section describes some standard network features such as the number of nodes andedges, and the weight distribution of the edges. For all the combinations of landscapetype and parameters, the measurements are the average of 30 independent landscapeinstances. When possible, we have also reported the data for the corresponding stan-dard
N K landscape [9, 10] in order to facilitate the comparison. In the figures, if notexplicitly stated, the thick curves labeled
N K stand for the standard, non-neutral case.
Figure 3 shows the average of the number of nodes in the optima networks of boththe
N K q (top) and N K p (bottom) landscapes with all the combinations of parametersstudied. Notice that the number of nodes increases rapidly as K increases. Clearly,for given N and K , the standard N K landscape always has more nodes than the cor-responding neutral version because the probability of changing fitness in non-neutrallandscapes is higher than in neutral ones. Therefore, for a given K , the number ofnodes decreases with increasing neutrality. All other things being equal, it is reason-able to assume that the search will be more difficult the larger the number of nodes.Therefore, as it is well known, the search is more difficult as K increases, and for agiven K , it will be more difficult when neutrality is low. In other words, an easier12earch will be expected for low K and high neutrality. N u m be r o f node s KNKq=10q=4q=2 0 2000 4000 6000 8000 10000 12000 14000 2 4 6 8 10 12 14 16 18 N u m be r o f node s KNKp=0.5p=0.8p=0.9
Figure 3: Average number of nodes in the networks for all the landscape parameterscombinations.
N K q landscapes (top), and N K p landscapes (bottom). Averages on 30independent landscapes. Results for the standard N K case are also shown for compar-ison (thick lines).
Similarly, Figure 4 illustrates the average number of edges in the networks for boththe
N K q and N K p families of landscapes. Notice that the number of connectionsincreases exponentially with increasing K . For the N K q landscape (Figure 4, bottom),the number of edges decreases with increasing neutrality for all K ; whereas for N K p landscapes, this is true only for K ≤ . In this case when K > the trend is theopposite, that is the number of edges increases with increasing neutrality. The weightdistribution results in the next subsection may help to clarify this finding. For weighted networks, the weights are characterized by both the weight distribution p ( w ) that any given edge has weight w , and the average of this distribution. In our13 N u m be r o f edge s KNKq=10q=4q=2 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 1.6e+06 2 4 6 8 10 12 14 16 18 N u m be r o f edge s KNKp=0.5p=0.8p=0.9
Figure 4: Average number of edges with weight greater than 0, for all the landscapeparameters combinations.
N K q landscapes (top) and N K p landscapes (bottom). Av-erages on 30 independent landscapes. The standard N K data are also reported (thicklines). Note the different scales on the y-axis.study, for each node i , the total sum of weights from i is equal to . Therefore, animportant measure is the weight w ii of self-connecting edges (i.e. configurations re-maining in the same node). We have the relation: w ii + s i = 1 . s i , the vertex strength ,is defined as s i = P j ∈ V ( i ) \{ i } w ij where the sum is over the set V ( i ) \ { i } of neigh-bors of i [45]. The strength of a node is a generalization of the node’s connectivitygiving information about the number and importance of the edges.Figure 5 shows the averages, over all the nodes in the network, of the weights w ii (i.e. the probabilities of remaining in the same basin after a hill-climbing from a muta-tion of one configuration in the basin). On the other hand, Figure 6 shows the empiricalaverage of weights w ij with i = j . It is clear from these results that jumping into an-other basin is much less likely than walking around in the same basin (approximatelyby an order of magnitude). Notice that for both types of neutral landscapes, the weightsto remain in the same basin, w ii (fig. 5), decrease with increasing K , which is also thetrend followed in standard N K landscapes. The weights to get to another basins (fig. 6)14lso decrease with increasing K up to K = 8 , thereafter they seem to remain constantor increase slightly. This can be explained as follows, as the number of basins increasesnon-linearly with increasing K , the probability to get to one particular basin decreases.The trend with regards to neutrality is more complex, and it is different for thetwo families of neutral landscapes. On the N K q landscape, for a fixed K , the aver-age weight to stay in the same basin decreases with increasing neutrality (fig. 5, top);whereas the opposite happens on the N K p landscape, that is, the average weight to stayin the same basin increases with neutrality (fig. 5, bottom). The trend of the weightsto get to another basin (fig. 6) is similar for both families of landscapes. It changeswhen K = 8 : for K < it increase with neutrality, while for K > it is nearly con-stant. Therefore, neutrality increases the probability that a given configuration escapesits basin and gets to another basin; but neutrality also increases the number of basins towhich the current configuration is linked. A v e r age W ii K NKq=10q=4q=2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 4 6 8 10 12 14 16 18 A v e r age W ii K NKp=0.5p=0.8p=0.9
Figure 5: Average weight w ii according to the parameters for N K q landscapes (top)and N K p landscapes (bottom). Averages on 30 independent landscapes.The general network features discussed in this section are related to the search dif-ficulty on the corresponding landscapes , since they reflect both the number of basins, The Appendix reports an empirical study exploring the effect of neutrality on the search difficulty for a A v e r age W ij K NKq=10q=4q=2 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 2 4 6 8 10 12 14 16 18 A v e r age W ij K NKp=0.5p=0.8p=0.9
Figure 6: Average of the outgoing weights w ij where i = j , for N K q landscapes (top)and N K p landscapes (bottom). Averages on 30 independent landscapes. P ( W ij = w ) w NKq=10q=4q=2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0001 0.001 0.01 0.1 1 P ( W ij = w ) w NKp=0.5p=0.8p=0.9 Figure 7: Probability distribution of the network weights w ij for outgoing edges with j = i in logscale on x-axis, for N K q landscapes (top) and N K p landscapes (bottom).Averages on 30 independent landscapes. standard evolutionary algorithm. Besides the local optima networks, it is useful to describe the associated basins ofattraction as they play a key role in heuristic search algorithms. Furthermore, somecharacteristics of the basins can be related to the local optima network features. Thenotion of the basin of attraction of a local maximum has been presented in section 3.We have exhaustively computed the size and number of the basins of all the neutrallandscapes under study.
Fig. 8 shows the average size (left) and standard deviation (right) of the basins forall the studied landscapes (averaged over the 30 independent instances in each case).Notice that size of basins decreases exponentially with increasing K . They also de-crease when neutrality decreases, being smallest for non-neutral N K landscapes, asone would expect intuitively. The standard deviations show the same behaviour as theaverage. It decreases exponentially with increasing K and also decreases when theneutrality decreases.Using the Shapiro-Wilk normality test [46] we confirmed that some distributionsof basin’s sizes can be fitted by a log-normal law when K is low. Fig 9 shows thenumber of landscape instances where the size distribution can be fitted by a log-normaldistribution according to the statistical test at level of . The number on the y-axis means that for all the instances studied the size distribution can be fitted by alog-normal. For the non-neutral N K landscapes when K ≤ , nearly all the sizedistribution are log-normal.For K ≥ , the neutrality increases the number of log-normal distributions. Againthe influence of neutrality on the two types of landscapes is not the same: for N K q landscapes, the number of log-normal distributions increases when there is more neu-trality whereas, the number of log-distribution is not maximal for the more neutral N K p landscapes. For large K , the average size of basins is very small (Fig. 8 left).17n this case, the size distributions are not log-normal, and become very narrow. Fewdifferent sizes exist and those are very small. This confirms the ruggedness of thelandscape when K is very large even when there is some neutrality. The log-normaldistribution implies that the majority of basins have a size close to average; and thatthere are few basins with larger than average size. We will see that this may be relatedto the search difficulty on the underlying landscape.
10 100 1000 10000 100000 2 4 6 8 10 12 14 16 18 A v e r age s i z e o f ba s i n s K NKq=10q=4q=2 10 100 1000 10000 100000 2 4 6 8 10 12 14 16 18 S t d . D e v . s i z e o f ba s i n s K NKq=10q=4q=2 10 100 1000 10000 100000 1e+06 2 4 6 8 10 12 14 16 18 A v e r age s i z e o f ba s i n s K NKp=0.5p=0.8p=0.9 10 100 1000 10000 100000 2 4 6 8 10 12 14 16 18 S t d . D e v . s i z e o f ba s i n s K NKp=0.5p=0.8p=0.9
Figure 8: Average (left) and standard deviation (right) of distribution of sizes for
N K q landscapes K = 4 (top) and for N K p landscapes K = 4 (bottom). Averages on 30independent landscapes. The scatter-plots in Fig. 10 (left) illustrate the correlation between the basin sizes (inlogarithmic scale) and their fitness values, for two representative landscape instances(with K = 6 , q = 3 and p = 0 . ). Fig. 10 (right) reports the correlation coefficients ρ for all combinations of landscape types and its parameters. Notice that the correlationsare positive and high, which implies that the larger basins have the higher fitness value.Therefore, the most interesting basins are also the larger ones! This may be surprising,18 N u m be r o f i n s t an c e s K NKq=10q=4q=2 0 5 10 15 20 25 30 2 4 6 8 10 12 14 16 18 N u m be r o f i n s t an c e s K NKp=0.5p=0.8p=0.9
Figure 9: Number of landscape instances (over the 30 independent instances) where thesize distribution is a log-normal distribution according to the Shapiro-Wilk normalitytest at level of for N K q landscapes (top) and for N K p landscapes (bottom).but consider that our results on basin sizes show that the size differences between largeand small decreases with increasing epistases. In consequence, with increasing rugged-ness the difficulty to find the basin with higher fitness, also increases. Notice also thatthe correlations increase with K , up to K = 8 and then they decrease. Fig. 10, alsoillustrates that neutrality decreases the correlation between basin sizes and their fitnessvalues. In other words, the size of basins is less related to the fitness of their localoptima when neutrality is present. But, as we have discussed before, basins are largerin size and smaller in number with increasing neutrality. In Fig. 11 we plot the average size of the basin corresponding to the global maximumfor all combinations of landscape types and its parameters. The results clearly showthat the size exponentially decreases when K increases. This agrees with our previousresults on standard N K landscapes [8, 9]. With respect to neutrality the size of the19
10 100 1000 10000 0.6 0.65 0.7 0.75 0.8 0.85 0.9 S i z e o f ba s i n Fitness of local optimum 0.76 0.78 0.8 0.82 0.84 0.86 0.88 2 4 6 8 10 12 14 16 18 ρ K NKq=10q=4q=2 1 10 100 1000 10000 100000 0.1 0.15 0.2 0.25 0.3 0.35 0.4 S i z e o f ba s i n Fitness of local optimum 0.76 0.78 0.8 0.82 0.84 0.86 0.88 2 4 6 8 10 12 14 16 18 ρ K NKp=0.5p=0.8p=0.9
Figure 10: Correlation between the fitness of local optima and their correspondingbasin sizes (in log) for
N K q landscapes (top) and N K p landscapes (bottom). Tworepresentative instances with K = 6 , q = 4 and p = 0 . (left) and the average of corre-lation coefficient on independent landscapes for each parameters (right). Averageson 30 independent landscapes.global maximum basin increases with increasing neutrality. In this section, we study the weighted clustering coefficient, the average path lengthbetween nodes, and the disparity of the local optima networks.
The standard clustering coefficient [47] does not consider weighted edges. We thus usethe weighted clustering measure proposed by [45], which combines the topologicalinformation with the weight distribution of the network:20 R e l a t i v e s i z e o f ba s i n K NKq=10q=4q=2 0.0001 0.001 0.01 0.1 1 2 4 6 8 10 12 14 16 18 R e l a t i v e s i z e o f ba s i n K NKp=0.5p=0.8p=0.9
Figure 11: Average of the relative size of the basin corresponding to the global maxi-mum for each K and neutral parameter over 30 independent landscapes (top
N K q andbottom N K p ). c w ( i ) = 1 s i ( k i − X j,h w ij + w ih a ij a jh a hi where s i = P j = i w ij , a nm = 1 if w nm > , a nm = 0 if w nm = 0 and k i = P j = i a ij .For each triple formed in the neighborhood of the vertex i , c w ( i ) counts the weightof the two participating edges of the vertex i . C w is defined as the weighted clusteringcoefficient averaged over all vertices of the network.Figure 12 shows the average values of the weighted clustering coefficients for allthe combinations of landscape parameters. On both the N K q and N K p landscapes, thecoefficient decreases with the degree of epistasis and increases with the degree of neu-trality. The decrease in the clustering coefficients with increasing epistasis is consistentwith our previous results on standard NK-landscapes [9]. For high epistasis and lowneutrality, there are fewer transitions between adjacent basins, and/or the transitionsare less likely to occur. 21 W e i gh t ed CC K NKq=10q=4q=2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 18 W e i gh t ed CC K NKp=0.5p=0.8p=0.9
Figure 12: Average (30 independent landscapes) of weighted clustering coefficient.
N K q landscapes (top) and N K p landscapes (bottom). The disparity measure proposed in [45], Y ( i ) , gauges the heterogeneity of the contri-butions of the edges of node i to the total weight (strength): Y ( i ) = X j = i (cid:18) w ij s i (cid:19) Figure 13 depicts the disparity coefficients as defined above. Again the measuresare consistent with our previous study on standard
N K landscapes [9]. Some interest-ing results with regards to neutrality can also be observed. For low values of K , a highdegree of neutrality increases the average disparity. When epistasis is high and regard-less of the neutrality degree, the basins are more uniformly connected, and thereforewe can picture the local optima network as more ”random” i.e. more uniform, whichhas implications on the search difficulty of the underlying landscape.22 D i s pa r i t y Y K NKq=10q=4q=2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2 4 6 8 10 12 14 16 18 D i s pa r i t y Y K NKp=0.5p=0.8p=0.9
Figure 13: Average disparity Y for N K q landscapes (top) and N K p landscapes (bot-tom). Averages of independent landscapes. Finally, as in [9, 10], in order to compute the shortest distance between two nodes onthe local optima network of a given landscape, we considered the expected numberof bit-flip mutations to go from one basin to the other. This expected number canbe computed by considering the inverse of the transition probabilities between basins(defined in 3). In other words, if we attach to the edges the inverse of the transitionprobabilities, this value would represent the average number of random mutations topass from one basin to another. More formally, the distance between two nodes isdefined by d ij = 1 /w ij where w ij = p ( b i → b j ) . Now, the length of a path betweentwo nodes is defined as being the sum of these distances along the edges that connectthe respective basins. The average path length of the whole network is the averagevalue of all the possible shortest paths.Fig. 14 is a graphical illustration of the average shortest path length between basinsfor all the neutral landscapes studied. The epistasis has the same influence on theresults whatever the family of landscapes and the level of neutrality. This path length23
50 100 150 200 250 300 350 2 4 6 8 10 12 14 16 18 P a t h l eng t h K NKq=10q=4q=2 50 100 150 200 250 300 350 2 4 6 8 10 12 14 16 18 P a t h l eng t h K NKp=0.5p=0.8p=0.9
Figure 14: Average shortest path lengths between local optima for
N K q landscapes(top), and N K p landscapes (bottom). Averages of 30 independent landscapes. A v e r age d i s t an c e t o g l oba l op t i m u m K NKq=10q=4q=2 0 50 100 150 200 250 2 4 6 8 10 12 14 16 18 A v e r age d i s t an c e t o g l oba l op t i m u m K NKp=0.5p=0.8p=0.9
Figure 15: Average path length to the optimum from all the other basins for
N K q land-scapes (top), and N K p landscapes (bottom). Averages of 30 independent landscapes.24ncreases until K = 12 and decreases thereafter. However, the degree of neutralityintroduces some differences between the families; whereas more neutrality decreasesthe shortest path length for the N K p family (bottom plot, Fig. 14); the minimal pathlength is obtained for the intermediate neutrality degrees q = 4 for N K q family (topplot, Fig. 14). The longest path length, in this case, is obtained for the largest degreeof neutrality ( q = 2 ). So, even though neutrality is high, the basins are more distant.This confirms that there are structural differences on the two types of landscapes thatinclude neutrality, and some of these structural differences are captured by the localoptima networks.Some paths are more relevant than others from the point of view of a stochasticlocal search algorithm following a trajectory over the local optima network. In orderto better illustrate the relationship of this network property with the search difficultyby heuristic methods such as stochastic local search, Fig. 15 shows the shortest pathlength to the global optimum from all the other basins in the landscape. The trend isclear, the path lengths to the optimum increase steadily with increasing K in all cases.With regards to neutrality, in both types of neutral landscapes, the higher the degree ofneutrality, the shortest the path length to the global optimum. This suggest, therefore,that the kind of neutrality introduced in the N K p and N K q landscapes could be apositive factor in the search of the global optimum . The fitness landscape concept has proved extremely useful in many fields, and it isespecially valuable for the description of the configuration spaces generated by dif-ficult combinatorial optimization problems. In previous work, we have introduced anetwork-based model that abstracts many details of the underlying landscape and com-presses the landscape information into a graph G w which we have named the localoptima network [9, 10]. The vertices of this weighted oriented graph are the local op- The empirical evaluation of search difficulty in NK p and NK q landscapes for a standard EA is studiedin the Appendix. It shows that the landscapes with more neutrality (search space size and parameters K being equal) are easier to solve for the EA. N K p and N K q , are neutral variants of the well known N K familyof landscapes. This choice also has the advantage of permitting a comparison betweenneutral and non-neutral variants of the same family of landscapes. We have measureda set of network and basin properties for these three classes. The general observation isthat there is a smooth variation with respect to standard
N K landscapes when neutralityis gradually introduced. This outcome was somewhat expected and it confirms that ourdefinitions for neutral landscapes are adequate.Our analysis of the local optima networks concentrates on the inherent structureof the studied landscapes rather than on the dynamics of a search algorithm on suchlandscapes. However, our findings, summarized below, support the view that neutralitymay enhance evolutionary search [17, 24, 29, 37, 38, 39, 40, 48, 49]. The empiricalstudy reported in the Appendix further corroborates this view. As discussed in [50],there is considerable controversy on whether neutrality helps or hinders evolutionarysearch. This is so, because many studies emphasize algorithm performance, insteadof providing an in-depth investigation of the search dynamics. Moreover, there is nota single definition of neutrality, nor an unified approach of adding redundancy to anencoding [50]. Our study, however, concentrates on specific model landscapes which26osses fitness correlation and selective neutrality. These model landscapes have beenfound to resemble the properties of biological RNA-folding landscapes. In particular,they feature neutral networks which have the “constant innovation” property [17]. Thisproperty raises the possibility that (given enough time) almost any possible fitness valuecan ultimately be attained by the population. The scenario of a population trapped ona local optima vanishes [39]. The detailed study by Barnett [38, 39], illustrates thedynamics of a simple evolutionary algorithm on several landscapes featuring neutralnetworks, and compares it with the dynamics on rugged landscapes without neutrality.The dynamics on both cases are strikingly different (Figs. 4 and 5 in [39]). On thenon-neutral landscape, the population climbs rapidly up the landscape until it reaches alocal optimum, at which higher optima are difficult to reach by mutation; the populationis effectively trapped. In the presence of percolating neutral networks, the scenarioof entrapment by local optima is evaded; adaptation is characterized by neutral driftpunctuated by transitions to higher fitness networks.We argue that our results are only relevant to optimization problems that featurepercolating neutral networks with similar statistical properties than those present inthe model landscapes studied. It is not possible to directly judge the impact of theresults for more realistic optimization problems. Therefore, it is important to analyzemore complex genotype-phenotype mappings in future work. It is worth noticing thatmassively redundant genotype-phenotype mappings, such as those used in CartesianGenetic Programming [35], have been found to be beneficial to evolutionary search.The application of the local optima network model in such scenarios is, therefore, aresearch direction worth exploring.Our results, which were at least partly unknown to our knowledge, can be summa-rized as follows.The optima networks for neutral
N K landscapes are smaller, in terms of the num-ber of nodes, with respect to standard
N K . Since the number of maxima (nodes) inthe landscape increases with N and K , search difficulty in general also increases. Butfor the same N , K pair, the search should be easier in neutral N K landscapes, and thedifficulty should decrease with increasing neutrality.The number of edges in the networks gives the average number of possible transi-27ions between maxima. However, it is more interesting to observe the average proba-bilities, which can be computed from the empirical distribution of the weights for theoutgoing edges. It is seen that neutrality increases the probability that a given local op-timal configuration escapes its present basin under the effect of a stochastic local searchoperator. This observation supports the idea that a heuristic search algorithm with anadequately set mutation rate could be more effective when neutrality is present, as theopportunity of finding a promising (adaptive) search path is increased [40].The statistics on the basins of attraction of the landscapes are particularly interest-ing. The trend is similar to what has been previously reported by the authors [9, 10]for the standard
N K family, but the size of the basins is larger the higher the degree ofneutrality, and it decreases exponentially with increasing K . Similarly, and as an im-portant particular case, the size of the global maximum basin decreases exponentiallywith K , and increases with increasing neutrality.The analysis of the clustering coefficient and the disparity, two useful local featuresof the optima networks, show that the clustering decreases with the degree of epistasis K while, for a fixed K , it tends to increase with increasing locality. This is an indirecttopological indication of the fact that maxima are more densely connected in the neutralcase, which again confirms the easier heuristic search of the corresponding landscapes.The disparity coefficient, on the other hand, says that for high K the basins tend to berandomly connected, independent of the degree of neutrality, a known result confirmedhere from the purely network point of view.Finally, we have statistically analyzed the average shortest paths between nodes inthe maxima networks. This is an important characterization of the landscape whichis easy to obtain from our maxima networks. It is relevant because it gives usefulindications on the average number of transitions that a stochastic local searcher will dobetween two maxima. In all cases the path length increases with K up to K = 12 andthen stays almost constant or decreases slightly. Neutrality decreases the mean pathlength in the N K p case, while it increases it for the N K q family. The same trend isobserved for the particular average path length from any maximum to the optimum.This last measure gives a rough approximation of the average number of steps a localsearcher would perform in the landscape to reach the optimum from any starting local28ptimal configuration, if it were “well-informed”, i.e. if it knew what would be theaverage best local optimum hop at each step. We have found that the topological observation of the local maxima networks of agiven fitness landscapes gives both useful information on the problem difficulty andmay suggest improved ways of searching them.However, although we think that our network methodology is promising as a de-scription of both neutral and non-neutral combinatorial landscapes, several issues mustbe addressed before it acquires practical usefulness. For example, we have limitedourselves to landscape sizes that can be fully enumerated in reasonable time by usingrelatively low values of N . Of course, this is not going to be possible for bigger spaces.Work is thus ongoing to sample the landscapes in a statistically significant way, a stepthat will allow us to extend the analysis to more interesting problem instances. Sec-ond, we plan to extend the present type of analysis to more significant combinatorialoptimization problems such as the TSP, SAT, knapsack problems, and several others inorder to better understand the relationships between problem difficulty and topologicalstructure of the corresponding networks. Additionally, the analysis of problems withmore complex genotype-phenotype mappings, would help to further enlighten the roleof neutrality in evolutionary search. A further step would be to incorporate and analyzethe dynamic aspects of search heuristics operating on these landscapes. The ultimategoal would be to try to improve the design of stochastic local search heuristics by usingthe information gathered in the present and future work on the local optima and basinnetworks of several problem classes. Acknowledgements
Gabriela Ochoa is supported by the Engineering and Physical Sciences Research Coun-cil, UK, under grant EPSRC (EP/D061571/1), in which Prof. Edmund K. Burke is theprincipal investigator. 29 ssessing the impact of neutrality on evolutionary search
Table 1: Evolutionary algorithm component choices and parameter settings.
Component Choice P arameter value ( s ) Population random initialisation size = 100Mutation bit-flip mutation rates = { . /N, . /N, . /N, /N, . /N, /N } Recombination 1-point crossover rates = { . , . , . , . , . , . } Selection tournament size = 2Stopping criteria fixed number of evaluations of search space size (26215 evaluations)Replacement generational with elitism
This appendix compares the search performance of a standard evolutionary algo-rithm (EA) running on
N K landscapes of equal size and ruggedness (epistasis) levelbut with different degrees of neutrality. The goal is to asses whether the presence ofneutrality in a landscape would enhance evolutionary search. Given that the fitnessvalue of the global optimum in a
N K landscape depends on its parameters ( N , K , p or q ), a comparison based on the average best fitness of a number of EA runs is not possi-ble. Therefore, we resort to the success rate as a performance measure. This is possibleon the small landscapes explored here as the global optimum is known after the ex-haustive exploration for extracting the optima networks. For our empirical study wechose the same landscape parameters as those used in the main sections of the article.Namely, N K landscapes with N = 18 and K = { , , , , , , , , } withand without neutrality, with three levels of (increasing) neutrality: q = { , , } and p = { . , . , . } for the N K q and N K p models, respectively. Table 1 summarizesthe evolutionary algorithm operator choices and parameter settings employed.A preliminary study was carried out to select the optimal combination of mutationand recombination rates for each N K model and neutrality level. The study exploredthe performance of the possible mutation and recombination rate pairs (see Table 1),on 30 independent randomly generated landscape instances of each type. The ‘optimal’combination was the one achieving the highest average success rate, which is simply30 A v e r age S u cc e ss R a t e K NKq=10q=4q=2 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 S t d . de v . S u cc e ss R a t e K NKq=10q=4q=2
Figure 16: Average (top) and standard deviation (bottom) of the success rate of a stan-dard EA searching on the
N K q landscapes. See table 1 for EA parameter settings.Averages on 100 independent landscapes.defined as the number of runs where the global optimum was found divided by thetotal number of runs. We found that the ‘optimal’ crossover rates were low (on average . over all landscape types) and the mutation rates per bit were around the well-known figure /N [51] (on average . /N ).To compute the search difficulty on each landscape type, the average and standarddeviation of success rates on runs were computed over independent landscapeinstances with the‘optimal’ parameter setting found as discussed above. Figures 16and 17show the average success rates and their standard deviations for the N K q andthe N K p models, respectively. As it is already known, the success rates were foundto decrease with increasing epistasis ( K values) in all the studied landscapes. Mostinterestingly, for a given ruggedness level (value of K ), the average success rates werefound to increase with the degree of neutrality (figures 16 and 17, top plots). Thesuccess rate standard deviations (figures 16 and 17, bottom plots) are higher for K values around 6 except for the N K q model with q = 2 , for which the standard deviation31 A v e r age S u cc e ss R a t e K NKp=0.5p=0.8p=0.9 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 S t d . de v . S u cc e ss R a t e K NKp=0.5p=0.8p=0.9
Figure 17: Average (top) and standard deviation (bottom) of the success rate of a stan-dard EA searching on the
N K p landscapes. See table 1 for EA parameter settings.Averages on 100 independent landscapes.was found to increase steadily with increasing K values.Since the distribution of success rates is not Normal, we conducted a Mann-Whitneytest to asses the statistical significance of the difference between the averages (see fig-ure 18). We compared the averages for various neutrality degrees with the same epis-tasis ( K value). A thick line between two neutral parameter values means that thedifference is significant with a p-value of ; whereas a thin line indicates that thedifference between the averages are not statistically significant. For N K q landscapes,the average differences are nearly always significant except between some non-neutral N K landscapes and
N K q with low neutrality ( q = 10 ). Similar results are found forthe N K p model, with the exception the highest epistasis values where there is nearlyno difference between the averages. Our results clearly suggest that, for the landscapemodels studied, neutrality increase the evolvability of rugged landscapes. More pre-cisely, N K landscapes of equal size and epistasis level, are easier to search for a simpleEA when neutrality is higher. 32 =NK NK Figure 18: Mann-Whitney test to compare the success rate averages of simple EA on
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