Emergent Mating Topologies in Spatially Structured Genetic Algorithms

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1 Emergent Matng Topologes n Spatally Structured Genetc Algorthms Joshua L. Payne Dept. of Computer Scence Unversty of Vermont Burlngton, VT jpayne@cems.uvm.edu Margaret J. Eppsten Dept. of Computer Scence Unversty of Vermont Burlngton, VT Magge.Eppsten@uvm.edu ABSTRACT The applcaton of network analyss to emergent matng topologes n spatally structured genetc algorthms s presented n ths prelmnary study as a framework for nferrng evolutonary dynamcs n recombnant evolutonary search. Emergent matng topologes of populatons evolvng on regular, scale-free, and small-world mposed spatal topologes are analyzed. When the populaton evolves on a scale-free mposed spatal topology, the topology of matng nteractons s also found to be scale-free. However, due to the random ntal placement of ndvduals n the spatal topology, the scale-free matng topology lacks correlaton between ftness and vertex connectvty, resultng n hghly varable convergence rates. Scale-free matng topologes are also shown to emerge on regular mposed spatal topologes under hgh selecton pressure. Snce these scale-free emergent matng topologes self-organze such that the most-ft ndvduals are nherently located n hghly connected vertces, such emergent matng topologes are shown to promote rapd convergence on the test problem consdered heren. The emergent matng topologes of populatons evolvng on small-world mposed spatal topologes are not found to possess scale-free or small-world characterstcs. However, due to the decrease n the characterstc path length of the emergent matng topology, the rate of populaton convergence s shown to ncrease as the mposed spatal topology s tuned from regular to small-world. Categores and Subject Descrptors I.2.8 Artfcal Intellgence [Problem Solvng, Control Methods, and Search]: Heurstc Methods General Terms Algorthms, Performance, Desgn, Expermentaton Keywords Emergence, Genetc Algorthms, Matng Topologes, Network Analyss, Scale-Free, Self-Organzaton, Small-World Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. GECCO'06, July 8 12, 2006, Seattle, Washngton, USA. Copyrght 2006 ACM /06/ $ INTRODUCTION There has been a recent surge of nterest n modelng and analyzng nteractons n complex systems as networks. Such analyss has provded an understandng of the mechansms by whch complex systems are generated and has offered useful nsght nto ther dynamcs and underlyng topologcal structure. Many seemngly dsparate systems, both natural and manmade, have been shown to possess small-world and/or scale-free topologcal characterstcs. The realzaton that the topologes of many real-world systems possess smlar attrbutes began wth the semnal work of Watts and Strogatz [22]. Pror to ths work, the connecton topologes of real-world systems were typcally modeled as ether regular graphs or completely random graphs. However, Watts and Strogatz found that the nteracton topologes of several bologcal, technologcal, and socetal systems could not be captured by ether of these models; these systems had characterstcs that left them somewhere n the mddle of the two extremes. In partcular, these topologes were found to have a hgh degree of clusterng among vertces, remnscent of a regular graph, but a short characterstc path length between vertces, remnscent of a random graph. In order to model such systems, Watts and Strogatz ntroduced a smple algorthm that produced networks wth these topologcal characterstcs and named them small-world. The neural network of the worm Caenorhabdts elegans, the collaboraton topology of flm actors, the power grd of the Western Unted States [22], emal networks [6], and the cerebral cortex of prmates [21] are just a few examples of systems that have been shown to possess small-world characterstcs. Shortly after the ntroducton of small-world networks, Albert and Barabás nvestgated the connecton topology of the World Wde Web (WWW) and found that the dstrbuton of vertex connectvty dd not follow the Posson dstrbuton predcted by the random and small-world models [2]. The connecton topology of the WWW obeyed a power law, where the probablty a gven vertex had k connectons was governed by the relatonshp P(k) ~ k -γ. Ths was an mportant fndng as t showed that whle the majorty of vertces had very few lnks, a few vertces possessed the majorty of lnks, actng as hubs n the network. These networks were termed scale-free and t was subsequently shown that ths connecton topology s qute ubqutous n natural and manmade systems. The nternet [20], human sexual nteractons

2 [14], metabolc networks [10], proten-proten nteractons [11], and semantc relatonshps between words n the Englsh language [17] have all been shown to possess a scale-free dstrbuton of vertex connectvty. Understandng the structural characterstcs of nteracton networks n complex systems has provded useful nsght nto ther dynamcs. The dscovery of the scale-free dstrbuton of vertex connectvty n the map of the nternet provded an understandng of both the robustness of ths system to random falure and ts vulnerablty to targeted attacks [3]. The fndng that the dstrbuton of human sexual nteractons n Sweden obeyed a power-law provded nsght nto how to possbly desgn more effectve methods for publc health nterventon and educatonal campagns [14]. The shortcuts found n small-world networks, such as socetal nteractons, provded nsght nto how nformaton spreads wthn a populaton [12], knowledge that has proved very useful to both epdemologsts and marketng strategsts n understandng the spread of dsease and consumer awareness of new products Analyss of the structural propertes of spatal nteracton networks n evolvng artfcal populatons has receved attenton n the evolutonary computaton and artfcal lfe communtes as well. There has been a growng apprecaton for the nfluence of populaton structure on evolutonary dynamcs n recent years; metapopulaton (.e. sland model) [5] and cellular [1],[9],[15],[18],[23] spatal structures have been thoroughly studed and have proven useful n mantanng populaton dversty and curbng premature convergence. More recently, the effect of small-world and scale-free populaton structures on the dynamcs of evolutonary algorthms has been examned, focusng prmarly on the structural characterstcs of the network of potental matng nteractons (.e. populaton structure). In partcular, the evolutonary dynamcs of populatons evolvng on scale-free mposed spatal topologes have been explored n the context of evolutonary game theory [13], and the evolutonary dynamcs of populatons evolvng on both small-world and scalefree mposed spatal topologes have been nvestgated wth genetc algorthms (GA) [8]. However, network analyss has yet to be appled to the topology of actual matng nteractons that emerge when a populaton evolves on a gven mposed spatal topology. Dstngushng between the mposed spatal topology (IST) upon whch the populaton evolves and the emergent matng topology (EMT) s mportant snce t s ultmately the EMT that governs the dynamcs of a populaton-based optmzaton algorthm n recombnant evolutonary search. Understandng the structure of the EMT may provde more drect nsght regardng the adaptablty of a populaton and the rate at whch genetc nformaton dssemnates throughout a populaton. Thus, the EMT may prove to be the more relevant topology to nvestgate. In the followng, the EMT of a generatonal GA s nvestgated on a varety of ISTs: regular graphs wth varous matng neghborhood szes (from nearest neghbor to panmctc), smallworld graphs, and scale-free graphs. The prmary goal of ths prelmnary study s to assess the structural characterstcs of these emergent matng networks. None of the EMTs found were smallworld, but EMTs from both scale-free and regular ISTs wth suffcent selecton pressure were found to be scale-free, although wth dfferent relatonshps between ftness and vertex connectvty. A few prelmnary results regardng some mplcatons on evolutonary dynamcs are dscussed, although the relatonshp between the structural characterstcs of the emergent matng networks and ther functon wll be addressed more fully n future research. 2. METHODS 2.1 Network Analyss of EMTs EMTs were modeled as a labeled graph, G, wth ndvduals represented as vertces v 1,v 2,,v µ where µ s the populaton sze. Matng nteractons between ndvduals were represented as edges between vertces, captured n a symmetrc adjacency matrx, A, such that a,j = 1 f ndvduals and j mate wth one another n a gven generaton and a,j = 0 otherwse. Edge multplcty was gnored n ths study, though ths nformaton could easly be ncorporated by weghtng each edge by the number of matng nteractons that took place between two ndvduals. Three metrcs were computed to assess the structural propertes of the EMT: the probablty dstrbuton of vertex connectvty, P(k), the clusterng coeffcent C, and the characterstc path length L. P(k) s a probablty dstrbuton functon that depcts the frequency wth whch a gven vertex has k connectons. For a gven vertex connected to k nodes, the clusterng coeffcent of vertex, C, s the rato between the number of edges, E, that actually exst between the k nodes and the number of edges that could potentally exst between the k nodes. Thus, C 2E = k ( k 1) and the clusterng coeffcent for the entre EMT s gven by, C C (1) EMT (2) = where N s the total number of vertces n the EMT. L s defned as the number of edges n the shortest path between two vertces, averaged over all pars of vertces. An EMT was consdered scale-free f a strong lnear correlaton between the logarthm of the probablty of vertex connectvty (P(k)) and the logarthm of vertex connectvty (k) was found. The lnearty of the correlaton was determned by vsual nspecton on a log-log plot and quantfed by the proporton of explaned varaton (R 2 ) between log(p(k)) and log(k). Based on prelmnary expermentaton, an R 2 > 0.96 was used as a threshold to delneate between scale-free dstrbutons of vertex connectvty and nonscale-free dstrbutons of vertex connectvty (for lower R 2, the log-log plots vsually appeared decdedly non-lnear, as n Fg. 1, open crcles). An EMT was consdered small-world f C >> C rand and L L rand, where C rand and L rand are the clusterng coeffcent and characterstc path length, respectvely, of a random graph wth the same number of vertces (N) and mean vertex connectvty (<k>) [22]. Metrcs for random graphs were approxmated analytcally [22] as C rand N k (3) N

3 and L rand ln ln 2.2 Expermental Desgn ( N ) ( k ) A generatonal GA was used to optmze a sngle 100-varable bnary knapsack problem n all experments, wheren ftness was mnmzed. Whle prelmnary expermentaton usng other bnary knapsack problems and alternatve benchmark ftness functons produced qualtatvely smlar results, attenton s restrcted heren to a sngle problem for the sake of clarty. The GA used snglepont crossover (p cross = 0.85) and btwse mutaton (p mut = 0.05). The sngle best ndvdual was allowed to survve each generaton wthout undergong any genetc operatons. Selecton probabltes were assgned usng a nonlnear rank-based functon. The probablty an ndvdual of rank was selected as a parent was calculated as p sel α () = N where N conssts of all the ndvduals n the matng neghborhood of ndvdual and 0 α 1. Thus, parent selecton was based solely on relatve ftness; no assortatve matng preferences were employed. Rank was ordered such that the most-ft ndvdual had the lowest rank and the least-ft ndvdual had the hghest. A te n rank resulted n equal probablty of selecton. Stochastc unversal samplng [7] was employed as the selecton mechansm. All experments were performed n Matlab v.7.0 [16]. EMTs were analyzed on the followng ISTs: regular graphs, scale-free graphs, and small-world graphs. These ISTs are defned as follows. The regular graph ISTs were represented as square lattces wth perodc boundary condtons, wth addtonal edges added between all vertces wthn rectangular neghborhoods of a specfed matng radus (r) centered on each vertex (ncludng edges from a vertex to tself). Nearest neghbor nteractons n a cellular genetc algorthm thus correspond to r = 1, k = 9, whle panmctc GAs correspond to r = µ/2 1, k = µ (.e., complete graphs). Increasng the sze of the matng radus has the effect of ncreasng selecton pressure. Scale-free ISTs were generated accordng to the preferental attachment model of Albert and Barabás [4]. The topology was created ncrementally such that new vertces were sequentally added to the IST wth a bas toward already hghly connected vertces. Formally, the topology was generated by addng new nodes to the IST one at a tme untl a maxmum number of vertces was reached. Here, the maxmum number of vertces was smply the populaton sze (µ) snce only one vertex was needed per ndvdual. When a new node was ntroduced to the IST, t attached to a node havng vertex connectvty k wth probablty ( k ) accordng to: k ( k ) = j IST α k j (4) (5) (6) Small-world ISTs were generated accordng to the rewrng algorthm of Watts and Strogatz [22], as follows. The IST was ntalzed as a rng of µ vertces where each vertex was connected to ts k = 10 nearest neghbors. Each vertex was then vsted one at a tme n a clockwse fashon and the edge that connected that vertex to ts nearest neghbor was rewred to a randomly chosen node n the topology wth probablty p. Duplcate edges were forbdden. The algorthm then consdered edges jonng more dstant neghbors n the same clockwse manner. Ths process repeated tself untl every edge had been consdered for rewrng exactly once. Snce edges are rewred at random, and from any gven vertex there are more dstant vertces than local vertces, such rewrng forms shortcuts n the topology. EMTs were analyzed from ISTs across the entre range of p [0,1], although only those n the mddle of ths range (e.g., p [5e-3,1e-1]) have small-world characterstcs. For experments on all types of ISTs, the ntal populaton was dstrbuted randomly on the IST wthout regard to ftness (as n [8]) for the same ten random ntal populatons for each gven populaton sze and experment (.e., wherever possble, pared replcatons were performed n whch experments were ntalzed wth dentcal ntal populatons). On each of these ISTs, ten replcatons were performed for each ntal populaton, for a total of 100 trals per IST per experment. Further detals of the varous experments are provded n the next secton. 3. EXPERIMENTAL RESULTS 3.1 EMTs from Regular ISTs EMTs were analyzed on regular ISTs wth matng neghborhood szes vared from nearest neghbor nteractons to panmctc. EMTs were examned for both scale-free and small-world characterstcs, as follows. For examnng whether or not the EMTs were scale-free, we used a large populaton sze of 102,400 ndvduals ( ) n order to encourage the dstrbuton, P(k), to span a greater range of k, snce P(k) 1/µ. (However, smaller populatons exhbted qualtatvely smlar topologes). Three values of α (α {1/5,1/3,1/2}) and 8 matng rad (r {1,2,3,4,5,10,50, panmctc} were nvestgated. Increasng ether α or the matng radus ncreases selecton pressure. Selecton pressure was found to affect the emergence of scale-free matng topologes when the populaton evolved on a regular IST. When α = 1/2, the EMTs from nearest neghbor nteractons (r =1, k = 9) were not scale-free, whle the EMTs from panmctc nteractons (r = µ/2-1, k = µ) were consstently scale-free (Fgure 1, Table 1). For matng neghborhoods larger than strct nearest neghbor nteractons (r 2, k 25), a scale-free topology consstently emerged for α = 1/2 and α = 1/3 (Fgure 2a). However, for α = 1/5, selecton pressure was too low to promote the emergence of a scale-free EMT for any matng radus (Fgure 2a). For all three values of α consdered, the proporton of explaned varaton (R 2 ) between the logarthm of P(k) and the logarthm of k brefly peaked for 2 r 5 (Fgure 2a), but the mportance of ths trend s uncertan. Increasng ether α or r decreased the parameter (γ) governng the scale-free vertex

4 Fgure. 1. Dstrbuton of vertex connectvty, P(k), of EMTs from regular ISTs wth nearest neghbor nteractons (r = 1, open crcles) and panmctc nteractons on a complete graph (r = µ/2-1, black dots). Data summarzes 10 replcatons for each of 10 dfferent ntal populatons wth α = 1/2, µ = 102,400. The best-ft lne of the scale-free EMT s offset to the rght for vsual clarty. The horzontal dotted lne represents the mnmum possble P(k), whch s equvalent to 1/µ. Note the log-log scale. connectvty dstrbuton ( P( k) k γ ) (Fgure 2b). That s, hgher selecton pressure caused hgher connectvty n the hubs of the emergent scale-free matng topologes. In addtonal experments (not shown), we found that as the absolute matng neghborhood sze ncreased, the selecton pressure ncreased, and the parameter governng the emergent power-law dstrbuton of connectvty, γ, decreased, regardless of the overall doman sze. Thus, t appears that the absolute matng neghborhood sze, rather than the relatve neghborhood to doman sze, governs the dstrbuton of connectvty, P(k), n the scale-free EMT, snce the absolute sze of the matng radus determnes how many potental matng nteractons a gven ndvdual has. For examnng whether the EMTs possessed small-world characterstcs, experments were lmted to populaton szes of 2500, α = 1/2, and r {1,2,panmctc}, for computatonal reasons. As the sze of the matng neghborhood ncreased, both the characterstc path length (L) and the clusterng coeffcent (C) decreased (Table 2). Wth small matng neghborhoods (r 2), L 2.4 L rand, so these cannot be consdered small-world, whle at the other extreme (panmxa) C 1.25 C rand, so these are also not small-world (Table 2). It s possble that for some ntermedate neghborhood sze the EMT would be small-world, but ths has not yet been demonstrated. Fgure 2. (A) Effect of ncreasng the matng radus (r) on the proporton of explaned varaton, R 2, between the log(p(k)) and the log(k) for α = 1/2, α = 1/3, and α = 1/5, wth µ=102,400. The dashed horzontal lne represents the threshold used to delneate between scale-free and nonscale-free dstrbutons of vertex connectvty. (B) Effect of ncreasng the matng radus on the power law parameter, γ, governng the scale-free dstrbuton of connectvty n the EMT. Data shown pertans to the EMT at the end of the frst generaton (t = 1). Note the log-scale on the x-axs. 3.2 EMTs from Scale-Free ISTs Scale-free ISTs were generated as descrbed n secton 2.2. In order to ncrease the speed of program executon, a sparse representaton of the adjacency matrx of the scale-free IST was kept n memory at all tmes. Due to these memory constrants, the populaton sze was lmted to 10,000 ndvduals. In the generated scale-free ISTs, the parameter n the power law dstrbuton of vertex connectvty was γ = 2.68 (R 2 >0.98).

5 Table 1. Topologcal characterstcs of emergent matng topologes (EMT) from varous mposed spatal topologes (IST), wth regard to whether or not the dstrbuton of node connectvty n the EMT s scale-free. An EMT s consdered scale-free f the proporton of explaned varaton (R 2 ) between log(p(k)) and log(k) s greater than For the scale-free EMTs, estmates of γ are shown as mean ± standard devaton over 100 trals. IST Pk ( ) = k γ EMT µ Scale-free? γ R 2 Regular (r = 1) 102,400 No Regular (r = 2) 102,400 Yes 3.79 ± 0.05 Complete 102,400 Yes 3.24 ± 0.02 Scale-Free (γ=2.68) 10,000 Yes 3.25 ± Small-World 10,000 No Table 2. Topologcal characterstcs of emergent matng topologes (EMT) from varous mposed spatal topologes (IST), wth regard to whether or not the EMT possesses small-world characterstcs. An EMT s consdered small-world f L L rand and C >> C rand. Estmates of connectvty (C) and characterstc path length (L) are shown as mean ± standard devaton over 100 trals wth µ=2,500. EMT IST Small-world? C C rand L L rand Regular (r = 1) No, L > L rand ±2.5e ±2.6e ± ±0.09 Regular (r = 2) No, L > L rand ±2.7e ±3.4e ± ±0.16 Complete No, C C rand ±8e ±3e ± ±0.10 Scale-Free No, C < C rand ±1.3e ±6.3e ± ±0.45 Small-World No, L > L rand ±3.2e-3 ±1.5e-6 ±7.0e-1 ±3.5e-4 As expected, EMTs from scale-free ISTs were also scale-free, although the slope of the average power law was steeper (γ=3.25, R 2 >0.97, Table 1) than n the IST. However, the relatonshp between the ftness of an ndvdual n a gven node and the connectvty of that node was markedly dfferent between the EMTs found on scale-free ISTs and complete ISTs. In order to make a far comparson of the evolutonary dynamcs between these two, we ran an addtonal set of pared experments usng 10 replcatons on each of the same 10 random ntal populatons, for µ = 10,000, α = 1/2. Fgure 3 depcts the mean ftness of each ndvdual wth connectvty k, over all 100 trals evolvng on complete ISTs and scale-free ISTs, at generatons 1 and In the frst generaton (t = 1), there s already a strong postve correlaton between good ftness and hgh vertex connectvty n the scale-free matng topologes that emerge from complete ISTs (Fg. 3, open crcles), but not for the scale-free EMTs from scalefree ISTs (Fg. 3, symbols). Ths occurs because the scale-free EMTs from complete ISTs are self-organzed, such that the more connected ndvduals nherently possess hgher ftness, whereas n the scale-free ISTs, the matng neghborhood of a gven ndvdual was determned by ts random ntal placement. Regardless of ther ftness, the hghly connected ndvduals n the scale-free ISTs had more matng opportuntes than the less connected ndvduals. Ths dfference n correlaton between ftness and vertex connectvty affected evolutonary dynamcs, as dscussed below. At t = 1000, the EMTs from scale-free ISTs have ncreased ther correlaton between ftness and vertex connectvty (Fg. 3, + symbols), as more-ft ndvduals have now had enough tme to nfltrate the hubs of the IST, and thus become hubs n the EMT. Note that the maxmum connectvty k also ncreases over tme (see how the + symbols go farther to the rght than the symbols n Fg. 3), because ftter ndvduals n the hubs are better able to explot the hgh connectvty of those hubs n the scale-free IST than ther less-ft counterparts at t = 1.

6 Fgure 3. Relatonshp between vertex connectvty, k, and ftness n EMTs from scale-free ISTs and complete ISTs at the end of generaton one (t = 1) and one thousand (t = 1000). Data summarzes ten replcatons for each of ten dfferent ntal populatons wth µ = 10,000. Ftness s beng mnmzed. Note the log-scale on the x-axs. In contrast, scale-free EMTs from complete ISTs mantan a hgh correlaton between good ftness and hgh vertex connectvty throughout the evoluton of the populaton (compare Fg. 3 open and closed crcles). Ths hgh correlaton reduced convergence tmes on the complete ISTs relatve to the scale-free ISTs. Surprsngly, n all 100 trals, populatons on both complete and scale-free ISTs found fnal solutons wth dentcal ftness (Fg. 3), so at least n ths test case the more rapd convergence on the complete IST was not detrmental relatve to the scale-free IST. However, t s not yet known f ths result s general. EMTs from scale-free ISTs were also examned for small-world characterstcs. As wth the regular ISTs, these experments were performed on populaton szes of 2,500 wth α = 1/2. Although the average characterstc path lengths were actually shorter than those from random graphs (L 0.5 L rand, Table 2), the clusterng coeffcents were also lower than n random graphs (C 0.04 C rand, Table 2), so they cannot be consdered small-world. 3.3 EMTs from Small-World ISTs To nvestgate the EMT of a populaton evolvng on a small-world IST, small-world ISTs wth 10,000 vertces were generated as descrbed n secton 2.2 wth each node ntally connected to ts k = 10 nearest neghbors. Once agan, ths populaton sze was chosen due to the constrants ncurred by keepng the adjacency matrx of the IST n memory at all tmes. In order to make a far comparson between the EMT and the IST upon whch the populaton evolved, ten parental parngs were made n each matng neghborhood n an attempt to keep the mean vertex connectvty consstent between the two topologes. No scale-free EMT was ever found when the populaton evolved on a small-world IST for any p. However, only small-world ISTs wth k = 10 were consdered n ths experment. Increasng the ntal vertex connectvty of the IST would ncrease selecton pressure and may promote the emergence of scale-free EMTs as was found on regular ISTs wth hgh vertex connectvty n secton 3.1. Fgure 4. Comparson between the topologcal characterstcs of small-world ISTs and the EMTs of populatons evolvng on these small-world ISTs for α = 1/2. Each data pont represents the mean of ten replcatons for each of ten dfferent ntal populatons. The characterstc path length (L) and the clusterng coeffcent (C) are normalzed by the L and C of a regular graph (p = 0). The characterstc path length of the EMT s consstently hgher than the characterstc path length of the IST upon whch the populaton evolved, though they both follow the same trend. The clusterng coeffcent of the EMT remans low for all p, even when the clusterng coeffcent of the IST s hgh. Data shown pertans to the EMT at the end of the frst generaton (t = 1). Note the log-scale on the x-axs. Surprsngly, the EMTs from small-world ISTs dd not possess small-world characterstcs (Fgure 4, Table 2). Ths occurred as not every lnk n the IST necessarly manfested tself n the EMT. That s, two ndvduals that had the potental to mate due to ther proxmty n the IST, dd not necessarly mate and form a lnk n the EMT. Ths resulted n the EMT havng fewer total connectons than the IST upon whch the populaton evolved, despte the fact that ten parental parngs were made n each matng neghborhood. Thus, the normalzed characterstc path length n the EMT s always greater, and the normalzed clusterng coeffcent n the EMT s always less than, those of the IST from whch the EMT arose (Fg. 4). Note that for small-world ISTs (e.g., p [5e-3,1e-1]), the hgh clusterng coeffcent of the IST does not result n a hgh clusterng coeffcent n the correspondng EMT, so these EMTs cannot be consdered small-world. For consstent comparson to the other ISTs studed, data for C and L are also reported for µ = 2,500, α = 1/2, p = 0.008, where t can be seen that L 5.5 L rand and C 8.4 C rand, (Table 2), ndcatng that the EMT from the small-world IST s not, tself, small-world. In a separate experment, the convergence rates of dentcal populatons of µ = 10,000 evolvng on regular ISTs wth nearest neghbor nteractons (r = 1, k = 9), small-world ISTs (p = 0.008), and complete ISTs (r = µ/2-1, k = µ) were compared. In all trals (data not shown), the convergence rates of populatons evolvng on small-world ISTs were more rapd than the convergence rates of the same populatons evolvng on regular ISTs (r = 1, k = 9). Ths result s consstent wth [8]. Interestngly, the convergence rates of populatons evolvng on small-world ISTs were also hgher than the

7 convergence rates of the same populatons evolvng on complete ISTs. Further, on ths test problem, the populatons evolvng on small-world ISTs dentfed fnal solutons wth better ftness than the fnal solutons obtaned by the same populatons evolvng on ether the regular or complete ISTs. Ths mples that the populatons evolvng on regular and complete ISTs were convergng prematurely on local optma, although t s unclear as to whether ths relatonshp s general. The relatonshp between the topologcal characterstcs of the EMT and convergence warrants further study. 4. DISCUSSION The goal of ths prelmnary study was to examne the network characterstcs of emergent matng topologes n spatally structured genetc algorthms, specfcally to see f they were small-world and/or scale-free. We found that the topologcal characterstcs of emergent matng topologes can be qute dfferent from the mposed spatal topologes upon whch the populaton evolves. When the mposed spatal topology s scale-free, the emergent matng topology s also scale-free, but good ftness s not ntally postvely correlated wth hgh connectvty. More nterestngly, scale-free matng topologes were shown to emerge from regular graph mposed spatal topologes, as long as selecton pressure was suffcently hgh, and these exhbted a strong postve correlaton between good ftness and hgh connectvty for at least 1000 generatons on the test problem. Scale-free topologes were never found to emerge when nteractons were lmted to nearest neghbors on a rectangular lattce, because the selecton pressure was too low. Emergent matng topologes from a varety of mposed spatal topologes (ncludng regular graphs wth small matng neghborhoods, complete graphs (panmxa), scale-free graphs, and small-world graphs) were never found to exhbt small-world characterstcs, although the reasons vared for the dfferent mposed spatal topologes. Trends n the data hnt that small-world characterstcs may arse from ntermedate neghborhood szes on regular graphs, but ths has not yet been demonstrated. These fndngs have mplcatons for the study of evolutonary dynamcs n both spatally structured GAs and spatally explct artfcal lfe smulatons. Prevous nvestgatons of evolutonary dynamcs on scale-free mposed spatal topologes [8],[13] have shown that f an ndvdual of hgh ftness could successfully nfltrate one of the hubs of the scale-free topology, then that ndvdual s genetc nformaton would dssemnate rapdly throughout the populaton. However, when the mposed spatal topology s scale-free, there s ntally no postve correlaton between good ftness and hgh vertex connectvty. Here, we have shown that self-organzng scale-free matng topologes spontaneously emerge from regular mposed spatal topologes, and nherently have a strong correlaton between good ftness and hgh vertex connectvty. Thus, n these selforganzng scale-free matng topologes, fortutous genetc combnatons are quckly communcated throughout the populaton. Such swft dssemnaton of advantageous genetc nformaton has mplcatons for rapd, possbly premature, convergence. In [8], t was also shown that the rate of convergence of populatons evolvng on small-world mposed spatal topologes ncreased as the probablty of rewrng (p) ncreased. Ths s due to the shortcuts that are formed n the spatal topology as edges are rewred, whch decreases the characterstc path length and allows for the spread of genetc nformaton over longer spatal scales. In the current study, we have shown that the emergent matng topologes from smallworld mposed spatal topologes are not, themselves, small-world due to ther hgh characterstc path lengths and low clusterng coeffcents. However, snce the characterstc path lengths of the emergent matng topologes do decrease wth ncreased probablty of rewrng, ncreasng p nevertheless has the effect of ncreasng the populaton convergence rate. We also found that the spatal scale of ndvdual matng nteractons drectly affected the structure of the emergent matng topology. As expected, our results confrm that the localzaton of ndvdual nteractons (commonly employed n both cellular genetc algorthms [1],[9],[15],[18],[23] and artfcal lfe smulatons [19]) gves rse to a long characterstc path length n the emergent matng topology. As a result, genetc nformaton remans qute localzed and travels slowly across longer spatal scales, gvng rse to fundamentally dfferent evolutonary dynamcs than found n randomly mxng populatons. Further, the results of ths study show that the connectvty dstrbuton, P(k), of scale-free emergent matng topologes from regular mposed spatal topologes s governed by absolute, as opposed to relatve, matng neghborhood sze. Snce dfferent absolute matng neghborhood szes produce emergent matng topologes whch yeld dramatcally dfferent evolutonary dynamcs, the results of ths study suggest that care should be taken n choosng bologcally meanngful matng neghborhoods n spatally explct artfcal lfe smulatons. 5. SUMMARY In recent years, there has been a growng apprecaton for the mportant nfluence of spatal relatonshps on evolutonary dynamcs. Consequently, a varety of spatally explct mposed populaton structures are beng explored for use n GAs [1],[8],[23]. Our results, whle prelmnary, ndcate that analyss of the network characterstcs of the emergent (rather than mposed) matng topologes may provde valuable nsght regardng evolutonary dynamcs n populatons, both natural and artfcal. Ths method can expose underlyng smlartes and dfferences n the evolutonary dynamcs produced on varous dsparate mposed spatal topologes. Whle the man focus of ths study was to understand the structural characterstcs of the matng topologes that emerge from varous mposed spatal topologes, future endeavors wll more fully explore the relatonshp between the structure of these emergent topologes and evolutonary dynamcs. Further, future work wll expand upon the results presented n ths prelmnary study by nvestgatng emergent matng topologes and ther dynamcs on a broader range of test problems, and usng alternatve rankng functons and selecton operators. We are partcularly nterested n the mplcatons for adaptablty n changng envronments. 6. Acknowledgments Ths work was supported n part by a graduate research assstantshp funded by DOE-FG02-00ER45828 awarded by the US Department of Energy through ts EPSCoR Program. 7. References [1] Alba, E. & Dorronsoro, B. 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