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Rebuilding Iberian motorways with slime mould
BioSystems 105 (2011) 89–100
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Rebuilding Iberian motorways with slime mould Andrew Adamatzky a,∗ , Ramon Alonso-Sanz b a b
University of the West of England, Bristol BS16 1QY, United Kingdom ETSI Agronomos (Estadistica, GSC), C. Universitaria, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 3 December 2010 Received in revised form 3 February 2011 Accepted 15 March 2011 Keywords: Bio-inspired computing Physarum polycephalum Pattern formation Spanish and Portuguese motorways Road planning
a b s t r a c t Plasmodium of a cellular slime mould Physarum polycephalum is a unique living substrate proved to be efficient in solving many computational problems with natural spatial parallelism. The plasmodium solves a problem represented by a configuration of source of nutrients by building an efficient foraging and intra-cellular transportation network. The transportation networks developed by the plasmodium are similar to transport networks built by social insects and simulated trails in multi-agent societies. In the paper we are attempting to answer the question “How close plasmodium of P. polycephalum approximates man-made motorway networks in Spain and Portugal, and what are the differences between existing motorway structure and plasmodium network of protoplasmic tubes?”. We cut agar plates in a shape of Iberian peninsula, place oat flakes at the sites of major urban areas and analyse the foraging network developed. We compare the plasmodium network with principle motorways and also analyse man-made and plasmodium networks in a framework of planar proximity graphs. © 2011 Elsevier Ireland Ltd. All rights reserved.
1. Introduction A cellular slime mould Physarum polycephalum has a sophisticated life cycle (Stephenson and Stempen, 2000), which includes fruit bodied, spores, single-cell amoebas, and syncytium. Plasmodium is a vegetative stage of P. polycephalum, is a syncytium – a single cell, where many nuclei share the same cytoplasm. The plasmodium consumes microscopic particles, and during its foraging behaviour the plasmodium spans scattered sources of nutrients with a network of protoplasmic tubes. The protoplasmic network is usually optimized to cover all sources of food and yet guarantees robust and speedy distribution of nutrients in the plasmodium body. Plasmodium’s foraging behaviour can be interpreted as computation, when data are represented by spatial of attractants and repellents, and results are represented by structure of protoplasmic network (Adamatzky, 2010). Plasmodium can solve computational problems with natural parallelism: shortest path (Nakagaki et al., 2001) and construction of hierarchies of planar proximity graphs (Adamatzky, 2009a), computation of plane tessellations (Shirakawa et al., in press), execution of basic logical computing schemes (Tsuda et al., 2004; Adamatzky, 2010), and natural implementation of spatial logic and process algebra (Schumann and Adamatzky, 2009); see overview of Physarum computing devices in Adamatzky (2010).
∗ Corresponding author. E-mail addresses: [email protected] (A. Adamatzky), [email protected] (R. Alonso-Sanz). 0303-2647/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2011.03.007
Previously (Adamatzky, 2007) we have evaluated a roadmodeling potential of P. polycephalum, however no conclusive results were presented back in 2007. A step forward biologicalapproximation, or evaluation, of man-made road networks was done in our previous papers on approximation of motorways in United Kingdom (Adamatzky and Jones, 2010), Mexico (Adamatzky et al., 2010) and the Netherlands (Adamatzky and Sloot, 2010) by plasmodium of P. polycephalum. For all three countries we found that, in principle, network of protoplasmic tubes developed by plasmodium matches, at least partly, network of man-made transport arteries. Shape of a county and exact spatial distribution of urban areas (represented by source of nutrients) may play a key role in determining exact structure of plasmodium network. Also we suspect that a degree of matching between Physarum networks and motorway networks is determined by original government designs of motorways in any particular country. This is why it is so important to collect data on development of plasmodium networks in all major countries, and then undertake a comparative analysis. In the present paper we have chosen the Iberian peninsula, i.e., the continental part of the Portugal and Spain states, as a test field for Physarum road building. Both Spain and Portugal states are members of the European Union (EU) that adopted the common EU currency (euro) and signed the Schengen agreement, which abolishes passport controls among most of the EU state members. Both states belong to the NATO military alliance. Thus, both countries committed themselves to join relevant supra-national entities, which, beyond pure geographical reasons, supports a common treatment of the Iberian peninsula, albeit such a being has not any real political entity. Let us stress here that when referring to
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Spain and Portugal in the text here, we should properly refer to continental Spain and Portugal. Thus, importantly, not including the Canary, Balearic, Azores and Madeira archipelagos. The paper is structured as follows. We present experimental setup in Section 2. Section 3 discusses principal experimental results. We compare Physarum-made and man-made transport networks with most known planar proximity graphs in Section 4. Reaction of Physarum-made network on drastic environmental conditions is studied in Section 5. We briefly overview our results in Section 7.
a wave-fragment in a sub-excitable non-linear media. Such plasmodium wave-fragment may travel for a long distance conserving its size and shape, without expanding or collapsing. After reaching and colonizing Valencia the plasmodium colonizes Alicante, Murcia, Granada, and then follows clock-wise along Iberia’s boundaries. In 30–35 h from the beginning of experiment the plasmodium colonizes almost all cities (Fig. 2c and d) and reaches Zaragoza. In the interval 34–58 h the plasmodium colonizes Tarragona and Barcelona, and also builds a link between Madrid and Valladolid (Fig. 2e and f).
2. Methods
Finding 1. Plasmodium choses clockwise and anti-clockwise directions of propagation equi-probably.
Plasmodium of P. polycephalum is cultivated in plastic container, on paper kitchen towels sprinkled with still drinking water and fed with oat flakes.1 For experiments we use 120 × 120 mm polyestyrene square and round Petri dishes. We use 2% agar gel (Select agar, Sigma Aldrich). Agar plates are cut in a shape of Iberian peninsula. We consider the 23 most populous urban areas in the Iberian peninsula (Fig. 1a): (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23)
A Coruna Gijon-Ovideo Santander Bilbao San Sebastian Vigo Valladolid Zaragoza Porto Tarragona Barcelona Pombal Madrid (capital of Spain) Valencia Lisboa (capital of Portugal) Alicante Cordoba Murcia Sevilla Faro Granada Malaga-Marbella Cadiz
Further we refer to the urban regions as U. To project regions of U onto agar gel we place oat flakes in the positions of the regions of U (Fig. 1b). At the beginning of each experiment a piece of plasmodium, usually already attached to an oat flake, is placed in Madrid (region 13 in Fig. 1a). The Petri dishes with plasmodium are kept in darkness, at temperature 22–25 ◦ C, except for observation and image recording. We undertook 30 experiments. Periodically (usually in 12 h or 24 h intervals) we scanned the dishes in Epson Perfection 4490. Scanned images of dishes, illustrated in the paper, are enhanced for higher visibility, saturation increased to 55, and contrast to 40. To ease readability of experimental images we provide complementary binary version of each image, where appropriate. Each pixel of a color image is assigned black color if red R and green G components of its RGB color exceed some specified thresholds, R > R , G > G and blue component B does not exceed some threshold value B < B ; otherwise, the pixel is assigned white color (exact values of the thresholds are indicated in the figure captions as = ( R , G , B ).
3. Transport Links Via Foraging A typical experiment is illustrated in Fig. 2. Plasmodium is inoculated on oat flake representing Madrid. The plasmodium initially produces a short-range scouting pattern, by propagating omnidirectionally (Fig. 2a and b). As soon as the plasmodium starts feeling attractants, emitted by oat flakes representing Valencia, the plasmodium forms a typical wave-like pattern propagating toward Valencia (Fig. 2a and b). This travelling pattern looks like and, as we demonstrated experimentally and in computational experiments in Adamatzky et al. (2009); Adamatzky (2009b), behaves like
1
Asda’s Smart Price Porridge Oats.
Plasmodium foraging patterns in Iberian peninsula are surprisingly (compare with our results on Physarum approximation of motorways in UK (Adamatzky and Jones, 2010) and Mexico (Adamatzky et al., 2010)) consistent in all experiments. The plasmodium reaches on the cities lying along the shore and then propagates along the shore. In 16 of 30 experiments plasmodium colonizes Iberian peninsula moving clockwise, in 14 anti-clockwise. We personally feel more comfortable with clock-wise direction, so we selected images only with clock-wise propagation. Finding 2. When colonising urban regions of Iberian peninsula plasmodium of P. polycephalum does not grow as a minimum spanning tree rooted in Madrid. The foraging activity of Physarum in experimental setup of Iberian peninsula surprised us. We expected – due to favourable location of Madrid – the plasmodium would first sprouts from Madrid to closest shore cities, e.g., will simultaneously build links from Madrid to Bilbao/San Sebastian on the north, Zaragoza and Valencia on the east, Cordoba in the south and Pombal on the west. This ideal expected scenario of plasmodium development is Fig. 3. This did not happen. The plasmodium did not follow classical growth of spanning tree but was rather choosing one shore region closes to Madrid and then grown along the shore. In some experiments the plasmodium does not span all cities. Three examples are shown in (Fig. 4), in these experiments the plasmodium had more than enough time (up to 4–5 days) to discover and colonize all oat flakes if the plasmodium ‘wanted to’. In experiment shown in Fig. 4a and b plasmodium develops two branches routed in Madrid. First branch grows from Madrid to Valladolid, then sprouts to Santander and Bilbao. Plasmodium propagates form Santander to Gijon-Ovideo and stops then, it does not propagate further to A Coruna. From Bilbao the plasmodium grows to San Sebastian and then to Zaragoza. At Zaragoza the plasmodium branches into protoplasmic tubes following to Tarragona and Barcelona, and a tube leading to Valencia and Alicante (Fig. 4a and b, eastern parts of plasmodium network). Second branch grows from Madrid to Cordoba, then protoplasmic tube follows to Sevilla and Granada. Part of plasmodium residing in Sevilla colonises Faro and Cadiz, while plasmodium which colonized Granada propagates to Malaga-Marbella region. Even after 58 h of foraging activity the plasmodium ‘refuses’ to colonise Murcia in the east and several urban regions in the west: A Coruna, Vigo, Porto, Pombal and Lisboa. The example (Fig. 4a and b) were so many cities remain uncolonized is rather unusual. For example, in experiment (Fig. 4c and d) plasmodium did not colonise Lisboa, and in experiment (Fig. 4e and f) the plasmodium misses Tarragona and Barcelona. Finding 3.
Plasmodium rarely ventures outside Iberia.
In our previous experiments with approximating transport links in United Kingdom (Adamatzky and Jones, 2010), Mexico (Adamatzky et al., 2010) and the Netherlands (Adamatzky and Sloot, 2010) we found that often plasmodium got carried away with
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Fig. 1. Experimental basics. (a) Contour map of Iberian peninsula with 19 sources of nutrients indicated. (b)–(d) Snapshot of typical setups: urban areas are represented by oat flakes, plasmodium is inoculated in Madrid, the plasmodium spans oat flakes by protoplasmic transport network.
his foraging performance and propagates outside country-shaped agar plate. We never observed such thing in experiments with Iberian peninsula: the plasmodium never ventured outside the Iberia-shape agar plate. One possible explanation could be because a shape of Iberia is more close to convex shapes then shapes of Mexico (Adamatzky et al., 2010), United Kingdom (Adamatzky and Jones, 2010) or the Netherlands (Adamatzky and Sloot, 2010). To generalise our experimental results we constructed a Physarum graph with weighted-edges. Physarum graph is a tuple P = U, E, w, where U is a set of urban areas, E is a set edges, and w : E → [0, 1] is a probability-weights of edges from E. For every two regions a and b from U there is an edge connected a and b if a plasmodium’s protoplasmic link is recorded at least in one of k experiments, and the edge (ab) has a probability calculated as a ratio of experiments where protoplasmic link (ab) occurred to the total number of experiments k, e.g., if we observed a protoplasmic tubes connection cities a and b in 5 experiments, he weight of edge (a, b) will be w(a, b) = 5/30. We do not take into account exact configuration of the protoplasmic tubes but merely their existence. Situations where protoplasmic tube originated in city a branched towards cities b and c were represented by two edges (a, b) and (a, c). Protoplasmic tubes to be selected as edges of Physarum graph were detected on binarized images of experimental arena.
Further we will be dealing with threshold Physarum graphs P() = U, T (E), w. The threshold Physarum graph is modified from Physarum graph by the transformation: T (E) = {e ∈ E : w(e) > }. That is all edges with weights less or equal to are removed. Graphs P() extracted from 30 laboratory experiments are shown in Fig. 5. An unconstrained graph P(0) is shown in Fig. 5a, it has an outer ‘shell’ of heavy-weight edges and mostly lightweighted interior. The graph becomes planar when we remove edges with weights below 5/30 (Fig. 5b). The graph becomes disconnected, with isolated edge (Valladolid, Madrid), when only edges weighting at least 16/30 are allowed to remain (Fig. 5c). The graph P() becomes acyclic when = 22/30. The acyclic graph P(22/30) consists of three isolated nodes: Madrid, Valladolid, and Barcelona, one isolated segment: (Zaragoza, Tarragona), and almost linear chain of nodes starting in San Sebastian and finishing in Valencia (Fig. 5c). The acyclic graph P(22/30) is a minimal structure, or a core, of Physarum graph. Let us check how well Physarum graphs approximate motorway network? A graph H derived from motorway network in Spain and Portugal is shown in Fig. 6. We construct the motorway graph H as follows. Let U be a set of urban regions, for any two regions a and b from U, the nodes a and b are connected by an edge (ab) if there is a motorway starting in vicinity of a and passing in vicinity of b and not passing in vicinity of any other urban area c ∈ U.
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Fig. 2. Typical plasmodium development: (a)–(e) scanned image of experimental Petri dish. Time elapsed from inoculation is shown in the sub-figure captions. (f)–(j) Binarized images, = (100, 100, 100).
Assuming edges which appear at least in two of 30 experiments are taken into account we compare motorway H with Physarum graph P(1/30) in Fig. 6b. We see that P(1/30) almost matches motorway graph H apart of few edges. The edges of H not represented in P(1/30) are mainly transport links from Valladolid and Madrid to peripheral cities of Portugal and Spain: (Valladolid, A Coruna), (Valladolid, Porto), (Madrid, Lisboa), (Madrid, Sevilla), (MalagaMarbella, Murcia), (Madrid, Bilbao), (Madrid, San Sebastian) (Fig. 6a and b). When we increase minimal allowed weight of edges to 2/30 intersection of motorway and Physarum graphs becomes discon-
nected (Fig. 6c). P(2/30) H consists of western component of transport network: starting in A Coruna and ending in MalagaMarbella area and Cadiz, and eastern component, which includes the rest of the cities. Eastern part belongs to Portugal, apart of few cities, and western part is entirely Spanish. Finding 4. Plasmodium of P. polycephalum segregates transport networks in Spain and Portugal. This is because a restriction on over-2/30 weight for Physarum graph edges means that these edges appear in over 6% of experiments, i.e., just above possible level of noise. Further increase of
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Fig. 3. Snapshots of a spanning tree of U rooted in Madrid. Vertices active at each time step are shown by black disc at snapshot.
leads to isolation of Granada and further simplification of the western part of network (Fig. 6d). For = 5/30 western network, A Coruna – Vigo – Porto – Pombal – Lisboa – Faro – Cadiz, Sevilla, remains undisturbed. The eastern network is transformed to a main cycle: Madrid – Valladolid – Gijon-Ovideo – Santander – Bilbao – San Sebastian – Zaragoza – Tarragona – Valencia – Madrid. The cycle include two embedded sub-cycles Bilbao – San Sebastian – Zaragoza – Bilbao and Zaragoza – Tarragona – Valencia – Zaragoza. Two small chains attached to the major cycle are Tarragona – Barcelona and Valencia – Alicante – Murcia (Fig. 6d). As soon as rises to 16/30 the intersection of motorway and Physarum graphs split into two isolated nodes: Malaga-Marbella and Granada, a two-node segment: (Madrid, Valladolid), and two trees. One tree is rooted in Sevilla and consists of three chains (Sevilla, Cordoba), (Sevilla, Cadiz) and Sevilla – Faro – Lisboa – Pombal – Porto – Vigo – A Coruna. Second tree is rooted in Tarragona and consists of three chains (Tarragon, Barcelona), (Tarragona, Valencia, Alicante, Murcia) and (Tarragona, Zaragoza, San Sebastian, Bilbao, Santander, Gijon-Ovideo) (Fig. 6e). Finding 5. Plasmodium of P. polycephalum does not consider Madrid (13) and Valladolid (7) as a part of connected transport network in Iberian peninsula. Why is it so? Plasmodium is never concerned with political games, it only tries to optimize its foraging activity. By keeping Madrid and Valladolid as a part of integrated network puts unnecessary strains on plasmodium functioning. Thus we can speculate that a star-shaped motorways network rooted in Madrid is rather an artificial, or man-made feature, than an intrinsically biological phenomenon. Motorway and Physarum graphs have the same labeling therefore we can compare them straightforwardly. Let diP () and diH be degrees of node i in graphs P and H. A degree mismatch between P() and H is a sum of absolute values of node mismatches. Correlation between and degree mismatch is illustrated in Fig. 7a. Lowest mismatches are observed for = 8/30, 9/30, 12/30, 13/30, and 16/30. Lowest degree mismatch in Physarum graphs P(16/30) bears particular importance because when threshold exceeds 16/30 the graph becomes disconnected yet fitting well the motorway graph in terms of node degrees. How spatial distribution of degree mismatches changes during transition from P(16/30) to P(22/30)? Degree mismatch in Madrid,
Vallidolid and Zaragosa slightly increases (Fig. 7b and c). Tarragona and Sevilla start showing some level of degree mismatch while Faro reverts to fully matching state. Finding 6. From man-made motorway perspective Physarum underdevelops transport links in the North-East of Iberia and overdevelops transport links in the South of Iberia. 4. Comparing with Proximity Graph In our previous papers we proposed, and experimentally demonstrated, that P. polycephalum constructs planar proximity graphs by its protoplasmic network (Adamatzky, 2009a). A protoplasmic network constructed in any particular experiment is planar, a generalised Physarum graph P may be non-planar, however becomes planar when we put constraints on a minimal weight of edges, P(5/30). A planar graph consists of nodes which are points of Euclidean plane and edges which are straight segments connecting the points. A planar proximity graph is a planar graph where two points are connected by an edge if they are close in some sense. A pair of points is assigned certain neighborhood, and points of the pair are connected by an edge if their neighborhood is empty. Relative neighborhood graph (Jaromczyk and Toussaint, 1992) and spanning tree are most known examples of proximity graphs. Points a and b are connected by an edge in RNG if no other point c is closer to a and b than dist(a, b) (Toussaint, 1980) (Fig. 8a). The Euclidean minimal spanning tree (MST) (Nesetril et al., 2001) is a connected acyclic graph which has minimal possible sum of edges’ lengths (Fig. 8b). MST is a sub-graph of RNG (Toussaint, 1980; Jaromczyk and Toussaint, 1992). RNG and MST are examples of minimum, cyclic in case of RNG and acyclic in case of MST, spanning of a finite planar set. Relative neighborhbood graphs are commonly considered to be the best approximations of road networks, see e.g., Watanabe (2005, 2008). By comparing Physarum and motorway graphs with RNG and MST we get an insight into minimality and redundancy of the man-made and slime mould-built transport networks. To calculate spanning trees we used classical Jaromczyk–Supowit method (Jaromczyk and Kowaluk, 1987; Supowit, 1988). Strictly speaking a spanning tree rooted in Madrid
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Fig. 4. Plasmodium does not always span all cities. Snapshots from three experiments: (a)–(c) scanned images of experimental Petri dish. Time elapsed from inoculation is shown in the sub-figure captions. (d)–(e) Binarized images, = (100, 100, 100). Time lapses from beginning of experiments (inoculation of plasmodium in Madrid) is shown in sub-figures captions.
(Fig. 8d) is not a minimum spanning tree. The minimal length tree is rooted in Alicante. However, differences in lengths are negligible: tree rooted in Madrid is just over 9% longer then MST rooted in Alicante (Table 1). Therefore further on we will address spanning tree rooted in Madrid as MST.
Finding 7. RNG(U) − ST(U) = {(A Coruna, Gijon-Ovideo), (Bilbao, San Sebastian), (Alicante, Murcia), (Lisboa, Faro)} This finding shows that acyclic minimal spanning of urban areas U is just three links shorter then cyclic spanning. Amongst the edges
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Fig. 5. Configurations Physarum-graph P() for various cut of values of . Thickness of each edge is proportional to the edge’s weight.
of RNG(U) − ST(U) only link (A Coruna, Gijon-Ovideo) is not presented in Physarum graph for highest value of = 16/30. We say a planar graph on a set U is optimal if it natches well minimum spanning tree constructed on U or at least relative neighborhood graph of U.
Cordoba), (Madrid, Zaragoza)}. Relative neighborhood graph is almost sub-graph of Physarum graphs because RNG − w ⊂ P(0) and RNG − L ⊂ P(5/30). Moreover, we see that MST − w ⊂ P(0) and MST − L ⊂ P(16/30).
Finding 8. Man-made motorway network in Iberian peninsula is not optimal while its approximation by Physarum is optimal.
5. Collapse of Infrastructure
The statement is validated as follows. Let us consider intersection of motorway graph and proximity graphs. Intersection of H and RNG has one isolated node Granada. Five edges of RNG are not represented in H: (A Coruna, Gijon-Ovideo), (Zaragoza, Tarragona), (Murcia, Granada), (Granada, Malaga-Marbella), (Faro, Cadiz). There are three isolated nodes – Granada, Murcia and Faro – in intersection of H and MST. Let w = (Sandander, Valladolid) and L = { (Santander, Valladolid), (Valladolid, Porto), (Madrid,
What would happen with transport networks in the Iberian peninsula if the whole infrastructure collapses? A fairly real and frequent kind of motorway crisis has been tried to be simulated: that of the relatively frequent traffic interruption caused by very heavy and intense storms which happen in the East coast of Spain. These punctual episodes associated with very heavy rain in sort a time, generate huge floods. The disastrous effects are induced not only by the rain itself, but also by the lack of vegetative cover (due to deforestation) and the occupation by human urbanization of many
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Fig. 6. Graph H of man-made motorway network is shown in (a). Intersection of Physarum graphs and motorway graphs is shown in (b).
of the ways of natural water evacuation (Bull et al., 2000; Camarasa et al., 2001). To imitate possible scenarios of infrastructure breakdown we deprived Physarum transport networks from further supply of nutrients and made its condition harsh by allowing substrate to dry. When substrate dries the plasmodium abandons its protoplasmic tubes and migrates to a single domain of a substrate, where it forms a hardened protoplasm. In natural conditions sclerotium can survive in such hibernation for a very long period of time, it can then come back to life when moisturized. Two examples of sclerotium are shown in Fig. 9a–d. In scanned images of dried agar gel sclerotia is visible as brownish con-
centrations (Fig. 9a and c), while in the binarized images the sclerotia are represented by areas with high density of black pixels (Fig. 9b and d). Finding 9. In case of infrastructure breakdown in Iberian peninsula the following scenarios may develop. Portugal may become isolated. Spain will concentrate to Madrid, Castilla La Mancha and Andalusia with probability 0.5 and it concentrates to a small region at the boundary between Castilla La Mancha and Andalusia with probability 0.8. We undertook 20 experiments, for each experiment we logged position and area of sclerotium and superpositions the sclerotia,
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Fig. 7. Mismatch between node degrees of Physarum and motorway graphs. (a) Degree mismatch vs. threshold . Values of mismatches are normalized by dividing them by minimal mismatch for the studied range of . (b and c) Spatial representation of degree mismatches for P() and H for = 16/30 (b) and = 22/30 (c). A node i has green (light gray) band if diP > diH and red (dark gray) band if diP < diH . Width of the band at i is proportional to |diP − diH |. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
represented by ellipses onto one image (Fig. 9e). From this image we extracted domains occupied by sclerotia for in 50% (Fig. 9f) and 80% (Fig. 9g) of experiments. 6. Physarum vs. Roman Road Networks To evaluate Physarum’s performance in imitating road development we also compared Physarum graphs with road networks as they were in Iberian peninsula in maps of Iberian peninsula in year 125 (Nacu, 2011). The following Iberian settlements were taken into account: Tarraco (Tarragona), Valentia (Valencia), Caesaraugusta (Zaragoza), Felicitas Iulia (Lisboa), Corduba (Cordoba), Gades (Gadiz). We assumed that due to proximity of Brignatium (Betanzos) to A Coruna, Tolentum (Toled) to Madrid, and Nova Carthago (Cartagena) to Murcia, they can be identified with these urban
areas. We did not incorporate Castra Legionis (Leon) and August Emerita (Merida) (Fig. 10a). We found that Physarum protoplasmic network represents 7 of 11 major roads (Fig. 10b), including Via Hercúlea and Via del Norte. Via del Atlantico is not represented in a straightforward way. However, by allowing intermediary cities the representation could be achieved. Incorporating mountains in experiments would change the resultant transport networks developed by plasmodium of P. polycephalum. This issue (that will be scrutinized in a further study) is particularly important in the case of the Iberian peninsula, whose transport network has been notably conditioned by important orographic accidents. Incidentally, although the land-based boundaries between Portugal and Spain are mostly not defined by geomorphologic entities (e.g., mountains and/or
Fig. 8. Proximity graphs constructed on regions U. (a) Relative neighborhood graph RNG. (b) Minimum spanning tree MST
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Fig. 9. Imitating collapse of infrastructure with Physarum polycephalum. (a)–(d) Two examples of dried substrate with sclerotium formed. Scanned images of experiments are shown in (a) and (c), their binarised versions in (b) and (d). Positions of sclerotium are overlapped from 20 experiments. Domains occupied by sclerotia in 50% of experiments (f) and 80% of experiments (g).
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Fig. 10. Motorway graph of roads in Iberian peninsula in year 125 (a) and its intersection with Physarum graph P(0) (b). Node numbering is as follows: (1) Brigantium, (8) Caesaraugusta, (10) Tarraco, (13) Toletum, (14) Valentia, (15) Felicitas Iulia, (17) Corduba, (18) Nova Carthago, (23) Gades. Other nodes are shown for completeness.
rivers), they have been notably stable in history (History of Portugal). 7. Discussion Approximation of shortest or even computation of a transportation network have been already a hot application for unconventional computing scientists. Nature-inspired computing paradigms and experimental implementations were successfully applied to calculation of a minimal-distance path between two given points in a space or a road network. Thus, computational models of ant-based optimization are proved to be efficient in developing novel approaches towards load-balancing of telecommunications (Dorigo and Stutzle, 2004), which indeed involves dynamical design of transport links for packets. A shortest-path problem is solved in experimental reaction–diffusion chemical systems (Adamatzky et al., 2005), gas-discharge analog systems (Reyes et al., 2002), spatially extended crystallization systems (Adamatzky, 2009c), formation of fungi mycelium networks (Jarrett et al., 2006) and plasmodium of P. polycephalum (Nakagaki et al., 2001). Amongst all experimental prototypes of path-computing devices slime mould P. polycephalum is the most cheap, userfriendly, easy to cultivate and observe biological substrate. This is Table 1 Ratios of lengths of spanning trees rooted in node of U to the length of minimum spanning tree (rooted in Alicante). Node
ST tree length
1. A Coruna 2. Gijon-Ovideo 3. Santander 4. Bilbao 5. San Sebastian 6. Vigo 7. Valladolid 8. Zaragoza 9. Porto 10. Tarragona 11. Barcelona 12. Pombal 13. Madrid 14. Valencia 15. Lisboa 16. Alicante 17. Cordoba 18. Murcia 19. Sevilla 20. Faro 21. Granada 22. Malaga-Marbella 23. Cadiz
1.04 1.02 1.08 1.04 1.05 1.10 1.08 1.02 1.09 1.06 1.06 1.09 1.10 1.05 1.02 1.00 1.09 1.05 1.12 1.10 1.07 1.06 1.05
the main reason we used the plasmodium of P. polycephalum as a computation substrate in present paper. “Is there a match between man-made motorways in Spain and Portugal and their biological analogs developed by slime mould?” To get an answer we employ a promising biological computing substrate (Adamatzky, 2010) – plasmodium of P. polycephalum. We represented major urban regions in Iberian peninsula with oat flakes, inoculated the plasmodium in Madrid and recorded formation of protoplasmic networks. Then we compared Physarum transport graph with man-made motorways and abstract proximity graphs. Experimenting with P. polycephalum we got more than we bargained for. Contrary to our expectations the plasmodium did not built a spanning tree rooted in Madrid, but instead spanned major Iberian cities by propagating (anti)-clockwise along the shore. We found that plasmodium likes to segregate transport networks of Spain and Portugal and prefers not to integrate Madrid (13) and Valladolid (7) into united transport network of Spain. In overall, we can claim that plasmodium offers alternative yet optimal transport network different from existing motorway networks. Additional indications on optimality of Physarum network of protoplasmic tubes are that minimum spanning tree is matched better by plasmodium than by motorway network, and that relative neighborhood graph is a subgraph of Physarum graph, while motorway graph is not. With regards to optimality and relationships between Physarum and motorway graphs it is a worth to compare results related to Iberia with outcomes of laboratory experiments with UK (Adamatzky and Jones, 2010) and Mexico (Adamatzky et al., 2010):
UK H⊂P
Mexico
Iberia
MST ⊂ P RNG − w ⊂ P
P = RNG RNG ⊂ P There are two traits in the Iberian peninsula that notably distinguishes it from previously studied countries. First, contrary to a fairly universal rule, the most populated area (Madrid) in the Iberian peninsula is not located near coast. Moreover, Madrid is located roughly in the geographical centre. Second, the considerable depopulation of the central region (besides Madrid), which reaches extreme low population densities in some regions, dramatically contrasts with a very high population densities in the coast, coming both from resident inhabitants and tourism (Desequilibrios demográficos). More experimental work is required to develop rigorous experimental approaches for validating man-made transport networks and their biological counterparts.
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Acknowledgemant The Spanish economist Jesus Garcia Miguel helped us in some aspects of the results interpretation. References Adamatzky, A., De Lacy Costello, B., Asai, T., 2005. Reaction–Diffusion Computers. Elsevier, Amsterdam. Adamatzky, A., 2007. From reaction–diffusion to Physarum computing. In: Invited Talk at Los Alamos Lab Workshop “Unconventional Computing: Quo Vadis?”, Santa Fe, NM, March. Adamatzky, A., 2009. Developing proximity graphs by Physarum polycephalum: does the plasmodium follow the Toussaint hierarchy? Parallel Process. Lett. 19 (1), 105–127. Adamatzky A., 2009. If BZ medium did spanning trees these would be the same trees as Physarum built. Phys. Lett. A 373 (10), 952–956. Adamatzky A., 2009. Hot ice computer. Phys. Lett. A 374 (2), 264–271. Adamatzky, A., De Lacy Costello, B., Shirakawa, T., 2009. Universal computation with limited resources: Belousov–Zhabotinsky and Physarum computers. Int. J. Bifurcat. Chaos 18, 2373–2389. Adamatzky A., Jones J., 2010. Road planning with slime mould: if Physarum built motorways it would route M6/M74 through Newcastle. Int. J. Bifurcat. Chaos 20 (12), 3065–3084. Adamatzky, A., 2010. Slime mould logical gates: exploring ballistic approach. http://arxiv.org/abs/1005.2301. Adamatzky, A., 2010. Physarum Machines: Making Computers from Slime Mould. World Scientific. Adamatzky, A., Martinez, G.J., Chapa-Vergara, S.V., Asomoza-Palacio, R., Stephens, C.R., 2010. Approximating Mexican highways with slime mould. arXiv:1010.0557v1 [nlin.PS] http://arxiv.org/abs/1010.0557. Adamatzky, A., Sloot, P., 2010. Bio-development of motorway networks in the Netherlands: slime mould approach. Bull, L.J., Kirkby, M.J., Shannon, J., Hooke, J.M., 2000. The impact of rainstorms on floods in ephemeral channels in southeast Spain. CATENA 38, 191–209. Camarasa, B., Ana, M., Segura Beltrán, F., 2001. Flood events in Mediterranean ephemeral streams (ramblas) in Valencia region, Spain. CATENA 45, 229–249, teasing.
Desequilibrios demográficos. Ministerio de Fomento. http://www.ign. es/espmap/desdem bach.htm. Dorigo, M., Stutzle, T., 2004. Ant Colony Optimization. MIT Press. History of Portugal. http://en.wikipedia.org/wiki/History of Portugal; History of Spain. http://en.wikipedia.org/wiki/History of Spain. Jaromczyk, J.W., Kowaluk, M., 1987. A note on relative neighbourhood graphs. In: Proc. 3rd Ann. Symp. Comput. Geometry , pp. 233–241. Jaromczyk, J.W., Toussaint, G.T., 1992. Relative neighborhood graphs and their relatives. Proc. IEEE 80, 1502–1517. Jarrett, T.C., Ashton, D.J., Fricker, M., Johnson, N.F., 2006. Interplay between function and structure in complex networks. Phys. Rev. E 74, 026116. Nacu A. Roman Empire 125.http://en.wikipedia.org/wiki/File:Iberian Peninsula in 125.svg. Nakagaki, T., Yamada, H., Toth, A., 2001. Path finding by tube morphogenesis in an amoeboid organism. Biophys. Chem. 92, 47–52. Nesetril, J., Milkova, E., Nesetrilova, H., 2001. Otakar Boruvka on minimum spanning tree problem. Discr. Math. 233, 3–36. Reyes, D.R., Ghanem, M.G., George, M., 2002. Glow discharge in micro fluidic chips for visible analog computing. Lab Chip 1, 113–116. Schumann, A., Adamatzky, A., 2009. Physarum spatial logic. In: Proc. 1th Int. Symp. Symbolic and Numeric Algorithms for Scientific Computing , Timisoara, Romania, September 26–29. Shirakawa, T., Adamatzky, A., Gunji, Y.-P., Miyake, Y. On simultaneous construction of Voronoi diagram and Delaunay triangulation by Physarum polycephalum. Int. J. Bifurcat. Chaos, in press. Stephenson, S.L., Stempen, H., 2000. Myxomycetes: A Handbook of Slime Molds. Timber Press. Supowit, K.J., 1988. The relative neighbourhood graph, with application to minimum spanning tree. J. ACM 30, 428–448. Toussaint, G.T., 1980. The relative neighborhood graph of a finite planar set. Pattern Recogn. 12, 261–268. Tsuda, S., Aono, M., Gunji, Y.-P., 2004. Robust and emergent Physarum logicalcomputing. BioSystems 73, 45–55. Watanabe, D., 2005. A study on analyzing the road network pattern using proximity graphs. J. City Plan. Inst. Jpn. 40, 133–138. Watanabe, D., 2008. Evaluating the configuration and the travel efficiency on proximity graphs as transportation networks. Forma 23, 81–87.
Contents lists available at ScienceDirect
BioSystems journal homepage: www.elsevier.com/locate/biosystems
Rebuilding Iberian motorways with slime mould Andrew Adamatzky a,∗ , Ramon Alonso-Sanz b a b
University of the West of England, Bristol BS16 1QY, United Kingdom ETSI Agronomos (Estadistica, GSC), C. Universitaria, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 3 December 2010 Received in revised form 3 February 2011 Accepted 15 March 2011 Keywords: Bio-inspired computing Physarum polycephalum Pattern formation Spanish and Portuguese motorways Road planning
a b s t r a c t Plasmodium of a cellular slime mould Physarum polycephalum is a unique living substrate proved to be efficient in solving many computational problems with natural spatial parallelism. The plasmodium solves a problem represented by a configuration of source of nutrients by building an efficient foraging and intra-cellular transportation network. The transportation networks developed by the plasmodium are similar to transport networks built by social insects and simulated trails in multi-agent societies. In the paper we are attempting to answer the question “How close plasmodium of P. polycephalum approximates man-made motorway networks in Spain and Portugal, and what are the differences between existing motorway structure and plasmodium network of protoplasmic tubes?”. We cut agar plates in a shape of Iberian peninsula, place oat flakes at the sites of major urban areas and analyse the foraging network developed. We compare the plasmodium network with principle motorways and also analyse man-made and plasmodium networks in a framework of planar proximity graphs. © 2011 Elsevier Ireland Ltd. All rights reserved.
1. Introduction A cellular slime mould Physarum polycephalum has a sophisticated life cycle (Stephenson and Stempen, 2000), which includes fruit bodied, spores, single-cell amoebas, and syncytium. Plasmodium is a vegetative stage of P. polycephalum, is a syncytium – a single cell, where many nuclei share the same cytoplasm. The plasmodium consumes microscopic particles, and during its foraging behaviour the plasmodium spans scattered sources of nutrients with a network of protoplasmic tubes. The protoplasmic network is usually optimized to cover all sources of food and yet guarantees robust and speedy distribution of nutrients in the plasmodium body. Plasmodium’s foraging behaviour can be interpreted as computation, when data are represented by spatial of attractants and repellents, and results are represented by structure of protoplasmic network (Adamatzky, 2010). Plasmodium can solve computational problems with natural parallelism: shortest path (Nakagaki et al., 2001) and construction of hierarchies of planar proximity graphs (Adamatzky, 2009a), computation of plane tessellations (Shirakawa et al., in press), execution of basic logical computing schemes (Tsuda et al., 2004; Adamatzky, 2010), and natural implementation of spatial logic and process algebra (Schumann and Adamatzky, 2009); see overview of Physarum computing devices in Adamatzky (2010).
∗ Corresponding author. E-mail addresses: [email protected] (A. Adamatzky), [email protected] (R. Alonso-Sanz). 0303-2647/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2011.03.007
Previously (Adamatzky, 2007) we have evaluated a roadmodeling potential of P. polycephalum, however no conclusive results were presented back in 2007. A step forward biologicalapproximation, or evaluation, of man-made road networks was done in our previous papers on approximation of motorways in United Kingdom (Adamatzky and Jones, 2010), Mexico (Adamatzky et al., 2010) and the Netherlands (Adamatzky and Sloot, 2010) by plasmodium of P. polycephalum. For all three countries we found that, in principle, network of protoplasmic tubes developed by plasmodium matches, at least partly, network of man-made transport arteries. Shape of a county and exact spatial distribution of urban areas (represented by source of nutrients) may play a key role in determining exact structure of plasmodium network. Also we suspect that a degree of matching between Physarum networks and motorway networks is determined by original government designs of motorways in any particular country. This is why it is so important to collect data on development of plasmodium networks in all major countries, and then undertake a comparative analysis. In the present paper we have chosen the Iberian peninsula, i.e., the continental part of the Portugal and Spain states, as a test field for Physarum road building. Both Spain and Portugal states are members of the European Union (EU) that adopted the common EU currency (euro) and signed the Schengen agreement, which abolishes passport controls among most of the EU state members. Both states belong to the NATO military alliance. Thus, both countries committed themselves to join relevant supra-national entities, which, beyond pure geographical reasons, supports a common treatment of the Iberian peninsula, albeit such a being has not any real political entity. Let us stress here that when referring to
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Spain and Portugal in the text here, we should properly refer to continental Spain and Portugal. Thus, importantly, not including the Canary, Balearic, Azores and Madeira archipelagos. The paper is structured as follows. We present experimental setup in Section 2. Section 3 discusses principal experimental results. We compare Physarum-made and man-made transport networks with most known planar proximity graphs in Section 4. Reaction of Physarum-made network on drastic environmental conditions is studied in Section 5. We briefly overview our results in Section 7.
a wave-fragment in a sub-excitable non-linear media. Such plasmodium wave-fragment may travel for a long distance conserving its size and shape, without expanding or collapsing. After reaching and colonizing Valencia the plasmodium colonizes Alicante, Murcia, Granada, and then follows clock-wise along Iberia’s boundaries. In 30–35 h from the beginning of experiment the plasmodium colonizes almost all cities (Fig. 2c and d) and reaches Zaragoza. In the interval 34–58 h the plasmodium colonizes Tarragona and Barcelona, and also builds a link between Madrid and Valladolid (Fig. 2e and f).
2. Methods
Finding 1. Plasmodium choses clockwise and anti-clockwise directions of propagation equi-probably.
Plasmodium of P. polycephalum is cultivated in plastic container, on paper kitchen towels sprinkled with still drinking water and fed with oat flakes.1 For experiments we use 120 × 120 mm polyestyrene square and round Petri dishes. We use 2% agar gel (Select agar, Sigma Aldrich). Agar plates are cut in a shape of Iberian peninsula. We consider the 23 most populous urban areas in the Iberian peninsula (Fig. 1a): (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23)
A Coruna Gijon-Ovideo Santander Bilbao San Sebastian Vigo Valladolid Zaragoza Porto Tarragona Barcelona Pombal Madrid (capital of Spain) Valencia Lisboa (capital of Portugal) Alicante Cordoba Murcia Sevilla Faro Granada Malaga-Marbella Cadiz
Further we refer to the urban regions as U. To project regions of U onto agar gel we place oat flakes in the positions of the regions of U (Fig. 1b). At the beginning of each experiment a piece of plasmodium, usually already attached to an oat flake, is placed in Madrid (region 13 in Fig. 1a). The Petri dishes with plasmodium are kept in darkness, at temperature 22–25 ◦ C, except for observation and image recording. We undertook 30 experiments. Periodically (usually in 12 h or 24 h intervals) we scanned the dishes in Epson Perfection 4490. Scanned images of dishes, illustrated in the paper, are enhanced for higher visibility, saturation increased to 55, and contrast to 40. To ease readability of experimental images we provide complementary binary version of each image, where appropriate. Each pixel of a color image is assigned black color if red R and green G components of its RGB color exceed some specified thresholds, R > R , G > G and blue component B does not exceed some threshold value B < B ; otherwise, the pixel is assigned white color (exact values of the thresholds are indicated in the figure captions as = ( R , G , B ).
3. Transport Links Via Foraging A typical experiment is illustrated in Fig. 2. Plasmodium is inoculated on oat flake representing Madrid. The plasmodium initially produces a short-range scouting pattern, by propagating omnidirectionally (Fig. 2a and b). As soon as the plasmodium starts feeling attractants, emitted by oat flakes representing Valencia, the plasmodium forms a typical wave-like pattern propagating toward Valencia (Fig. 2a and b). This travelling pattern looks like and, as we demonstrated experimentally and in computational experiments in Adamatzky et al. (2009); Adamatzky (2009b), behaves like
1
Asda’s Smart Price Porridge Oats.
Plasmodium foraging patterns in Iberian peninsula are surprisingly (compare with our results on Physarum approximation of motorways in UK (Adamatzky and Jones, 2010) and Mexico (Adamatzky et al., 2010)) consistent in all experiments. The plasmodium reaches on the cities lying along the shore and then propagates along the shore. In 16 of 30 experiments plasmodium colonizes Iberian peninsula moving clockwise, in 14 anti-clockwise. We personally feel more comfortable with clock-wise direction, so we selected images only with clock-wise propagation. Finding 2. When colonising urban regions of Iberian peninsula plasmodium of P. polycephalum does not grow as a minimum spanning tree rooted in Madrid. The foraging activity of Physarum in experimental setup of Iberian peninsula surprised us. We expected – due to favourable location of Madrid – the plasmodium would first sprouts from Madrid to closest shore cities, e.g., will simultaneously build links from Madrid to Bilbao/San Sebastian on the north, Zaragoza and Valencia on the east, Cordoba in the south and Pombal on the west. This ideal expected scenario of plasmodium development is Fig. 3. This did not happen. The plasmodium did not follow classical growth of spanning tree but was rather choosing one shore region closes to Madrid and then grown along the shore. In some experiments the plasmodium does not span all cities. Three examples are shown in (Fig. 4), in these experiments the plasmodium had more than enough time (up to 4–5 days) to discover and colonize all oat flakes if the plasmodium ‘wanted to’. In experiment shown in Fig. 4a and b plasmodium develops two branches routed in Madrid. First branch grows from Madrid to Valladolid, then sprouts to Santander and Bilbao. Plasmodium propagates form Santander to Gijon-Ovideo and stops then, it does not propagate further to A Coruna. From Bilbao the plasmodium grows to San Sebastian and then to Zaragoza. At Zaragoza the plasmodium branches into protoplasmic tubes following to Tarragona and Barcelona, and a tube leading to Valencia and Alicante (Fig. 4a and b, eastern parts of plasmodium network). Second branch grows from Madrid to Cordoba, then protoplasmic tube follows to Sevilla and Granada. Part of plasmodium residing in Sevilla colonises Faro and Cadiz, while plasmodium which colonized Granada propagates to Malaga-Marbella region. Even after 58 h of foraging activity the plasmodium ‘refuses’ to colonise Murcia in the east and several urban regions in the west: A Coruna, Vigo, Porto, Pombal and Lisboa. The example (Fig. 4a and b) were so many cities remain uncolonized is rather unusual. For example, in experiment (Fig. 4c and d) plasmodium did not colonise Lisboa, and in experiment (Fig. 4e and f) the plasmodium misses Tarragona and Barcelona. Finding 3.
Plasmodium rarely ventures outside Iberia.
In our previous experiments with approximating transport links in United Kingdom (Adamatzky and Jones, 2010), Mexico (Adamatzky et al., 2010) and the Netherlands (Adamatzky and Sloot, 2010) we found that often plasmodium got carried away with
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Fig. 1. Experimental basics. (a) Contour map of Iberian peninsula with 19 sources of nutrients indicated. (b)–(d) Snapshot of typical setups: urban areas are represented by oat flakes, plasmodium is inoculated in Madrid, the plasmodium spans oat flakes by protoplasmic transport network.
his foraging performance and propagates outside country-shaped agar plate. We never observed such thing in experiments with Iberian peninsula: the plasmodium never ventured outside the Iberia-shape agar plate. One possible explanation could be because a shape of Iberia is more close to convex shapes then shapes of Mexico (Adamatzky et al., 2010), United Kingdom (Adamatzky and Jones, 2010) or the Netherlands (Adamatzky and Sloot, 2010). To generalise our experimental results we constructed a Physarum graph with weighted-edges. Physarum graph is a tuple P = U, E, w, where U is a set of urban areas, E is a set edges, and w : E → [0, 1] is a probability-weights of edges from E. For every two regions a and b from U there is an edge connected a and b if a plasmodium’s protoplasmic link is recorded at least in one of k experiments, and the edge (ab) has a probability calculated as a ratio of experiments where protoplasmic link (ab) occurred to the total number of experiments k, e.g., if we observed a protoplasmic tubes connection cities a and b in 5 experiments, he weight of edge (a, b) will be w(a, b) = 5/30. We do not take into account exact configuration of the protoplasmic tubes but merely their existence. Situations where protoplasmic tube originated in city a branched towards cities b and c were represented by two edges (a, b) and (a, c). Protoplasmic tubes to be selected as edges of Physarum graph were detected on binarized images of experimental arena.
Further we will be dealing with threshold Physarum graphs P() = U, T (E), w. The threshold Physarum graph is modified from Physarum graph by the transformation: T (E) = {e ∈ E : w(e) > }. That is all edges with weights less or equal to are removed. Graphs P() extracted from 30 laboratory experiments are shown in Fig. 5. An unconstrained graph P(0) is shown in Fig. 5a, it has an outer ‘shell’ of heavy-weight edges and mostly lightweighted interior. The graph becomes planar when we remove edges with weights below 5/30 (Fig. 5b). The graph becomes disconnected, with isolated edge (Valladolid, Madrid), when only edges weighting at least 16/30 are allowed to remain (Fig. 5c). The graph P() becomes acyclic when = 22/30. The acyclic graph P(22/30) consists of three isolated nodes: Madrid, Valladolid, and Barcelona, one isolated segment: (Zaragoza, Tarragona), and almost linear chain of nodes starting in San Sebastian and finishing in Valencia (Fig. 5c). The acyclic graph P(22/30) is a minimal structure, or a core, of Physarum graph. Let us check how well Physarum graphs approximate motorway network? A graph H derived from motorway network in Spain and Portugal is shown in Fig. 6. We construct the motorway graph H as follows. Let U be a set of urban regions, for any two regions a and b from U, the nodes a and b are connected by an edge (ab) if there is a motorway starting in vicinity of a and passing in vicinity of b and not passing in vicinity of any other urban area c ∈ U.
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Fig. 2. Typical plasmodium development: (a)–(e) scanned image of experimental Petri dish. Time elapsed from inoculation is shown in the sub-figure captions. (f)–(j) Binarized images, = (100, 100, 100).
Assuming edges which appear at least in two of 30 experiments are taken into account we compare motorway H with Physarum graph P(1/30) in Fig. 6b. We see that P(1/30) almost matches motorway graph H apart of few edges. The edges of H not represented in P(1/30) are mainly transport links from Valladolid and Madrid to peripheral cities of Portugal and Spain: (Valladolid, A Coruna), (Valladolid, Porto), (Madrid, Lisboa), (Madrid, Sevilla), (MalagaMarbella, Murcia), (Madrid, Bilbao), (Madrid, San Sebastian) (Fig. 6a and b). When we increase minimal allowed weight of edges to 2/30 intersection of motorway and Physarum graphs becomes discon-
nected (Fig. 6c). P(2/30) H consists of western component of transport network: starting in A Coruna and ending in MalagaMarbella area and Cadiz, and eastern component, which includes the rest of the cities. Eastern part belongs to Portugal, apart of few cities, and western part is entirely Spanish. Finding 4. Plasmodium of P. polycephalum segregates transport networks in Spain and Portugal. This is because a restriction on over-2/30 weight for Physarum graph edges means that these edges appear in over 6% of experiments, i.e., just above possible level of noise. Further increase of
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Fig. 3. Snapshots of a spanning tree of U rooted in Madrid. Vertices active at each time step are shown by black disc at snapshot.
leads to isolation of Granada and further simplification of the western part of network (Fig. 6d). For = 5/30 western network, A Coruna – Vigo – Porto – Pombal – Lisboa – Faro – Cadiz, Sevilla, remains undisturbed. The eastern network is transformed to a main cycle: Madrid – Valladolid – Gijon-Ovideo – Santander – Bilbao – San Sebastian – Zaragoza – Tarragona – Valencia – Madrid. The cycle include two embedded sub-cycles Bilbao – San Sebastian – Zaragoza – Bilbao and Zaragoza – Tarragona – Valencia – Zaragoza. Two small chains attached to the major cycle are Tarragona – Barcelona and Valencia – Alicante – Murcia (Fig. 6d). As soon as rises to 16/30 the intersection of motorway and Physarum graphs split into two isolated nodes: Malaga-Marbella and Granada, a two-node segment: (Madrid, Valladolid), and two trees. One tree is rooted in Sevilla and consists of three chains (Sevilla, Cordoba), (Sevilla, Cadiz) and Sevilla – Faro – Lisboa – Pombal – Porto – Vigo – A Coruna. Second tree is rooted in Tarragona and consists of three chains (Tarragon, Barcelona), (Tarragona, Valencia, Alicante, Murcia) and (Tarragona, Zaragoza, San Sebastian, Bilbao, Santander, Gijon-Ovideo) (Fig. 6e). Finding 5. Plasmodium of P. polycephalum does not consider Madrid (13) and Valladolid (7) as a part of connected transport network in Iberian peninsula. Why is it so? Plasmodium is never concerned with political games, it only tries to optimize its foraging activity. By keeping Madrid and Valladolid as a part of integrated network puts unnecessary strains on plasmodium functioning. Thus we can speculate that a star-shaped motorways network rooted in Madrid is rather an artificial, or man-made feature, than an intrinsically biological phenomenon. Motorway and Physarum graphs have the same labeling therefore we can compare them straightforwardly. Let diP () and diH be degrees of node i in graphs P and H. A degree mismatch between P() and H is a sum of absolute values of node mismatches. Correlation between and degree mismatch is illustrated in Fig. 7a. Lowest mismatches are observed for = 8/30, 9/30, 12/30, 13/30, and 16/30. Lowest degree mismatch in Physarum graphs P(16/30) bears particular importance because when threshold exceeds 16/30 the graph becomes disconnected yet fitting well the motorway graph in terms of node degrees. How spatial distribution of degree mismatches changes during transition from P(16/30) to P(22/30)? Degree mismatch in Madrid,
Vallidolid and Zaragosa slightly increases (Fig. 7b and c). Tarragona and Sevilla start showing some level of degree mismatch while Faro reverts to fully matching state. Finding 6. From man-made motorway perspective Physarum underdevelops transport links in the North-East of Iberia and overdevelops transport links in the South of Iberia. 4. Comparing with Proximity Graph In our previous papers we proposed, and experimentally demonstrated, that P. polycephalum constructs planar proximity graphs by its protoplasmic network (Adamatzky, 2009a). A protoplasmic network constructed in any particular experiment is planar, a generalised Physarum graph P may be non-planar, however becomes planar when we put constraints on a minimal weight of edges, P(5/30). A planar graph consists of nodes which are points of Euclidean plane and edges which are straight segments connecting the points. A planar proximity graph is a planar graph where two points are connected by an edge if they are close in some sense. A pair of points is assigned certain neighborhood, and points of the pair are connected by an edge if their neighborhood is empty. Relative neighborhood graph (Jaromczyk and Toussaint, 1992) and spanning tree are most known examples of proximity graphs. Points a and b are connected by an edge in RNG if no other point c is closer to a and b than dist(a, b) (Toussaint, 1980) (Fig. 8a). The Euclidean minimal spanning tree (MST) (Nesetril et al., 2001) is a connected acyclic graph which has minimal possible sum of edges’ lengths (Fig. 8b). MST is a sub-graph of RNG (Toussaint, 1980; Jaromczyk and Toussaint, 1992). RNG and MST are examples of minimum, cyclic in case of RNG and acyclic in case of MST, spanning of a finite planar set. Relative neighborhbood graphs are commonly considered to be the best approximations of road networks, see e.g., Watanabe (2005, 2008). By comparing Physarum and motorway graphs with RNG and MST we get an insight into minimality and redundancy of the man-made and slime mould-built transport networks. To calculate spanning trees we used classical Jaromczyk–Supowit method (Jaromczyk and Kowaluk, 1987; Supowit, 1988). Strictly speaking a spanning tree rooted in Madrid
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Fig. 4. Plasmodium does not always span all cities. Snapshots from three experiments: (a)–(c) scanned images of experimental Petri dish. Time elapsed from inoculation is shown in the sub-figure captions. (d)–(e) Binarized images, = (100, 100, 100). Time lapses from beginning of experiments (inoculation of plasmodium in Madrid) is shown in sub-figures captions.
(Fig. 8d) is not a minimum spanning tree. The minimal length tree is rooted in Alicante. However, differences in lengths are negligible: tree rooted in Madrid is just over 9% longer then MST rooted in Alicante (Table 1). Therefore further on we will address spanning tree rooted in Madrid as MST.
Finding 7. RNG(U) − ST(U) = {(A Coruna, Gijon-Ovideo), (Bilbao, San Sebastian), (Alicante, Murcia), (Lisboa, Faro)} This finding shows that acyclic minimal spanning of urban areas U is just three links shorter then cyclic spanning. Amongst the edges
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Fig. 5. Configurations Physarum-graph P() for various cut of values of . Thickness of each edge is proportional to the edge’s weight.
of RNG(U) − ST(U) only link (A Coruna, Gijon-Ovideo) is not presented in Physarum graph for highest value of = 16/30. We say a planar graph on a set U is optimal if it natches well minimum spanning tree constructed on U or at least relative neighborhood graph of U.
Cordoba), (Madrid, Zaragoza)}. Relative neighborhood graph is almost sub-graph of Physarum graphs because RNG − w ⊂ P(0) and RNG − L ⊂ P(5/30). Moreover, we see that MST − w ⊂ P(0) and MST − L ⊂ P(16/30).
Finding 8. Man-made motorway network in Iberian peninsula is not optimal while its approximation by Physarum is optimal.
5. Collapse of Infrastructure
The statement is validated as follows. Let us consider intersection of motorway graph and proximity graphs. Intersection of H and RNG has one isolated node Granada. Five edges of RNG are not represented in H: (A Coruna, Gijon-Ovideo), (Zaragoza, Tarragona), (Murcia, Granada), (Granada, Malaga-Marbella), (Faro, Cadiz). There are three isolated nodes – Granada, Murcia and Faro – in intersection of H and MST. Let w = (Sandander, Valladolid) and L = { (Santander, Valladolid), (Valladolid, Porto), (Madrid,
What would happen with transport networks in the Iberian peninsula if the whole infrastructure collapses? A fairly real and frequent kind of motorway crisis has been tried to be simulated: that of the relatively frequent traffic interruption caused by very heavy and intense storms which happen in the East coast of Spain. These punctual episodes associated with very heavy rain in sort a time, generate huge floods. The disastrous effects are induced not only by the rain itself, but also by the lack of vegetative cover (due to deforestation) and the occupation by human urbanization of many
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Fig. 6. Graph H of man-made motorway network is shown in (a). Intersection of Physarum graphs and motorway graphs is shown in (b).
of the ways of natural water evacuation (Bull et al., 2000; Camarasa et al., 2001). To imitate possible scenarios of infrastructure breakdown we deprived Physarum transport networks from further supply of nutrients and made its condition harsh by allowing substrate to dry. When substrate dries the plasmodium abandons its protoplasmic tubes and migrates to a single domain of a substrate, where it forms a hardened protoplasm. In natural conditions sclerotium can survive in such hibernation for a very long period of time, it can then come back to life when moisturized. Two examples of sclerotium are shown in Fig. 9a–d. In scanned images of dried agar gel sclerotia is visible as brownish con-
centrations (Fig. 9a and c), while in the binarized images the sclerotia are represented by areas with high density of black pixels (Fig. 9b and d). Finding 9. In case of infrastructure breakdown in Iberian peninsula the following scenarios may develop. Portugal may become isolated. Spain will concentrate to Madrid, Castilla La Mancha and Andalusia with probability 0.5 and it concentrates to a small region at the boundary between Castilla La Mancha and Andalusia with probability 0.8. We undertook 20 experiments, for each experiment we logged position and area of sclerotium and superpositions the sclerotia,
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Fig. 7. Mismatch between node degrees of Physarum and motorway graphs. (a) Degree mismatch vs. threshold . Values of mismatches are normalized by dividing them by minimal mismatch for the studied range of . (b and c) Spatial representation of degree mismatches for P() and H for = 16/30 (b) and = 22/30 (c). A node i has green (light gray) band if diP > diH and red (dark gray) band if diP < diH . Width of the band at i is proportional to |diP − diH |. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
represented by ellipses onto one image (Fig. 9e). From this image we extracted domains occupied by sclerotia for in 50% (Fig. 9f) and 80% (Fig. 9g) of experiments. 6. Physarum vs. Roman Road Networks To evaluate Physarum’s performance in imitating road development we also compared Physarum graphs with road networks as they were in Iberian peninsula in maps of Iberian peninsula in year 125 (Nacu, 2011). The following Iberian settlements were taken into account: Tarraco (Tarragona), Valentia (Valencia), Caesaraugusta (Zaragoza), Felicitas Iulia (Lisboa), Corduba (Cordoba), Gades (Gadiz). We assumed that due to proximity of Brignatium (Betanzos) to A Coruna, Tolentum (Toled) to Madrid, and Nova Carthago (Cartagena) to Murcia, they can be identified with these urban
areas. We did not incorporate Castra Legionis (Leon) and August Emerita (Merida) (Fig. 10a). We found that Physarum protoplasmic network represents 7 of 11 major roads (Fig. 10b), including Via Hercúlea and Via del Norte. Via del Atlantico is not represented in a straightforward way. However, by allowing intermediary cities the representation could be achieved. Incorporating mountains in experiments would change the resultant transport networks developed by plasmodium of P. polycephalum. This issue (that will be scrutinized in a further study) is particularly important in the case of the Iberian peninsula, whose transport network has been notably conditioned by important orographic accidents. Incidentally, although the land-based boundaries between Portugal and Spain are mostly not defined by geomorphologic entities (e.g., mountains and/or
Fig. 8. Proximity graphs constructed on regions U. (a) Relative neighborhood graph RNG. (b) Minimum spanning tree MST
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Fig. 9. Imitating collapse of infrastructure with Physarum polycephalum. (a)–(d) Two examples of dried substrate with sclerotium formed. Scanned images of experiments are shown in (a) and (c), their binarised versions in (b) and (d). Positions of sclerotium are overlapped from 20 experiments. Domains occupied by sclerotia in 50% of experiments (f) and 80% of experiments (g).
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Fig. 10. Motorway graph of roads in Iberian peninsula in year 125 (a) and its intersection with Physarum graph P(0) (b). Node numbering is as follows: (1) Brigantium, (8) Caesaraugusta, (10) Tarraco, (13) Toletum, (14) Valentia, (15) Felicitas Iulia, (17) Corduba, (18) Nova Carthago, (23) Gades. Other nodes are shown for completeness.
rivers), they have been notably stable in history (History of Portugal). 7. Discussion Approximation of shortest or even computation of a transportation network have been already a hot application for unconventional computing scientists. Nature-inspired computing paradigms and experimental implementations were successfully applied to calculation of a minimal-distance path between two given points in a space or a road network. Thus, computational models of ant-based optimization are proved to be efficient in developing novel approaches towards load-balancing of telecommunications (Dorigo and Stutzle, 2004), which indeed involves dynamical design of transport links for packets. A shortest-path problem is solved in experimental reaction–diffusion chemical systems (Adamatzky et al., 2005), gas-discharge analog systems (Reyes et al., 2002), spatially extended crystallization systems (Adamatzky, 2009c), formation of fungi mycelium networks (Jarrett et al., 2006) and plasmodium of P. polycephalum (Nakagaki et al., 2001). Amongst all experimental prototypes of path-computing devices slime mould P. polycephalum is the most cheap, userfriendly, easy to cultivate and observe biological substrate. This is Table 1 Ratios of lengths of spanning trees rooted in node of U to the length of minimum spanning tree (rooted in Alicante). Node
ST tree length
1. A Coruna 2. Gijon-Ovideo 3. Santander 4. Bilbao 5. San Sebastian 6. Vigo 7. Valladolid 8. Zaragoza 9. Porto 10. Tarragona 11. Barcelona 12. Pombal 13. Madrid 14. Valencia 15. Lisboa 16. Alicante 17. Cordoba 18. Murcia 19. Sevilla 20. Faro 21. Granada 22. Malaga-Marbella 23. Cadiz
1.04 1.02 1.08 1.04 1.05 1.10 1.08 1.02 1.09 1.06 1.06 1.09 1.10 1.05 1.02 1.00 1.09 1.05 1.12 1.10 1.07 1.06 1.05
the main reason we used the plasmodium of P. polycephalum as a computation substrate in present paper. “Is there a match between man-made motorways in Spain and Portugal and their biological analogs developed by slime mould?” To get an answer we employ a promising biological computing substrate (Adamatzky, 2010) – plasmodium of P. polycephalum. We represented major urban regions in Iberian peninsula with oat flakes, inoculated the plasmodium in Madrid and recorded formation of protoplasmic networks. Then we compared Physarum transport graph with man-made motorways and abstract proximity graphs. Experimenting with P. polycephalum we got more than we bargained for. Contrary to our expectations the plasmodium did not built a spanning tree rooted in Madrid, but instead spanned major Iberian cities by propagating (anti)-clockwise along the shore. We found that plasmodium likes to segregate transport networks of Spain and Portugal and prefers not to integrate Madrid (13) and Valladolid (7) into united transport network of Spain. In overall, we can claim that plasmodium offers alternative yet optimal transport network different from existing motorway networks. Additional indications on optimality of Physarum network of protoplasmic tubes are that minimum spanning tree is matched better by plasmodium than by motorway network, and that relative neighborhood graph is a subgraph of Physarum graph, while motorway graph is not. With regards to optimality and relationships between Physarum and motorway graphs it is a worth to compare results related to Iberia with outcomes of laboratory experiments with UK (Adamatzky and Jones, 2010) and Mexico (Adamatzky et al., 2010):
UK H⊂P
Mexico
Iberia
MST ⊂ P RNG − w ⊂ P
P = RNG RNG ⊂ P There are two traits in the Iberian peninsula that notably distinguishes it from previously studied countries. First, contrary to a fairly universal rule, the most populated area (Madrid) in the Iberian peninsula is not located near coast. Moreover, Madrid is located roughly in the geographical centre. Second, the considerable depopulation of the central region (besides Madrid), which reaches extreme low population densities in some regions, dramatically contrasts with a very high population densities in the coast, coming both from resident inhabitants and tourism (Desequilibrios demográficos). More experimental work is required to develop rigorous experimental approaches for validating man-made transport networks and their biological counterparts.
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Acknowledgemant The Spanish economist Jesus Garcia Miguel helped us in some aspects of the results interpretation. References Adamatzky, A., De Lacy Costello, B., Asai, T., 2005. Reaction–Diffusion Computers. Elsevier, Amsterdam. Adamatzky, A., 2007. From reaction–diffusion to Physarum computing. In: Invited Talk at Los Alamos Lab Workshop “Unconventional Computing: Quo Vadis?”, Santa Fe, NM, March. Adamatzky, A., 2009. Developing proximity graphs by Physarum polycephalum: does the plasmodium follow the Toussaint hierarchy? Parallel Process. Lett. 19 (1), 105–127. Adamatzky A., 2009. If BZ medium did spanning trees these would be the same trees as Physarum built. Phys. Lett. A 373 (10), 952–956. Adamatzky A., 2009. Hot ice computer. Phys. Lett. A 374 (2), 264–271. Adamatzky, A., De Lacy Costello, B., Shirakawa, T., 2009. Universal computation with limited resources: Belousov–Zhabotinsky and Physarum computers. Int. J. Bifurcat. Chaos 18, 2373–2389. Adamatzky A., Jones J., 2010. Road planning with slime mould: if Physarum built motorways it would route M6/M74 through Newcastle. Int. J. Bifurcat. Chaos 20 (12), 3065–3084. Adamatzky, A., 2010. Slime mould logical gates: exploring ballistic approach. http://arxiv.org/abs/1005.2301. Adamatzky, A., 2010. Physarum Machines: Making Computers from Slime Mould. World Scientific. Adamatzky, A., Martinez, G.J., Chapa-Vergara, S.V., Asomoza-Palacio, R., Stephens, C.R., 2010. Approximating Mexican highways with slime mould. arXiv:1010.0557v1 [nlin.PS] http://arxiv.org/abs/1010.0557. Adamatzky, A., Sloot, P., 2010. Bio-development of motorway networks in the Netherlands: slime mould approach. Bull, L.J., Kirkby, M.J., Shannon, J., Hooke, J.M., 2000. The impact of rainstorms on floods in ephemeral channels in southeast Spain. CATENA 38, 191–209. Camarasa, B., Ana, M., Segura Beltrán, F., 2001. Flood events in Mediterranean ephemeral streams (ramblas) in Valencia region, Spain. CATENA 45, 229–249, teasing.
Desequilibrios demográficos. Ministerio de Fomento. http://www.ign. es/espmap/desdem bach.htm. Dorigo, M., Stutzle, T., 2004. Ant Colony Optimization. MIT Press. History of Portugal. http://en.wikipedia.org/wiki/History of Portugal; History of Spain. http://en.wikipedia.org/wiki/History of Spain. Jaromczyk, J.W., Kowaluk, M., 1987. A note on relative neighbourhood graphs. In: Proc. 3rd Ann. Symp. Comput. Geometry , pp. 233–241. Jaromczyk, J.W., Toussaint, G.T., 1992. Relative neighborhood graphs and their relatives. Proc. IEEE 80, 1502–1517. Jarrett, T.C., Ashton, D.J., Fricker, M., Johnson, N.F., 2006. Interplay between function and structure in complex networks. Phys. Rev. E 74, 026116. Nacu A. Roman Empire 125.http://en.wikipedia.org/wiki/File:Iberian Peninsula in 125.svg. Nakagaki, T., Yamada, H., Toth, A., 2001. Path finding by tube morphogenesis in an amoeboid organism. Biophys. Chem. 92, 47–52. Nesetril, J., Milkova, E., Nesetrilova, H., 2001. Otakar Boruvka on minimum spanning tree problem. Discr. Math. 233, 3–36. Reyes, D.R., Ghanem, M.G., George, M., 2002. Glow discharge in micro fluidic chips for visible analog computing. Lab Chip 1, 113–116. Schumann, A., Adamatzky, A., 2009. Physarum spatial logic. In: Proc. 1th Int. Symp. Symbolic and Numeric Algorithms for Scientific Computing , Timisoara, Romania, September 26–29. Shirakawa, T., Adamatzky, A., Gunji, Y.-P., Miyake, Y. On simultaneous construction of Voronoi diagram and Delaunay triangulation by Physarum polycephalum. Int. J. Bifurcat. Chaos, in press. Stephenson, S.L., Stempen, H., 2000. Myxomycetes: A Handbook of Slime Molds. Timber Press. Supowit, K.J., 1988. The relative neighbourhood graph, with application to minimum spanning tree. J. ACM 30, 428–448. Toussaint, G.T., 1980. The relative neighborhood graph of a finite planar set. Pattern Recogn. 12, 261–268. Tsuda, S., Aono, M., Gunji, Y.-P., 2004. Robust and emergent Physarum logicalcomputing. BioSystems 73, 45–55. Watanabe, D., 2005. A study on analyzing the road network pattern using proximity graphs. J. City Plan. Inst. Jpn. 40, 133–138. Watanabe, D., 2008. Evaluating the configuration and the travel efficiency on proximity graphs as transportation networks. Forma 23, 81–87.