Gráfelmélet és biológiai alkalmazásai Jordán Ferenc

Gráfelmélet és biológiai alkalmazásai Jordán Ferenc jordan.ferenc@gmail.com www.cosbi.eu A hálózatelméleti megközelítés Gráfelméleti alapfogalmak, gráftípusok Hálózatok lokális és globális sajátságai, szerkezet és viselkedés Biológiai alkalmazások: ökológiai, molekuláris, társas kapcsolathálózatok Hasznos szoftverek (?)

Steps of network analysis 1. Network data collection 2. Network construction 3. Network analysis BC i 2 j k ( g g jk ( n ) / g 1)( g 2) i jk

Steps of network analysis 0. Determine whether the problem is a network-problem 1. Network data collection 2. Network construction 3. Network analysis 4. Testing predictions

The network approach specific, detailed tactic water molecule general, not detailed, strategic alkanes Sylvester, J.J. 1878. Chemistry and Algebra. Nature 17: 284.

The network approach specific, detailed tactic water molecule chemical reaction general, not detailed, strategic alkanes biochemical network

The network approach specific, detailed tactic water molecule chemical reaction prey and predator general, not detailed, strategic alkanes biochemical network food web

Network analysis: when and when not? A classroom of children Crowd celebrating Beşiktaş Community (system): structure infuences behaviour

Graph theory some basics Graph: node set V(G), link set E(G), relation

Graph theory some basics Graph: node set V(G), link set E(G), relation Directed: ordered pair of nodes (digraph)

Graph theory some basics Graph: node set V(G), link set E(G), relation Directed: ordered pair of nodes (digraph) Weighted: number assigned to each link

Graph theory some basics Graph: node set V(G), link set E(G), relation Directed: ordered pair of nodes (digraph) Weighted: number assigned to each link Signed graph: each edge is of either + or - sign + - + - +

Graph theory some basics Graph: node set V(G), link set E(G), relation Directed: ordered pair of nodes (digraph) Weighted: number assigned to each link Signed graph: each edge is of either + or - sign Component: connected subgraph Complete graph: all nodes are neighbours Regular graph: all nodes have the same D(i) Bipartite: nodes in 2 subsets + - + - +

Ökoszisztéma: fajok = gráfpontok lehetséges relációk = gráfélek

Reláció #1: az i faj ugyanolyan színű, mint a j faj i j fekete-fehér vöröses sárgás irányítatlan élek klikkek ekvivalencia

Reláció #2: az i fajt megeszi a j faj i j irányított élek loopok ritkák nem ekvivalencia

Systems thinking in ecology: early abstractions Summerhayes, V.S. and Elton, C. 1923. Contributions to the ecology of Spitsbergen and Bear Island. Journal of Ecology, 11:214-268.

Graph indices from the local to the global

Degree: the most local view In a flow network: Source: D(in)=0 Sink: D(out)=0 degree distribution degree = in-degree + out-degree density

Degree: the most local view In a flow network: Source: D(in)=0 Sink: D(out)=0 degree distribution degree = in-degree + out-degree L/M*N bipartite density

Clustering coefficient i = 4/12 links of node i links between the neighbours of node i non-existing

Clique: complete subgraph

Hálózati modulok / motívumok A B C A B C D E

Most frequent, empirically-defined modules in food webs Menge, B.A. 1995. Ecological Monographs, 65:21-74.

Milo, R. et al. 2002. Science, 298: 824-827.

Distance: length of shortest path i and j are not reachable: d ij = i j d ij values in D distance matrix d ij -1 values in R reciprocal distance matrix average distance

Gráfpontok távolsága és indirekt populációs kölcsönhatások ants rodents + + + small-seeded plant - large-seeded plant

Gráfpontok távolsága és indirekt populációs kölcsönhatások removing the rodents indirect effects on ants: exploitative competition + - - ants rodents + + + indirect commensalism small-seeded plant - large-seeded plant + + - -

Gráfpontok távolsága és indirekt populációs kölcsönhatások removing the rodents indirect effects on ants: exploitative competition 2 steps 3 steps + - - indirect commensalism + + - -

Small world networks: * high clustering (like regular) * small distance (like random) SW measure: clustering / distance Six degrees / two degrees Watts, D.J. and Strogatz, S.H. 1998. Collective dynamics of small-world networks. Nature, 393:440-442.

How to understand complexity?

degree distribution

0,4 density degree distribution

0,4 density 2 degree distribution distance / diameter

0,4 density 2 degree distribution distance / diameter aggregation

0,4 density 2 degree distribution distance / diameter aggregation subgraph

0,4 density 2 degree distribution distance / diameter aggregation centrality subgraph

Centrality measures: status - contrastatus = net status positions in hierarchies 9-0=9 1 1 1 4-1=3 2 2 1 1 s = summed distance: d ix for all i s is the same after reversing the signs 3 2 Harary, F. 1961. General Systems 6: 41-44.

Quantifying indirect effects in food webs: parasitoid overlap graphs Godfray, H.C.J., Lewis, O.T. and Memmott, J. 1999. Phil. Trans. Roy. Soc. 354:1811-1824.

TI (WI)-index C P BA * P CB + P DA * P CD = P CA P CD = 1/D C C C P CB P CD D B D P CB P BA P DA B A P BA * P CB = P CA P BA A Jordán, F., Liu, W.-C. and van Veen, F.J.F. 2003. Community Ecology, 4:79-88.

interaction matrix 1 2 3 4 5 6 7 8 9 10 1 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 2 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 3 0,09 0,03 0,06 0,09 0,31 0,19 0,09 0,09 0,03 0,03 4 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 5 0,09 0,03 0,06 0,09 0,24 0,26 0,09 0,09 0,03 0,03 6 0,08 0,05 0,03 0,08 0,18 0,32 0,08 0,08 0,05 0,05 7 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 8 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 9 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 10 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 0,83 0,42 0,41 0,83 2,03 2,97 0,83 0,83 0,42 0,42

Relative importance of nodes:

strength 1 2 3 4 5 6 7 8 9 10 1 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 2 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 3 0,09 0,03 0,06 0,09 0,31 0,19 0,09 0,09 0,03 0,03 4 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 5 0,09 0,03 0,06 0,09 0,24 0,26 0,09 0,09 0,03 0,03 6 0,08 0,05 0,03 0,08 0,18 0,32 0,08 0,08 0,05 0,05 7 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 8 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 9 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 10 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 0,83 0,42 0,41 0,83 2,03 2,97 0,83 0,83 0,42 0,42

Relative effects of node 5 on others:

Relative effects of others on node 5:

symmetry 1 2 3 4 5 6 7 8 9 10 1 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 2 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 3 0,09 0,03 0,06 0,09 0,31 0,19 0,09 0,09 0,03 0,03 4 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 5 0,09 0,03 0,06 0,09 0,24 0,26 0,09 0,09 0,03 0,03 6 0,08 0,05 0,03 0,08 0,18 0,32 0,08 0,08 0,05 0,05 7 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 8 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 9 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 10 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 0,83 0,42 0,41 0,83 2,03 2,97 0,83 0,83 0,42 0,42

feedback 1 2 3 4 5 6 7 8 9 10 1 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 2 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 3 0,09 0,03 0,06 0,09 0,31 0,19 0,09 0,09 0,03 0,03 4 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 5 0,09 0,03 0,06 0,09 0,24 0,26 0,09 0,09 0,03 0,03 6 0,08 0,05 0,03 0,08 0,18 0,32 0,08 0,08 0,05 0,05 7 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 8 0,08 0,04 0,04 0,08 0,22 0,28 0,08 0,08 0,04 0,04 9 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 10 0,08 0,05 0,03 0,08 0,13 0,37 0,08 0,08 0,05 0,05 0,83 0,42 0,41 0,83 2,03 2,97 0,83 0,83 0,42 0,42

Effects spreading out from the black node: its trophic field Jordán, F. 2001. Community Ecology, 2:181-185.

Hálózatelemzés és betegséggének detektálása obesity /yellow the rest autism / green cancer heart diabetes

obesity /yellow autism / green the rest cancer heart diabetes

obesity P08254 P08588 P16671 P17302 P18825 P78504 Q14524 Q9UGJ0 Q9Y4J8 heart O \ P O00253 O75056 P06241 P41240 P01189 P07550 P49407, P62993, Q5JY77 P12931 P12931, P17252, P41240 Q14232 P13945 P12931 P12931 P25874 P29120 P32245 P37231 P28482 P41159 P41968 P48357 P52895 P55851 P63104 P55916 P63104 P81133 Q15466 Q16620 O14908 P06241 Q86YN6 Q9UBU3 P54646 Two-step mediators

obesity P08254 P08588 P16671 P17302 P18825 P78504 Q14524 Q9UGJ0 Q9Y4J8 heart O \ H O00253 0 O75056 0.01538 0.01549 0.03087 P01189 0 P07550 0.08666 0.00046 0.02222 0.12301 0.23235 P13945 0.00133 0.00128 0.00261 P25874 0 P29120 0 P32245 0 P37231 0.00707 0.00707 P41159 0.05079 0.05079 P41968 0 P48357 0 P52895 0 P55851 0.05833 0.05833 P55916 0.05833 0.05833 P81133 0 Q15466 0 Q16620 0.05745 0.01204 0.06949 Q86YN6 0 Q9UBU3 0 0 0.14411 0.02921 0.04606 0.23967 0 0 0.05079 0 Strengths of two-step mutual relationships

nd in PPI Indirect mediators between diseases in the human PPI network 3.5 3 2.5 2 1.5 1 0.5 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 M 2 in DPIP Nguyen, T.P. and Jordán, F. 2010. BMC Systems Biology.

Regular and structural equivalence Equivalence 1 2 3 4 5 6 7 8 9 structural 1 2 3 4 5, 6 7 8, 9 Luczkovich, J.J. et al. 2003. J. Theor. Biol., 220:303-321.

Regular and structural equivalence Equivalence 1 structural regular 2 3 4 5 6 7 8 9 1 1 2 2, 3 3 4 4 5, 6, 7 5, 6 8, 9 7 8, 9 Luczkovich, J.J. et al. 2003. J. Theor. Biol., 220:303-321.

Regular and structural equivalence Equivalence 1 2 3 4 5 6 7 8 9 structural regular max. regular 1 1 1 2 2, 3 2, 3, 4 3 4 5, 6, 7, 8, 9 4 5, 6, 7 5, 6 8, 9 7 8, 9 Luczkovich, J.J. et al. 2003. J. Theor. Biol., 220:303-321.

Regular and structural equivalence Luczkovich, J.J. et al. 2003. J. Theor. Biol., 220:303-321.

Image graph Luczkovich, J.J. et al. 2003. J. Theor. Biol., 220:303-321.

El Verde rainforest, Puerto Rico Coachella valley desert, USA Luczkovich, J.J. et al. 2003. J. Theor. Biol., 220:303-321.

Cliques and hypergraphs Predator (niche) overlap graph Prey overlap (resource) graph clique: preys of one predator, top predator: isolated node Sugihara, G. 1984 Graph theory, homology and food webs. Proc. Symp. Appl. Math., 30:83-101.

Triangulation non-triangulated patterns are missing (~ prohibited): 1 2 3 4

Előjeles gráfok: indirekt hatások eredő hatása (loop analízis) seal - human - + fish

Előjeles gráfok: indirekt hatások eredő hatása (loop analízis) seal + human - + fish - +

Testing the results of network analysis Ecosystem experiments? Natural experiments? Microcosm studies? Time series analysis?

Testing the results of network analysis: simulations Prince William Sound, Alaska Okey, T. A. 2004. PhD thesis, University of British Columbia, Vancouver

Comparison of structural to functional indices of importance D KI BC CLS

KI CLS BC D Dall`s porpoise (Phocoenoides dalli) Group name Transient orca Avian raptors Porpoise Seabirds Sea otter Invert-eating birds Juvenile pollock (0) Juvenile salmon (0-1) Pinnipeds Sleeper shark Salmon shark Octopods Juv. Arrowtooth Baleen Whales Resident orca Spiny dogfish Offshore phytoplankton Deep demersal fishes Sablefish Pacific cod Lingcod Adult arrowtooth Adult herring Eulachon Shallow lg epibenthos Nearshore demersals Capelin Squid Keystoneindex 66330,8 28064,8 1299,7 869,7 812,1 689,2 385 367,1 293,4 265,3 161,5 143,9 110,6 66,3 54,4 38,2 34,9 32,8 27,9 26,1 25,9 23,9 21,7 14,1 12,3 10,9 10,3 8,8 Group name Deep lg infauna Deep epibenthos Omnivorous zooplank Sleeper shark Adult herring Adult arrowtooth Meiofauna Macrophytes Halibut Shallow lg epibenthos Adult salmon Near phytoplankton Salmon shark Shallow sm infauna Spiny dogfish Squid Shallow sm epibenthos Near herbiv zooplank Octopods Juvenile pollock (0) Sandlance Juvenile herring Near omnivorous zoops Porpoise Pacific cod Deep sm infauna Lingcod Juvenile salmon (0-1) CIb 83 31,1 24,9 22,6 19,5 18,7 15,2 13,2 12,8 10,9 10,3 9,7 5,2 4,6 4,3 1,5 1,4 1 0,7 0,5 0,2 0,1 0-0,2-0,2-0,2-0,4-1,9 Group name Pinnipeds Macrophytes Squid Invert-eating birds Adult herring Sandlance Pacific cod Halibut Seabirds Juvenile herring Octopods Shallow sm epibenthos Adult salmon Juvenile pollock (0) Deep lg infauna Sea otter Adult Pollock (1+) Herbivorous zooplank Offshore phytoplankton Spiny dogfish Shallow sm infauna Adult arrowtooth Lingcod Juv. Arrowtooth Sablefish Deep epibenthos Deep demersal fishes Resident orca Betweenness 404,208252 186,016129 88,918518 63,669628 56,079964 49,625996 45,295261 42,522026 38,392803 37,897614 36,198444 31,864403 31,008747 30,488354 29,44025 28,546709 27,281027 25,47077 23,081909 22,865673 21,281664 20,717108 20,325266 20,032566 17,910799 16,959192 15,823001 13,851252 Group name Near herbiv zooplank Shallow sm infauna Porpoise Offshore phytoplankton Octopods Adult salmon Lingcod Capelin Near omnivorous zoops Eulachon Pacific cod Adult herring Near phytoplankton Juvenile pollock (0) Juvenile salmon (0-1) Invert-eating birds Juv. Arrowtooth Squid Adult Pollock (1+) Juvenile herring Sleeper shark Adult arrowtooth Shallow sm epibenthos Sea otter Deep epibenthos Salmon shark Macrophytes Deep demersal fishes D 29 28 28 28 27 26 26 26 25 24 23 22 22 22 22 22 22 21 20 19 18 18 18 18 17 17 16 16 Adult salmon 7,8 Sea otter -2,2 Deep sm infauna 12,541736 Halibut 15 Halibut 7,4 Nearshore demersals -3 Avian raptors 12,189554 Shallow lg epibenthos 15 Sandlance 6,2 Baleen Whales -3,3 Omnivorous zooplank 11,573871 Nearshore demersals 14 Near herbiv zooplank 6 Invert-eating birds -3,5 Eulachon 8,547655 Seabirds 13 Herbivorous zooplank 6 Deep demersal fishes -5 Rockfish 8,438546 Herbivorous zooplank 13 Near phytoplankton 5,7 Eulachon -6,1 Capelin 7,833774 Rockfish 13 Adult Pollock (1+) 5,4 Sablefish -12,6 Salmon shark 7,809402 Meiofauna 12 Rockfish 4,9 Seabirds -14,2 Near phytoplankton 7,753927 Sablefish 11 Juvenile herring 4,5 Adult Pollock (1+) -16,4 Porpoise 7,730037 Omnivorous zooplank 9 Deep epibenthos 4,4 Offshore phytoplankton -20,2 Sleeper shark 6,648723 Spiny dogfish 9 Jellies 4,2 Herbivorous zooplank -20,6 Meiofauna 5,204166 Sandlance 9 Near omnivorous zoops 3,4 Resident orca -27,7 Transient orca 4,897946 Deep lg infauna 8 Shallow sm infauna 3,2 Avian raptors -31,9 Near omnivorous zoops 2,45219 Resident orca 8 Omnivorous zooplank 2,1 Capelin -42,7 Nearshore demersals 2,375 Pinnipeds 8 Shallow sm epibenthos 1,9 Transient orca -46,2 Shallow lg epibenthos 2,119158 Avian raptors 7 Deep sm infauna 1,4 Rockfish -53,3 Near herbiv zooplank 0,598232 Transient orca 7 Shallow lg infauna 1,4 Juv. Arrowtooth -53,8 Juvenile salmon (0-1) 0 Deep sm infauna 6 Meiofauna 0,6 Pinnipeds -57,6 Baleen Whales 0 Baleen Whales 5 Deep lg infauna 0,6 Jellies -77,5 Jellies 0 Jellies 5 Macrophytes 0,2 Shallow lg infauna -133,7 Shallow lg infauna 0 Shallow lg infauna 5

function Testing topological predictions (Spearman) structure D BC wd ubc CC TI1 WI1 TI2 WI2 TI3 WI3 TI8 WI8 CI 0.17 0.16 0.28 0.22 0.17 0.24 0.28 0.21 0.29 0.21 0.30 0.18 0.30 CLS 0.02-0.01 0.16 0.06 0.06 0.04 0.20 0.05 0.20 0.06 0.19 0.06 0.18 ISI 0.00-0.08 0.44 0.08 0.01 0.07 0.55 0.04 0.54 0.03 0.53 0.00 0.49 KI -0.11-0.18-0.75-0.09-0.06-0.15-0.54-0.13-0.63-0.12-0.66-0.09-0.71 Jordán, F., Okey, T.A., Bauer, B. and Libralato, S. 2008. Ecological Modelling, 216: 75-80.

Conclusions: Local structural indices poorly correlate with functional/dynamical ones. Unweighted indices correlate with nothing. Network analysis must consider indirect indices on weighted networks.

UCINET

Opening view

Input formats

Graph imported and the four basic windows; graph properties shown

Node properties

Edge properties

Aggregation

Layout: grid

Layout: spiral

Layout: circle

Pictures as node attributes + saveable manual layout

Actions

Node size reflecting D

Node size reflecting TO 3 0.06

Node size reflecting IH(st)

Node size reflecting I H(st) + pictures