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Biological networks and statistical physics Said Business School, University of Oxford, UK Diego Garlaschelli Dipartimento di Fisica, Università di Siena, ITALY BioPhys09, Arcidosso, ITALY

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Biological networks: from cells to ecosystems

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Metabolic networks Vertices = cellular substrates (products or educts) Links = biochemical reactions (enzyme-mediated) (part of E. coli’s metabolic network ) educt enzyme product complex

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Protein-protein interaction networks Vertices = proteins Links = interactions within the cell

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Neural networks Vertices = neurons Links = synapses ← single neuron ↑ web of synaptic connections

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Vascular networks Vertices = tissues Links = blood vessels

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Ecological networks (food webs) Vertices = coexisting species Links = predator-prey interactions

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Real networks versus regular graphs Two problems: 1) characterization of network structure (and complexity) 2) network modelling Protein-protein interaction network (Saccharomyces cerevisiae) Regular graphs

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Average vertex-vertex distance: ijij ij ij ij Graph Theory Directed GraphUndirected Graph ij i j corresponds to “Graph”≡ G(V,E) V: N vertices E: L links Adjacency Matrix: Clustering coefficient: Degree (number of links) of vertex i

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Short mean distance D: “it’s a small world, after all!” Efficient information transport (and fast disease spreading too!) Small-world character of (most) real networks: Large clustering coefficient C: “my friends are friends of each other” High robustness under vertex removal

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Degree distribution in (most) real networks: Power-law distribution P(k) k - 2< <3 No characteristic scale (scale-free)! (a) Archaeoglobus fulgidus (archea); (b) E. coli (bacterium); (c) Caenorhabditis elegans (eukaryote); (d) 43 different organisms together. Few highly connected vertices Many poorly connected vertices

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Scale-free networks: P(k) decays as a power law Few vertices have a degree much larger that the average value Finite-scale versus scale-free networks Finite-scale networks: P(k) decays exponentially No vertex has a degree much larger than the average value

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Scale-free networks: P(k) decays as a power law Finite-scale networks: P(k) decays exponentially (in both cases N=130 and L=215: same average degree) 5 vertices with largest degree vertices connected to the red ones (random 27%, scale-free 60%) other vertices Finite-scale versus scale-free networks

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Degree distribution P(k): (Poisson) ● Start with a set of N isolated vertices; ● For each pair of vertices draw a link with uniform probability p. Average vertex-vertex distance: Clustering coefficient p=0p=0.1 p=0.5p=1 RANDOM GRAPH model (Erdös, Renyi 1959)

15 The interesting feature of the random graph model is the presence of a critical probability p c marking the appearance of a giant cluster: When p

p c there are few very small clusters and one giant one; at p=p c the cluster size distribution has a power-law form: P(s) s - Percolation threshold p c 1/N Connected components in random graphs

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SMALL-WORLD model (Watts, Strogatz Nature 1998) p =0 0 < p < 1 p = 1 Regular Small-world Random P(k) k Degree distribution ● Start with a regular d-dimensional lattice, connected up to q nearest neighbours; ● With probability p, an end of each link is rewired to a new randomly chosen vertex. C(p)/C(0) D(p)/D(0) small-world regime Average distance and clustering coefficient

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P(k) k - =3 After a certain number of iterations, the degree distribution approaches a power-law distribution: P(k) k - =3 Growth and preferential attachment are both necessary! ● Start with m 0 vertices and no link; ● at each timestep add a a new vertex with m links, connected to preexisting vertices chosen randomly with probability proportional to their degree k (preferential attachment). SCALE-FREE model (Barabási, Albert Science 1999)

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● Each vertex i is assigned a fitness value x i drawn from a given distribution (x) ; ● A link is drawn between each pair of vertices i and j with probability f(x i,x j ) depending on x i and x j. FITNESS model (Caldarelli et al. Phys. Rev. Lett. 2002) Power-law degree distributions are obtained by chosing (x) x α f(x i,x j ) x i x j or (x)= e x f(x i,x j ) (x i +x j –z)

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Exponential random graphs

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Reciprocity of directed networks

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Do reciprocated links (pairs of mutual links between two vertices) occur more or less often than expected by chance in a directed network? Adjacency matrix (NxN): Important aspect of many networks: Mutuality of relationships (friendship, acquaintance, etc.) in social networks Reversibility of biochemical reactions in cellular networks Symbiosis in food webs Synonymy in word association networks Economic/financial interdependence in trade/shareholding networks … Link reciprocity: the problem

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Reciprocity = fraction of reciprocated links in the network reciprocity Number of reciprocated links: Total number of directed links: ( and WWW) (WTW) Standard definition of reciprocity

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- is not an absolute quantity, to be compared to - as a consequence, networks with different density cannot be compared - self-loops should be excluded when computing and New definition of reciprocity: correlation coefficient between reciprocal links reciprocal areciprocal antireciprocal avoiding the aforementioned problems. Conceptual problems with the standard definition: A new definition of reciprocity D. Garlaschelli, M.I. Loffredo Phys. Rev. Lett.93,268701(2004)

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Results: reciprocity classifies real networks WTW WWW Neural Metabolic Food Webs Words Financial D. Garlaschelli, M.I. Loffredo Phys. Rev. Lett.93,268701(2004)

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World Trade Web Food Webs Metabolic networks Size dependence of the reciprocity

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We introduce a multi-species formalism where reciprocated and non-reciprocated links are regarded as two different ‘chemical species’, each governed by the corresponding chemical potential ( and ) ‘particles’ of type distributed among ‘states’ Decomposition of the adjacency matrix: where Graph Hamiltonian: Garlaschelli and Loffredo, PHYSICAL REVIEW E 73, (R) 2006 A general model of reciprocity

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Grand Partition Function: Grand Potential: Conditional connection probability: Occupation probabilities: A general model of reciprocity

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Models of weighted networks

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Structural correlations in complex networks In order to detect patterns in networks, one needs (one or more) null model(s) as a reference. Examples of null models for unweighted networks : -the random graph (Erdos-Renyi) model (number of links fixed), -the configuration model (degree sequence fixed), -etc. Problem of structural correlations: When a low-level constraint is fixed, patterns may be generated at a higher level, even if they do not signal ‘true’ high-level correlations. A null model is obtained by fixing some topological constraint(s), and generating a maximally random network consistent with them.

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The (solved) problem for unweighted networks Maslov et al. Problem : specifying the degree sequence alone generates anticorrelations between k nn i and k i ( disassortativity ) and between c i and k i ( hierarchy ). Solution : in unweighted networks, structural correlations can be fully characterized analytically in terms of exponential random graphs : Park & Newman Correct prediction:

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Model 3: Local weighted rewiring (fixed strengths) Model 4: Local weighted rewiring (fixed strengths and degrees) Model 1: Global weight reshuffling (fixed topology) Model 2: Global weight & tie reshuffling (fixed degrees) Some null models for weighted networks Is it possible to characterize these models analytically?

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Exponential formulation of the four null models Model 3: Local weighted rewiring (fixed strengths) Note: H 1, H 2, H 3 and H 4 are particular cases of: Model 4: Local weighted rewiring (fixed strengths and degrees) Model 1: Global weight reshuffling (fixed topology) Model 2: Global weight & tie reshuffling (fixed degrees)

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Analytic solution of the general null model: Solution : the probability of a link of weight w between i and j is

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Models 1 and 2 (global weight reshuffling): Fermionic correlations This means that weighted measures ( except the disparity ) display a satisfactory behaviour under these null models (but they inherit purely topological correlations!) The expectations are confirmed, however implies

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Model 3 (fixed strength): Bosonic correlations Now all weighted measures are uninformative!

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Model 4 (fixed strength+degree): mixed Bose-Fermi statistics We still have as in model 3: All weighted measures are uninformative in this case too!

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Particular case: the Weighted Random Graph (WRG) model See a Mathematica demonstration of the model (by T. Squartini) at:

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The Weighted Random Graph (WRG) model

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Largest connected component in the WRG after weak (+) and strong (-) edge removal

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Clustering coefficient in the WRG after weak (+) and strong (-) edge removal

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Food webs

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Networks of predation relationships among N biological species i is eaten by j ij

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Only property similar to other networks: small distance D Dunne, Williams, Martinez Proc. Natl. Acad. Sci. USA 2002 C/C random N N C/C random P > (k’) k’=k/ Not scale-free! Peculiar (problematic?) aspects of food webs The connectance c=L/N 2 varies across different webs (fraction of directed links out of the total possible ones) Not small-world! C/C random =1

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A modest proposal: food webs as transportation networks Resource transfer along each food chain: Flux of matter and energy form prey to predators, in more and more complex forms: directionality Species ultimately feed on the abiotic resources (light, water, chemicals): connectedness Almost 10% of the resources are transferred from the prey to the predator: energy dispersion

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Minimum-energy subgraphs: minimum spanning trees Minimum spanning trees can be obtained as zero-temperature ensembles where l i is the trophic level (shortest distance to abiotic resources) of species i

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Spanning trees and allometric scaling Structure minimizing each species’ distance from the “environment vertex” A i C i Spanning tree: all links from a species at level ℓ to species at levels ℓ’≤ℓ are removed. Allometric relations: C i (A i ) → C (A) A C(A) Power-law scaling: C(A) A η Trophic level ℓ of a species i: minimum distance from the environment to i. ℓ=ℓ= ℓ=ℓ= ℓ=ℓ= ℓ=ℓ=

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Allometric scaling in river networks Banavar, Maritan, Rinaldo Nature 1999 C(A) A η η = 3/2 A i = drainage area of site i C i = water in the basin of i

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Allometric scaling in vascular systems West, Brown, Enquist Science 1999; Banavar, Maritan, Rinaldo Nature 1999 A 0 = metabolic rate (B) C 0 = nutrient volume (M) C(A) A η η = 4/3 General case (dimension d): η = (d+1)/d maximum efficiency Kleiber’s law of metabolism: B(M) M 3/4

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Allometric scaling in food webs Garlaschelli, Caldarelli, Pietronero Nature 423, (2003) C(A) A η η = The resource transfer is universal and efficient (common organising principle?)

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C(A) A star efficient C(A) A 2 chain inefficient C(A) A η 1<η<2 competition Transport efficiency in food webs The constraint limiting the efficiency is not the geometry, but the competition!

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Tree-forming links: 1) Determine the degree of transportation EFFICIENCY 2) Measured by the allometric exponent η 3) η is universal! (Common evolutionary principle?) Summary: food web structure decomposition Spanning trees and loops: complementary properties and roles Source Species Loop-forming links: 1) Determine the STABILITY under species removal 2) Measured by the directed connectance c 3) c varies! (Web-specific organization?)

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Out-of-equilibrium statistical mechanics of networks

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Restoring the feedback Dynamical process Topological evolution We focus on the case when topology and dynamics evolve over comparable timescales: As a result, the process is self-organized and a non-equilibrium stationary state is reached, independently of (otherwise arbitrary) initial conditions. We choose the simplest possible dynamical rule: Bak-Sneppen model and the simplest possible network formation mechanism: Fitness model

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Coupling the Bak-Sneppen and the fitness model Bak-Sneppen model on fixed graphs (Bak, Sneppen PRL 1993 – Flyvbjerg, Sneppen, Bak PRL 1993 – Kulkarni, Almaas, Stroud cond-mat/ – Moreno, Vazquez EPL Lee, Kim PRE Masuda, Goh, Kahng PRE 2005) 1) Specify graph, and keep it fixed; 2) assign each vertex i a fitness x i drawn uniformly in (0,1); 3) draw anew fitnesses of least fit vertex and its neighbours; 4) evolve fitnesses iterating 3). Fitness network model with quenched fitnesses (Caldarelli et al. PRL 2002 – Boguna, Pastor-Satorras PRE 2003) 1) Specify fitness distribution (x); 2) assign each vertex i a fitness x i drawn from (x), and keep it fixed; 3) draw network by joining i and j with probability f(x i, x j ); 4) repeat realizations and perform ensemble average. Coupled (Self-organized) model: 1) Assign each vertex i a fitness x i drawn from what you like; 2) draw network by joining i and j with probability f(x i, x j ); 3) draw anew fitnesses of least fit vertex and its neighbours, uniformly in (0,1); 4) draw anew links of least fit vertex and its neighbours with probability f(x i, x j ); 5) repeat from 3).

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Typical iteration of the model:

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Analytical solution for arbitrary f(x,y) Stationary fitness distribution: Critical threshold obtained from normalization condition: novel result: depends on x (not uniform) uniform, as in standard BS Distribution of minimum fitness: uniform D. Garlaschelli, A. Capocci, G. Caldarelli, Nature Physics 3, (2007)

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Analytical solution for arbitrary f(x,y) Degree versus fitness: Similarly, all other topological properties are derived as in the static fitness model Stationary degree distribution:

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Particular choices of f(x,y) Null case: random graph (“grandcanonically” equivalent to random-neighbor BS model) Stationary fitness distribution: Step-like, as in random-neighbor BS model Critical threshold: subcritical sparse dense dynamical regimes rooted in an underlying percolation transition, located at (if sparse)

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Particular choices of f(x,y) Simplest nontrivial (and unbiased) case: configuration model Stationary fitness distribution: Zipf (but normalizable!) Critical threshold: subcritical sparse dense conjecture (verified later): underlying percolation transition, located at see Garlaschelli and Loffredo, Phys. Rev. E 78, (R) (2008).

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Stationary fitness distribution In the self-organized model, it is no longer step-like (as in the BS model on fitness-independent networks) but power-law:

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Theoretical results against simulations Power-law fitness distribution (above ):

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Check the percolation transition conjecture Power-law cluster size distribution at the transition

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Check the percolation transition conjecture

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Degree versus fitness The “saturation” reflects repulsion between large degrees: implies disassortativity and hierarchy (not shown)

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Cumulative degree distribution Scale-free degree distribution (above )

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Average fitness versus threshold

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References Reciprocity Weighted networks Food web scaling Out-of-equilibrium model D. Garlaschelli, New Journal of Physics 11, (2009) D. Garlaschelli, M.I. Loffredo, Phys. Rev. Lett. 102, (2009) D. Garlaschelli, A. Capocci, G. Caldarelli, Nature Physics 3, (2007) G. Caldarelli, A. Capocci, D. Garlaschelli, Eur. Phys. J. B 64, (2008) D. Garlaschelli, M. I. Loffredo, Phys. Rev. E 73, (R) (2006) D. Garlaschelli, M. I. Loffredo, Phys. Rev. Lett. 93, (2004) D. Garlaschelli, G. Caldarelli, L. Pietronero, Nature 423, (2003) D. Garlaschelli, Eur. Phys. J. B 38 (2), 277 (2004)

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