8 Real networks versus regular graphs Protein-protein interaction network(Saccharomyces cerevisiae)Regular graphsTwo problems:1) characterization of network structure (and complexity)2) network modelling
9 Graph Theory Directed Graph Undirected Graph corresponds to jcorresponds to“Graph”≡ G(V,E)V: N verticesE: L linksAdjacency Matrix:ijDegree (number of links) of vertex iAverage vertex-vertex distance:Clustering coefficient:
10 Small-world character of (most) real networks: Short mean distance D:“it’s a small world, after all!”Efficient information transport(and fast disease spreading too!)Large clustering coefficient C:“my friends are friends of each other”High robustnessunder vertex removal
11 Degree distribution in (most) real networks: Power-law distributionP(k) k -2< <3No characteristic scale (scale-free)!Many poorly connected verticesFew highly connected vertices(a) Archaeoglobus fulgidus (archea);(b) E. coli (bacterium);(c) Caenorhabditis elegans (eukaryote);(d) 43 different organisms together.
12 Finite-scale versus scale-free networks Finite-scale networks:P(k) decays exponentiallyNo vertex has a degree muchlarger than the average valueScale-free networks:P(k) decays as a power lawFew vertices have a degree muchlarger that the average value
13 Finite-scale versus scale-free networks Finite-scale networks:P(k) decays exponentiallyScale-free networks:P(k) decays as a power law(in both cases N=130 and L=215: same average degree)5 vertices with largest degreevertices connected to the red ones (random 27%, scale-free 60%)other vertices
14 RANDOM GRAPH model (Erdös, Renyi 1959) ● Start with a set of N isolated vertices;● For each pair of vertices draw a link with uniform probability p.p=0p=0.1Degree distribution P(k):(Poisson)p=0.5p=1Average vertex-vertex distance:Clustering coefficient
15 Connected components in random graphs The interesting feature of the random graph model is the presence of acritical probability pc marking the appearance of a giant cluster:Percolation thresholdpc 1/NWhen p<pc the network is made of many small clustersand P(s) decays exponentially;when p>pc there are few very small clusters and one giant one;at p=pc the cluster size distribution has a power-law form: P(s) s -
16 SMALL-WORLD model (Watts, Strogatz Nature 1998) ● 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.p = <p< p = 1Regular Small-world RandomP(k)10 -110 -210 -310 -4kDegree distributionC(p)/C(0)D(p)/D(0)small-worldregimeAverage distance and clustering coefficient
17 SCALE-FREE model (Barabási, Albert Science 1999) ● Start with m0 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).P(k) k - =3After a certain number of iterations, the degree distribution approaches a power-law distribution:Growth and preferential attachment are both necessary!
18 distributions are obtained by FITNESS model (Caldarelli et al. Phys. Rev. Lett. 2002)● Each vertex i is assigned a fitness value xi drawn from a given distribution r(x) ;● A link is drawn between each pair of vertices i and j with probability f(xi,xj) depending on xi and xj .Power-law degreedistributions are obtained bychosingr(x) x-αf(xi,xj) xi xjorr(x)= e-xf(xi,xj) (xi +xj –z)
21 Important aspect of many networks: Link reciprocity: the problemDo 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):126453Important aspect of many networks:Mutuality of relationships (friendship, acquaintance, etc.) in social networksReversibility of biochemical reactions in cellular networksSymbiosis in food websSynonymy in word association networksEconomic/financial interdependence in trade/shareholding networks…
22 Standard definition of reciprocity Reciprocity = fraction of reciprocated links in the networkTotal number of directed links:reciprocityNumber of reciprocated links:( and WWW)(WTW)
23 A new definition of reciprocity Conceptual problems with the standard definition: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 andNew definition of reciprocity:correlation coefficient between reciprocal linksreciprocalareciprocalantireciprocalavoiding the aforementioned problems.D. Garlaschelli, M.I. Loffredo Phys. Rev. Lett.93,268701(2004)
25 Size dependence of the reciprocity Metabolic networksFood WebsWorld Trade Web
26 A general model of reciprocity We introduce a multi-species formalism where reciprocated andnon-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:whereGraph Hamiltonian:• Garlaschelli and Loffredo, PHYSICAL REVIEW E 73, (R) 2006
27 A general model of reciprocity Grand Partition Function:Grand Potential:Occupation probabilities:Conditional connection probability:
29 Structural correlations in complex networks In order to detect patterns in networks,one needs (one or more) null model(s) as a reference.A null model is obtained by fixing some topological constraint(s),and generating a maximally random network consistent with them.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.
30 The (solved) problem for unweighted networks Problem: specifying the degree sequence alonegenerates anticorrelations between knni and ki (disassortativity)and between ci and ki (hierarchy).Maslov et al.Solution: in unweighted networks, structural correlations can be fully characterized analytically in terms of exponential random graphs:Park & NewmanCorrect prediction:
31 Some null models for weighted networks Model 1: Global weight reshuffling (fixed topology)Model 2: Global weight & tie reshuffling (fixed degrees)Model 3: Local weighted rewiring (fixed strengths)Model 4: Local weighted rewiring (fixed strengths and degrees)Is it possible to characterize these models analytically?
32 Exponential formulation of the four null models Model 1: Global weight reshuffling (fixed topology)Model 2: Global weight & tie reshuffling (fixed degrees)Model 3: Local weighted rewiring (fixed strengths)Model 4: Local weighted rewiring (fixed strengths and degrees)Note: H1, H2, H3 and H4 are particular cases of:
33 Analytic solution of the general null model: Solution: the probability of a link of weight w between i and j is
34 (but they inherit purely topological correlations!) Models 1 and 2 (global weight reshuffling):Fermionic correlationsThe expectationsare confirmed, howeverimpliesThis means that weighted measures (except the disparity)display a satisfactory behaviour under these null models(but they inherit purely topological correlations!)
35 Now all weighted measures are uninformative! Model 3 (fixed strength): Bosonic correlationsNow all weighted measures are uninformative!
36 All weighted measures are uninformative in this case too! Model 4 (fixed strength+degree):mixed Bose-Fermi statisticsWe still have as in model 3:All weighted measures are uninformative in this case too!
37 the Weighted Random Graph (WRG) model Particular case:the Weighted Random Graph (WRG) modelSee a Mathematica demonstration of the model (by T. Squartini) at:
43 Networks of predation relationships among N biological species Food websNetworks of predation relationships among N biological speciesi is eaten by jij
44 Peculiar (problematic?) aspects of food webs P>(k’)k’=k/<k>Not scale-free!C/CrandomC/CrandomNNot small-world!C/Crandom=1NThe connectance c=L/N2 varies across different webs(fraction of directed links out of the total possible ones)Only property similar to other networks: small distance DDunne, Williams, Martinez Proc. Natl. Acad. Sci. USA 2002
45 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: directionalitySpecies ultimately feed on the abiotic resources (light, water, chemicals): connectednessAlmost 10% of the resources are transferred from the prey to the predator: energy dispersion
46 Minimum-energy subgraphs: minimum spanning trees Minimum spanning trees can be obtained as zero-temperature ensembleswhere li is the trophic level (shortest distance to abiotic resources) of species i
47 Spanning trees and allometric scaling Structure minimizing each species’ distance from the “environment vertex”Allometric relations:Ci (Ai) → C (A)AC(A)Trophic level ℓ of a species i:minimum distance from the environment to i.ℓ=Ai CiSpanning tree:all links from a species at level ℓ to species at levels ℓ’≤ℓ are removed.Power-law scaling:C(A) Aη
48 Ai = drainage area of site i Ci = water in the basin of i Allometric scaling in river networksC(A) Aηη = 3/2Ai = drainage area of site iCi = water in the basin of iBanavar, Maritan, Rinaldo Nature 1999
49 Allometric scaling in vascular systems C(A) Aηη = 4/3Kleiber’s law of metabolism:B(M) M 3/4A0= metabolic rate (B)C0= nutrient volume (M)General case (dimension d): η = (d+1)/d maximum efficiencyWest, Brown, Enquist Science 1999; Banavar, Maritan, Rinaldo Nature 1999
50 Allometric scaling in food webs The resource transfer is universal and efficient (common organising principle?)C(A) Aη η =Garlaschelli, Caldarelli, Pietronero Nature 423, (2003)
51 Transport efficiency in food webs The constraint limiting the efficiency is not the geometry, but the competition!C(A) A2chaininefficientC(A) Aη1<η<2competitionC(A) Astarefficient
52 Summary: food web structure decomposition Spanning trees and loops: complementary properties and rolesTree-forming links:1) Determine the degree of transportation EFFICIENCY2) Measured by the allometric exponent η3) η is universal! (Common evolutionary principle?)Loop-forming links:1) Determine the STABILITY under species removal2) Measured by the directed connectance c3) c varies! (Web-specific organization?)SourceSpecies
53 Out-of-equilibrium statistical mechanics of networks
54 Restoring the feedback We focus on the case when topology and dynamicsevolve over comparable timescales:Dynamical processTopological evolutionAs a result, the process is self-organizedand a non-equilibrium stationary state is reached,independently of (otherwise arbitrary) initial conditions.We choose the simplest possible dynamical rule: Bak-Sneppen modeland the simplest possible network formation mechanism: Fitness model
55 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 EPLLee, Kim PRE Masuda, Goh, Kahng PRE 2005)1) Specify graph, and keep it fixed;2) assign each vertex i a fitness xi 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 xi drawn from (x), and keep it fixed;3) draw network by joining i and j with probability f(xi, xj);4) repeat realizations and perform ensemble average.Coupled (Self-organized) model:1) Assign each vertex i a fitness xi drawn from what you like;2) draw network by joining i and j with probability f(xi, xj);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(xi, xj);5) repeat from 3).
57 Analytical solution for arbitrary f(x,y) Stationary fitness distribution:uniform, as in standard BSnovel result:depends on x(not uniform)Distribution of minimum fitness:uniformCritical threshold obtained from normalization condition:D. Garlaschelli, A. Capocci, G. Caldarelli, Nature Physics 3, (2007)
58 Analytical solution for arbitrary f(x,y) Degree versus fitness:Stationary degree distribution:Similarly, all other topological properties are derivedas in the static fitness model
59 Particular choices of f(x,y) Null case: random graph(“grandcanonically” equivalent to random-neighbor BS model)Stationary fitness distribution:Step-like, as inrandom-neighbor BS model(if sparse)Critical threshold:subcriticalsparsedensedynamical regimes rooted in an underlyingpercolation transition, located at
60 Particular choices of f(x,y) Simplest nontrivial (and unbiased) case: configuration modelsee Garlaschelli and Loffredo, Phys. Rev. E 78, (R) (2008).Stationary fitness distribution:Zipf(but normalizable!)Critical threshold:subcriticalsparsedenseconjecture (verified later): underlyingpercolation transition, located at
61 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:
62 Theoretical results against simulations Power-law fitness distribution (above ):
63 Check the percolation transition conjecture Power-law cluster size distribution at the transition
68 References Reciprocity Weighted networks Food web scaling D. Garlaschelli, M. I. Loffredo, Phys. Rev. Lett. 93, (2004)D. Garlaschelli, M. I. Loffredo, Phys. Rev. E 73, (R) (2006)Weighted networksD. Garlaschelli, M.I. Loffredo, Phys. Rev. Lett. 102, (2009)D. Garlaschelli, New Journal of Physics 11, (2009)Food web scalingD. Garlaschelli, G. Caldarelli, L. Pietronero, Nature 423, (2003)D. Garlaschelli, Eur. Phys. J. B 38(2), 277 (2004)Out-of-equilibrium modelD. Garlaschelli, A. Capocci, G. Caldarelli, Nature Physics 3, (2007)G. Caldarelli, A. Capocci, D. Garlaschelli, Eur. Phys. J. B 64, (2008)