Topological phase transitions in equilibrium network ensembles Collegium Budapest, June 2004 Networks and Risks Thematic Institute How do the properties of equilibrium networks change as a function of the ”temperature” (perturbations, volatility, etc) ? Imre Derényi, Illés Farkas, Gergely Palla, Tamás Vicsek Thanks to: Albert-László Barabási, Gábor Tusnády

natural system subunit interaction adjacency matrix Lattice Gas restructuring Lattice site: possible edge Particle: edge of the graph empty lattice site graph vertex edge

Numerical methods Monte-Carlo simulations Berg et.al.: Phys. Rev. Lett. 89, (2002) Baiesi et.al.: Phys. Rev. E (2003) Palla et.al.: Phys. Rev. E (2004) based on the energy of graphs moves single edges of the graph search: stationary graph ensemble

Enumeration of graphs find all graphs with N unlabeled vertices and M unlabeled edges (number of graphs grows faster than exp. with N) build graphs from connected graphs connected graphs with M edges from connected graphs with M­1 edges how to compare two graphs: identical degree sequence permutations among vertices with identical degrees compare two adjacency matrices example

Lattice gas analogy z 2 = N(N-1)/2–1–2(N-2) O(N 2 ) second neighbors NO 3rd, 4th, etc. neighbors z 3 = z 4 = … = 0 N: # of vertices M: # of edges = 2N / M: average degree z 1 = 2 (N–2) O(N) first neighbors Topology of the lattice

Lattice gas analogy (cont’d) ? order of transitions, exponents 1 st order transition, hysteresis gap grows with system size energy extensive in the dispersed state, non-extensive in the star state

I. Derényi et.al., Physica A 334 (3-4) (2004) G. Palla et.al., Phys. Rev. E (2004) I. Farkas et.al., In: Lecture Notes in Physics (2004) Enumeration spinodal curve for the lattice gas energy Lattice gas analogy (cont’d)

Further single-vertex energies motivation: Weber-Fechner logarithmic law of sensation first transition: continuous with infinite exponent (dynamics: broad degree dist.) second transition: compactification

Enumeration Most probable graphs at T=0.65 at T=0.3 the compactification transition

Component-size energies continuous transition, exponent: 1 phase diagram: connected and disconnected

Component-size energies (cont’d) 1 st order transition relative size of the largest component conditional free energy

Component-size energies (cont’d) distribution of the size of the largest component (the order parameter) discontinuous transition

Topological phase transitions in equilibrium network ensembles Collegium Budapest, June 2004 Networks and Risks Thematic Institute How do the properties of equilibrium networks change as a function of the ”temperature” (perturbations, volatility, etc) ? Imre Derényi, Illés Farkas, Gergely Palla, Tamás Vicsek Thanks to: Albert-László Barabási, Gábor Tusnády

Topológiai fázisátalakulások egyensúlyi gráf sokaságokban Statisztikus Fizikai Nap, 2004 egyensúlyi hálózatok tulajdonságainak éles változása a „hőmérséklet” (perturbációk, volatilitás) változásának függvényében Derényi Imre, Farkas Illés, Palla Gergely, Vicsek Tamás Köszönet: Barabási Albert-László, Tusnády Gábor

egyszerű gráf: bármely két csúcs között 0 vagy 1 él cimkézett gráf:csúcsok megkülönböztethetőek, élek is topológia:cimkézetlen gráf (csúcsok nem megkül.) energia:skalár, a topológiához rendelve egyensúlyi gráf sokaság: stacionárius gráf sokaság + átkötözési folyamat részletes egyensúly, ergodicitás Példa:1) átkötözési folyamat energia alapján: 2) rácsgáz analógia: ? átalakulások rendje exponensek (pl. Berg et.al, PRL 89, )