A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Dynamical and spectral properties of complex networks Albert D´ıaz-Guilera Universitat de Barcelona http://albert.diaz.guilera.googlepages.com
Mallorca February 2008 Juan A. Almendral (URJC) A. Arenas (URV) J. G´ omez-Gardenes (Catania) Y. Moreno (BIFI) Conrad J. P´ erez (UB) Phys. Rev. Lett. 96 (2006) 114102 Physica D 224 (2006) 27 Euro. Phys. J. ST 143 (2007) 19 New J. Phys. 9 (2007) 187 Int. J. Bif. Chaos (submitted) Re. Mod. Phys. (submitted)
Outline A. D´ıaz-Guilera —– Dynamics on Networks
1
Networks
—————– Mallorca, Feb 2008 —————–
2
Communities in Networks
3
Synchronization Dynamics
4
Spectral Properties
5
Time to synchronize
6
Conclusions
Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Characterizing networks A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
Small scale: roles of nodes centrality
Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Large scale: statistical properties of the network Degree distribution Clustering Correlations
Communities in Networks A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Analysis of modular structures in networks Definition: subsets of nodes that are more densely linked, when compared with the rest of the network Community detection: From computer scientists To statistical physicists (Girvan-Newman, PNAS 99, 7821, 2002)
Evaluating Community Identification A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Modularity Q=
X
eii − ai2
i
eij : fraction of total links starting at a node in partition i and ending at a node in partition j ai : fraction of links connected to i ai2 : number of intracommunity links
Methods of Community Identification A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
L. Danon, J. Duch, A.D-G, A. Arenas J. Stat. Mech. (2005) P09008 Link removal methods Agglomerative methods Maximizing modularity Spectral analysis methods ...
Comparing Algorithms A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
ad-hoc networks (Newman-Girvan, PRE 69, 026113, 2004) 128 nodes 4 communities of 32 nodes each Each node has 16 links: zin internal nodes within the community zout nodes out of its community
Hierarchical structure A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
Communities are hierarchically organized Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Communities within communities and so on Individuals ⇒ neighborhood ⇒ cities ⇒ countries ⇒ . . ..
Dynamics A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline
Dynamical evolution at all scales
Networks
From bottom to top
Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Synchro in nature A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline
Flashing fireflies Hearts beats Cricket chirps
Networks
Laser
Communities in Networks
Superconductivity
Synchronization Dynamics
Menstrual synchrony in women living together
Spectral Properties Time to synchronize Conclusions
Synchro in nature A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline
Flashing fireflies Hearts beats Cricket chirps
Networks
Laser
Communities in Networks
Superconductivity
Synchronization Dynamics
Menstrual synchrony in women living together
Spectral Properties Time to synchronize Conclusions
Synchro in the lab A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
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Synchronization dynamics A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
X dθi = ωi + σ Aij sin(θj − θi ) dt j
i = 1, ..., N
Synchronization dynamics A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
X dθi = ωi + σ Aij sin(θj − θi ) dt j
Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Kuramoto applet
i = 1, ..., N
Synchronization dynamics A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
X dθi = ωi + σ Aij sin(θj − θi ) dt
i = 1, ..., N
j
Outline Networks Communities in Networks
Kuramoto applet
Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
X dθi =σ Aij sin(θj − θi ) dt j
i = 1, ..., N
Hierarchical structure of communities A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Time evolution of the correlation matrix A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
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Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
ρij =< cos[θi (t) − θj (t)] >
(1)
New graphical representations A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline
Two nodes are connected if they are synchronized Dynamic connectivity matrix Dt (T )ij =
1 0
if ρij (t) > T if ρij (t) > T
(2)
Networks Communities in Networks Synchronization Dynamics Spectral Properties
Fixed time - moving threshold Fixed threshold - time evolution of the network Dt (T ) ⇒ DT (t)
Time to synchronize Conclusions
Structure at different time scales
(3)
Number of connected components A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Dynamics and modularity A. D´ıaz-Guilera
—————– Mallorca, Feb 2008 —————– Outline
0.7 modularity Q
0.6 0.5 0.4 0.3 0.2 0.1
number of communities
—– Dynamics on Networks
100 13_4 10
1 10
100
Networks
time
Communities in Networks
0.8
Time to synchronize Conclusions
0.6 0.5 0.4 0.3 0.2 0.1
number of communities
Spectral Properties
modularity Q
0.7
Synchronization Dynamics
100 15_2
10
1
10
100 time
Spectral Properties A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Spectrum of the Laplacian matrix Relative stability of the synchronized state Ratio λN /λ2 We order the eigenvalues 0 = λ 1 ≤ λ2 ≤ . . . ≤ λ N Number of connected components Differences in time scales
Spectral versus dynamics A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Spectral versus dynamics A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Networks without community structure A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Random Geographic Graph (RGG)
Networks without community structure A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Random Geographic Graph (RGG)
Eigenvalues and eigenvectors of the Laplacian matrix A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
Linearized system X dθi = −k Lij θj dt
i = 1, ..., N
(4)
j
Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
whose solution is X ϕi (t) = Bij θj = ϕi (0)e −λi t
i = 1, ..., N
j
This set of equations has to be verified at any time t Form larger to smaller eigenvalues, kind of threshold
(5)
Time to synchronize A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
For the linear model we get: 1 1 ln C − ln ε . Tsync ∼ λ2 2
(6)
Other dynamics A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Kuramoto
spin (discrete) dynamics)
Spin system A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
+1 if hi (n) > 0 s (t) if hi (n) = 0 si (t + 1) = i −1 if hi (n) < 0
Spin system A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
+1 if hi (n) > 0 s (t) if hi (n) = 0 si (t + 1) = i −1 if hi (n) < 0
Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
X Aij sj (t) + µsi (t) si (t+1) = Θ j
Spin system A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————–
+1 if hi (n) > 0 s (t) if hi (n) = 0 si (t + 1) = i −1 if hi (n) < 0
Outline Networks Communities in Networks
X Aij sj (t) + µsi (t) si (t+1) = Θ j
Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Aij +
µ δij ki
Conclusions A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties New emergent applications in many different fields
Conclusions A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties New emergent applications in many different fields
Conclusions A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties New emergent applications in many different fields
Conclusions A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties New emergent applications in many different fields
Conclusions A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties New emergent applications in many different fields
Conclusions A. D´ıaz-Guilera —– Dynamics on Networks —————– Mallorca, Feb 2008 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties New emergent applications in many different fields
