D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics
Dynamical and Spectral Properties of Complex Networks Albert D´ıaz-Guilera Universitat de Barcelona http://complex.ffn.ub.es
Net-Works. Aranjuez Sept. 10-11, 2007
Spectral Properties Time to synchronize Conclusions
Juan A. Almendral (Universidad Rey Juan Carlos) A. Arenas (Universitat Rovira i Virgili) Conrad J. P´ erez (Universitat de Barcelona) Phys. Rev. Lett. 96 (2006) 114102 Euro. Phys. J. ST 143 (2007) 19
Physica D 224 (2006) 27 New J. Phys. 9 (2007) 187
Outline D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline
1
Networks
2
Communities in Networks
3
Synchronization Dynamics
4
Spectral Properties
5
Time to synchronize
6
Conclusions
Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Network components D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
nodes: individuals, groups, computers, chemical components. links: some kind of relationship.
Characterizing networks D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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
D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
zin = 15
Outline Networks Communities in Networks Synchronization Dynamics
zin = 12
Spectral Properties Time to synchronize Conclusions
zin = 8
D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
zin = 15
Outline Networks Communities in Networks Synchronization Dynamics
zin = 12
Spectral Properties Time to synchronize Conclusions
zin = 8
Hierarchical structure D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Synchronization dynamics D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Time evolution of the correlation matrix D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
ρij =< cos[θi (t) − θj (t)] >
(1)
New graphical representations D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Dynamics and modularity D´ıaz-Guilera
Outline
modularity Q
—————– Net-Works. Aranjuez, 2007 —————–
0.7 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Spectrum of the Laplacian matrix Up to now: relative stability of the synchronized state Ratio λN /λ2 We order the eigenvalues λ 0 ≤ λ1 ≤ λ 2 ≤ . . . ≤ λ N Number of connected components Differences in time scales
Spectral versus dynamics D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Spectral versus dynamics D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Eigenvalues and eigenvectors of the Laplacian matrix D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
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 syncrhonize D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Kuramoto
spin (discrete) dynamics)
Spin system D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– 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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
+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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————–
+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 D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics
Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties
Conclusions D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics
Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties
Conclusions D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics
Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties
Conclusions D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics
Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties
Conclusions D´ıaz-Guilera —– Dynamics on Networks —————– Net-Works. Aranjuez, 2007 —————– Outline
Communities can be hierarchically organized Dynamics reflects community structure Community structure affects dynamics
Networks Communities in Networks Synchronization Dynamics Spectral Properties Time to synchronize Conclusions
Relation with spectral graph analysis (topological characterization of the network) New dynamics need new spectral properties