Complex Networks: Complex Networks: Structures and Dynamics Changsong Zhou AGNLD, Institute für Physik Universität Potsdam

Summary Part - I –Characterization of Complex Networks. Part - II –Dynamics on Complex Networks. Part - III –Relevance to Neurosciences.

1. INTRODUCTION Reductionism and complexity Brain in ``DynamicsLand´´

1. INTRODUCTION Connection topology Reductionism and complexity Crystal Lattices

1. INTRODUCTION Internet All-to-all interactionsCrystal Lattices Reductionism and complexity Connection topology

1. INTRODUCTION Internet All-to-all interactionsCrystal Lattices Reductionism and complexity Connection topology Diffusion Mean field ?

1. INTRODUCTION Technological Networks World-Wide Web Power Grid Internet

1. INTRODUCTION Social Networks Friendship Net Sexual Contacts Citation Networks Movie Actors Collaboration Networks

1. INTRODUCTION Transportation Networks Airport Networks Road Maps Local Transportation

1. INTRODUCTION Biological Networks Neural Networks Genetic Networks Protein interaction Ecological Webs Metabolic Networks

2. NETWORKS... A food web A Unified Approach towards the Connection Topology of various Complex Systems

2. NETWORKS... Networks Approach Basic Graphs Symmetrical Adjacency Matrix A ij =

2. NETWORKS... Basic GraphsDiGraphs Non-Symmetrical Adjacency Matrix A ij = Networks Approach

2. NETWORKS Basic GraphsDiGraphsWeighted Graphs Networks Approach

2. NETWORKS... Characterization Vertex degree: k(v) Basic Graphs Friendship

2. NETWORKS... Clustering Coeficient: C(v) Basic Graphs Characterization Friendship

2. NETWORKS... Clustering Coeficient: C(v) Number of existing connections: 2 Basic Graphs Characterization Friendship

2. NETWORKS... Clustering Coeficient: C(v) Number of existing connections: 2 Total Number of possible connections: ½·k v ·(k v -1) = ½·(4·3) = 6 Simple Graphs Characterization Friendship

2. NETWORKS... Clustering Coeficient: C(v) Number of existing connections: 2 Total Number of possible connections: ½·k v ·(k v -1) = ½·(4·3) = 6 C v = 2 / 6 = Basic Graphs Characterization Friendship

2. NETWORKS... Clustering Coeficient: C(v) Number of existing connections: 2 Total Number of possible connections: ½·k v ·(k v -1) = ½·(4·3) = 6 C v = 2 / 6 = Basic Graphs How well are the neighbours connected ! Characterization Friendship

2. NETWORKS... Distance (pathlength) Basic Graphs Characterization i j Friendship

2. NETWORKS... Distance (pathlength) Basic Graphs Characterization i j Friendship

2. NETWORKS... Distance (pathlength) Basic Graphs Characterization i j Friendship

2. NETWORKS... Distance Basic Graphs Characterization Friendship

2. NETWORKS... Distance Basic Graphs Characterization

2. NETWORKS... Distance Simple Graphs Characterization

2. NETWORKS... Distance Simple Graphs 0 Characterization

2. NETWORKS... Distance Simple Graphs Characterization

2. NETWORKS... Distance Simple Graphs Characterization

2. NETWORKS... Characterization Distance Simple Graphs Distance: length of the shortest paths

2. NETWORKS... All-to-all distance matrix: Length of the shortest paths L ij = Characterization

2. NETWORKS... Other measures Simple Graphs Characterization Neighbours` degree Average degree of the neighbouring nodes Betweenness (load) The number of shortest paths passing a node or an edge

2. NETWORKS... General Features of Real Networks Scale-free structure Power-law distribution of degrees

2. NETWORKS... General Features of Real Networks Small world structure Small distance High clustering

2. NETWORKS... ERDOS - RÉNYI MODELL (E-R) Random Network Models Connecting a pair of nodes with probability p

2. NETWORKS... Degree distribution: Poissonian! ERDOS - RÉNYI MODELL (E-R) Random Network Models Mean degree K=NP

2. NETWORKS... Degree distribution: Giant Component: Poissonian! ERDOS - RÉNYI MODELL (E-R) Random Network Models

2. NETWORKS... WATTS - STROGATZ MODELL (W-S): Degree? Clustering? Pathlength? Random Network Models

2. NETWORKS... WATTS - STROGATZ MODELL (W-S): Random Network Models Rewiring a link with probability p

2. NETWORKS... WATTS - STROGATZ MODELL (W-S): Random Network Models Having shortcuts now!

2. NETWORKS... SMALL - WORLD NETS = –High clustering –Short distance Watts, Strogatz. Nature 393/4, 1998 WATTS - STROGATZ MODELL (W-S): Random Network Models

2. NETWORKS... Regular LatticeSmall-World NetRandom Graph P(k) = δ(k-Z) : Z= number of neighbours Poissonian! Comparison

2. NETWORKS... Comparison Average Pathlength Average Clustering

2. NETWORKS... EVOLVING NETWORKS, Barabási-Albert model (B-A) Ingredients: –Growing AND –Preferential attachment Random Network Models

2. NETWORKS... Ingredients: –Growing AND –Preferential attachment Results: –Richer-Gets-Richer – k distribution: Scale Invariant! EVOLVING NETWORKS, Barabási-Albert model (B-A) Random Network Models

2. NETWORKS... Barabási, Albert. Science 286 (1999) SCALE - FREE NETWORKS EVOLVING NETWORKS, Barabási-Albert model (B-A) Random Network Models

2. NETWORKS... Properties of the models Small-WorldLatticeScale-FreeRandom Pathlength Clustering LongShort Large Small > Large in many real scale-free networks !

2. NETWORKS... Small-WorldLatticeScale-FreeRandom Pathlength Clustering LongShort Large Small > Properties of the models

2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. Significant Impacts

2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. Significant Impacts

2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. –Highly vulnerable to deliberate attack on HUBS. Significant Impacts

2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. –Highly vulnerable to deliberate attack on HUBS. Significant Impacts

2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. –Highly vulnerable to deliberate attack on HUBS. Applications: –Inmunization in computer networks and populations Cohen et al PRL, (2000, 2002) Significant Impacts

2. NETWORKS... Cat cortico-cortical connections Physics collaboration network Palla et al. Nature 435, 9 (2005) Communities and Overlapping Nodes

3.... AND MORE REALISTIC CHARACTERIZATION Weighted and Directed Networks GraphsWeightedDirected degree, k( ) = 4 out-degree, out-k( ) = 3 in- degree, in-k( ) = 2 intensity. S( ) = 21.7out-intensity, out-S( ) =24.9 in-intensity, in-S( ) = 12.3 Degree k Intensity In/out-degree In/out-intensity

3.... AND MORE REALISTIC CHARACTERIZATION Weighted Networks Are weights correlated with degrees? –NO Scientific Collaborations (SCN) –YES World-Airport-Networks (WAN)

3.... AND MORE REALISTIC CHARACTERIZATION Weighted Networks Are weights correlated with degrees? –NO SCN –YES WAN

3.... AND MORE REALISTIC CHARACTERIZATION Weighted Networks Weighted Clustering Coeficient: (WAN) Barrat et al. (2004) PNAS vol.101, 11 v v j h