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

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

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

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1. INTRODUCTION Connection topology Reductionism and complexity Crystal Lattices

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

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

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

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

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1. INTRODUCTION Transportation Networks Airport Networks Road Maps Local Transportation

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

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

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

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

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2. NETWORKS Basic GraphsDiGraphsWeighted Graphs Networks Approach

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

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

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

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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

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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

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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

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

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

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

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2. NETWORKS... Distance Basic Graphs Characterization Friendship

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2. NETWORKS... Distance Basic Graphs Characterization

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2. NETWORKS... Distance Simple Graphs Characterization

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2. NETWORKS... Distance Simple Graphs 0 Characterization

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2. NETWORKS... Distance Simple Graphs Characterization

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2. NETWORKS... Distance Simple Graphs Characterization

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

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

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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

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

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

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

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

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

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

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

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

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

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

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2. NETWORKS... Comparison Average Pathlength Average Clustering

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

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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

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

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

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

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

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

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

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

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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

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

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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

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

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

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

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