Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set...
Transcript of Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set...
![Page 1: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/1.jpg)
Lecture I Introduction to complex networks
Santo Fortunato
![Page 2: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/2.jpg)
ReferencesEvolution of networksS.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002), cond-mat/0106144
Statistical mechanics of complex networksR. Albert, A-L BarabasiReviews of Modern Physics 74, 47 (2002), cond-mat/0106096
The structure and function of complex networksM. E. J. Newman, SIAM Review 45, 167-256 (2003), cond-mat/0303516
Complex networks: structure and dynamicsS. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. HwangPhysics Reports 424, 175-308 (2006)
Community detection in graphsS. Fortunato arXiv: 0906.0612
![Page 3: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/3.jpg)
Plan of the course
I. Networks: definitions, characteristics, basic concepts in graph theory
II. Real world networks: basic properties. Models IIII. Models IIIV. Community structure IV. Community structure IIVI. Dynamic processes in networks
![Page 4: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/4.jpg)
What is a network?Network or graph=set of vertices joined by edges
very abstract representation
very general
convenient to describemany different systems
![Page 5: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/5.jpg)
Some examples
Nodes Links
Social networks Individuals Social relations
Internet RoutersAS
CablesCommercial agreements
WWW Webpages Hyperlinks
Protein interactionnetworks
Proteins Chemical reactions
and many more (email, P2P, foodwebs, transport….)
![Page 6: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/6.jpg)
Interdisciplinary science
Science of complex networks:
-graph theory
-sociology
-communication science
-biology
-physics
-computer science
![Page 7: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/7.jpg)
Interdisciplinary science
Science of complex networks:
Empirics
Characterization
Modeling
Dynamical processes
![Page 8: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/8.jpg)
Graph TheoryOrigin: Leonhard Euler (1736)
![Page 9: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/9.jpg)
Graph theory: basics
Graph G=(V,E)
V=set of nodes/vertices i=1,…,n
E=set of links/edges (i,j), m
Undirected edge:
Directed edge:
i Bidirectional communication/interaction
j
i j
![Page 10: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/10.jpg)
Graph theory: basicsMaximum number of edges
Undirected: n(n-1)/2
Directed: n(n-1)
Complete graph:
(all to all interaction/communication)
![Page 11: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/11.jpg)
Adjacency matrix
0 1 2 30 0 1 1 11 1 0 1 12 1 1 0 13 1 1 1 0
0
3
1
2
n vertices i=1,…,n
aij=1 if (i,j) E0 if (i,j) E
![Page 12: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/12.jpg)
Adjacency matrix
0 1 2 30 0 1 0 01 1 0 1 12 0 1 0 13 0 1 1 0
Symmetricfor undirected networks
n vertices i=1,…,n
aij=1 if (i,j) E0 if (i,j) E
0
3
1
2
![Page 13: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/13.jpg)
Adjacency matrix
0 1 2 30 0 1 0 11 0 0 0 02 0 1 0 03 0 1 1 0
0
3
1
2
Non symmetricfor directed networks
n vertices i=1,…,n
aij=1 if (i,j) E0 if (i,j) E
![Page 14: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/14.jpg)
Sparse graphs
Density of a graph D=|E|/(n(n-1)/2)
Number of edges
Maximal number of edgesD =
Representation: lists of neighbours of each node
Sparse graph: D <<1 Sparse adjacency matrix
l(i, V(i))
V(i)=neighbourhood of i
![Page 15: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/15.jpg)
PathsG=(V,E)
Path of length l = ordered collection of
l+1 vertices i0,i1,…,il ε V
l edges (i0,i1), (i1,i2)…,(il-1,il) ε E
i2i0 i1
i5
i4i3
Cycle/loop = closed path (i0=il) with all other vertices and edges distinct
![Page 16: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/16.jpg)
Paths and connectednessG=(V,E) is connected if and only if there exists a pathconnecting any two nodes in G
is connected
•is not connected•is formed by two components
![Page 17: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/17.jpg)
Trees
n nodes, n-1 links
Maximal looplessgraph
Minimal connected graph
A tree is a connected graph without loops/cycles
![Page 18: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/18.jpg)
Paths and connectedness
G=(V,E)=> distribution of components’ sizes
Giant component= component whose size scaleswith the number of vertices n
Existence of a giantcomponent
Macroscopic fraction of the graph is connected
![Page 19: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/19.jpg)
Paths and connectedness:directed graphs
Tube TendrilTendrils
Giant SCC: StronglyConnected Component
Giant OUT Component
Giant IN Component
Disconnectedcomponents
Paths are directed
![Page 20: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/20.jpg)
Shortest paths
i
j
Shortest path between i and j: minimum number oftraversed edges
distance l(i,j)=minimum numberof edges traversed on a pathbetween i and j
Diameter of the graph= max[l(i,j)]Average shortest path= ∑ij l(i,j)/(n(n-1)/2)
Complete graph: l(i,j)=1 for all i,j“Small-world” “small” diameter
![Page 21: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/21.jpg)
Graph spectraSpectrum of a graph: set of eigenvalues of adjacencymatrix A
If A is symmetric (undirected graph), n real eigenvalueswith real orthogonal eigenvectors
If A is asymmetric, some eigenvalues may be complex
Perron-Frobenius theorem: any graph has (at least) one real eigenvalue μn with one non-negative eigenvector, such that |μ|≤ μn for any eigenvalue μ. If the graph is connected, the multiplicity of μn is one.
Consequence: on an undirected graph there is onlyone eigenvector with positive components, the othershave mixed-signed components
![Page 22: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/22.jpg)
Graph spectra
Spectral density
Continuous function in the limit
k-th moment of spectral density
![Page 23: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/23.jpg)
Wigner’s semicircle lawFor real symmetric uncorrelated random matriceswhose elements have finite moments in the limit
![Page 24: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/24.jpg)
Centrality measuresHow to quantify the importance of a node?
Degree=number of neighbours=∑j aij
iki=5
• Closeness centrality
gi= 1 / ∑j l(i,j)
For directed graphs: kin, kout
![Page 25: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/25.jpg)
Betweenness centralityfor each pair of nodes (l,m) in the graph, there are
slm shortest paths between l and msi
lm shortest paths going through i
bi is the sum of silm / slm over all pairs (l,m)
NB: similar quantity= load li=∑ σilm
NB: generalization to edge betweenness centrality
Path-based quantity
jbi is largebj is small
i
![Page 26: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/26.jpg)
Eigenvector centrality
ix3
x4
x1 xi
x5
x2
Basic principle = the importance of a vertex is proportionalto the sum of the importances of its neighbors
Solution: eigenvectors of adjacency matrix!
![Page 27: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/27.jpg)
Eigenvector centralityNot all eigenvectors are good solutions!
Requirement: the values of the centrality measure have tobe positive
Because of Perron-Frobenius theorem only the eigenvectorwith largest eigenvalue (principal eigenvector) is a goodsolution!
The principal eigenvector can be quickly computed withthe power method!
![Page 28: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/28.jpg)
Structure of neighborhoods
Clustering: My friends will know each other with high probability! (typical example: social networks)
k
C(i) =# of links between 1,2,…n neighbors
k(k-1)/2
Clustering coefficient of a node
i
![Page 29: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/29.jpg)
Structure of neighborhoods
Average clustering coefficient of a graph
C=∑i C(i)/n
![Page 30: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/30.jpg)
Statistical characterizationDegree distribution
•List of degrees k1,k2,…,kn Not very useful!
•Histogram:nk= number of nodes with degree k
•Distribution:P(k)=nk/n=probability that a randomly chosen
node has degree k
•Cumulative distribution:P>(k)=probability that a randomly chosennode has degree at least k
![Page 31: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/31.jpg)
Statistical characterizationCumulative degree distribution
Conclusion: power laws and exponentials can be easily recognized
![Page 32: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/32.jpg)
Statistical characterizationDegree distribution
P(k)=nk/n=probability that a randomly chosennode has degree k
Average=< k > = ∑i ki/n = ∑k k P(k)=2|E|/n
Fluctuations: < k2 > - < k > 2< k2 > = ∑i k2
i/n = ∑k k2 P(k)< kn > = ∑k kn P(k)
Sparse graphs: < k > << n
![Page 33: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/33.jpg)
Statistical characterizationMultipoint degree correlations
P(k): not enough to characterize a network
Large degree nodes tend toconnect to large degree nodesEx: social networks
Large degree nodes tend toconnect to small degree nodesEx: technological networks
![Page 34: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/34.jpg)
Statistical characterizationMultipoint degree correlations
Measure of correlations:P(k’,k’’,…k(n)|k): conditional probability that a node ofdegree k is connected to nodes of degree k’, k’’,…
Simplest case:P(k’|k): conditional probability that a node of degreek is connected to a node of degree k’
often inconvenient (statistical fluctuations)
![Page 35: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/35.jpg)
Statistical characterizationMultipoint degree correlations
Practical measure of correlations:
Average degree of nearest neighbors
ki=4knn,i=(3+4+4+7)/4=4.5
![Page 36: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/36.jpg)
Statistical characterizationAverage degree of nearest neighbors
Correlation spectrum:
putting together verticeshaving the same degree
class of degree k
![Page 37: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/37.jpg)
Statistical characterizationCase of random uncorrelated networks
P(k’|k)•independent of k•prob. that an edge points to a vertex of degree k’
proportionalto k’ itselfPunc(k’|k)=k’P(k’)/< k >
number of edges from nodes of degree k’number of edges from nodes of any degree
![Page 38: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/38.jpg)
Typical correlationsAssortative behaviour: growing knn(k)
Example: social networks
Large sites are connected with large sites
Disassortative behaviour: decreasing knn(k)
Example: internet
Large sites connected with small sites, hierarchical structure
![Page 39: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/39.jpg)
![Page 40: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/40.jpg)
Correlations:Clustering spectrum
•P(k’,k’’|k): cumbersome, difficult to estimate from data•Average clustering coefficient C=average over nodes withvery different characteristics
Clustering spectrum:
putting together nodes whichhave the same degree
class of degree k(link with hierarchical structures)
![Page 41: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/41.jpg)
MotifsMotifs: subgraphs occurring more often than on random versions of the graph
Significance of motifs:Z-score!
![Page 42: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/42.jpg)
Weighted networksReal world networks: links
carry traffic (transport networks, Internet…)
have different intensities (social networks…)
General description: weights
i jwij
aij: 0 or 1wij: continuous variable
![Page 43: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/43.jpg)
● Scientific collaborations: number of common papers
● Internet, emails: traffic, number of exchanged emails
● Airports: number of passengers
● Metabolic networks: fluxes
● Financial networks: shares
● …
Weights: examples
usually wii=0symmetric: wij=wji
![Page 44: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/44.jpg)
Weighted networksWeights: on the links
Strength of a vertex:
si = ∑j ε V(i) wij
=>Naturally generalizes the degree to weighted networks
=>Quantifies for example the total traffic at a node
![Page 45: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/45.jpg)
Weighted clustering coefficient I
si=16ci
w=0.625 > ci
ki=4ci=0.5
si=8ci
w=0.25 < ci
wij=1
wij=5
i i
A. Barrat, M. Barthélemy, R. Pastor-Satorras, A. Vespignani, PNAS 101, 3747 (2004)
![Page 46: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/46.jpg)
Weighted clustering coefficient II
J. Saramäki, M. Kivela, J.-P. Onnela, K. Kaski, J. Kertész, Phys. Rev. E 75, 027105 (2007)
Definition based on subgraph intensity
![Page 47: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/47.jpg)
Weighted clustering coefficient
Random(ized) weights: C = Cw
C < Cw : more weights on cliquesC > Cw : less weights on cliques
wik
Average clustering coefficientC=∑i C(i)/nCw=∑i Cw(i)/n
Clustering spectra
![Page 48: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/48.jpg)
Weighted assortativity
ki=5; knn,i=1.8
![Page 49: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/49.jpg)
Weighted assortativity
ki=5; knn,i=1.8
511
1
1i
![Page 50: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/50.jpg)
Weighted assortativity
ki=5; si=21; knn,i=1.8 ; knn,iw=1.2: knn,i > knn,i
w
155
5
5i
![Page 51: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/51.jpg)
ki=5; si=9; knn,i=1.8 ; knn,iw=3.2: knn,i < knn,i
w
511
1
1i
Weighted assortativity
![Page 52: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/52.jpg)
Participation ratio
1/ki if all weights equal
close to 1 if few weights dominate
![Page 53: Lecture I Introduction to complex networks · 2010-06-03 · Graph spectra Spectrum of a graph: set of eigenvalues of adjacency matrix A If A is symmetric (undirected graph), n real](https://reader033.fdocuments.net/reader033/viewer/2022060301/5f08736d7e708231d4221349/html5/thumbnails/53.jpg)
Plan of the courseI. Networks: definitions, characteristics, basic concepts
in graph theoryII. Real world networks: basic properties. Models IIII. Models IIIV. Community structure IV. Community structure IIVI. Dynamic processes in networks