Graphs - why should we care?
- network of companies & board-of-directors members
- 'viral'marketing
- web-log(‘blog’) news propagation
- computer network security: email/IP traffic and anomaly detection
Problem#1:How do real graphs look like?
Degree distributions
Power law in the degree distribution
Eigenvalues
Power law in the eigenvalues of the adjacency matrix
Triangles
Real social networks have a lot of triangles,eg Friends of friends are friends
X-axis: #of Triangles a node participates in Y-axis:count of such nodes
But triangles are expensive to compute,Can we do that quickly?
#triangles= 1/6 Sum (λi)(and,because of skewness,we only need the top few eigenvalues!)
Weights
How do the weights of nodes relate to degree?
Snapshot Power Law: At any time,total incoming weight of a node is proportional to in-degree with PL exponent 'iw': i.e. 1.01 < iw <1.26,super-linear
Problem#2:How do they evolve?
Diameter
Diameter shrinks over time
Temporal Evolution of the Graphs
N(t) … nodes at time t E(t) … edges at time t
Suppose that N(t+1) = 2 * N(t)
Q: what is your guess for E(t+1) =? 2 * E(t)
A: over-doubled! But obeying the Densification Power Law
GCC and NLCC
Q1: How does the GCC emerge?
Most real graphs display a gelling point.
After gelling point,they exhibit typical behavior. This is marked by a spike in diameter.
Q2: How do NLCC emerge and join with the GCC?
(NLCC= non-largest conn. components)
Do they continue to grow in size? or do they shrink? or stabilize?
After the gelling point,the GCC takes off,but NLCC's remain ~constant (actually,oscillate).
Blogs,linking times,cascades
Q1:popularity-decay of a post?
Post popularity drops-off – POWER LAW!
Exponent?-1.6 (close to -1.5: Barabasi's stack model)
Q2:degree distributions?
44,356nodes,122,153 edges. Half of blogsbelong to largest connected component.
References：
Mining Graphs and Tensors------Christos Faloutsos CMU
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