![Triangles in networks
How to count the number of node-disjoint
triangles?
Algorithms need exhaustive enumeration.
def triangles(G, A):
triangles = 0
for n1, n2, n3 in zip(G.nodes(), G.nodes(), G.nodes()):
if (A[n1,n2] and A[n2,n3] and A[n3,n1]):
triangles += 1
return triangles / 6
In this example we have 4 triangles.
Why are we interested in triangles anyway?
Important metric to classify networks.
Clustering coefficient C =
3 × number of triangles
number of connected triplets of nodes
3](https://image.slidesharecdn.com/symbolicregressiononnetworkproperties-170424115305/85/Symbolic-Regression-on-Network-Properties-3-320.jpg)
Symbolic Regression on Network Properties
- 1. Symbolic Regression on Network Properties Marcus Märtens, Fernando Kuipers and Piet Van Mieghem Follow this presentation on your device: Network Architectures and Services https://goo.gl/SLr7Gx 1
- 2. Network representations Number of nodes: Number of links: Adjacency matrix Laplacian matrix N L A = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 1 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ Q = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 4 −1 −1 −1 −1 0 −1 3 0 0 −1 −1 −1 0 3 −1 −1 0 −1 0 1 2 0 0 −1 −1 −1 0 4 −1 0 −1 0 0 −1 2 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ In this example: and .N = 6 L = 9 2
- 3. Triangles in networks How to count the number of node-disjoint triangles? Algorithms need exhaustive enumeration. def triangles(G, A): triangles = 0 for n1, n2, n3 in zip(G.nodes(), G.nodes(), G.nodes()): if (A[n1,n2] and A[n2,n3] and A[n3,n1]): triangles += 1 return triangles / 6 In this example we have 4 triangles. Why are we interested in triangles anyway? Important metric to classify networks. Clustering coefficient C = 3 × number of triangles number of connected triplets of nodes 3
- 4. Triangles in networks Spectral decomposition of the Adjacency Matrix: are the eigenvalues of the network. are the Laplacian eigenvalues (We will need them later) A = X ⎡ ⎣ ⎢ ⎢ λ1 0 ⋱ 0 λN ⎤ ⎦ ⎥ ⎥ X T ≥ … ≥λ1 λN ≥ … ≥μ1 μN In this example the eigenvalues are: 3.182, 1.247, −1.802, −1.588, −0.445, −0.594 Computing the number of triangles becomes simple, once you know the eigenvalues.▲ ▲ = 1 6 ∑ i=1 N λ 3 i 4
- 5. Network spectra Random network Barbasi-Albert ( = 5000)N Network of AS tech-WHOIS, Mahadevan, P., Krioukov, D. et al. ( = 7476)N The adjacency and the Laplacian spectra might not be as intuitive for humans as the graph representation, but both contain the complete information about the topology of the network. 5
- 6. Can we extract the hidden relations between the properties of a network and their corresponding spectra? 6
- 7. Proof of concept: counting triangles Idea: use to evolve symbolic expression! 1. generate all networks with nodes 2. compute as target 3. compute as features Does it work? 1. allow operations , do not bother with the constant in 99%! Okay, that was too easy... 2. allow operations , give constant as input node in 15%! Probably still expected? 3. allow operations , only constants as input nodes in 2%! Maybe this could really work!? Cartesian Genetic Programming (CGP) N = 6 ▲ , … ,λ1 λ6 {+, }⋅ 3 1 6 6▲ = ∑ i=1 6 λ 3 i {+, ×, }⋅ 3 1 6 ▲ = 1 6 ∑ i=1 6 λ 3 i {+, ×, ÷, }⋅ 3 {1, 2, 3} ▲ = 1 6 ∑ i=1 6 λ 3 i 7
- 8. Rediscover a known formula for a network property 8
- 9. Discover an unknown formula for a network property 9
- 10. Network diameter Length of the longest of all shortest paths in a network. The exact formula for the network diameter implies this exhaustive enumeration: def diameter(G): diam = 0 for src, dest in zip(G.nodes(), G.nodes()): path = shortest_path(src, dest) diam = max(len(path), diam) return diam ρ ρ = min dist(src, dest)max src,dest In this example the diameter is 3. Why are we interested in the diameter anyway? Diameter = worst-case lower bound on number of hops i.e. limits the speed of spreading/ ooding in a network 10
- 11. Symbolic regression for the diameter Idea: Try (systematically) different con gurations with CGP! 1. generate all networks with nodes 2. compute as the target 3. compute as features 4. include and some basic constants as features as well 5. use different sets of operators like or Pick the expression that minimizes the sum of absolute errors: Does it work? What happened? N = 6 ρ , … ,λ1 λ6 N, L {1, 2, 3} {+, −, ×, }⋅ 2 {+, −, ×, ÷, }⋅√ f( ) = ∣ ρ(G) − ∣ρ^ ∑ G ρ(G) ˆ = 2ρ^ 11
- 12. Target distributions Triangles ▲ Diameter ρ Exhaustive learning on all networks is no longer feasible. Networks need to be sampled to create a good (diverse) learning set. 12
- 13. Sampling networks Augmented path graphs ρ = 11 ρ = 10 ρ = 8 sparse networks varies by keeping constantρ N Barbell graphs ρ = 6 ρ = 8 ρ = 8 dense networks varies by keeping constantN ρ Mixed graphs 50% augmented path + 50% barbell 13
- 15. CGP parametrization parameter value tness function sum of absolute errors evolutionary strategy 1+4 mutation type and rate probabilistic (0.1) node layout 1 row 200 columns levels-back unrestricted number of generations 200000 operators constants featuresets A) B) +, −, ×, ÷, , , , log⋅ 2 ⋅ 3 ⋅√ 1, 2, 3, 4, 5, 6, 7, 8, 9 N , L, , , ,λ1 λ2 λ3 λN N , L, , , ,μ1 μN−1 μN−2 μN−3 15
- 16. Automatically generated formulas = + + 3ρ^ log (2L +6)+6μ N−3 log (L + )μ N−3 5√ 1 μ N−1 √ 5 – √ 1 μ N−1 − −−− √ 82 −− √ 1 729L −5μ N−2 μ N−3 − −−−−−−−−−− √ = N − − 2 −ρ^ 1− 1 (L−N) 3 2 +4 6− 6 (L−N) 3 2 L−N√ L−N√ L − N − −−−− √ 1 L−N√ =ρ^ + + log ( ) − +N −− √ 45μ N−3 ( + )μ N−1 μ N−3 2 216 ( + )μ N−1 μ N−3 2 16 9μ N−3 8 μ N−3√ 4 Lμ N−1 μ N−2 − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− √ 16
- 17. Bounds on the diameter = + + 3ρ^ log (2L +6)+6μ N−3 log (L + )μ N−3 5√ 1 μ N−1 √ 5 – √ 1 μ N−1 − −−− √ 82 −− √ 1 729L −5μ N−2 μ N−3 − −−−−−−−−−− √ Comparison to known analytical results from graph theory: Upper bound by Chung, F.R., Faber, V. and Manteuffel, T.A.: "An Upper bound on the diameter of a graph from eigenvalues associated with its Laplacian." (See also Van Dam, E.R., Haemers, W.H.: "Eigenvalues and the diameter of graphs.") ρ ≤ ⌊ ⌋ + 1 (N−1)cosh −1 ( )cosh −1 +μ 1 μ N−1 −μ 1 μ N−1 1 /ρρ^ 17
- 18. Diameter on real networks parameter value 379 914 true diameter 17 approximated diameter 21.48 upper bound 160.09 N L ρ ρ^ ca-netscience source: http://www.networkrepository.com 18
- 19. Diameter on real networks parameter value 4941 6594 true diameter 46 approximated diameter 97.52 upper bound 749.49 N L ρ ρ^ inf-power source: http://www.networkrepository.com 19
- 20. Diameter on real networks parameter value 1458 1948 true diameter 19 approximated diameter 18.59 upper bound 207.52 N L ρ ρ^ bio-yeast source: http://www.networkrepository.com 20
- 21. Diameter on real networks 21
- 22. Can we extract the hidden relations between the properties of a network and their corresponding spectra? 22
- 23. Closing thoughts Automatically evolving approximate equations for network properties is possible. Equations are unbiased by human preconceptions. Unexpected results may aid and stimulate research in spectral graph theory and network science in general. Challenges and possible (?) improvements Sampling network space for stronger diversity. Make complexity of formula part of the objective function. Automatically evolve constants rather than using them as inputs. Apply more advanced feature-selection methods to increase quality of formulas. 23