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Intrinsic Metrics on Graphs & Graph Geometry
- D. J. Klein
Intrinsic Metrics on Graphs & Graph Geometry D. J. Klein Texas - - PDF document
Intrinsic Metrics on Graphs & Graph Geometry D. J. Klein Texas - - PDF document
Intrinsic Metrics on Graphs & Graph Geometry D. J. Klein Texas A&M University @ Galveston, Galveston, TX 77553-1675 kleind@tamug.edu Abstract Graphs are a cosmopolitan representation of a wide range of things: group networks in
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G ≡ graph * G is a set V of vertices & set E of edges between unordered pairs of vertices
* realized as:
( )
molecules in chemistry electric circuits in physics & electrical engineering Feynman interaction diagrams in physics reaction networks in chemistry & biochemistry respiratory & circulatory networks in biology food webs in eco-biology migratory routes in ecology psychological & sociological interaction patterns transportation networks in geography & business evolutionary patterns in biology neural networks
( )
(as in the brain) communication networks like www etc ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ * graphs are ubiquitous with numerous mathematical papers, & texts
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Graph Distances
* Is there an intrinsic geometry of graphs? * Is there an intrinsic distance function on graphs? * If there is more than one, what might be their characteristics & differences? * Are there natural graph invariants with a geometric interpretation? * distance functions (or metrics) on graphs ( : : V V i j i j ( , ) ρ ρ × → ×
- )
example ( D ρ = ): ( , ) { } {
- }
{ } {
- }
{minimum # edges along path } D i j chemical distance through bond distance topological distance shortest path distance i j ≡ ≡ ≡ ≡ ≡ →
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Lots of work − most only on shortest-path metric − as in the >300 page book: * Are there any other intrinsic graph metrics?
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Candidate combinatorial metric
comb
Ω
* define a tree as a connected acyclic graph * define a bi-tree as an acyclic graph with 2 components * a subgraph of G is spanning if it includes all vertices of G
comb
(# spanning bi-trees, & in different components) ( , ) (# spanning trees) i j i j Ω ≡
* the denominator (# is just a normalization spanning trees) * in the numerator, such spanning bi-trees can be generated from spanning trees by deleting an edge “between” i & j – it being more or less plausible that the farther apart i & j are, the more choices for such a deletion (and consequent bi-tree) there will be
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Candidate random-walk metric
walk
Ω
* define (degree of vertex )
i
d i ≡ ∈V * probability of a random walk leaving 1/ going to any particular one of its neighbors
i
i d ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ * define probability of a random walk ( ) leaving to arrive at before p i j i j ⎛ ⎞ → ≡ ⎜ ⎟ ⎝ ⎠ i
walk( , )
/ ( )
j
i j d p i j ≡ →
Ω
* the factor just makes
j
d
walk( , )
i j Ω symmetric in its arguments – since ( ) p i j → involves a probability 1 at its first step, but no similar 1/ for its arriving site j /
i
d
j
d * the plausibility of this as a metric occurs since the farther apart i & j are, the less likely it should be for a walker from to end up at i j before returning to i
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Candidate wave-amplitude metric
wave
Ω
* 1 , { , } discretized version of , Laplacian operator 0 , otherwise
ij i
i j E d i j − ∈ ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ ≡ = ⇒ = ⎨ ⎬ ⎜ ⎟ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ L L * (discretized) standing waves
ε
ψ with energy ε :
ε ε
ψ εψ = L * longer wave-length waves should be lower energy ε
2 wave( , )
| | /
i j
i j
ε ε ε
ψ ψ ε
>
Ω ≡ −
∑
* this is a correlation function on wave amplitudes, weighting the low-energy (long-wavelength) standing wave patterns more heavily – it is plausibly a metric since the nearer 2 sites i & j are, the more similar should be the corresponding longer-wavelength amplitudes
i ε
ψ &
j ε
ψ
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Candidate electric metric
elec
Ω
* let G correspond to an electric network, with unit resistors placed on each edge
elec
(when battery connected between & )
effective resistance between & ( , )
i j
i j i j
⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭
Ω ≡
* plausibly a metric since the farther apart i & j are, the greater (one imagines) should be the effective resistance between i & j
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Candidate linear-algebraic metric
lin-alg
Ω
*
( )
1 , { , } , "Laplacian" matrix 0 , otherwise
ij i
i j E d i j − ∈ ⎧ ⎫ ⎪ ⎪ ≡ = ⇒ = ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ L L * L nonnegative definite since for arbitrary vector x
† * , { , }
| |
E i ij j i j i j i j 2
x x x x
∈
= = −
∑ ∑
x x L L * (G connected) ⇒ (one 0-eigenvalue with eigenvector φ , all components
i
φ equal) * 0 on -space & inverse of on space orthogonal to
generalized inverse of
⎛ ⎞ ≡ ⎜ ⎟ ⎝ ⎠
Γ
φ φ L
L
lin-alg( , ) ii ij ji jj
i j Ω = Γ −Γ −Γ +Γ
* perhaps a metric because L correlates to “neighborliness”, so that one expects the “inverse” Γ plausibly correlates to “non-neighborliness” (or distance)
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Metrics ? * ( , ) ( : metric) ( , ) ( , ) 0, , , ( , ) ( , ) ( , ) i i V V i j j i i j i j V i j j k i k Ω = ⎧ ⎫ ⎪ ⎪ Ω × → ⇔ Ω = Ω > ≠ ∈ ⎨ ⎬ ⎪ ⎪ Ω + Ω ≥ Ω ⎩ ⎭
- * Are any of
, ,
comb
Ω
walk
Ω
wave
Ω
,
elec
Ω
,
lin-alg
Ω
metrics?
Yes! All of them are metrics! & All of them are the same Ω!
- ld: ~~~ G. Kirchoff, Ann. Phys. Chem. 72 (1847) 497-501
elec lin-alg
Ω = Ω
- P. G. Doyle & L. J. Snell, Random Walks & Electric Networks (MAA, 1984).
elec walk
Ω = Ω
- L. W. Shapiro, Math. Mag. 60 (1987) 36-38.
elec comb
Ω = Ω = metric DJK & M. Randic, J. Math. Chem. 12 (1993) 81-95.
elec
Ω ≡ Ω
G.E. Sharpe, Electron. Lett. 3 (1967) 444–445 & 543–544. A.D. Gvishiani, V.A. Gurvich,, Russ. Math. Surveys 42 (1987) 235–236.
DJK, Commun. Math. Chem. (MATCH) 35 (1996) 7-27.
lin-alg wave
Ω = Ω
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Resistance Distance
This distance (or electric metric) has several other nice properties – it attends to
multiple pathways of interconnection, even of different lengths. Classical Electrical-Circuit Theorems & Results (Kirchoff, Maxwell, etc) Consequent Graph Invariants H.-Y. Zhu, DJK, & I. Lukovits, J. Chem. Inf. & Comp. Sci. 36 (1996) 420 & 1067. DJK & H.-Y. Zhu, J. Math. Chem. 23 (1998) 179-195.
- Y. Yang & DJK, Disc. Appl. Math. (2014)
Sum Rules: DJK, Croatica Chem. Acta 75 (2002) 633-649. M.A. Jafarizadeh, R. Sufiani & S. Jafarizadeh, J. Phys. A. 40 (2007) 4949–4972.
- Y. Yang & H. Zhang, J. Phys. A 41 (2008) 445203.
- M. A. Jafarizadeh, R. Sufiani & S. Jafarizadeh, J. Math. Phys. 50 (2009) 023302.
- H. Chen, F. Zhang, J. Math. Chem. 44 (2008) 405–417
- H. Chen, Disc. Appl. Math. 158 (2010) 1691-1700.
- S. Jafarizadeh , R. Sufiani & M.A. Jafarizadeh, J Stat. Phys. 139 (2010) 177–199.
Recursion
- Y. Yang & DJK, Disc. Appl. Math. 161 (2013) 2702–2715
Ecological uses:
B H McRae, Evolution, 60 (2006) 1551–1561 B H McRae et al, Ecology 89 (2008) 2712–2724
Foundational work: A.D. Gvishiani & V.A. Gurvich, Russian Math. Surveys 42 (1987) 235-236.
- N. L. Biggs, Comb., Prob. & Comp. 2 (1993) 243-255.
- P. Chebotarev & E. Shamis, Automation & Remote Control 58 (1997) 1505-1514.
- R. Lyons, R. Pemantle, & Y. Peres, J. Comb. Theory A 86 (1999) 158-168.
- P. Chebotarev & E. Shamis, Elec. notes Discrete Math. 11 (2002) 98-107.
- P. Chebotarev & R. Agaev, Lin. Alg. & Its Appl. 356 (2002) 253-274.
F.Y. Wu, J. Phys. A : Math. Gen. 37 (2004) 6653–6673.
- H. Chen & F. Zhang, Disc. Appl. Math. 165 (2007) 654-61514.
- V. Gurvich, Disc. Appl. Math. 158 (2010) 1496-1505.
- P. Cheberatov, Disc. Appl. Math. 159 (2010) 295-302.
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Graph Cyclicity * , with equality iff there is a single path between i & j ( , ) ( , ) i j D i j Ω ≤
{ } { }
1 { , } 1 { , }
( , ) ( , ) 1 ( ) natural cyclicity : ( , ) ( , ) 1 ( ) measures
E i j E i j
D i j i j resistance deficit i j D i j conductance excess μ μ
∈ − ∈ −
⎧ ≡ ⋅Ω − ≡ ⎫ ⎪ ⎪ ⎪ ⎬ ⎨ ⎪ ⎪ ≡ Ω ⋅ − ≡ ⎭ ⎪ ⎩
∑ ∑
- theorem:
( , ) with (| | | | ) ( components G V E E V k cyclomatic k μ μ = ⎧ ⎫ ⇒ ≤ ≤ = − + ≡ ⎨ ⎬ ⎩ ⎭
- #)
* μ decreases with increasing size of cycles, unlike μ e.g., 1/ ( 1) n μ = −
- ,
- cycle
G n = * and several other characteristics DJK & O. Ivanciuc, J. Math. Chem. 30 (2002) 271-287.
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“Crumpledness” of Embeddings
for an embedding of in a Euclidean space
Euclidean distance between & ( , )
E
G
i j i j ρ ⎛ ⎞ ≡ ⎜ ⎟ ⎝ ⎠
* distance ratio matrices with 0 diagonals, and off diagonals &
( , ) / ( , )
ij E
i j i j ρ
+ ≡ Ω
R
( , ) / ( , )
ij E i j
i j ρ
− ≡
Ω R
* maximum eigenvalue & associated eigenvector:
λ
± ± ±
=
±
ψ ψ R
* components
ij ± ≥
R
⇒
i
ψ ± ≥
can take probability normalization
1
V i i ψ ∈ ± =
∑
*
† ,
for to cover , case ( ) average dilation for to cover , case
V E i j ij i j E
ρ λ ψ ψ ρ
∈ ± ± ± ± ± ±
Ω + ⎧ ⋅ = ⋅ ≡ ⎨ Ω − ⎩
∑
ψ ψ R
*
† † sim
- f embedding of
( , ) log{ ( ) ( )}
crumpledness
E
G
d ρ λ λ
+ + + − − −
⎧ ⎫ Ω ≡ ⋅ ≡ ⎨ ⎬ ⎩ ⎭ ψ ψ ψ ψ
*
is a semi-metric on the space of scale classes of metrics on V
sim
d
general theory of (posetic) comparisons of different metrics: DJK, J. Math. Chem. 18 (1995) 321-348. DJK & D. Babic, J. Chem. Inf. & Comp. Sci. 37 (1997) 656-671.
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Foldedness of Polymers
(path) (polymer chain) (crumpledness) (foldedness) G = = ⇒ =
* foldedness computed for different toy fractal polymer chains: * foldedness
- vs. fractal dimension
:
sim
d
fractal
d
dsim dfractal
- L. Bytautas, DJK, M. Randic, & T. Pisanski, Disc. Math. & Theor. Comp. Sci. 51 (2000) 39-61.
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Centrality in Graphs
* Wiener-Bavelas ρ -sum:
,
1 2
( ) ( , )
V x y
W G x y
ρ
ρ
∈
≡ ∑
for metric ρ of G * ρ -centrality measure for { , } u v E ∈ is
1 1
( ) ( , )
uv uv
W G A c u v n A
ρ ρ − −
∂ ≡ ∂
(where A is a weighted adjacency matrix of ) G * ρ = shortest-path metric gives a form of “betweenness” centrality (of L. C. Freeman, Sociometry 40, 35-41 & others) * ρ = Ω gives a (new) centrality measure
2 2 2 2 2
( , ) {( ) ( ) ( ) ( ) } ( ) /
uv uu uv vu vv uv u v
c u v A A
ε ε ε 2
ψ ψ ε
> Ω
= ⋅ − − + = ⋅ −
∑
Γ Γ Γ Γ interpretable as electrical power dissipated in edge { , } u v E ∈ & with other nice characteristics DJK, J. Math. Chem. 47 (2010) 1209-1223
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collaborators: F.A. Matsen, W.A. Seitz, T.G. Schmalz, S. Alexander, G. Hite, L. Griffin, W.C. Herndon, D.C. Foyt, A.H. Cowley, B.R. Junker, R.W. Kramling, A.A. Cantu, H. Pickett, J.C. Browne, E.M. Greenawalt, E. Rodriguez, C. Folden, M. Lesko, R. Pepper(Texas) R.D. Poshusta, F. Harary, F.T. Wall, R.P. Hurst, A.A. Cantu, P.v.R. Schleyer, C.H. Sah,
- D. Hankins, J. Michl, C. Larson (USA)
- M. Randic, N. Trinajstic, D. Babic, T.P. Zivkovic, T. Doslic, S. Nikolic, D. Vukecevic,
- D. Plavsic, A. Graovac, Z. Mihalic (Croatia)
A.T.Balaban, O. Ivanciuc, & T. Ivanciuc (Romania)
- X. Liu, H-Y. Zhu, J. Wu, F. Zhang, X.F. Guo, W. Yan, Y. Yang, D. Ye (China)
M.A.Garcia-Bach, J. Oliva, L. Serrano-Andres, A. Ayuela, R.Valenti,
- J. A. Alonso, A. Rubio, A. Penaranda (Spain)
- A. Misra, K. Chilikamarri, B. Mandal, A. Panda, D. Bhattacharya, V. Subramaniam,
- S. N. Datta, S. Shil, P. Ghosh T. Goswami & S. Basak (India)
- T. Morikawa, S. Narita, H. Hosoya (Japan)
V.O. Cheranovskii, L. Bytautas, A. Ryzhov, & V. Rosenfeld (former USSR)
- N. H. March, R. B. Mallion, E. Kirby, & P. Pollack (Britain)
- E. Ruch, W. Hasselbarth, N. Flocke, H. Sachs, & R. Bruggemann (Germany)
- L. Pogliani, G. G. N. Angilella, C. Amovilli, R. Pucci, & M. P. Tosi (Italy)
- Z. G. Soos, I. Lukovits, P. G. Mezey, J. Szucs, & A. Nagy (Hungary)
- S. J. Cyvin, B. N. Cyvin, & J. Brunvoll (Norway)
- D. Bonchev & N. Tyutyulkov (Bulgaria)
- I. Howard, C. Van Alsenoy, V. E. Van Doren, & F. E. Leys (Belgium)
- T. H. Seligman, I. Gutman, G. Restrepo, E. Yi, T. Pisanski, J. L. Palacios, S. El-Basil,
- A. Metropoulos