Topological implications of negative curvature for biological and - - PowerPoint PPT Presentation
Topological implications of negative curvature for biological and social networks Bhaskar DasGupta Department of Computer Science University of Illinois at Chicago Chicago, IL 60607 bdasgup@uic.edu November 29, 2014 Joint work with Rka
Topological implications of negative curvature for biological and social networks
Bhaskar DasGupta
Department of Computer Science University of Illinois at Chicago Chicago, IL 60607 bdasgup@uic.edu
November 29, 2014 Joint work with
Réka Albert (Penn State) Nasim Mobasheri (UIC)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 1 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 2 / 52
Introduction
Various network measures
Graph-theoretical analysis leads to useful insights for many complex systems, such as
World-Wide Web social network of jazz musicians metabolic networks protein-protein interaction networks
Examples of useful network measures for such analyses
degree based , e.g.
⊲ maximum/minimum/average degree, degree distribution, ......
connectivity based , e.g.
⊲ clustering coefficient, largest cliques or densest sub-graphs, ......
geodesic based , e.g.
⊲ diameter, betweenness centrality, ......
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 3 / 52
Introduction
network curvature as a network measure
network measure for this talk
network curvature via (Gromov) hyperbolicity measure
⊲ observed that many results concerning the fundamental group of a Riemann
surface hold true in a more general context
⊲ defined for infinite continuous metric space with bounded local geometry via
properties of geodesics
⊲ can be related to standard scalar curvature of Hyperbolic manifold
adopted to finite graphs using a so-called 4-node condition
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 4 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 5 / 52
Basic definitions and notations
Graphs, geodesics and related notations
Graphs, geodesics and related notations G = (V ,E) G = (V ,E) G = (V ,E) connected undirected graph of n ≥ 4 n ≥ 4 n ≥ 4 nodes u
P
v
u
P
v
u
P
v
path P ≡
=u,u1,...,uk−1,uk =v
=u,u1,...,uk−1,uk =v
=u,u1,...,uk−1,uk =v
u u and v v v ℓ(P ) ℓ(P ) ℓ(P ) length (number of edges) of the path u
P
v
u
P
v
u
P
v
ui
P
uj
ui
P
uj
ui
P
uj
sub-path
P P between nodes ui ui ui and uj uj uj u
s
v
u
s
v
u
s
v
a shortest path between nodes u u u and v v v du,v du,v du,v length of a shortest path between nodes u u u and v v v
u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 u5 u5 u5 u6 u6 u6
u2
P
u6
u2
P
u6
u2
P
u6 is the path P ≡
ℓ(P ) ℓ(P )= 3 = 3 = 3 du2,u6 du2,u6 du2,u6 = 2 = 2 = 2
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 6 / 52
Basic definitions and notations
4 node condition (Gromov, 1987)
Consider four nodes u1,u2,u3,u4 u1,u2,u3,u4 u1,u2,u3,u4 and the six shortest paths among pairs of these nodes u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 du1,u2 du1,u2 du1,u2 du3,u4 du3,u4 du3,u4 du1,u3 du1,u3 du1,u3 du2,u4 du2,u4 du2,u4 du1,u4 du1,u4 du1,u4 du2,u3 du2,u3 du2,u3
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 7 / 52
Basic definitions and notations
4 node condition (Gromov, 1987)
Consider four nodes u1,u2,u3,u4 u1,u2,u3,u4 u1,u2,u3,u4 and the six shortest paths among pairs of these nodes u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 du1,u2 du1,u2 du1,u2 du3,u4 du3,u4 du3,u4 du1,u3 du1,u3 du1,u3 du2,u4 du2,u4 du2,u4 du1,u4 du1,u4 du1,u4 du2,u3 du2,u3 du2,u3 Assume, without loss of generality, that
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
+ + + ≥ ≥ ≥ + + + ≥ ≥ ≥ + + +
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 7 / 52
Basic definitions and notations
4 node condition (Gromov, 1987)
Consider four nodes u1,u2,u3,u4 u1,u2,u3,u4 u1,u2,u3,u4 and the six shortest paths among pairs of these nodes u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 du1,u2 du1,u2 du1,u2 du3,u4 du3,u4 du3,u4 du1,u3 du1,u3 du1,u3 du2,u4 du2,u4 du2,u4 du1,u4 du1,u4 du1,u4 du2,u3 du2,u3 du2,u3 Assume, without loss of generality, that
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
+ + + ≥ ≥ ≥ + + + ≥ ≥ ≥ + + +
Let δu1,u2,u3,u4 = L−M
2
δu1,u2,u3,u4 = L−M
2
δu1,u2,u3,u4 = L−M
2
+ + +
− − − ( ( (
+ + +
) ) )
2 2 2
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 7 / 52
Basic definitions and notations
4 node condition (Gromov, 1987)
Consider four nodes u1,u2,u3,u4 u1,u2,u3,u4 u1,u2,u3,u4 and the six shortest paths among pairs of these nodes u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 du1,u2 du1,u2 du1,u2 du3,u4 du3,u4 du3,u4 du1,u3 du1,u3 du1,u3 du2,u4 du2,u4 du2,u4 du1,u4 du1,u4 du1,u4 du2,u3 du2,u3 du2,u3 Assume, without loss of generality, that
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
du1,u4 +du2,u3
≥ du1,u3 +du2,u4
≥ du1,u2 +du3,u4
+ + + ≥ ≥ ≥ + + + ≥ ≥ ≥ + + +
Let δu1,u2,u3,u4 = L−M
2
δu1,u2,u3,u4 = L−M
2
δu1,u2,u3,u4 = L−M
2
+ + +
− − − ( ( (
+ + +
) ) )
2 2 2
Definition (hyperbolicity of G)
δ(G) = max
u1,u2,u3,u4
max
u1,u2,u3,u4
max
u1,u2,u3,u4
Negative curvature for networks November 29, 2014 7 / 52
Basic definitions and notations
Hyperbolic graphs (graphs of negative curvature)
Definition (∆ ∆ ∆-hyperbolic graphs)
G G G is ∆ ∆ ∆-hyperbolic provided δ(G) ≤ ∆ δ(G) ≤ ∆ δ(G) ≤ ∆
Definition (Hyperbolic graphs)
If ∆ ∆ ∆ is a constant independent of graph parameters, then a ∆ ∆ ∆-hyperbolic graph is simply called a hyperbolic graph
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 8 / 52
Basic definitions and notations
Hyperbolic graphs (graphs of negative curvature)
Definition (∆ ∆ ∆-hyperbolic graphs)
G G G is ∆ ∆ ∆-hyperbolic provided δ(G) ≤ ∆ δ(G) ≤ ∆ δ(G) ≤ ∆
Definition (Hyperbolic graphs)
If ∆ ∆ ∆ is a constant independent of graph parameters, then a ∆ ∆ ∆-hyperbolic graph is simply called a hyperbolic graph
Example (Hyperbolic and non-hyperbolic graphs)
Tree: ∆(G) = 0 ∆(G) = 0 ∆(G) = 0 hyperbolic graph Chordal (triangulated) graph: ∆(G) = 1/2 ∆(G) = 1/2 ∆(G) = 1/2 hyperbolic graph Simple cycle: ∆(G) = ⌈n/4⌉ ∆(G) = ⌈n/4⌉ ∆(G) = ⌈n/4⌉ non-hyperbolic graph
b b b b b b b b b b
n = 10 n = 10 n = 10
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 8 / 52
Basic definitions and notations
Hyperbolicity of real-world networks
Are there real-world networks that are hyperbolic?
Yes, for example:
Preferential attachment networks were shown to be scaled hyperbolic
⊲ [Jonckheere and Lohsoonthorn, 2004; Jonckheere, Lohsoonthorn and Bonahon, 2007]
Networks of high power transceivers in a wireless sensor network were empirically
⊲ [Ariaei, Lou, Jonckeere, Krishnamachari and Zuniga, 2008]
Communication networks at the IP layer and at other levels were empirically
⊲ [Narayan and Saniee, 2011]
Extreme congestion at a very limited number of nodes in a very large traffic network
was shown to be caused due to hyperbolicity of the network together with minimum length routing
⊲ [Jonckheerea, Loua, Bonahona and Baryshnikova, 2011]
Topology of Internet can be effectively mapped to a hyperbolic space
⊲ [Bogun, Papadopoulos and Krioukov, 2010] Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 9 / 52
Basic definitions and notations
Average hyperbolicity measure, computational issues
Definition (average hyperbolicity)
δave(G) = 1 n
4
δu1,u2,u3,u4 δave(G) = 1 n
4
δu1,u2,u3,u4 δave(G) = 1 n
4
δu1,u2,u3,u4
expected value of δu1,u2,u3,u4 δu1,u2,u3,u4 δu1,u2,u3,u4 if u1,u2,u3,u4 u1,u2,u3,u4 u1,u2,u3,u4 are picked uniformly at random
Computation of δ(G) δ(G) δ(G) and δave(G) δave(G) δave(G)
Trivially in O
O
O
time
⊲ Compute all-pairs shortest paths
Floyd–Warshall algorithm O
O
O
time
⊲ For each combination u1,u2,u3,u4
u1,u2,u3,u4 u1,u2,u3,u4, compute δu1,u2,u3,u4 δu1,u2,u3,u4 δu1,u2,u3,u4 O
O
O
time
Open problem: can we compute in O
O
O
time?
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 10 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 11 / 52
Computing hyperbolicity for real networks
Direct calculation
Real networks used for empirical validation
20 well-known biological and social networks
11 biological networks that include 3 transcriptional regulatory, 5 signalling, 1
metabolic, 1 immune response and 1 oriented protein-protein interaction networks
9 social networks range from interactions in dolphin communities to the social
network of jazz musicians
hyperbolicity of the biological and directed social networks was computed by
ignoring the direction of edges
hyperbolicity values were calculated by writing codes in C using standard
algorithmic procedures Next slide: List of 20 networks
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 12 / 52
Computing hyperbolicity for real networks
Direct calculation
11 biological networks
# nodes # edges
311 451
512 1047
418 544
58 135
transcriptional 690 1082
453 2040
segment polarity (6 cells) 78 132
55 88
network 18 42
signaling 94 138
786 2445
9 social networks
# nodes # edges
62 160
College Football 115 612
34 78
US politics 105 442
communication network 36 62
network 198 2742
in San Juan 75 144
Data, Paris 1998 35 118
characters 77 251
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 13 / 52
Computing hyperbolicity for real networks
Direct calculation
Biological networks
Average degree δave δave δave δ δ δ
1.45 1.45 1.45 0.132 0.132 0.132 2 2 2
2.04 2.04 2.04 0.013 0.013 0.013 3 3 3
1.30 1.30 1.30 0.043 0.043 0.043 2 2 2
2.32 2.32 2.32 0.297 0.297 0.297 2 2 2
1.56 1.56 1.56 0.004 0.004 0.004 3 3 3
4.50 4.50 4.50 0.010 0.010 0.010 1.5 1.5 1.5
1.69 1.69 1.69 0.676 0.676 0.676 4 4 4
1.60 1.60 1.60 0.302 0.302 0.302 2 2 2
2.33 2.33 2.33 0.286 0.286 0.286 1.5 1.5 1.5
1.46 1.46 1.46 0.323 0.323 0.323 3 3 3
3.11 3.11 3.11 0.001 0.001 0.001 2 2 2
social networks
Average degree δave δave δave δ δ δ
5.16 5.16 5.16 0.262 0.262 0.262 2 2 2
10.64 10.64 10.64 0.312 0.312 0.312 2 2 2
4.58 4.58 4.58 0.170 0.170 0.170 1 1 1
8.41 8.41 8.41 0.247 0.247 0.247 2 2 2
3.44 3.44 3.44 0.162 0.162 0.162 1 1 1
27.69 27.69 27.69 0.140 0.140 0.140 1.5 1.5 1.5
3.84 3.84 3.84 0.422 0.422 0.422 3 3 3
1998 1998 3.37 3.37 3.37 0.270 0.270 0.270 2.5 2.5 2.5
6.51 6.51 6.51 0.278 0.278 0.278 2 2 2
Hyperbolicity values of almost all networks are small For all networks δave
δave δave is one or two orders of magnitude smaller than δ δ δ
⊲ Intuitively, this suggests that value of δ
δ δ may be a rare deviation from typical values of δu1,u2,u3,u4 δu1,u2,u3,u4 δu1,u2,u3,u4 for most combinations of nodes {u1,u2,u3,u4} {u1,u2,u3,u4} {u1,u2,u3,u4}
No systematic dependence of δ
δ δ on number of nodes/edges or average degree
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 14 / 52
Computing hyperbolicity for real networks
Direct calculation
Definition (Diameter of a graph) D = maxu,v
Fact δ ≤ D/2 δ ≤ D/2 δ ≤ D/2
small diameter implies small hyperbolicity
We found no systematic dependence of δ δ δ on D D D
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 15 / 52
Computing hyperbolicity for real networks
Direct calculation
Definition (Diameter of a graph) D = maxu,v
Fact δ ≤ D/2 δ ≤ D/2 δ ≤ D/2
small diameter implies small hyperbolicity
We found no systematic dependence of δ δ δ on D D D
For more rigorous checks of hyperbolicity of finite graphs and for evaluation of statistical significance of the hyperbolicity measure see our paper
Topological implications of negative curvature for biological and social networks. Physical Review E 89(3), 032811 (2014)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 15 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 16 / 52
Implications of hyperbolicity
We discuss topological implications of hyperbolicity somewhat informally Precise Theorems and their proofs are available in our paper
Topological implications of negative curvature for biological and social networks. Physical Review E 89(3), 032811 (2014)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 17 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 18 / 52
Implications of hyperbolicity
Hyperbolicity and crosstalk in regulatory networks
Definition (Path chord and chord)
p a t h - c h o r d
v u4 u5 u3 u0 u2 u1
c h o r d
u4 u5 u3 u0 u2 u1
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 19 / 52
Implications of hyperbolicity
Hyperbolicity and crosstalk in regulatory networks
Definition (Path chord and chord)
p a t h - c h o r d
v u4 u5 u3 u0 u2 u1
c h o r d
u4 u5 u3 u0 u2 u1
Theorem (large cycle without path-chord imply large hyperbolicity)
G G G has a cycle of k k k nodes which has no path-chord ⇒
⇒ ⇒
δ ≥ ⌈k/4⌉ δ ≥ ⌈k/4⌉ δ ≥ ⌈k/4⌉
Corollary
Any cycle containing more than 4δ 4δ 4δ nodes must have a path-chord
Example
δ < 1 δ < 1 δ < 1 ⇒
⇒ ⇒ G
G G is chordal graph
Next slide: implications for regulatory networks Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 19 / 52
Implications of hyperbolicity
Hyperbolicity and crosstalk in regulatory networks
An example of a regulatory network
Network associated to the Drosophila segment polarity
Nature 406, 188-192 (2000)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 20 / 52
Implications of hyperbolicity
Hyperbolicity and crosstalk in regulatory networks
Hyperbolicity and crosstalk in regulatory networks short-cuts in long feedback loops
node regulates itself through a long feedback loop
⇒ ⇒ ⇒
this loop must have a path-chord
⇒ ⇒ ⇒
a shorter feedback cycle through the same node
a node
interpreting chord or short path-chord as crosstalk
“source” regulates “target” through two long paths
⇒ ⇒ ⇒
must exist a crosstalk path between these two paths
c r
s t a l k
target source
number of crosstalk paths increases at least linearly with total length of two paths
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 21 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 22 / 52
Implications of hyperbolicity
Geodesic triangles and crosstalk paths
Geodesic triangles and crosstalk paths u0 u0 u0 u2 u2 u2 u1 u1 u1
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity
Geodesic triangles and crosstalk paths
Geodesic triangles and crosstalk paths
shortest path
u0 u0 u0 u2 u2 u2 u1 u1 u1
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity
Geodesic triangles and crosstalk paths
Geodesic triangles and crosstalk paths
shortest path
u0 u0 u0 u2 u2 u2 u1 u1 u1
u0,1 u0,1 u0,1 u0,2 u0,2 u0,2 u1,2 u1,2 u1,2
du0,u0,1 = du0,u1 +du0,u2 −du1,u2 2
du1,u2 +du1,u0 −du2,u0 2
du2,u0,2 =du2,u1,2
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity
Geodesic triangles and crosstalk paths
Geodesic triangles and crosstalk paths
shortest path
u0 u0 u0 u2 u2 u2 u1 u1 u1
u0,1 u0,1 u0,1 u0,2 u0,2 u0,2 u1,2 u1,2 u1,2
du0,u0,1 = du0,u1 +du0,u2 −du1,u2 2
du1,u2 +du1,u0 −du2,u0 2
du2,u0,2 =du2,u1,2
d
v,v′
dv,v′ dv,v′
✎ ✍ ☞ ✌
∀v ∀v ∀v in one path ∃v ′ ∃v ′ ∃v ′ in the other path such that dv,v ′ ≤ max
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity
Implications of geodesic triangles and crosstalk paths for regulatory networks
Implications of geodesic triangles for regulatory networks Consider feedback or feed-forward loop formed by the shortest paths among three nodes We can expect short cross-talk paths between these shortest paths
Feedback/feed-forward loop is nested with additional feedback/feed-forward loops
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 24 / 52
Implications of hyperbolicity
Implications of geodesic triangles and crosstalk paths for regulatory networks
Implications of geodesic triangles for regulatory networks Consider feedback or feed-forward loop formed by the shortest paths among three nodes We can expect short cross-talk paths between these shortest paths
Feedback/feed-forward loop is nested with additional feedback/feed-forward loops Empirical evidence [R. Albert, Journal of Cell Science 118, 4947-4957 (2005)] Network motifsa are often nested
Two generations of nested assembly
for a common E. coli motif
[DeDeo and Krakauer, 2012]
ae.g., feed-forward or feedback loops of small number of nodes
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 24 / 52
Implications of hyperbolicity
Hausdorff distance between shortest paths
Definition (Hausdorff distance between two paths P1
P1 P1 and P2 P2 P2) dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
small Hausdorff distance implies every node of either path is close to some node of the
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity
Hausdorff distance between shortest paths
Definition (Hausdorff distance between two paths P1
P1 P1 and P2 P2 P2) dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
small Hausdorff distance implies every node of either path is close to some node of the
u0 u0 u0 u2 u2 u2 dv,v′ dv,v′
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity
Hausdorff distance between shortest paths
Definition (Hausdorff distance between two paths P1
P1 P1 and P2 P2 P2) dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
small Hausdorff distance implies every node of either path is close to some node of the
P2 P2 P2 u0 u0 u0 u2 u2 u2 P1 P1 P1 dv,v′ dv,v′
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity
Hausdorff distance between shortest paths
Definition (Hausdorff distance between two paths P1
P1 P1 and P2 P2 P2) dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
dH (P1,P2) def = max
v1 ∈P1
min
v2 ∈P2
max
v2 ∈P2
min
v1 ∈P1
small Hausdorff distance implies every node of either path is close to some node of the
P2 P2 P2 u0 u0 u0 u2 u2 u2 P1 P1 P1 dv,v′ dv,v′
dH (P1,P2) ≤ max{ 6δ, 2 } dH (P1,P2) ≤ max{ 6δ, 2 } dH (P1,P2) ≤ max{ 6δ, 2 }
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity
Hausdorff distance between shortest paths
this result versus our previous path-chord result path-chord result
P2 P2 P2 u0 u0 u0 u2 u2 u2 P1 P1 P1
long cycle ⇒ ⇒ ⇒ there is a path chord
this result
P2 P2 P2 u0 u0 u0 u2 u2 u2 P1 P1 P1 dv,v′ dv,v′
dH (P1,P2) ≤ max{ 6δ, 2 } dH (P1,P2) ≤ max{ 6δ, 2 } dH (P1,P2) ≤ max{ 6δ, 2 }
Which result is more general in nature ?
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 26 / 52
Implications of hyperbolicity
Hausdorff distance between shortest paths
this result versus our previous path-chord result path-chord result
P2 P2 P2 u0 u0 u0 u2 u2 u2 P1 P1 P1
long cycle ⇒ ⇒ ⇒ there is a path chord
this result
P2 P2 P2 u0 u0 u0 u2 u2 u2 P1 P1 P1 dv,v′ dv,v′
dH (P1,P2) ≤ max{ 6δ, 2 } dH (P1,P2) ≤ max{ 6δ, 2 } dH (P1,P2) ≤ max{ 6δ, 2 }
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 26 / 52
Implications of hyperbolicity
A notational simplification
A notational simplification unless G is a tree or a complete graph (Kn Kn Kn), δ > 0 δ > 0 δ > 0 δ > 0 ≡ δ ≥ 1/2 δ > 0 ≡ δ ≥ 1/2 δ > 0 ≡ δ ≥ 1/2 δ ≥ 1/2 ⇒ max{ 6δ, 2 } = 6δ δ ≥ 1/2 ⇒ max{ 6δ, 2 } = 6δ δ ≥ 1/2 ⇒ max{ 6δ, 2 } = 6δ Hence, we will simply write 6δ 6δ 6δ instead of max{ 6δ, 2 } max{ 6δ, 2 } max{ 6δ, 2 }
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 27 / 52
Implications of hyperbolicity
Distance between geodesic and arbitrary path
Distance from a shortest path u0
s
u1
u0
s
u1
u0
s
u1 to another arbitrary path u0
P
u1
u0
P
u1
u0
P
u1
n n n is the number of nodes in the graph ℓ(P ) ℓ(P ) ℓ(P ) is length of path P P P u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
shortest path u0
s
u1
u0
s
u1
u0
s
u1
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 28 / 52
Implications of hyperbolicity
Distance between geodesic and arbitrary path
Distance from a shortest path u0
s
u1
u0
s
u1
u0
s
u1 to another arbitrary path u0
P
u1
u0
P
u1
u0
P
u1
n n n is the number of nodes in the graph ℓ(P ) ℓ(P ) ℓ(P ) is length of path P P P u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
v ′ v ′ v ′ dv,v ′ dv,v ′ dv,v ′
∃v ′ ∃v ′ ∃v ′ dv,v ′ ≤ 6δ log2 ℓ(P )
dv,v ′ ≤ 6δ log2 ℓ(P )
dv,v ′ ≤ 6δ log2 ℓ(P )
O(logn) (logn) (logn) if δ δ δ is constant
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 28 / 52
Implications of hyperbolicity
Distance between geodesic and arbitrary path
An interesting implication of this bound u0 u0 u0 u1 u1 u1 v v v P P P
shortest path
v ′ v ′ v ′ dv,v ′ dv,v ′ dv,v ′
∃v ′ dv,v ′ ≤ 6δ log2 ℓ(P ) ∃v ′ dv,v ′ ≤ 6δ log2 ℓ(P ) ∃v ′ dv,v ′ ≤ 6δ log2 ℓ(P )
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 29 / 52
Implications of hyperbolicity
Distance between geodesic and arbitrary path
An interesting implication of this bound
= γ = γ = γ
assume ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ u0 u0 u0 u1 u1 u1 v v v P P P
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 29 / 52
Implications of hyperbolicity
Distance between geodesic and arbitrary path
An interesting implication of this bound
= γ = γ = γ
assume ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ⇒ ⇒ ⇒ ℓ(P ) ≥ 2
γ 6δ =
Ω
if δ δ δ is constant
ℓ(P ) ≥ 2
γ 6δ =
Ω
if δ δ δ is constant
ℓ(P ) ≥ 2
γ 6δ =
Ω
if δ δ δ is constant
u0 u0 u0 u1 u1 u1 v v v P P P
Next: better bounds for approximately short paths
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 29 / 52
Implications of hyperbolicity
Approximately short path
Why consider approximately short paths ? Regulatory networks
Up/down-regulation of a target node is mediated by two or more “close to shortest” paths starting from the same regulator node Additional “very long” paths between the same regulator and target node do not contribute significantly to the target node’s regulation
target source
Definition ε ε ε-additive-approximate short path P P P ℓ(P )
length of P P P ≤
ℓ(P )
length of P P P ≤
ℓ(P )
length of P P P ≤ length of shortest path +ε
+ε +ε
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 30 / 52
Implications of hyperbolicity
Approximately short path
Why consider approximately short paths ? Algorithmic efficiency reasons
Approximate short path may be faster to compute as opposed to exact shortest path
Routing and navigation problems (traffic networks)
Routing via approximate short path
Definition µ µ µ-approximate short path u0
P
uk =
P
uk =
P
uk =
P
uj
P
uj
P
uj
from ui to uj
≤ µ ≤ µ ≤ µ dui ,u j dui ,u j dui ,u j
distance between ui and uj
for all 0 ≤ i < j ≤ k 0 ≤ i < j ≤ k 0 ≤ i < j ≤ k
u0=v0 u0=v0 u0=v0 u1 u1 u1 u2 u2 u2 u3=v2 u3=v2 u3=v2 u4 u4 u4 u5 u5 u5 u6 u6 u6 u7=v4 u7=v4 u7=v4 v1 v1 v1 v3 v3 v3
2 2 2-approximate path u0
P
u7 =
P
u7 =
P
u7 =
s
u7
u0
s
u7
u0
s
u7
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 31 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path Distance from shortest path to an approximately short
ε ε-additive approximate
µ µ-approximate
path u0
P
u1
u0
P
u1
u0
P
u1
u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
shortest path u0
s
u1
u0
s
u1
u0
s
u1
v ′ v ′ v ′ dv,v ′ dv,v ′ dv,v ′
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 32 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path Distance from shortest path to an approximately short
ε ε-additive approximate
µ µ-approximate
path u0
P
u1
u0
P
u1
u0
P
u1
u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
shortest path u0
s
u1
u0
s
u1
u0
s
u1
v ′ v ′ v ′ dv,v ′ dv,v ′ dv,v ′
u0
P
u1
u0
P
u1
u0
P
u1 is ε
ε ε-additive approximate ∀v ∃v ′ dv,v ′ ≤ ∀v ∃v ′ dv,v ′ ≤ ∀v ∃v ′ dv,v ′ ≤
2
2
2
δ δ and ε ε ε
short crosstalk path for small ε ε ε and δ δ δ
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 32 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path Distance from shortest path to an approximately short
ε ε-additive approximate
µ µ-approximate
path u0
P
u1
u0
P
u1
u0
P
u1
u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
shortest path u0
s
u1
u0
s
u1
u0
s
u1
v ′ v ′ v ′ dv,v ′ dv,v ′ dv,v ′
u0
P
u1
u0
P
u1
u0
P
u1 is µ
µ µ-approximate ∀v ∃v ′ dv,v ′ ≤ ∀v ∃v ′ dv,v ′ ≤ ∀v ∃v ′ dv,v ′ ≤
6δ+2
δ δ and µ µ µ
short crosstalk path for small µ µ µ and δ δ δ
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 32 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Contrast the new bounds with the old bound of dv,v ′ = O
du0,u1 du0,u1 is the length of a shortest path between u0 u0 u0 and u1 u1 u1 u0
P
u1
u0
P
u1
u0
P
u1 is ε
ε ε-additive approximate ℓ(P ) ≤ du0,u1 +ε ℓ(P ) ≤ du0,u1 +ε ℓ(P ) ≤ du0,u1 +ε Old bound O
O
no dependency
du0,u1 du0,u1 u0
P
u1
u0
P
u1
u0
P
u1 is µ
µ µ-approximate ℓ(P ) ≤ µdu0,u1 ℓ(P ) ≤ µdu0,u1 ℓ(P ) ≤ µdu0,u1 Old bound O
O
no dependency
du0,u1 du0,u1
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 33 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path Distance from an approximately short
ε ε-additive approximate
µ µ-approximate
path u0
P
u1
u0
P
u1
u0
P
u1 to a shortest path
for simplified exposition, we show bounds only in asymptotic O(·) (·) (·) notation please refer to our paper for more precise bounds
u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
shortest path u0
s
u1
u0
s
u1
u0
s
u1
v ′ v ′ v ′ dv ′,v dv ′,v dv ′,v
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 34 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path Distance from an approximately short
ε ε-additive approximate
µ µ-approximate
path u0
P
u1
u0
P
u1
u0
P
u1 to a shortest path
for simplified exposition, we show bounds only in asymptotic O(·) (·) (·) notation please refer to our paper for more precise bounds
u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
shortest path u0
s
u1
u0
s
u1
u0
s
u1
v ′ v ′ v ′ dv ′,v dv ′,v dv ′,v
u0
P
u1
u0
P
u1
u0
P
u1 is ε
ε ε-additive approximate ∀v ′ ∃v dv ′,v ≤ ∀v ′ ∃v dv ′,v ≤ ∀v ′ ∃v dv ′,v ≤ O
δ δ and ε ε ε
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 34 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path Distance from an approximately short
ε ε-additive approximate
µ µ-approximate
path u0
P
u1
u0
P
u1
u0
P
u1 to a shortest path
for simplified exposition, we show bounds only in asymptotic O(·) (·) (·) notation please refer to our paper for more precise bounds
u0 u0 u0 u1 u1 u1 v v v P ≡ P ≡ P ≡u0
P
u1
u0
P
u1
u0
P
u1
arbitrary node
shortest path u0
s
u1
u0
s
u1
u0
s
u1
v ′ v ′ v ′ dv ′,v dv ′,v dv ′,v
u0
P
u1
u0
P
u1
u0
P
u1 is µ
µ µ-approximate ∀v ′ ∃v dv ′,v ≤ ∀v ′ ∃v dv ′,v ≤ ∀v ′ ∃v dv ′,v ≤O
δ δ and µ µ µ
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 34 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Distance from approximate short path P1 P1 P1
v v
to approximate short path P2 P2 P2
v ′ v ′ u0 u0 u0 u1 u1 u1 P2 P2 P2 P1 P1 P1
arbitrary node
v v v
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 35 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Distance from approximate short path P1 P1 P1
v v
to approximate short path P2 P2 P2
v ′ v ′ u0 u0 u0 u1 u1 u1 v ′′ v ′′ v ′′ P2 P2 P2 P1 P1 P1
arbitrary node
v v v
shortest path
go to any shortest path
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 35 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Distance from approximate short path P1 P1 P1
v v
to approximate short path P2 P2 P2
v ′ v ′ u0 u0 u0 u1 u1 u1 v ′′ v ′′ v ′′ P2 P2 P2 P1 P1 P1
arbitrary node
v v v v ′ v ′ v ′
shortest path
continue to the other path
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 35 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Distance from approximate short path P1 P1 P1
v v
to approximate short path P2 P2 P2
v ′ v ′
we sometimes overestimate quantities to simplify expression
P1 P1 P1 is ε1 ε1 ε1-additive approximate P2 P2 P2 is ε2 ε2 ε2-additive approximate
O
P1 P1 is ε ε ε-additive approximate P2 P2 P2 is µ µ µ-approximate
O
P1 P1 is µ µ µ-approximate P2 P2 P2 is ε ε ε-additive approximate
O
P1 P1 is µ1 µ1 µ1-approximate P2 P2 P2 is µ2 µ2 µ2-approximate
O
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 36 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Interesting implications of these improved bounds u0 u0 u0 u1 u1 u1 v v v
approximately short path P P P
shortest path
v ′ v ′ v ′ dv,v ′ dv,v ′ dv,v ′
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Interesting implications of these improved bounds
= γ = γ = γ
assume ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ u0 u0 u0 u1 u1 u1 v v v
approximately short path P P P
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Interesting implications of these improved bounds
= γ = γ = γ
assume ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ⇒ ⇒ ⇒ if P P P is ε ε ε-additive-approximate short then ε = Ω
δ
− logδ
δ
− logδ
δ
− logδ
u0 u0 u1 u1 u1 v v v
approximately short path P P P
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path
Interesting implications of these improved bounds
= γ = γ = γ
assume ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ∀v ′ ∈ P dv,v ′ ≥ γ ⇒ ⇒ ⇒ if P P P is µ µ µ-approximate short then µ = Ω
γ
γ
γ
u0 u0 u1 u1 u1 v v v
approximately short path P P P
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path To wrap it up, approximate shortest paths look like the following cartoon
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 38 / 52
Implications of hyperbolicity
Distance between geodesic and approximately short path To wrap it up, approximate shortest paths look like the following cartoon
Interpretation for regulatory networks
It is reasonable to assume that, when up- or down-regulation of a target node is
mediated by two or more approximate short paths starting from the same regulator node, additional very long paths between the same regulator and target node do not contribute significantly to the target node’s regulation
We refer to the short paths as relevant, and to the long paths as irrelevant Then, our finding can be summarized by saying that
almost all relevant paths between two nodes have crosstalk paths between each other
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 38 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 39 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3(κ−2)δ r > 3(κ−2)δ r > 3(κ−2)δ
Example: 5 1 9δ+1 9δ+1 9δ+1
u0 u0 u0
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3(κ−2)δ r > 3(κ−2)δ r > 3(κ−2)δ
Example: 5 1 9δ+1 9δ+1 9δ+1
u0 u0 u0 r u1 u1 u1 u2 u2 u2 ≥ 3κδ ≥ 3κδ ≥ 3κδ
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3(κ−2)δ r > 3(κ−2)δ r > 3(κ−2)δ
Example: 5 1 9δ+1 9δ+1 9δ+1
r + α r + α r + α u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 ≥ 3κδ ≥ 3κδ ≥ 3κδ α α α α α α
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3(κ−2)δ r > 3(κ−2)δ r > 3(κ−2)δ
Example: 5 1 9δ+1 9δ+1 9δ+1
u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 ≥ 3κδ ≥ 3κδ ≥ 3κδ α α α α α α consider any shortest path P P P between u3 u3 u3 and u4 u4 u4 P P P must look like this
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3(κ−2)δ r > 3(κ−2)δ r > 3(κ−2)δ
Example: 5 1 9δ+1 9δ+1 9δ+1
u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 v v v α α α α α α γ γ γ consider any shortest path P P P between u3 u3 u3 and u4 u4 u4 P P P must look like this P P P
γ = du0,v ≤ γ = du0,v ≤ γ = du0,v ≤ r − 3
2κ−1
r − 3
2κ−1
r − 3
2κ−1
r −Θ(κδ) r −Θ(κδ) r −Θ(κδ)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3(κ−2)δ r > 3(κ−2)δ r > 3(κ−2)δ
Example: 5 1 9δ+1 9δ+1 9δ+1
u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 v v v α α α α α α γ γ γ consider any shortest path P P P between u3 u3 u3 and u4 u4 u4 P P P must look like this P P P
γ = du0,v ≤ γ = du0,v ≤ γ = du0,v ≤ r − 3
2κ−1
r − 3
2κ−1
r − 3
2κ−1
r −Θ(κδ) r −Θ(κδ) r −Θ(κδ) ℓ(P ) ≥ ℓ(P ) ≥ ℓ(P ) ≥ (3κ−2)δ+2α (3κ−2)δ+2α (3κ−2)δ+2α Ω(κδ+α) Ω(κδ+α) Ω(κδ+α)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
Corollary (of previous results)
u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 ≥ 3κδ ≥ 3κδ ≥ 3κδ α α α α α α P P P consider any path P P P between u3 u3 u3 and u4 u4 u4 suppose that P P P does not intersect the shaded region
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
Corollary (of previous results)
u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 ≥ 3κδ ≥ 3κδ ≥ 3κδ α α α α α α P P P very long path consider any path P P P between u3 u3 u3 and u4 u4 u4 suppose that P P P does not intersect the shaded region ℓ(P ) ≥ ℓ(P ) ≥ ℓ(P ) ≥ 2 α 6δ + κ 4 2 α 6δ + κ 4 2 α 6δ + κ 4
2Ω α
δ +κ
α
δ +κ
α
δ +κ
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
Corollary (of previous results)
u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 ≥ 3κδ ≥ 3κδ ≥ 3κδ α α α α α α P P P very long path consider any path P P P between u3 u3 u3 and u4 u4 u4 suppose that P P P does not intersect the shaded region ℓ(P ) ≥ ℓ(P ) ≥ ℓ(P ) ≥ 2 α 6δ + κ 4 2 α 6δ + κ 4 2 α 6δ + κ 4
2Ω α
δ +κ
α
δ +κ
α
δ +κ
P P ε ε ε-additive-approximate ⇒ ⇒ ⇒
ε > 2
α 6δ+κ 4
48δ − log2(48δ) Ω
if δ δ δ is constant
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Influence of a node on the geodesics between other pair of nodes
Corollary (of previous results)
u0 u0 u0 r u1 u1 u1 u2 u2 u2 u3 u3 u3 u4 u4 u4 ≥ 3κδ ≥ 3κδ ≥ 3κδ α α α α α α P P P very long path consider any path P P P between u3 u3 u3 and u4 u4 u4 suppose that P P P does not intersect the shaded region ℓ(P ) ≥ ℓ(P ) ≥ ℓ(P ) ≥ 2 α 6δ + κ 4 2 α 6δ + κ 4 2 α 6δ + κ 4
2Ω α
δ +κ
α
δ +κ
α
δ +κ
P P ε ε ε-additive-approximate ⇒ ⇒ ⇒
ε > 2
α 6δ+κ 4
48δ − log2(48δ) Ω
if δ δ δ is constant
P P P µ µ µ-approximate ⇒ ⇒ ⇒ µ ≥
2 α 6δ + κ 4 12α+6δ
δ +κ)
α+κδ
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
shortest path usource usource usource utarget utarget utarget
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
umiddle umiddle umiddle usource usource usource utarget utarget utarget
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
umiddle umiddle umiddle
ξ = O(δ) ξ = O(δ) ξ = O(δ)
usource usource usource utarget utarget utarget
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
ξ = O(δ) ξ = O(δ) ξ = O(δ)
s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h
umiddle umiddle umiddle usource usource usource utarget utarget utarget All shortest paths between usource usource usource and utarget utarget utarget must intersect the ξ ξ ξ-neighborhood Therefore, “knocking out” nodes in ξ ξ ξ-neighborhood cuts off all shortest regulatory paths between usource usource usource and utarget utarget utarget
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
ξ = O(δ) ξ = O(δ) ξ = O(δ)
s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h
umiddle umiddle umiddle usource usource usource utarget utarget utarget But, it gets even more interesting !
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h
usource usource usource utarget utarget utarget But, it gets even more interesting ! shifting the ξ ξ ξ-neighborhood does not change claim
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
ξ = O(δ) ξ = O(δ) ξ = O(δ)
s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h
umiddle umiddle umiddle usource usource usource utarget utarget utarget how about enlarging the ξ ξ ξ-neighborhood ?
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h s h
t e s t p a t h
approximately short paths start intersecting the neighborhood
a p p r
i m a t e l y s h
t a p p r
i m a t e l y s h
t 2 ξ 2 ξ 2 ξ
umiddle umiddle umiddle usource usource usource utarget utarget utarget how about enlarging the ξ ξ ξ-neighborhood ?
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
umiddle umiddle umiddle usource usource usource utarget utarget utarget Consider a ball (neighborhood) of radius ξlogn ξlogn ξlogn (n n n is the number of nodes)
ξ logn ξ logn ξ logn
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Interesting implications of these bounds for regulatory networks
umiddle umiddle umiddle usource usource usource utarget utarget utarget
All paths intersect the neighborhood
Consider a ball (neighborhood) of radius ξlogn ξlogn ξlogn (n n n is the number of nodes)
ξ logn ξ logn ξ logn
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Empirical estimation of neighborhoods and number of essential nodes We empirically investigated these claims on relevant paths passing through a neighborhood of a central node for the following two biological networks:
T-LGL signaling
by selecting a few biologically relevant source-target pairs Our results show much better bounds for real networks compared to the worst-case pessimistic bounds in the mathematical theorems see our paper for further details
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 43 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions
a p p r
i m a t e g e
e s i c
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions
a p p r
i m a t e g e
e s i c
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions
a p p r
i m a t e g e
e s i c
the further we move from the central node the more a shortest path bends inward towards the central node
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
eavesdropper need not be a hub
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Implications of hyperbolicity
Identifying essential edges and nodes in regulatory networks
Traffic network
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Outline of talk
1
Introduction
2
Basic definitions and notations
3
Computing hyperbolicity for real networks
4
Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 45 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks Visual illustration of a well-known social network
Zachary’s Karate Club (http://networkdata.ics.uci.edu/data.php?id=105)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 46 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
Structural hole in a social network [Burt, 1995; Borgatti, 1997]
Definition (Adjacency matrix of an undirected unweighted graph)
u u u ... ... ... ... ... . . . . . . . . . . . . . . . v v v ... ... au,v au,v au,v ... ... . . . . . . . . . . . . . . . ... ... ... ... ...
au,v = 1, if {u,v} is an edge 0,
Definition (measure of structural hole at node u
u u [Burt, 1995; Borgatti, 1997]) (assume u u u has degree at least 2)
Mu
def
= =
au,v + av,u max
x=u
[ 1−
y=u,v
au,y + ay,u
av,y + ay,v
max
z=y
too complicated
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 47 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
Structural hole in a social network [Burt, 1995; Borgatti, 1997]
Definition (Adjacency matrix of an undirected unweighted graph)
u u u ... ... ... ... ... . . . . . . . . . . . . . . . v v v ... ... au,v au,v au,v ... ... . . . . . . . . . . . . . . . ... ... ... ... ...
au,v = 1, if {u,v} is an edge 0,
Definition (measure of structural hole at node u
u u [Burt, 1995; Borgatti, 1997]) (assume u u u has degree at least 2)
Let Nbr(u) Nbr(u) Nbr(u) be set of nodes adjacent to u u u
Mu =
av,y
av,y
av,y
Mu Mu
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 47 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
An intuitive interpretation of Mu
Mu Mu
Definition (weak dominance ≺ρ,λ
weak
≺ρ,λ
weak
≺ρ,λ
weak )
Nodes v, y v, y v, y are weakly (ρ,λ) (ρ,λ) (ρ,λ)-dominated by node u u u provided
ρ < du,v ,du,y ≤ ρ +λ
ρ < du,v ,du,y ≤ ρ +λ ρ < du,v ,du,y ≤ ρ +λ, and
for at least one shortest path P
P P between v v v and y y y, P P P contains a node z z z such that du,z ≤ ρ du,z ≤ ρ du,z ≤ ρ
ρ=1 λ=2
u
y v
Definition (strong dominance ≺ρ,λ
strong
≺ρ,λ
strong
≺ρ,λ
strong )
Nodes v, y v, y v, y are strongly (ρ,λ) (ρ,λ) (ρ,λ)-dominated by node u u u provided
ρ < du,v ,du,y ≤ ρ +λ
ρ < du,v ,du,y ≤ ρ +λ ρ < du,v ,du,y ≤ ρ +λ, and
for every shortest path P
P P between v v v and y y y, P P P contains a node z z z such that du,z ≤ ρ du,z ≤ ρ du,z ≤ ρ
ρ=1 λ=2
u
y v
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 48 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
An intuitive interpretation of Mu
Mu Mu
Notation (boundary of the ξ ξ ξ-neighborhood of node u u u) Bξ (u) =
ξ ξ from u u u
Observation
Mu Mu Mu
= = = E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is weakly (0
ρ,1 λ
) (0
ρ,1 λ
) (0
ρ,1 λ
)-dominated by u u u
v v is selected uniformly ran- domly from
λ
Bj (u)
λ
Bj (u)
λ
Bj (u) ≥ ≥ ≥ E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is strongly (0,1)-dominated by u u u
v v is selected uniformly ran- domly from
Bj (u)
Bj (u)
Bj (u) always true equality does not hold in general
u u u y y y v v v
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 49 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
Generalize Mu
Mu Mu to Mu,ρ,λ Mu,ρ,λ Mu,ρ,λ for larger ball of influence of a node
replace (0,1) (0,1) (0,1) by (ρ,λ) (ρ,λ) (ρ,λ)
Mu Mu Mu =
= = E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is weakly (0
ρ,1 λ
) (0
ρ,1 λ
) (0
ρ,1 λ
)-dominated by u u u
v v is selected uniformly ran- domly from
λ
Bj (u)
λ
Bj (u)
λ
Bj (u)
Mu,ρ,λ Mu,ρ,λ Mu,ρ,λ =
= = E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is weakly (ρ,λ) (ρ,λ) (ρ,λ)-dominated by u u u
v v is selected uniformly ran- domly from
Bj (u)
Bj (u)
Bj (u)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 50 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
Generalize Mu
Mu Mu to Mu,ρ,λ Mu,ρ,λ Mu,ρ,λ for larger ball of influence of a node
replace (0,1) (0,1) (0,1) by (ρ,λ) (ρ,λ) (ρ,λ)
Lemma (equivalence of strong and weak domination)
If λ ≥ 6δlog2 n λ ≥ 6δlog2 n λ ≥ 6δlog2 n then Mu,ρ,λ Mu,ρ,λ Mu,ρ,λ
def
= =
def
= =
def
= = E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is weakly (ρ,λ) (ρ,λ) (ρ,λ)-dominated by u u u
v v is selected uniformly ran- domly from
Bj (u)
Bj (u)
Bj (u) = = = E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is strongly (ρ,λ) (ρ,λ) (ρ,λ)-dominated by u u u
v v is selected uniformly ran- domly from
Bj (u)
Bj (u)
Bj (u) equality holds now
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 50 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
Lemma (equivalence of strong and weak domination)
If λ ≥ 6δ log2 n λ ≥ 6δ log2 n λ ≥ 6δ log2 n then Mu,ρ,λ Mu,ρ,λ Mu,ρ,λ
def
= =
def
= =
def
= = E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is weakly (ρ,λ) (ρ,λ) (ρ,λ)-dominated by u u u
v v is selected uniformly ran- domly from
Bj (u)
Bj (u)
Bj (u) = = = E number of pairs of nodes v, y v, y v, y such that v, y v, y v, y is strongly (ρ,λ) (ρ,λ) (ρ,λ)-dominated by u u u
v v is selected uniformly ran- domly from
Bj (u)
Bj (u)
Bj (u)
What does this lemma mean intuitively ?
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 51 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
What does this lemma mean intuitively ?
u u u Bρ(u) Bρ(u) Bρ(u) ρ ρ ρ λ ≥ 6δ log2 n λ ≥ 6δ log2 n λ ≥ 6δ log2 n Bρ+λ(u) Bρ+λ(u) Bρ+λ(u)
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 51 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
What does this lemma mean intuitively ?
u u u Bρ(u) Bρ(u) Bρ(u) ρ ρ ρ either all the shortest paths are completely inside Bρ+λ(u) Bρ+λ(u) Bρ+λ(u) λ ≥ 6δ log2 n λ ≥ 6δ log2 n λ ≥ 6δ log2 n Bρ+λ(u) Bρ+λ(u) Bρ+λ(u) v v v
y y y
u u u
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 51 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
What does this lemma mean intuitively ?
u u u Bρ(u) Bρ(u) Bρ(u) ρ ρ ρ
completely outside of Bρ+λ(u) Bρ+λ(u) Bρ+λ(u) λ ≥ 6δ log2 n λ ≥ 6δ log2 n λ ≥ 6δ log2 n Bρ+λ(u) Bρ+λ(u) Bρ+λ(u) v v v
y y y
u u u
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 51 / 52
Implications of hyperbolicity
Effect of hyperbolicity on structural holes in social networks
What does this lemma mean intuitively ?
u u u Bρ(u) Bρ(u) Bρ(u) ρ ρ ρ
but not both !
λ ≥ 6δ log2 n λ ≥ 6δ log2 n λ ≥ 6δ log2 n Bρ+λ(u) Bρ+λ(u) Bρ+λ(u) v v v
y y y
u u u
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 51 / 52
Final slide Thank you for your attention
Questions??
Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 52 / 52