Information Spreading in Social Edge-MEGs
The quest for a good model
Francesco Pasquale
Dipartimento di Informatica “R. Capocelli”
Università di Salerno
joint work-in-progress by
IIT-CNR di Pisa, “Sapienza” Università di Roma,
Università di Roma “Tor Vergata”, Università di Salerno
Roma, April 4-5, 2011
Information Spreading in Social Edge-MEGs
Roma, April 4, 2011
Information Spreading in Social Networks
Fact
Information spreading in Social Networks is fast
Why?
◮
Expansion;
◮
Dynamicity.
How to model dynamic networks?
Evolving graphs
G = {Gt = ([n], Et ) : t ∈ N}
Sequence of graphs with the same set of nodes.
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Information Spreading in Social Edge-MEGs
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Edge-MEG vs. Social Edge-MEG
Edge-MEG [Clementi et al. PODC’08]
◮
Every edge can be in two states exists / does not exist
◮
Transition given by a two-state Markov chain
p
0
non
edge
1
q
edge
- In this talk: death-rate q = 1 − p;
- On first approximation this is nearly w.l.o.g.
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Edge-MEG vs. Social Edge-MEG
Edge-MEG [Clementi et al. PODC’08]
◮
Every edge can be in two states exists / does not exist
◮
Transition given by a two-state Markov chain
- In this talk: death-rate q = 1 − p;
- On first approximation this is nearly w.l.o.g.
Social Edge-MEG
◮
C1 , . . . , Ck :
communities;
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Edge-MEG vs. Social Edge-MEG
Edge-MEG [Clementi et al. PODC’08]
◮
Every edge can be in two states exists / does not exist
◮
Transition given by a two-state Markov chain
- In this talk: death-rate q = 1 − p;
- On first approximation this is nearly w.l.o.g.
Social Edge-MEG
◮
C1 , . . . , Ck : communities;
◮
pin : Edge birth-rate
inside communities;
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Edge-MEG vs. Social Edge-MEG
Edge-MEG [Clementi et al. PODC’08]
◮
Every edge can be in two states exists / does not exist
◮
Transition given by a two-state Markov chain
- In this talk: death-rate q = 1 − p;
- On first approximation this is nearly w.l.o.g.
Social Edge-MEG
◮
C1 , . . . , Ck : communities;
◮
pin : Edge birth-rate inside
communities;
◮
pout : Edge birth-rate
between communities;
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Information Spreading
Question
Information spreading faster or slower?
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Information Spreading
Question
Information spreading faster or slower?
Flooding Time
◮
Start from one informed node (source node);
◮
At each step, neighbors of informed nodes get informed;
◮
Flooding Time = Time needed to have all nodes informed.
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Information Spreading
Question
Information spreading faster or slower?
Flooding Time
◮
Start from one informed node (source node);
◮
At each step, neighbors of informed nodes get informed;
◮
Flooding Time = Time needed to have all nodes informed.
Density = Expected fraction of edges in the graph
δ=
1
n
2
X
pi ,j
[n]
2
{i ,j}∈(
)
In Edge-MEGs δ = p.
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
Theorem (Clementi et al. PODC’08)
The Flooding Time of Edge-MEGs with n nodes and density p is
w.h.p.
log n
FT (Gn,p ) = Θ
log(1 + np)
Question
Given density p, Flooding Time of Social Edge-MEG is faster
or slower?
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
Theorem (Clementi et al. PODC’08)
The Flooding Time of Edge-MEGs with n nodes and density p is
w.h.p.
log n
FT (Gn,p ) = Θ
log(1 + np)
Question
Given density p, Flooding Time of Social Edge-MEG is faster or
slower?
Answer
It can be arbitrarily slower!
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
Example
√
For p = 1/ n Flooding Time FT (Gn,p ) = O(1).
Observation
For every (arbitrarily increasing) function f = f (n) a family of
√
social edge-MEGs G exists with density δ = 1/ n and flooding
time T (G) = Ω(f ).
Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
Example
√
For p = 1/ n Flooding Time FT (Gn,p ) = O(1).
Observation
For every (arbitrarily increasing) function f = f (n) a family of
√
social edge-MEGs G exists with density δ = 1/ n and flooding
time T (G) = Ω(f ).
◮
B
|A| = 1 and |B| = n − 1;
Source in A;
Social Edge-MEGs
A
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Information Spreading in Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
Example
√
For p = 1/ n Flooding Time FT (Gn,p ) = O(1).
Observation
For every (arbitrarily increasing) function f = f (n) a family of
√
social edge-MEGs G exists with density δ = 1/ n and flooding
time T (G) = Ω(f ).
◮
|A| = 1 and |B| = n − 1;
Source in A;
◮
(1 − pout )(n−1)t ≈ e −pout nt
> 1/e for t < 1/(npout );
Social Edge-MEGs
B
A
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Information Spreading in Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
Example
√
For p = 1/ n Flooding Time FT (Gn,p ) = O(1).
Observation
For every (arbitrarily increasing) function f = f (n) a family of
√
social edge-MEGs G exists with density δ = 1/ n and flooding
time T (G) = Ω(f ).
◮
|A| = 1 and |B| = n − 1;
Source in A;
◮
(1 − pout )(n−1)t ≈ e −pout nt
> 1/e for t < 1/(npout );
◮
Choose pout ≈ 1/(nf (n))
√
pin ≈ 1/ n − 1/(n2 f (n))
Social Edge-MEGs
B
A
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Information Spreading in Social Edge-MEGs
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Information Spreading in Social Edge-MEGs
Example
√
For p = 1/ n Flooding Time FT (Gn,p ) = O(1).
Observation
For every (arbitrarily increasing) function f = f (n) a family of
√
social edge-MEGs G exists with density δ = 1/ n and flooding
time T (G) = Ω(f ).
◮
|A| = 1 and |B| = n − 1;
Source in A;
◮
(1 − pout )(n−1)t ≈ e −pout nt
> 1/e for t < 1/(npout );
◮
Choose pout ≈ 1/(nf (n))
√
pin ≈ 1/ n − 1/(n2 f (n))
√
so that δ ≈ 1/ n
◮
Social Edge-MEGs
B
A
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Information Spreading in Social Edge-MEGs
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Leaders
Idea: Leaders (or Hubs)
Leaders are special nodes in a community that are highly
connected to leaders in other communities.
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Idea: Leaders (or Hubs)
Leaders are special nodes in a community that are highly
connected to leaders in other communities.
Social Edge-MEG with
Leaders
◮
C1 , . . . , Ck :
communities;
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Idea: Leaders (or Hubs)
Leaders are special nodes in a community that are highly
connected to leaders in other communities.
Social Edge-MEG with
Leaders
◮
C1 , . . . , Ck : communities;
◮
ℓ1 , . . . , ℓk : leaders;
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Idea: Leaders (or Hubs)
Leaders are special nodes in a community that are highly
connected to leaders in other communities.
Social Edge-MEG with
Leaders
◮
C1 , . . . , Ck : communities;
◮
ℓ1 , . . . , ℓk : leaders;
◮
pin , pout : Edge
birth-rates in and
between communities;
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
Roma, April 4, 2011
Leaders
Idea: Leaders (or Hubs)
Leaders are special nodes in a community that are highly
connected to leaders in other communities.
Social Edge-MEG with
Leaders
◮
C1 , . . . , Ck : communities;
◮
ℓ1 , . . . , ℓk : leaders;
◮
pin , pout : Edge birth-rates
in and between
communities;
◮
plea : Edge birth-rate
between leaders.
Question
Do leaders speed-up information spreading?
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
Leaders
Example 1:
√
n communities ×
Roma, April 4, 2011
√
n nodes per community
√
√
Social Edge-MEG with n communities, n nodes, 1 leader for
each community. Suppose pout = 0.
Question
Can we find pin and plea such that
1. Social Edge-MEG with the same density of Edge-MEG Gn,p
√
n · npin + nplea ≈ n2 p
log n
2. Flooding time smaller than FT (Gn,p ) = Θ log(1+np)
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
Leaders
Example 1:
√
n communities ×
Roma, April 4, 2011
√
n nodes per community
Answer: No!
◮
Observe that
FT (Gn,pin ,plea ) = Θ
Social Edge-MEGs with Leaders
log n
log n
√
√
+
log(1 + n · pin ) log(1 + n · plea )
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Information Spreading in Social Edge-MEGs
Leaders
Example 1:
√
n communities ×
Roma, April 4, 2011
√
n nodes per community
Answer: No!
◮
Observe that
◮
log n
log n
√
√
FT (Gn,pin ,plea ) = Θ
+
log(1 + n · pin ) log(1 + n · plea )
√
√
Density constraint: n · npin + nplea ≈ n2 p ⇒ npin 6 np;
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
Leaders
Example 1:
√
n communities ×
Roma, April 4, 2011
√
n nodes per community
Answer: No!
◮
Observe that
◮
◮
log n
log n
√
√
FT (Gn,pin ,plea ) = Θ
+
log(1 + n · pin ) log(1 + n · plea )
√
√
Density constraint: n · npin + nplea ≈ n2 p ⇒ npin 6 np;
Time needed to fill the first community is
(asymptotically) the same time needed to fill the whole
homogeneous graph.
Idea
We need smaller communities.
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Example 2: k communities × h nodes per community
Social Edge-MEG with Leaders
◮
◮
k · h = n;
Density δ ≈ k · h2 pin + k 2 plea ;
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Example 2: k communities × h nodes per community
Social Edge-MEG with Leaders
◮
◮
k · h = n;
Density δ ≈ k · h2 pin + k 2 plea ;
Question
Take density p = 1/n so that
FT (Gn,p ) = Θ
log n
log(1 + np)
= Θ(log n)
Can we find h, k, pin , plea such that density δ = p and flooding
time FT (Gh,k,pin ,plea ) = o(log n)?
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Example 2: k communities × h nodes per community
◮
◮
Density constraint: k · h2 pin + k 2 plea ≈ n2 p = n;
Flooding time:
log h
log k
FT (Gh,k,pin ,plea ) = Θ log(1+hp
+
log(1+kplea ) ;
in )
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Example 2: k communities × h nodes per community
◮
◮
Density constraint: k · h2 pin + k 2 plea ≈ n2 p = n;
Flooding time:
log h
log k
FT (Gh,k,pin ,plea ) = Θ log(1+hp
+
log(1+kplea ) ;
in )
Density constraint ⇒
Social Edge-MEGs with Leaders
hpin = O(1)
kplea = O(h)
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Information Spreading in Social Edge-MEGs
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Leaders
Example 2: k communities × h nodes per community
◮
◮
Density constraint: k · h2 pin + k 2 plea ≈ n2 p = n;
Flooding time:
log h
log k
FT (Gh,k,pin ,plea ) = Θ log(1+hp
+
log(1+kplea ) ;
in )
Density constraint ⇒
◮
hpin = O(1)
kplea = O(h)
If we take pin and plea as large as possible we get
log n
FT (Gh,k,pin ,plea ) = Θ log h +
log h
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Leaders
Example 2: k communities × h nodes per community
◮
◮
Density constraint: k · h2 pin + k 2 plea ≈ n2 p = n;
Flooding time:
log h
log k
FT (Gh,k,pin ,plea ) = Θ log(1+hp
+
log(1+kplea ) ;
in )
Density constraint ⇒
hpin = O(1)
kplea = O(h)
◮
If we take pin and plea as large as possible we get
log n
FT (Gh,k,pin ,plea ) = Θ log h +
log h
◮
Find h that minimizes f (h) = log h +
Social Edge-MEGs with Leaders
log n
log h :
h=e
√
log n ;
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Information Spreading in Social Edge-MEGs
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Leaders
Example 2: k communities × h nodes per community
◮
◮
Density constraint: k · h2 pin + k 2 plea ≈ n2 p = n;
Flooding time:
log h
log k
FT (Gh,k,pin ,plea ) = Θ log(1+hp
+
log(1+kplea ) ;
in )
Density constraint ⇒
◮
◮
◮
hpin = O(1)
kplea = O(h)
If we take pin and plea as large as possible we get
log n
FT (Gh,k,pin ,plea ) = Θ log h +
log h
√
log n
log n ;
Find h that minimizes f (h) = log h + log
h: h = e
For such an h we get
p
FT (Gh,k,pin ,plea ) = Θ
log n = o(log n)
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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High probability problem
Ooops!
√
Size of communities: h = e log n
√
ω(logc n)
for any arbitrary large constant c
log n
=
e
o(nε )
for any arbitrary small constant ε > 0
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
Roma, April 4, 2011
High probability problem
Ooops!
√
Size of communities: h = e log n
√
ω(logc n)
for any arbitrary large constant c
log n
e
=
ε
o(n )
for any arbitrary small constant ε > 0
Result in PODC’08 cannot be directly used to guarantee
flooding completed in all communities w.h.p.
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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High probability problem
Ooops!
√
Size of communities: h = e log n
√
ω(logc n)
for any arbitrary large constant c
log n
e
=
o(nε )
for any arbitrary small constant ε > 0
Result in PODC’08 cannot be directly used to guarantee flooding
completed in all communities w.h.p.
A dog chasing its tail
The same argument needed to have fast flooding (i.e. small
communities) prevents proving that flooding is fast!
Social Edge-MEGs with Leaders
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Information Spreading in Social Edge-MEGs
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Where from here?
Conclusions
◮
Is Flooding Time of Social Edge-MEGs with leaders
smaller than Flooding Time of Edge-MEGs?
◮
◮
We still don’t know;
If yes, then proving it is not straightforward;
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Information Spreading in Social Edge-MEGs
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Where from here?
Conclusions
◮
Is Flooding Time of Social Edge-MEGs with leaders smaller
than Flooding Time of Edge-MEGs?
◮
◮
◮
We still don’t know;
If yes, then proving it is not straightforward;
Refine results in PODC’08 to achieve overwhelming
probability?
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Information Spreading in Social Edge-MEGs
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Where from here?
Conclusions
◮
Is Flooding Time of Social Edge-MEGs with leaders smaller
than Flooding Time of Edge-MEGs?
◮
◮
We still don’t know;
If yes, then proving it is not straightforward;
◮
Refine results in PODC’08 to achieve overwhelming
probability?
◮
Does the notion of leader we used capture the intuitive
idea of leader? Unknown leaders vs Known leaders
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Information Spreading in Social Edge-MEGs
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Where from here?
Conclusions
◮
Is Flooding Time of Social Edge-MEGs with leaders smaller
than Flooding Time of Edge-MEGs?
◮
◮
We still don’t know;
If yes, then proving it is not straightforward;
◮
Refine results in PODC’08 to achieve overwhelming
probability?
◮
Does the notion of leader we used capture the intuitive idea of
leader? Unknown leaders vs Known leaders
Thank you!
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