Download Duration dependent communication graphs

Institutions do not die
Networks & Communication
What is a network?
How are the local and global
properties related to each other?
Local, immediate
properties
Geometry Topology
Global, long-time,
large-scale
properties
Transportation metrics on networks & databases
Monge – Kantorovich transport problem
Y
K  X , Y   inf  cost x, y dL
L
L
-- the transportation plan L;
X
-- X , Y probability measures on
a compact space;
Α
T, Α  0,  Tij  1,
j
-- the stochastic automorphism /
a diffusion generator;
 X,Y T
 X , G Y 
-- a scalar product of probability
measures;
K → the transportation metric
G
n
1


T

1
T

"
L
",

1
n 
-- the Green function (propagator)
X
2
T
  X , GX 
-- the (squared) norm of a distribution;
The Green function of diffusion process defines the transportation metric on the data manifold
Transportation (Ricci) curvature
T by local roads
T by a highway
T by a highway
  1
T by local roads
A geometric method for data analysis & representation
G major is based on the pitches
G, A, B, C, D, E, and F♯.
V.A. Mozart, Eine Kleine Nachtmusik
T
G
 Tn  1-T  " L1" ,
1
n 
C, “do”:
“
1
0
“ =
First-passage time (
Recurrence time (
) T2 =
(
) =1/p
,
,G
)
pp G
Ricci curvature:
First - passage time
  1
Recurrence time
Anticipation is possible within the data neighborhood of positive Ricci curvature
Tax assessment value of land ($)
Can we see the first-passage times?
Manhattan, 2005
(Mean) First passage time
(Mean) first-passage times in the city graph of Manhattan
SoHo
Federal Hall
10
East Village
100
1,000
Bowery
East Harlem
5,000
10,000
Why are mosques located close to railways?
NEUBECKUM:
Isolation Moschee   10  log
first - passage time (Moschee)
 12 dB
Min first - passage time
first - passage time (Kirche)
Isolation Kirche   10  log
 3 dB
Min first - passage time
Social isolation vs. structural isolation
Organizations: why do we think that
the majority is always right?
• We like to be a part of a majority, as ”if many
believe so, it is so” – The drive towards a
majority is that of an insurance policy against
the risks with which the daily life is fraught.
The position of the majority is always
perceived as fair.
Social time is unfolding through the switching
between communication and withdrawing
”We can only preserve our unity by being able to
’open and close’, to participate in and
withdraw from the flow of messages. It
therefore becomes vital to find a rhythm of
entry and exit that allows each of us to
communicate meaningfully without nullifying
our inner being.”
Melucci, A., ”Inner Time and Social Time in a World of Uncertainty”, Time Society 7, 179
(1998).
Communication
Patterns in Organizations
• We have presented the first study integrating the
analysis of temporal patterns of interaction, interaction
preferences and the local vs. global structure of
communication in two organizations over a period of
three weeks.
• The data suggest that there are three regimes of
interaction arising from the organizational context of
our observations: casual, spontaneous (or deliberate)
and institutional interaction.
• We show that institutions never die, as once
interrupted communication can be resumed anytime
Data collection was carried out in
June and July 2010
H-art
~71 assigned to multiple projects.
employees → functions
H-farm
~ 75 employees and hosted 9
start-ups
employees → start-ups
We analyze face-to-face interactions in two organizations over a period of four weeks.
Data on interactions among ca 140 individuals have been collected through a wearable
sensors study carried on two start-up organizations in the North-East of Italy.
The radio-frequency identification sensors
reported on occasions of physical proximity
Smaller groups communicate more frequently than
larger groups; brief communications are much
more common than longer ones.
employees → start-ups
The impression of a power law can result from the superposition of
different behaviors.
A simple probability model describing
the communication behavior < 20 min
-- each interruption of a communication is a statistically independent event;
-- the probability to interrupt an ongoing communication act pT > 0 depends only on the total
expected duration of communication T.
20 min
The distribution of communication durations averaged
over the observation period is a weighted sum of different
exponentials featured by various durations of speaking.
20 min;
difficult to
interrupt
Every communication is potentially an extremely time consuming
action (“sticky”), time spent in communications has to be invested
prudently
Intervals between sequent interactions
Probability
The distribution of intervals between the
sequent communication events is
remarkably skewed, indicating a
significant proportion of the abnormally
long periods of inactivity.
Duration of intervals between sequent communications (min)

exp  t  2  21.78
2
2

2p 1.78
2
Probability
Casual
The normal distribution can be interpreted
as an average outcome of many
statistically independent processes that
determine the majority of casual
interactions characterized by the very
short intervals between them lasting not
longer than 4 mins.
Duration of intervals between sequent communications (min)

exp  t  2  21.78
2
2

2p 1.78
2
Probability
Casual
The most probable interval
between sequent
communications lasts 2 min.
Although time is a scarce resource,
short time intervals go largely
unmanaged in organizations.
Duration of intervals between sequent communications (min)

exp  t  2  21.78
2
2

2p 1.78
2
Casual
Probability
1 t  1t  2   1
Deliberate
(spontaneous)
Uniformly
random
t
c
t2
Fixed
during
the day
Duration of intervals between sequent communications (min)

exp  t  2  21.78
2
2

2p 1.78
2
Casual
Probability
1 t  1t  2   1
Deliberate
(spontaneous)
Uniformly
random
t
c
t2
Fixed
during
the day
The simplest time management strategy is
to postpone or avoid unimportant or
unwanted meetings
Duration of intervals between sequent communications (min)

exp  t  2  21.78
2
2

2p 1.78
2
Casual
Probability
1 t  1t  2   1
Deliberate
(spontaneous)
Uniformly
random
t
c
Fixed
during
the day
t2
1
t
1
,   103
Mandatory
(institutional)
c  1
Duration of intervals between sequent communications (min)

exp  t  2  21.78
2
2

2p 1.78
2
Casual
Probability
Mandatory institutional
communications may include Deliberate
(spontaneous)
urgent, exigent contacts
made in emergency, as well
c
as some common rites andUniformly
random
t
rituals that serve important
functions for all team
members.


1 t  1t  2   1
Fixed
during
the day
t2
1
t
1
,   103
Mandatory
(institutional)
A logic of institutional interaction prevails,
where top-down, almost mandatory
interaction occurs
c  1
Duration of intervals between sequent communications (min)
Duration dependent communication graphs
1
2
3
4
5
Communication durations (min)
6
Duration dependent communication graphs
In order to analyze how interaction propensities and the duration of interaction
affect each other, we use mutual information as a statistical measure of pairwise
interaction propensities.
If during the observation period A and B
participated in meetings independently,
1
2
3
4
5
Communication durations (min)
6
Duration dependent communication graphs
The degree of communication
selectivity:
I  X , Y  

X ,Y 
P  X , Y  log 2
P  X , Y 
P  X P Y 
How much knowing the fact of that X is
communicating during time t would
reduce uncertainty about that Y is
communicating
1
2
3
4
5
Communication durations (min)
6
Duration dependent communication graphs
The degree of communication
selectivity:
I  X , Y  

X ,Y 
P  X , Y  log 2
P  X , Y 
P  X P Y 
How much knowing the fact of that X is
communicating during time t would
reduce uncertainty about that Y is
communicating
• The performed analysis of mutual information shows that the degree of selectivity
in both companies monotonously increases with the interaction duration, until their
maximum values are attained;
• People are essentially selective in choosing partners for communications lasting
1
3
4
5
6
between
10 and220 min;
• For particularly long interactions, perhaps involving many group members at once,
the values of mutual information is particularly small.
Communication durations (min)
Duration dependent communication graphs
The degree of communication
selectivity:
I  X , Y  

X ,Y 
P  X , Y  log 2
P  X , Y 
P  X P Y 
How much knowing the fact of that X is
communicating during time t would
reduce uncertainty about that Y is
communicating
• The functional structure of organization matters essentially for the
communications of short duration (1-5 min):
departmental
structure
evokes 4
more selectivity
1
2
3
5 in short
communications
Communication durations (min)
6
Time-dependent interaction graphs
(Networks)
The main objective of analysis is to understand how the ”local”, individual
interaction propensities described by the connectivity of subjects as nodes of a
communication graph determine the ”global”, connectedness property of the
entire communication process described by the ensemble of communication
graphs for all communication durations.
Time-dependent interaction graphs
(Networks)
In order to address this problem in relation to all communication graphs, let us
consider a model of simple random walks, a statistical metaphor of message
transmission in a working team. We suppose that a message (requiring t time
units to be transmitted) is passed on by each subject X to another one – Y ,
selected at random among all available companions accordingly to the connection
probability T(t)XY determined by the communication graph of communication
duration t.
Interaction synchronization: could they all
speak altogether?
Then the minimal amount of information required to record a single random
transition of a message in the entire communication graph correspondent to
the duration t is defined by the entropy rate of random walks
Interaction synchronization: could they all
speak altogether?
Interaction synchronization: could they all
speak altogether?
Excess entropy/ complexity/ past-future mutual information:
Interaction synchronization: could they all
speak altogether?
Interaction synchronization: could they all
speak altogether?
• The functional structure of organization matters essentially for individuals of low connectivity
(subordinates):
departmental structure evokes more schedules/organization shaping interactions
between subordinates
Interaction synchronization: could they all
speak altogether?
• Individuals communicating with 10-12 people are evolved in the maximum number of
communicating groups;
• Individuals communicating with more than 10-12 people organize meetings themselves
How is the individual communication propensity (a
local property) related to global properties?
, a permutatio n matrix
if ,       0,
then   Aut  
T,   0,
T
ij
j
T  D 1
 1, a stochastic matrix
How is the individual communication propensity (a
local property) related to global properties?
, a permutatio n matrix
if ,       0,
then   Aut  
πT  π,
Ri 
Fi 
1
pi
ci
pi
i, π i  0

2E
, recurrence time
deg i 
, first - passage time
ci  1, perfectly integrated ;
ci  1, balanced;
ci  1, isolated;
T,   0,
T
ij
j
T  D 1
 1, a stochastic matrix
How is the individual communication propensity (a
local property) related to global properties?
, a permutatio n matrix
if ,       0,
then   Aut  
πT  π,
Ri 
1
pi
T,   0,
T
ij
T  D 1
 1, a stochastic matrix
j
i, π i  0

2E
, recurrence time
deg i 
A local property (connectivity)
Fi 
ci
pi
, first - passage time
A global property (connectedness)
ci  1, perfectly integrated ;
ci  1, balanced;
ci  1, isolated;
Connectedness exceeds connectivity
a “positive Ricci curvature”
Conclusion 1: Multilevel communication protocol
• Rule of thumb for
belonging:
Intervals between
communications that
last twice as long, occur
twice as rare;
→ Simply respect the
group discipline
Sample size
Conclusion 1: Multilevel communication protocol
• Rule of thumb for
belonging:
Intervals between
communications that
last twice as long, occur
twice as rare;
→ Simply respect the
group discipline
Sample size
t
t 1
t
1
 
t 1
t
t  1t  2  
t 

Institutions do not die!
Institutional & Spontaneous communications can be resumed at any time!
Conclusion 2: Structure affects selectivity
employees → functions
I  X , Y  
employees → start-ups
P  X , Y  log

 
X ,Y
2
P  X , Y 
P  X P Y 
Conclusion 2: Structure affects selectivity
employees → functions
more schedules/ structure
employees → start-ups
Conclusion 3: A team exists when
connectedness exceeds connectivity
“Positive Ricci curvature”: Directed intentional messages
traverse the team faster than rumors