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Social Capital as Engagement
and Belief Revision
Gaurav Koley[0000−0002−3918−0892],
Jayati Deshmukh[0000−0002−1144−2635],
Srinath Srinivasa?[0000−0001−9588−6550]
International Institute of Information Technology, Bangalore
Abstract. Social Capital or “goodwill” is an essential ingredient of any
collective activity– be it commercial, cultural or administrative activity.
In online environments, several models have been pursued for recording
and utilizing social capital based on signals including likes or upvotes.
Such explicitly stated signals are susceptible to impulsive behavior and
hyperinflation. In this paper, we develop an implicit model for social cap-
ital based on the extent of engagement generated by any participant’s
activities, and the way this engagement leads to a belief revision about
the participant from other members of the community. Two kinds of
social capital measures are proposed: an authority score that indicates
engagement, and a citizenship score that calibrates value-addition made
by a user as a result of engaging with others’ content. The proposed
model is implemented in two online communities showing different kinds
of content authorities, supported by a strong community of engaged cit-
izens.
Keywords: Social capital, Engagement, Belief Revision, Social Networks
1 Introduction
Social capital or “goodwill” is an important element of social interactions. Con-
ventionally, social transactions were analyzed from the lens of rational utility
functions. However, rational utility is itself known to be affected by cognitive
factors like trust, reputation, goodwill, aversion to risk, etc. (15; 14; 2).
In order to account for such factors, online social spaces typically have some
form of a reputation or rating model based on explicitly elicited signals like
star ratings, upvotes/downvotes, karma points, user endorsements, etc. The un-
derlying paradigm for interpreting these signals, is based on the assumption of
cross-rating where, stakeholders with conflicting objectives, balance out biases
in the ratings on an aggregate level (12). Sometimes, the base signals are further
used as input for some form of spreading activation algorithm to diffuse and
aggregate the scores across the population.
?Corresponding author
2 Gaurav Koley, Jayati Deshmukh, Srinath Srinivasa
However, there are some recurring challenges with explicitly stated signals.
Explicit signal models would typically result in a dearth of representative data,
because many users may simply ignore providing a rating. In addition, users
tend to speak up with bad ratings and other forms of impulsive responses to
bad experiences; and tend to take good experiences for granted, resulting in an
overall bias in the rating. Other forms of racial, gender and ethnic biases may
also influence explicit rating mechanisms. There is also a danger of “mob rating”
by groups of disgruntled participants, or even by a posse of fans, that can greatly
skew actual signals from that of the population.
There is hence, a need for computational modeling of goodwill, that is based
on implicit evidence and better theoretical underpinnings of what constitutes a
signal for social capital.
It is generally acknowledged that social capital can be seen as a function of
positive engagement (5; 22; 26). People having goodwill towards one another,
tend to be “strongly engaged”– they easily disclose information to one another,
entrust one another with something of value, and even develop a shared sense
of identity.
Engagement can be of a variety of forms (18; 27; 23). It represents people
investing something of value to them, in the work of somebody else. Online social
networks typically have their own custom signals to indicate engagement. For
instance, engagement on Twitter is often measured using metrics like number of
mentions, retweets and likes.
However, to build our theory of social capital, we first develop an applica-
tion agnostic model of engagement based on the extent of sustained attention
generated by any user’s activities. Sustained attention indicates a recurrent and
immersive form of interest. This becomes a precursor for other forms of engage-
ment. Sustained attention may be contrasted with cursory attention, which may
only elicit impulsive responses.
Based on this model of engagement, we use Degroot Belief revision to propose
two new network metrics: Authority Rank and Citizen Rank. Authority Rank
shows the relative command of an actor over its network for a topic. This can
be used to find influencers in a network. Citizen Rank ranks actors in a network
based on their engagement or interaction characteristics. This can be used to
find highly engaged users in a network.
2 Related Work
Several scholars have proposed the use of social network analysis to study the
social capital of groups and individuals; to emphasise the value of relationships
and networks to maintain social capital (13; 7).
The individualist approach explores the idea that an actor’s network or their
position within the network affects their social capital. Individuals benefit from
the following structural features of their networks: size of an actor’s ego net-
work, where small networks with strong ties provide material and emotional
support and large networks with weak ties provide access to information and
Social Capital as Engagement and Belief Revision 3
resources (16; 25); and actor’s position within the network, where having a
broker position leads to ability to access a greater diversity of resources than
others (6; 7).
The groupist perspective focuses on a network’s structural features and how
the rise and maintenance of reciprocity and trust is enabled by that network
structure. Coleman (8) posits that trust and the feeling of mutual obligation is
higher among members of a complete network where all the actors are connected
to each other.
Empirical research has shown social capital to correlate with increased per-
ceived credibility of the flow of information within a network and people trusting
each other on a long-term basis (7). In addition, Granovetter (9) notes that this
closed network structure persuades friends to behave honestly amongst them-
selves.
In recent times, researchers have looked at Sen’s Capability Approach (20; 21)
to explain certain properties exhibited by social capital. Sen calls social capital
as an endowment– or, “a set of means to achieve a life people reason to value.”
The potential positive effects of social capital can be seen through the in-
fluence of social connections. Friends and family helps individuals in lots of
ways– economically, socially and emotionally. Several social scientists including
Granovetter (9) emphasise the role of close friends and family, as well as ca-
sual acquaintances, in finding jobs. Burt (6) focuses on the lack of close ties
as a motivator for knowledge sharing and mobility of individuals. Mwangi and
Ouma (17) show that social capital increases the network reach of individuals
and thereby enhances financial inclusion through increased access to informal
loans in Kenya. Social capital encourages social trust and membership, reduces
health risks amongst children and adolescents (28) and discourages individuals
from engaging in harmful activities like smoking and binge drinking (4). Social
capital is also linked with greater well-being according to self-reported survey
measures (10) and reduced crime (1). Social capital also helps businesses– for
example in Bowling Alone (19), Putnam observes that the formal and informal
cooperation between startup companies in the Silicon Valley has led to their
collective success.
However highly entrenched networks can hinder people as well, despite having
high extents of social capital. Close knit communities usually have strong social
bonds, with the individuals relying heavily on relatives and others of the same
community for support. The lack of social bridges that can connect them to
the wider society, can also turn them into outsiders and hinder their social
development and economic upliftment (10). Social capital can also be put to
harmful use as well. The trust and reciprocity that allows close-knit networks
like mafia, criminal gangs and cults to operate is also a form of social capital (10).
3 Approach and Model
In this work, our focus is on building a computational model for measuring
social capital. As noted earlier, a common aspect in the disparate theories of
4 Gaurav Koley, Jayati Deshmukh, Srinath Srinivasa
social capital surveyed, is the element of engagement generated by a person’s
activities. Engagement in turn, leads to other elements like reciprocity, trust,
credibility, etc. We define engagement itself, in terms of consistent receipt of an
abstract concept called “stroke”– that refers to positive affinity received from
others.
3.1 Engagement and Strokes
Engagement manifests with a creator who creates resources, and one or more
consumers who consume the resources. A specific instance of consumption by
consumer uengaging with a resource created by creator vis called a stroke from u
to v. The term “stroke” is borrowed from Transaction Analysis in psychology (3),
which in our case, represents an episode of psychological engagement from the
recipient to the creator of a resource.
Each element of a stroke is formally represented as follows:
s=hu, v, t, u→vi
where uis the stroke initiator, vis the stroke receiver, tis the time of stroke
and u→vis the amount of engagement in the stroke. In different scenarios, the
act of engagement or stroke refers to different kind of social actions, e.g., reading
a research paper in scholarly networks and (optionally) subsequently citing it, or
responding to a tweet on Twitter. The action represents a directional influence
and we consider that there is an attention flow from the consumer to the creator.
The model is heavily dependent on what is meant by engagement u→vand
how it is measured. Any measurable resource can be used in our model that has
following properties:
Limited: An agent should have only a finite quantity of resources that charac-
terize the given mode of engagement.
Conserved: The resource should also be conserved i.e. once the measurable
resource is allocated to someone as engagement, the same cannot be assigned
to anyone else.
Renewable: The available resource should be renewed after some fixed duration
for all users.
There are several examples of resources which have the above properties and
can be used as a measure of engagement. One of the simplest is time spent.
The amount of time an actor spends engaging with another actor through their
tweets, posts, etc. can be tracked and used as a measure of engagement since
the amount of time an agent can spend is limited in a day and is also renewed
everyday. Another similar measure on Twitter can be the number of retweets as
a user can only post a maximum of 2400 tweets including retweets and this limit
is reset everyday.
The amount of engagement u→vis typically a function of contiguous ex-
pending of scarce resources. In the case of time spent, engagement is a function
of contiguous units of time that was spent, leading to sustained attention by the
Social Capital as Engagement and Belief Revision 5
attention-giver. A user spending nunits of time contiguously on a resource is
said to be more engaged than a user spending nunits of time on a resource in
several smaller chunks.
The above nature of engagement is also corroborated by studies from prospect
theory, specifically the cognitive models of System 1 and System 2 (24). Engage-
ment is a characteristic feature of System 2 cognition, that indulges in deliber-
ative and conscious reasoning. Merely glancing through the content and/or re-
acting impulsively to a post, cannot be considered as engagement. Engagement
requires deeper cognitive involvement of the attention-giver. Hence, engagement
also results in saturation, fatigue, or diminishing returns, when sustained for too
long a time.
Given these, we characterize the engagement function as an S-shaped sig-
moid curve, on the extent of contiguous time spent. This is elaborated in the
subsections on computing Authority and Citizenship ranks.
A Social Network can now be represented as a directed graph depicting
engagement pathways between users. A social graph can be formulated as a
weighted graph (V, E, w), where Vis the set of users and Eis the pair-wise
engagement between users. The weight on an edge (u, v) is the total engagement
obtained from uto vacross all strokes from uto v.
3.2 Belief Revision
Strokes and engagement between pairs of actors, changes the nature of their
relationship. A strong engagement from one actor to another would result in
greater familiarity, influence, affinity, trust, and other possibilities. This change
in the nature of relationship between any two actors also diffuses through the
network, affecting to different extents, others’ beliefs about these actors. We
model this dynamic as a social learning process using the DeGroot Belief Re-
vision model (11). Let us take a social network of nagents where everybody
has an opinion or belief about other actors at any given time t. In the simplest
sense, suppose we are considering a specific topic or interest p, this belief is
represented by a probability representing the belief holder’s confidence about
the other actor, regarding interest p. A belief bp
u7→vrepresents the goodwill or
credibility that uhas towards v, regarding topic p. Unless uis directly stroking
v,udoes not receive any new information to change its opinions. But, actors
can interact and communicate with their neighbors due to which, beliefs diffuse
through the network. Each acquaintance link between actors is represented as a
belief matrix T, where Ti,j represents the belief actor ihas about actor j. The
belief matrix is also modeled as a row-stochastic matrix, where the belief values
are all non-negative, and the belief values by one actor about all other actors in
the network, add up to 1.
As engagements change over time due to strokes between interacting actors,
belief values changes across the population. Let bt
irepresent column iindicating
6 Gaurav Koley, Jayati Deshmukh, Srinath Srinivasa
incoming belief values for actor ifrom all other actors, at time t. Belief revision
for actor ihappens as follows:
bt+1
i=T·bt
i(1)
Unravelling the recursion, the tth period opinions can be computed by
bt
i=Tt×b0
i(2)
Since we assume that every actor has a belief about every other actor for
any given topic, the network represented by Tis fully connected, aperiodic and
irreducible. This makes the stationary distribution of Tindependent of the initial
distributions of belief vectors and as tbecomes very large, all columns converge
to the same belief vector b.
b= lim
t→∞ Tt×b0
i(3)
We use the above DeGroot Belief Revision model to compute two kinds of
scores: Authority Rank and Citizen Rank for any actor.
3.3 Authority and Citizen Rank
Authority Rank is a measure of the social impact of an actor in the social net-
work. This can be interpreted as the network’s belief about an actor’s credibility
about a given topic. For example, AR(5) = 0.09 can be understood as the net-
work’s belief that the social capital of actor 5 extends up to 9% of the network.
We say that an actor v’s influence on actor udepends on the relative slice
of attention paid by uto v. For example, let us consider actor A who spends 4
hours engaging equally with actor C and 3 other actors on a social networking
site and actor B who engages with only actor C for 1 hour. Although, the time
(engagement measure in this scenario) spent by A and B on C is the same, A
will be less influenced by C as compared to B.
We construct the belief matrix Tfrom the engagement strokes as defined
earlier. Relative engagement is computed as per Equation 4 where the values
are normalized such that Ti,j 7→ [0,1].
Ti,j =w(i, j)
X
∀k,i→k
w(i, k)(4)
We want to reduce the influence of low engagement and dampen the effect
of extremely high engagement as well. This is due to the law of diminishing
returns, as well as entrenchment effects arising due to very high engagement.
For someone who is already highly engaged, any higher engagement doesn’t lead
to a linear increase in influence. Therefore, we use a softmax function to dampen
Social Capital as Engagement and Belief Revision 7
lower engagement values and skew the engagement to give more weight to higher
engagement values using Equation 5, that gives us the authority matrix.
Ai,j =(exp(λ·Ti,j )
Pk6=iexp(λ·Ti,k)if i6=j
0 if i=j(5)
Here Ai,j 7→ [0,1] and λis a scaling parameter that is representative of the
level of engagement required to judge another actor’s worth as an authority. λ
therefore depends on the measure of engagement being used as well as the net-
work structure. Higher the median value of engagement for a kind of engagement
measure, higher should be the λ. For example, if time is being used a measure
of engagement, for a social network of videos, average time of engagement will
be in 10s of minutes while for a twitter like network it will be lesser. Therefore,
λwill be smaller for a twitter like network as compared to a video network.
Thus Arepresents a strongly connected, aperiodic and irreducible graph,
therefore, a convergence point for the network’s belief exists.
A∗= lim
i→∞ Ai(6)
and the left diagonal of A∗gives us the Authority Rank (AR) for the social
network as:
AR(i) = A∗×1(7)
Here, Ais a non-Ergodic system, thus the initial state of A determines the
stable point of the system at the end of Equation 6. Therefore, by varying λ,
different results could be found for AR.
To find the ideal λ, we search the space [1,∞) and settle for a value of λfor
which subsequent increase in value doesn’t change the AR values.
We pick the λfor which Authority Ranks stabilize. This stable value of
λchanges across different social networks and different underlying notions of
engagement. For a network, the value at which λstabilizes also indicates the
depth at which people prefer engaging with others on the network. A small
value of stable λrepresents that actors prefer a number of connections with
cursory engagement, over deep engagement. A large value of stable λmeans
that actors prefer engaging deeply than merely interacting with lots of other
actors. The stable value for λhence represents a characteristic feature of the
network dynamics.
Citizen Rank is a measure of the perceived value accrued due to an actor’s
participation in the social network. It can be interpreted as the network’s belief
about the actor’s contributions to preserving and improving the network. For
example, CR(5) = 0.09 can be understood as the network’s belief that actor 5
is a stakeholder of the network with their stake being worth 9%.
The Citizen Rank is computed in an analogous fashion to that of Authority
Rank, but with some differences. We define participation matrix as follows:
pi,j =w(j, i)
X
∀k,i→k
w(k, i)(8)
8 Gaurav Koley, Jayati Deshmukh, Srinath Srinivasa
Here pi,j is the participation or engagement ihas received from j. We run
it through a similar softmax scaling treatment to give higher weightage to high
engagement as follows:
Ci,j =(exp(β·pi,j )
Pk6=iexp(β·pi,k)if i6=j
0 if i=j(9)
Although both Aand Cmatrices look similar, there are some fundamental
differences between the interpretation of the scores. The value Ai,j represents
the total proportion of engagement given by ithat was directed at j. In contrast,
the value Ci,j represents the engagement given by ito j, as a proportion of the
total engagement received by jfrom all others. Hence, imay have given all its
strokes to j, but if jwere to be receiving a lot of strokes from several others, its
citizenship contribution would still be low, while its contribution to the authority
of iwould be high.
As with the authority equation, the term βis the scaling parameter that rep-
resents point of valuable participation through engagement. It controls the point
at which the creator believes that the reader is engaged. βtherefore depends on
the measure of engagement being used, the kind of content on the network and
the network structure. Similar to λ, a higher median value of engagement should
reflect a higher β. Thus βwould be smaller for a Twitter-like network than for
a video network.
Similar to A,Cis also strongly connected, aperiodic and irreducible, there-
fore, a convergence point for the network’s belief exists at:
C∗= lim
i→∞ Ci(10)
and the left diagonal of C∗gives us the Citizen Rank (CR) for the social
network as:
CR(i) = C∗×1(11)
Here, C, like A, describes a non-Ergodic system. Therefore, we search the
space [1,∞) and settle for a value of βfor which subsequent increase in value
doesn’t change the CR values. This stable value of βchanges across different
social networks and different measures of engagement. For a network, the stable
point of βvalue indicates whether for the community the level of engagement
is valuable compared to the number of interactions. A small βrepresents that
more interactions are better for network maintenance over deep engagement. A
large stable value of βmeans that rich meaningful, highly engaging interactions
are preferred.
The complete algorithm for computing Authority Rank and Citizen Rank
from engagement values along with the ideal λand βvalues is described in
Algorithm 1 in the Appendix section.
Social Capital as Engagement and Belief Revision 9
4 Experiments
Our measure of engagement is based on the amount of time a consumer spends on
a creator either reading, viewing or responding to their works. We design several
experiments to capture user engagement in different online social networks and
use this data to show the value derived from using the AR and CR metrics. We
also compare our engagement model across different networks as well as with
other importance metrics like PageRank.
4.1 Gratia
The proposed model were tested on two online social environments, to which
we had access. The first was called Gratia, which is an academic pre-print man-
agement system1, where users can share pre-prints of their papers as well as
read pre-prints of others’ technical reports and papers, in an online reader. We
capture the amount of time spent by the users viewing each paper, which gives
us an estimate of the sustained attention and hence, the engagement received by
the said paper. We have designed mechanisms to attribute the attention received
by papers to its creators.
The portal revolves around resources in the form of PDF documents. Each
resource has one or more “creators” who are the owners of that resource and
any engagement with that resource will be attributed to them.
Each resource is also indexed with one or more topical tags representing its
contents. When users view resources, the portal captures the amount of time
users spend on engaging with a resource. This amount of time is then attributed
to the creators of that resource by organising them as strokes as defined earlier. In
the experimental models, the total amount of engagement received by a resource,
was divided equally among all its creators.
The time spent is tracked on the server side, with restrictions in place to
ensure that a particular user can have only one resource open at a time. It is
also ensured that the amount of time being counted towards engagement is only
when the user is actively interacting with the open resource. This prevents the
system from being gamed by false engagement attacks.
For the experiments, the data was collected from the activities of 68 users.
A total of 1534 minutes of time were spent by users, collectively, engaging with
15 users across 23 resources leading to a graph made up of 253 strokes.
4.2 CircuitVerse
The second online portal where the proposed model was tested, is called Cir-
cuitVerse2. CircuitVerse is an easy to use digital logic circuit simulator which
provides a platform to create, share and learn digital circuits.
1https://goodwill.zense.co.in
2https://circuitverse.org/
10 Gaurav Koley, Jayati Deshmukh, Srinath Srinivasa
For our data collection procedure, the same backend and data model as for
Gratia was used. For the experiments, data was collected over a sample of 711
users. This led to 1540 strokes across 554 resources. A total of 343 users spent
6928 minutes engaging with 456 users.
4.3 Twitter
The proposed model was also tested on Twitter. Here, tweets were scraped for
a particular hashtag and then for each twitter handle, a retweet or mention was
taken as 1 stroke. For the experiments, data was collected over a sample of 2251
users. This dataset consisted of 6278 strokes over 6873 sampled tweets.
(a) Gratia - λ= 51 (b) CircuitVerse - λ= 35
(c) Sampled Twitter net-
work network - λ= 4
Fig. 1: Authority Ranks for networks, sorted in decreasing order along with its
stable λvalue
(a) Gratia - β= 46 (b) CircuitVerse - β= 42
(c) Sampled Twitter net-
work - β= 9
Fig. 2: Citizen Ranks for networks, sorted in decreasing order along with its
stable βvalue
5 Results
Figure 1 shows the plots of Authority Rank across the different social networks.
We observe that CircuitVerse has a much more skewed distribution of AR values
than Gratia, while Gratia took much longer for its λvalue to stabilize. This in-
dicates that the Gratia community tended to spend more time on its resources,
than CircuitVerse. The skewed nature of CircuitVerse indicates the formation of
few authorities or participants who generated a high impact across the commu-
nity.
Figure 2 shows the plots of Citizen Rank for the networks. Even here, Cir-
cuitVerse has a much more skewed distribution of CR values than Gratia.
Social Capital as Engagement and Belief Revision 11
(a) Gratia - λ= 10 (b) Gratia - λ= 51
(c) CircuitVerse - λ= 10 (d) CircuitVerse - λ= 35
Fig. 3: PageRank and Authority Ranks comparison with low and stable λvalues
5.1 Comparing across Networks
Hypothesis H1:Average value of engagement is correlated to stable λvalue
across different networks.
Table 1 shows the correlation seen between average time per stroke and
the number of iterations it took for λto stabilize. Gratia has a higher average
time per stroke/interaction and also has the highest λvalue. Comparatively,
CircuitVerse has a lower average time and also a much lower stable λvalue.
This suggests that Gratia’s users are more engaged with the other user’s they
interact with than CircuitVerse’s users.
Inference:The stabilized values of λcan be seen as an indicator of the levels
of engagement present in the population.
5.2 Comparing with PageRank
Hypothesis H2:Authority Ranks closely mimic PageRank for λvalues which
are much smaller than the stable λ, but diverge from PageRank as the number
of iterations increase.
Figures 3a and 3b compare the PageRank for a user to their Authority Rank
for the network of users of Gratia. When we supply a low value for λ= 10 in
Figure 3a, the AR values closely mimic the PageRank values for all the users.
Table 1: Comparing λacross networks
Network Avg Time per stroke stable λ
Gratia 6.06 51
CircuitVerse 4.49 35
Twitter – 4
12 Gaurav Koley, Jayati Deshmukh, Srinath Srinivasa
The exact values might be different but the relative distribution is similar. On
the other hand, for the observed stable value of λ= 51 in Figure 3b, the AR
values follow a very different trend than the PageRank.
Several users having the same PageRank, have different AR; while some users
with low PageRank have a high AR and some other users have a high PageRank
but low AR. This shows the stark difference between PageRank and AR. Users
with high AR have engaged audiences who pay a lot of sustained attention, while
PageRank only indicates total attention– which could very well be from a large
audience most of who give only cursory attention. PageRank fails to make the
distinction between sustained and cursory attention, whereas AR does.
A similar trend can be seen for CircuitVerse. For a low value for λ= 10 in
Figure 3c, the AR values closely mimic the PageRank values for almost all the
users, whereas, for the observed stable value of λ= 35 in Figure 3d, most users
with high AR have very low PageRank and several users with high PageRank
have low AR. This indicates several local authorities with a loyal audience, that
are usually masked when PageRank is computed across the population. Some
specific examples, in Fig 3d with a stable value of λ: PR(user 1) = 0.07 and
AR(user 1) 0.00 whereas PR(user 531) = 0.005 and AR(user 531) = 0.25. Here
User 531 doesn’t have many followers in the network (hence the low PageRank)
but receives a lot of engagement from their followers (high AR). Conversely,
User 1 has a many followers but doesn’t get much engagement from them (hence
the low AR but high PageRank value).
Inference:Hence, while AR starts off similar to PageRank, it diverges from
it to reveal significantly different semantics, by the time its λvalues stabilize. AR
computations can be used not only to get a sense of the level of engagement in
a given social network, but also local authorities with a loyal engaged audience,
within the population.
6 Conclusions
To the best of our knowledge, the concept of social capital has lacked compu-
tational underpinnings, particularly in the field of social network analysis. This
work is an effort to address this gap and model social capital in terms of engage-
ment and belief revision.
Tracking the dynamics of engagement can help us understand not just how
communities are formed, but also how influence and tractable social changes
happen in populations. The proposed model provides quantifiable metrics like λ
and β, as well as the AR and CR scores, using which, we can reason about levels
of engagements and their underlying dynamics.
Acknowledgments.
We thank Prof. Sridhar Mandyam K (IIIT Bangalore) for his help and work on
Belief revision model; Dr. Prasad Ram (Gooru) whose work inspired AR and CR
metrics; Anshumaan Agrawal (IIIT Bangalore) for countless fruitful discussions
and Satvik Ramaprasad (CircuitVerse) for providing data from CircuitVerse.
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A Appendix: Algorithm for Computing Authority and
Citizen Ranks
Social Capital as Engagement and Belief Revision 15
Algorithm 1: Computing AR and CR using Engagement
1let T be a |V| × |V|matrix of engagement.
2T[u][v] = Pu→v∀strokes between u and v; u, v ∈V
3A=T,C=T>
4for i←1to |V|do
5for j←1to |V|do
6A[i][j] = A[i][j]
PjA[i][j]
7C[i][j] = C[i][j]
PjC[i][j]
8end
9end
10 AR old = [], C R old = []
11 for λ←1to ∞do
12 for i←1to |V|do
13 for j←1to |V|do
14 if i== jthen
15 A[i][j] = 0
16 else
17 A[i][j] = exp(λ·A[i][j])
Pk6=iexp(λ·A[i][k])
18 end
19 end
20 end
21 A∗= limi→∞ Ai
22 AR =diagonal(A∗)
23 if AR =AR old then
24 break
25 else
26 AR old =AR
27 end
28 end
29 for β←1to ∞do
30 for i←1to |V|do
31 for j←1to |V|do
32 if i== jthen
33 C[i][j] = 0
34 else
35 C[i][j] = exp(β·C[i][j])
Pk6=iexp(C[i][k])
36 end
37 end
38 end
39 C∗= limi→∞ Ci
40 CR =diagonal(C∗)
41 if CR =C R old then
42 break
43 else
44 CR old =CR
45 end
46 end
