Modelling and Analysing Social Networks through Formal Methods
and Heuristic Searches
Antonella Santone
1
and Gigliola Vaglini
2
1
Dipartimento di Ingegneria, University of Sannio, Benevento, Italy
2
Dipartimento di Ingegneria della Informazione, University of Pisa, Pisa, Italy
Keywords:
Social Networks, Formal Methods, Heuristic Searches.
Abstract:
The paper presents a process algebraic approach to formal specification and verification of social networks.
They are described using the Calculus of Communicating Systems and we reason and verify such formal sys-
tems by using directed model checking, which uses AI-inspired heuristic search strategies in order to improve
model checking techniques.
1 INTRODUCTION
Recently, great interest arose for analysis methods of
complex social networks, from communication net-
works, to friendship networks, to professional and or-
ganisational networks. A social network is a set of
people (or organizations or other social entities) con-
nected by a set of social relationships, such as friend-
ship, co-working or information exchange. Social
network analysis focuses on the analysis of patterns of
relationships among people, organizations, states and
such social entities. Social network analysis provides
both a visual and a mathematical analysis of human
relationships. The Web can also be considered as a
social network. Social networks are formed between
Web pages by hyperlinking to other Web pages.
Heuristic search (Pearl, 1984) is one of the classi-
cal techniques in Artificial Intelligence and has been
applied to a wide range of problem-solving tasks in-
cluding puzzles, two player games, and path finding
problems. A key assumption of heuristic search is that
we can assign a utility or cost to each state. This cost
guides the search suggesting the next state to expand;
in this way the most promising paths are considered
first.
Model checking (Clarke et al., 2001) is a method
to formally and automatically verify the correctness
of finite-state concurrent and distributed systems. As
model checking can be seen as a search in a state
space, heuristics can be exploited to explore state
spaces and verify properties. This approach is known
as directed model checking (Edelkamp et al., 2001;
Santone, 2003).
In this paper an application of directed model
checking to social networks is presented. First social
networks are described using the Calculus of Com-
municating Systems (CCS) of Milner (Milner, 1989).
Then, we consider some interesting properties of so-
cial network and we show how these properties can
be expressed in a temporal logic and verified using
either model checking or heuristic search. As an ex-
ample of heuristic search, we analyse the social dis-
tance: “how far (in terms of social distance) an actor
is from others”. In fact, the connections of an actor’s
social neighbours can be very important, even if the
actor is not directly connected to them. An admissi-
ble heuristic function is defined which is syntactically
defined, i.e., based on the CCS specification only, and
can be automatically computed. With an admissible
heuristic function, the A* algorithm is guaranteed to
find the minimal social distance. This is a prelimi-
nary work to formalise a social network in a simple
way. Our interest is in showing how formal methods
can be applied in this field. As future work we want
to enrich this model to capture more aspects.
2 THE PROPOSED METHOD
The term network has different meanings in different
disciplines. In the social sciences, a network is usu-
ally defined as a set of actors (or agents, or nodes,
or vertices) that may have relationships (or links, or
edges, or ties) with one another see Fig. 1, which is a
running example of this paper.
336
Santone A. and Vaglini G..
Modelling and Analysing Social Networks through Formal Methods and Heuristic Searches.
DOI: 10.5220/0004068903360339
In Proceedings of the 7th International Conference on Software Paradigm Trends (ICSOFT-2012), pages 336-339
ISBN: 978-989-8565-19-8
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

A
B
C
a
a
bc
Figure 1: A simple example of network.
The two most common ways of representing social
network data are by drawing the network and by us-
ing matrices. This section presents a representation of
social networks through CCS and interesting proper-
ties are formalised by using the selective mu-calculus
logic (Barbuti et al., 1999). The purpose is using for-
mal verification environments also to analyse social
networks. In this preliminary work social networks
are formalised in a simple way, as discussed below,
as future work the model can be enriched to capture
more aspects. Each node N, with k outgoing arcs, is
represented as the following CCS process:
N
def
=
k
∑
i=1
n.M
i
where M
i
, for i ∈ [1..k], are the immediate successors
of N. Note that all arcs of a node N are labelled with
the corresponding lower-case letter n.
For example, the CCS process rep-
resenting the network in Fig. 1 is:
A
def
=
a.B+ a.C B
def
=
b.C C
def
=
c.B.
Clearly, many networks can be composed using
the CCS parallel operator “|”.
2.1 Basic Properties of Networks
In this section we first recall some important proper-
ties of networks and we formalise them using selec-
tive mu-calculus logic.
Size of a network. The size of a network can be
determined in terms of the number of nodes of the
network (as stated in (Hanneman and Riddle, 2005))
or, alternatively, as the number of edges in the net-
work. In the network of our running example (shown
in Fig. 1) the number of nodes is 3, while the number
of edges is 4. Using any verification environment it is
sufficient to exploit specific commands and we obtain
the number of both states and edges.
Social distance. The connections of an actor’s so-
cial neighbours can be very important, even if the ac-
tor is not directly connected to them. In other words,
sometimes being a friend of a friend may be quitecon-
sequential. To capture this aspect of how individuals
are embedded in networks, one approach is to exam-
ine how far (in terms of social distance) an actor is
from others. The distance between two actors is the
minimum number of edges that takes to go from one
to another.
Reachability. Reachability between nodes is es-
tablished by the existence of a path between the
nodes. In simpler words, an actor is reachable by an-
other if there exists a set of connections by which we
can go from the source to the target actor, regardless
of how many others fall between them. In general, the
property: “node Y is reachable from node X” can be
expressed with the following logic formula:
hxi
/
0
hyi
/
0
tt (1)
On the other hand, the property: “node Y is reachable
from node X, without crossing nodes in the set S” can
be expressed with the following logic formula:
hxi
/
0
hyi
S
tt (2)
Recall the network shown in Fig. 1:
• “C is reachable from A”. The logic formula is:
ψ
1
= hai
/
0
hci
/
0
tt. It holds that A |= ψ
1
(instantia-
tion of (1)).
• “it is possible that C is reachable from A not
crossing node B. The logic formula is: ψ
2
=
hai
/
0
hci
{b}
tt. It holds that A |= ψ
2
(instantiation
of (2)).
Cycle. A cycle is a walk where the beginning and end
point of the walk are the same actor. In general, the
property: “a cycle on actor X” can be expressed with
the following logic formula:
νZ. hxi
/
0
Z (3)
We formally check, on the CCS process A, the follow-
ing property, which is an instantiation of the previous
formula (3):
• “there exists a cycle on actor B”: ϕ
1
= νZ. hbi
/
0
Z.
It holds that A |= ϕ
1
.
3 HEURISTIC BASED METHOD
FOR CALCULATING SOCIAL
DISTANCE
The problem of finding the minimal distance between
two actors can be seen as a search problem: there-
fore, heuristic search (Pearl, 1984) taken from Artifi-
cial Intelligence can be used. We assume the reader
be familiar with heuristic search algorithms, like A*
and Greedy. A* returns a minimal-cost solution path
whether the heuristic estimate function
b
h satisfies the
so-called admissibility condition, i.e.,
b
h is optimistic.
Now, we define an admissible heuristic function able
ModellingandAnalysingSocialNetworksthroughFormalMethodsandHeuristicSearches
337

to find the minimum social distance from p to the
node N. The heuristic we present is based on counting
the number of actions that a process can perform be-
fore reaching a node N. Remember that in the CCS
formalisation a node N has outgoing arcs labelled
with n. We suppose that each edge has cost equal to
1. We formally define the function
b
h
n
(p).
Definition 1 (
b
h
n
(p)). Let p be a CCS process, S , S
′
sets of visible actions, n the visible action of the node
N to reach, and C a set of pairs {hx,S
′
i}, where x is
a constant occurring in p. First, we define the auxil-
iary function
b
h
n
(p,S ,C ), inductively on p, as follows.
Then,
b
h
n
(p) =
b
h
n
(p,
/
0,
/
0).
R1.
b
h
n
(nil, S , C ) = ∞.
For p = nil the function
b
h
n
returns ∞ as this is not the
node N.
R2. if α ∈ S ∪ {n} then
b
h
n
(α.p,S ,C ) = 0
else
b
h
n
(α.p,S ,C ) = 1+
b
h
n
(p,S ,C ).
When applied to α.p the function returns 0 if α is ei-
ther a restricted action (i.e., α ∈ S ) or the desired ac-
tion n, otherwise we recursively apply the function to
find, if any, the node N. Roughly speaking, if α is re-
stricted by S then α.p could not be able to move; thus,
we optimistically return 0.
R3.
b
h
n
(p+ q,S ,C ) = min(
b
h
n
(p,S ,C ),
b
h
n
(q,S ,C )).
R4.
b
h
n
(p|q, S ,C ) = min(
b
h
n
(p,S ,C ),
b
h
n
(q,S ,C )).
When either the choice or the parallel composition of
two processes is encountered the minimum number of
actions between the two components is returned.
R5.
b
h
n
(p\L, S ,C ) =
b
h
n
(p,S ∪ L∪
L,C ).
The function is initially applied to a process with S =
/
0 which is modified when the function is applied to
p\L adding the actions in L∪
L, to S .
R6.
b
h
n
(p[ f],S ,C ) =
b
h
n
(p, f
−1
(S ∪ {n}),C ).
When considering a relabelled process we must take
as set of actions the set f
−1
(ρ) = {α | f(α) ∈ ρ},
since now the interesting actions are also those re-
labelled by f into actions in S or in n.
R7. if x
def
= p
x
, hx,S i ∈ C then
b
h
n
(x,S ,C ) = ∞
else
b
h
n
(x,S ,C ) =
b
h
n
(x,S ,C ∪ {hx,S i})
.
We expand the body of each constant x only once since
each constant already expanded is stored in C . Ini-
tially, C is equal to the empty set. We return ∞ when
we encounter a constant already expanded.
The following theorem ensure the admissibility of
the heuristic function.
Theorem 1. Let p be a CCS process and s be a state.
It holds that:
b
h
n
(s) ≤ h
∗
(s), where h
∗
(s) is the actual
cost of a preferred path from s to a goal node.
4 AN EXAMPLE
Let us apply our approach to a social network to ob-
tain the minimal social distance between two actors.
The node expansion terminates when a goal node is
reached. The CCS definition of the social network p
is the parallel composition of two sub-nets A and X
with synchronisation on the action c.
A
def
= a.B B
def
= b.F + b.C C
def
= c.D+ c.H + c.E
H
def
= h.I F
def
= f.M M
def
= m.N
N
def
= n.Q Q
def
= q.E D
def
= d.C
I
def
= i.E E
def
= e.I X
def
= x.Z
Z
def
= z.E p
def
=(A|X[
c/z])\{c}
(A | X [c/z ])\{c}
a x
b
(B | X [c/z ])\{c} (A | Z [c/z ])\{c}
(B | Z [c/z ])\{c}
(F | Z [c/z ])\{c}(C | Z [c/z ])\{c}
(D | E [c/z ])\{c}
(H | E [c/z ])\{c} (E | E [c/z ])\{c}
(D | I [c/z ])\{c}
a
b
τ
e
τ τ
=1
h
e
^
=0
h
e
^
=0
h
e
^
=0
h
e
^
=0
h
e
^
=0
h
e
^
=0
h
e
^
=0
h
e
^
Figure 2: A simple example with Greedy and
b
h
e
.
Suppose that we want to obtain the minimal social
distance between A and E. We apply the Greedy strat-
egy. The application of A
∗
is immediate: it is suffi-
cient to use the evaluationfunction f(s) = g(s)+
b
h(s).
In Fig. 2 the
b
h
e
-value of each node is reported and the
shaded nodes represent expanded states. It holds that
actor E can be reached with the minimal path with
length equal to 5 (i.e. x− a− b− τ− e). There exists
another path reaching actors E with length equal to 7.
We have obtained this result generating only 10 states
while the standard transition system of p has 24 states
and 39 transitions.
The method has been successfully applied to a real
case study where several functionalities of social net-
works have been modelled such as: registration, add
as friend, news feed.
ICSOFT2012-7thInternationalConferenceonSoftwareParadigmTrends
338

5 RELATED WORK AND
CONCLUSIONS
The paper propose several contributions: (i) an alge-
braic description of social networks through CCS; (ii)
the definition of several properties of social networks
(minimum distance, reachability, cycles) using a tem-
poral logic, so that they can be verified trough model
checking; (iii) an admissible heuristic function to ver-
ify social distance easy to compute and automatically
calculated. A consequence of point (i) is that all the
efficient techniques developed in CCS verification en-
vironments can be used, as for example compositional
analysis (Clarke et al., 1989; Santone, 2002). More-
over, the result of point (ii) is that we offer a query
language based on temporal logic instead of just a set
of fixed properties; thus the approach allows the spec-
ification and verification of any interesting property
(also not built-in in traditional tools).
At present, the most popular tool for Network
analysis is UCINET
1
based on mathematical opera-
tions on matrices. For example, to prove the prop-
erty “X is reachable from Y not crossing node Z”,
UCINET must found all paths from Y to X and check
that Z does not occur in any path. In the presented
approach the property is defined in the temporal logic
and then automatically verified in the model checking
environment.
To evaluate the effectiveness of the proposed
method, we have to consider both complexity and
scalability. From the complexity point of view, it is
easy to show that the complexity of the calculation of
b
h(n) is linear in the length of the CCS specification.
The heuristic function is simple to calculate since it
is syntactically defined, i.e., based on the CCS syntax
only; moreover, there is no need for user intervention
or manual efforts to compute it. From the scalabil-
ity point of view, in the work (Gradara et al., 2005)
an experimental study has been carried out to prove
the better scalability of the heuristic-based method
with respect to other techniques generally used (BFS,
DFS, etc.). Even if in that work a different property
(deadlock-freeness in concurrent systems) has been
considered, the results can be useful for a valid com-
parison. In some case studies a reduction of the state
space of 55% has been reached and a consequently
reduction in time with respect to BFS, for instance.
Moreover, while it is well-known that most current
formal methods are successfully applicable to small-
scale systems, but do not scale up well, in this paper,
directed model checking proposed in (Santone, 2003)
has been proved able to allow formal methods to scale
1
http://www.analytictech.com/ucinet/
up.
Recently, the authors in (He et al., 2007) also pro-
pose a process algebraic approach to modeling and
verifying the collective behaviors in social networks
using MWB (Victor and Moller, 1994) with no con-
sideration of the scalability. In (Jamali and Abolhas-
sani, 2006) a state of the art survey of the works done
on social network analysis, ranging from pure mathe-
matical analyses in graphs to analyzing the social net-
works in Semantic Web, is given. The main goal is to
provide a road map for researchers working on differ-
ent aspects of Social Network Analysis.
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