Financial Market - A Network Perspective

Abstract

We construct a weighted financial network for a subset of NYSE traded stocks, in which the nodes correspond to stocks and edges to interactions between them. We identify clusters of stocks in the network, based on the Forbes business sector classification, and study their intensity and coherence. Our approach indicates to what extent the business sector classifications are visible in market prices, enabling us to gauge the extent of group-behaviour exhibited by stocks belonging to a given business sector.

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Financial Market - A Network Perspective
Jukka-Pekka Onnela1, Jari Saram ¨aki1, Kimmo Kaski1, and J ´anos Kert´esz2
1Laboratory Computational Engineering, Helsinki University of Technology, P.O.Box 9203,
FIN-02015 HUT. jonnela@lce.hut.fi
2Department of Theoretical Physics, Budapest University of Technology and Economics,
Budafoki ´ut 8, H-1111 Budapest, Hungary.
We construct a weighted financial network for a subset of NYSE traded stocks, in
which the nodes correspond to stocks and edges to interactions between them. We
identify clusters of stocks in the network, based on the Forbes business sector clas-
sification, and study their intensity and coherence. Our approach indicates to what
extent the business sector classifications are visible in market prices, enabling us to
gauge the extent of group-behaviour exhibited by stocks belonging to a given busi-
ness sector.
1 Introduction
Complex networks provide a very general framework, based on the concepts of sta-
tistical physics, for studying systems with large numbers of interacting agents [1].
The nodes of the network represent the agents and a link connecting two nodes indi-
cates an interaction between them. In the complex networks framework, interactions
have typically been considered to be binary in nature, meaning that either two nodes
interact (are connected) or they do not (are not connected). Imposing a binary interac-
tion requires setting a threshold value for interaction strength, such that interactions
falling below it are discarded. Although this approach is a suitable first approxima-
tion, thresholding can lead to a loss of information. Consequently, a natural step
forward is to assign weights on the links to reflect the strengths of interactions.
In a financial market the performance of a company is compactly characterised
by a single number, the stock price, which results from a large number of interactions
between different market participants. Although the exact nature of these interactions
is not known, they are certainly reflected in the equal-time return correlations. In this
paper we study a financial network in which the nodes correspond to stocks and links
to return correlation based interactions between them. Mantegna [2] was the first to
construct such networks and the idea was followed and extended by others [3,4, 5,
6, 7].

2 Jukka-Pekka Onnela, Jari Saram¨aki, Kimmo Kaski, and J´anos Kert´esz
2 Methods
2.1 Constructing the Network
We start by considering a price time series for a set of Nstocks and denote the daily
closing price of stock iat time τ(an actual date) by Pi(τ). Since investors work in
terms of relative as opposed to absolute returns, logarithmic returns are commonly
used in studies, and thus we denote the daily logarithmic return of stock iby ri(τ)=
ln Pi(τ)−ln Pi(τ−1). We extract a time window of width T, measured in days and
in this paper set to T=1000 (equal to four years, assuming 250 trading days a year),
and obtain a return vector rt
ifor stock i, where the superscript tenumerates the time
window under consideration. Then equal time correlation coefficients between assets
iand jcan be written as
ρt
i j =
"rt
irt
j# − "rt
i#"rt
j#
!["rt
i
2# − "rt
i#2]["rt
j
2# − "rt
j#2]
,(1)
where "...#indicates a time average over the consecutive trading days included in
the return vectors. These correlation coefficients between Nassets form a symmetric
N×Ncorrelation matrix Ctwith elements ρt
i j . The different time windows are
displaced by δT, where we have used a step size of one week, i.e. δT=5 days.
Next we define interaction strengths, or link weights, based on the correlation
coefficients. One of the simplest alternatives is to use the absolute values of the cor-
relation coefficients, in which case the interaction strength reflects the strength of
linear coupling between the logarithmic returns of stocks iand jin time window t.
If we use wt
i j to denote the weight on the link connecting node iand node j, with this
choice we have wt
i j = |ρt
i j |,or in matrix form Wt= |Ct|. Because the correlation
coefficients ρt
i j vary between −1 and 1, the interaction strengths wt
i j are naturally
limited to the [0,1]interval. In the correlation matrix Ctwe have estimated the cor-
relations between all the assets. Thus, the resulting network will be fully connected
consisting of Nnodes and N(N−1)/2 links, corresponding to the elements in the
upper (or lower) triangular part of the the weight matrix.3
2.2 Characterising Network Clusters
Let us now consider any cluster or subgraph gin the above defined network. To
characterise how compact or tight the subgraph is, we use the concept of subgraph
intensity I (g)introduced in [8]. Put differently, subgraph intensity allows us to char-
acterise the interaction patterns within clusters. If we use vgto denote the set of nodes
and $gthe set of links in the subgraph with weights wi j , we can express subgraph
intensity as the geometric mean of its weights:
3It is possible, using some heuristic, to insert only a fraction of all the links in the network,
but this would result in an additional parameter to be determined.

Financial Market - A Network Perspective 3
I(g)=
$
(ij)∈!g
wi j 

1/|!g|
.(2)
Due to the nature of the geometric mean, the subgraph intensity I(g)may be
low because one of the weights is very low, or it may result from all of the weights
being low. In order to distinguish between these two extremes, we use the concept of
subgraph coherence Q(g)[8]. It assumes values from the interval [0,1]and is close
to unity only if the subgraph weights do not differ much, i.e. are internally coherent.
Subgraph coherence is defined as the ratio of the geometric to the arithmetic mean
of the weights as
Q(g)=I|$g|/'
(ij)∈!g
wi j .(3)
In order to compare intensity and coherence values, we need to establish a refer-
ence. A very natural reference system is obtained by considering the entire market.
In other words, we take all of the Nnodes and N(N−1)/2 links making up the net-
work G, and then using the above definitions compute I(G)and Q(G). We can also
use relative cluster intensity for cluster g,given by I(g)/I(G), and relative cluster
coherence,given by Q(g)/Q(G), if instead of absolute values we wish to examine
the cluster intensity or coherence relative to the reference system.
3 Results
In this section we consider a subset of 116 NYSE-traded stocks from the S&P 500
index from 1.1.1982 to 31.12.2000. We deal with the closing price, resulting in a total
of 4787 price quotes for each stock. To divide the stocks into clusters, we obtained
the Forbes business sector labels for each stock [9]. The stocks in our dataset fall into
12 business sectors, such as Energy and Utilities. Given these labels for each stock,
we use the concepts of subgraph intensity and coherence to gauge howhow similarly
stocks belonging to a given business behave as a function of time.
Let us consider a cluster g, constructed such that all of its nodes vgbelong to
the same business sector, and let ndenote the number of nodes in this cluster. Then
we add all the n(n−1)/2 links corresponding to the interaction strengths between
any pair of nodes within g.In one extreme, if all the link weights are equal to unity,
every node participating in ginteracts maximally with its n−1 neighbours. In the
other extreme, if one or more of the weights are zero, the subgraph intensity for the
fully connected subgraph gntends to zero because the original topological structure
no longer exists.
In Figure 1, we show the relative cluster intensity as a function of time for se-
lected business sector clusters. Values above unity indicate that the intensity of the
cluster is higher than that of the market. This implies that in most cases stocks be-
longing to a given business sector are tied together in the sense that intra-cluster in-
teraction strengths are considerably stronger than those of the market on the whole.

4 Jukka-Pekka Onnela, Jari Saram¨aki, Kimmo Kaski, and J´anos Kert´esz
It is also worth noting the high value for the absolute cluster intensity for the mar-
ket roughly between 1986 and 1990. This elevated value is due to the 1987 stock
market crash (Black Monday), which caused the market to behave in a unified man-
ner4. The crash also compresses the relative cluster intensities, which means that
the cluster-specific behaviour is temporarily suppressed by the crash, and after the
market recovers the clusters regain their characteristic behaviour.
1984 1986 1988 1990 1992 1994 1996 1998 2000
0.5
1
1.5
2
2.5
3
3.5
Time
Relative cluster intensity
Basic Materials
Conglomerates
Energy
Financial
Utilities
Market
1984 1988 1992 1996 2000
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time
Cluster intensity
Fig. 1. Relative (to the market) cluster intensity as a function of time for select clusters. Inset:
The (absolute) cluster intensity for the market used for normalisation.
Business sector clusters are also more coherent than the market, as shown in
Figure 2, except for Basic Materials. One explanation is obtained from the industry
classifications, which is a finer classification scheme, of stocks comprising the BM
cluster. These include Metal Mining, Paper, Gold & Silver and Forestry & Wood
Products. Therefore, it is clear that the Basic Materials business sector is extremely
diverse. Also, the price of some of these items is determined, at least partially, out-
side the stock market. Consequently, it is not so surprising that the cluster intensity
remains low, at times even falling below the market reference. Similarly, the low co-
herence values indicate that there are stocks in this cluster with very high correlations
(those belonging to the same industry, such as gold mining), but also very low (com-
panies belonging to different industries). In conclusion, our results indicate that, in
most cases, stocks belonging to the same business sector have higher intensity and
more coherent intra-cluster than inter-cluster interactions.
4The length of this elevated period is related to the window width parameter.

Financial Market - A Network Perspective 5
1984 1986 1988 1990 1992 1994 1996 1998 2000
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Time
Relative cluster coherence
Basic Materials
Conglomerates
Energy
Financial
Utilities
Market
Fig. 2. Relative (to the market) cluster coherence as a function of time.
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9. The website of Forbes at www.forbes.com