RESEARCH ARTICLE
Topology of Innovation Spaces in the
Knowledge Networks Emerging through
Questions-And-Answers
Miroslav Andjelković1, Bosiljka Tadić2*, Marija Mitrović Dankulov3, Milan Rajković1,
Roderick Melnik4,5
1 Institute of Nuclear Sciences, Vinča, University of Belgrade, Belgrade, Serbia, 2 Department of Theoretical
Physics, Jožef Stefan Institute, Ljubljana, Slovenia, 3 Scientific Computing Laboratory, Institute of Physics
Belgrade, University of Belgrade, Zemun-Belgrade, Serbia, 4 MS2Discovery Interdisciplinary Research
a11111 Institute, M2NeT Laboratory and Department of Mathematics, Wilfrid Laurier University, Waterloo, ON,
Canada, 5 BCAM–Basque Center for Applied Mathematics, E48009 Bilbao, Basque Country–Spain
*
Abstract
OPEN ACCESS
Citation: Andjelković M, Tadić B, Mitrović Dankulov
The communication processes of knowledge creation represent a particular class of human
M, Rajković M, Melnik R (2016) Topology of dynamics where the expertise of individuals plays a substantial role, thus offering a unique
Innovation Spaces in the Knowledge Networks possibility to study the structure of knowledge networks from online data. Here, we use the
Emerging through Questions-And-Answers. PLoS
empirical evidence from questions-and-answers in mathematics to analyse the emergence
ONE 11(5): e0154655. doi:10.1371/journal.
pone.0154655 of the network of knowledge contents (or tags) as the individual experts use them in the pro-
cess. After removing extra edges from the network-associated graph, we apply the methods
Editor: Matjaz Perc, University of Maribor,
SLOVENIA of algebraic topology of graphs to examine the structure of higher-order combinatorial
spaces in networks for four consecutive time intervals. We find that the ranking distributions
Received: February 16, 2016
of the suitably scaled topological dimensions of nodes fall into a unique curve for all time
Accepted: April 15, 2016
intervals and filtering levels, suggesting a robust architecture of knowledge networks. More-
Published: May 12, 2016 over, these networks preserve the logical structure of knowledge within emergent communi-
Copyright: © 2016 Andjelković et al. This is an open ties of nodes, labeled according to a standard mathematical classification scheme. Further,
access article distributed under the terms of the we investigate the appearance of new contents over time and their innovative combinations,
Creative Commons Attribution License, which permits
which expand the knowledge network. In each network, we identify an innovation channel
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are as a subgraph of triangles and larger simplices to which new tags attach. Our results show
credited. that the increasing topological complexity of the innovation channels contributes to net-
Data Availability Statement: The complete data are work’s architecture over different time periods, and is consistent with temporal correlations
available on Stack Exchange site Mathematics http:// of the occurrence of new tags. The methodology applies to a wide class of data with the suit-
math.stackexchange.com/. The exact data set which able temporal resolution and clearly identified knowledge-content units.
is used in this study is deposited into Figshare:
https://figshare.com/articles/MathQuestions_data/
3153145/MathQuestions.data.
Funding: This work was supported by Ministry of
Education, Science, and Technological Development
of the Republic of Serbia (http://www.mpn.gov.rs/), Introduction
under the projects ON174014, ON171017; Research
Agency of the Republic of Slovenia (https://www.arrs. The knowledge creation through online social interactions represents an emerging area of
gov.si/en/agencija/), under the Program P1-0044; and increased interest both for technological advances and the society [1] where the collective
PLOS ONE | DOI:10.1371/journal.pone.0154655 May 12, 2016
, Knowledge Networks Topology and Innovation
Natural Sciences and Engineering Research Council knowledge is recognised as a social value [2–4]. Recently studied examples include the knowl-
of Canada, (http://www.nserc-crsng.gc.ca/), under the edge accumulation in systems with direct questions-and-answers [5], crowdsourcing scientific
project code 213904.
knowledge production [6, 7] and scientific discovery games [8]. Similar phenomena can be
Competing Interests: The authors have declared observed in business/economics-associated online social networking [9–11]. On the other
that no competing interests exist. hand, the study of the collective knowledge creation opens new topics of research interests. In
particular, it provides ground to examine a novel type of collective dynamics in social systems
in which each actor possesses certain limited expertise. In the course of the collaborative social
efforts to solve a problem, such as communications through questions-and-answers that we
consider here, the tacit knowledge and the expertise of individual actors are externalised and
dynamically shared with other participants who take part in the process. When a systematic
tagging applies to the shared cognitive contents, the process leads to an explicit knowledge [3]
as the output value (the network of knowledge contents), from which others can learn. Further-
more, the dynamics underlying knowledge creation exemplifies multi-scale phenomena related
to the cognitive recognition, which may occur in a wider class of systems, social, biological and
physical [17].
By the nature of the underlying stochastic processes, the knowledge networks that emerge
through the collaborative social endeavours necessarily reflect the expertise and the activity
patterns of the involved participants. Furthermore, these networks tend to capture the logical
relationship among the used cognitive contents as it resides in the mind of each participating
individual. In this regard, these networks substantially differ from the commonly studied
knowledge networks, which are produced in ontological initiatives [12–14] such as those from
the online bibliographic data and Wikipedia, or the mapping citation relationships between
journal articles [15], to name a few. Also, the stochastic process of knowledge creation through
questions and answers are different from the spreading dynamics of scientific memes, whose
inheritance patterns are identified in citation networks [16].
In recent work [5], we have shown that the knowledge creation by questions-and-answers
involve two-scale dynamics, in which the constitutive social and cognitive elements (individual
experts or actors and the knowledge contents that they use) interact and influence each other
on the original scale. This complex system evolves in a self-organised manner leading to the
emergence of socio-technological structures where the involved actors share the accumulated
knowledge. These structures are visualised as communities on the related bipartite network of
actors and their artefacts [5]. Furthermore, the advance of innovation in this process, which
builds on the expertise of the involved participants, leads to the expansion of the knowledge
space by adding new cognitive contents. The central question for the research and applications
of the collective knowledge creation is how these stochastic processes work and potentially can
be controlled to converge towards the desired outcome. Furthermore, what is the structure of
the emergent knowledge that can be used by others?
A part of the answer relies on the structure of the networks, co-evolving with the knowl-
edge-sharing processes among the actors possessing the required expertise. In [5] the empirical
data from the Stack Exchange site Mathematics (http://math.stackexchange.com/) were down-
loaded and analysed, as a prototypal example. The sequence of events in the process of ques-
tions-and-answers (Q&A) suitably maps onto a growing bipartite network of actors, as one
partition, and their questions and answers, as another partition. The emergent communities on
these networks have been identified, consisting of the involved actors and the connected ques-
tions-and-answers. As a rule, in each community a dominant actor is found, representing an
active user with a broad expertise. The knowledge elements of each question are specified
according to the standard mathematical classification scheme by one to five tags (for instance,
“functional analysis”, “general topology”, “differential geometry”, “abstract algebra”, “algebraic
number theory”). Consequently, the expertise of the actor can be specified as a combination of
PLOS ONE | DOI:10.1371/journal.pone.0154655 May 12, 2016
, Knowledge Networks Topology and Innovation
tags that the actor had frequently used. Assuming that a minimal matching applies among the
actor’s expertise and the contents of the answered question, and using theoretical modelling
based on the empirical data, it was shown [5] that the emergent communities and the knowl-
edge that they share strongly depend on the population of the involved experts and their activ-
ity patterns.
In this work, using the same empirical dataset, our focus is on the networks of cognitive ele-
ments (tags) that emerge in these processes with questions-and-answers. Different from the
aforementioned bipartite networks, these emergent knowledge networks contain subelements
of both partitions, namely, knowledge contents of questions as well as a measure of the users’
expertise. Such networks, supported by the current information and computer technology
(ICT) systems, embody the collective knowledge that emerges via the cooperative social efforts
and can be used by others to learn. Moreover, the relevance and speed of knowledge acquisition
from these networks may be more efficient than from the networks generated through wide-
scale ontological plans and efforts. We apply the techniques of algebraic topology of graphs
[18–22] to investigate higher-order structures that characterise the connection complexity
between knowledge elements in the emergent networks. Specifically, we aim to determine
• the metrics to quantify the higher-order combinatorial structures which contain the logical
units of knowledge as the actors use them in communication;
• the role of innovative contents brought over time by the experts in building the network
architecture.
In addition to the standard graph-theoretic metrics and community detection in the emer-
gent networks of knowledge units, we describe their hierarchical organisation using several
algebraic topology measures. Further, we identify the appearance of new tags over time and
investigate the subgraphs (innovation channels) where these new cognitive elements attach to
the existing network. By tracking topology measures over the consecutive time periods for the
innovation channel together with the topology of the entire network, we quantify the impact of
the new-added contents. Our main findings indicate that the networks of cognitive elements
map to a nontrivial hierarchical architecture which contains aggregates of high-order cliques.
The increasing structural complexity of these networks over time, owing to the innovation
expansion, is consistent with the logical structure of knowledge that they contain and temporal
correlations in the appearance of new cognitive contents.
In the following, the networks of tags are built from the empirical data for four successive
one-year periods. At the initial stage, the networks are filtered to remove redundant links. At
the next stage, network measures are obtained at the graph level, and the community structure
is determined. At the final stage, the algebraic topology analysis of these networks for different
periods and filtering levels is performed. The analysis is focused on the subgraphs, which are
related to the appearance of new tags, representing the innovation channels of these networks.
Emergence of the tags networks
The Q&A process and structure of the empirical data
In this work, we have constructed knowledge networks from the empirical data, which are col-
lected and described in Ref. [5]. In the data, the knowledge contents are mathematical tags
used in the communications on Q&A system Mathematics Stack Exchange. In particular, the
content of each question is specified (tagged) by one or more (maximum five) tags according
to the standard mathematical classification scheme. While in Ref. [5] we investigated the role
of expertise in the social process taking part on the co-evolving bipartite network of users-and-
PLOS ONE | DOI:10.1371/journal.pone.0154655 May 12, 2016
Topology of Innovation Spaces in the
Knowledge Networks Emerging through
Questions-And-Answers
Miroslav Andjelković1, Bosiljka Tadić2*, Marija Mitrović Dankulov3, Milan Rajković1,
Roderick Melnik4,5
1 Institute of Nuclear Sciences, Vinča, University of Belgrade, Belgrade, Serbia, 2 Department of Theoretical
Physics, Jožef Stefan Institute, Ljubljana, Slovenia, 3 Scientific Computing Laboratory, Institute of Physics
Belgrade, University of Belgrade, Zemun-Belgrade, Serbia, 4 MS2Discovery Interdisciplinary Research
a11111 Institute, M2NeT Laboratory and Department of Mathematics, Wilfrid Laurier University, Waterloo, ON,
Canada, 5 BCAM–Basque Center for Applied Mathematics, E48009 Bilbao, Basque Country–Spain
*
Abstract
OPEN ACCESS
Citation: Andjelković M, Tadić B, Mitrović Dankulov
The communication processes of knowledge creation represent a particular class of human
M, Rajković M, Melnik R (2016) Topology of dynamics where the expertise of individuals plays a substantial role, thus offering a unique
Innovation Spaces in the Knowledge Networks possibility to study the structure of knowledge networks from online data. Here, we use the
Emerging through Questions-And-Answers. PLoS
empirical evidence from questions-and-answers in mathematics to analyse the emergence
ONE 11(5): e0154655. doi:10.1371/journal.
pone.0154655 of the network of knowledge contents (or tags) as the individual experts use them in the pro-
cess. After removing extra edges from the network-associated graph, we apply the methods
Editor: Matjaz Perc, University of Maribor,
SLOVENIA of algebraic topology of graphs to examine the structure of higher-order combinatorial
spaces in networks for four consecutive time intervals. We find that the ranking distributions
Received: February 16, 2016
of the suitably scaled topological dimensions of nodes fall into a unique curve for all time
Accepted: April 15, 2016
intervals and filtering levels, suggesting a robust architecture of knowledge networks. More-
Published: May 12, 2016 over, these networks preserve the logical structure of knowledge within emergent communi-
Copyright: © 2016 Andjelković et al. This is an open ties of nodes, labeled according to a standard mathematical classification scheme. Further,
access article distributed under the terms of the we investigate the appearance of new contents over time and their innovative combinations,
Creative Commons Attribution License, which permits
which expand the knowledge network. In each network, we identify an innovation channel
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are as a subgraph of triangles and larger simplices to which new tags attach. Our results show
credited. that the increasing topological complexity of the innovation channels contributes to net-
Data Availability Statement: The complete data are work’s architecture over different time periods, and is consistent with temporal correlations
available on Stack Exchange site Mathematics http:// of the occurrence of new tags. The methodology applies to a wide class of data with the suit-
math.stackexchange.com/. The exact data set which able temporal resolution and clearly identified knowledge-content units.
is used in this study is deposited into Figshare:
https://figshare.com/articles/MathQuestions_data/
3153145/MathQuestions.data.
Funding: This work was supported by Ministry of
Education, Science, and Technological Development
of the Republic of Serbia (http://www.mpn.gov.rs/), Introduction
under the projects ON174014, ON171017; Research
Agency of the Republic of Slovenia (https://www.arrs. The knowledge creation through online social interactions represents an emerging area of
gov.si/en/agencija/), under the Program P1-0044; and increased interest both for technological advances and the society [1] where the collective
PLOS ONE | DOI:10.1371/journal.pone.0154655 May 12, 2016
, Knowledge Networks Topology and Innovation
Natural Sciences and Engineering Research Council knowledge is recognised as a social value [2–4]. Recently studied examples include the knowl-
of Canada, (http://www.nserc-crsng.gc.ca/), under the edge accumulation in systems with direct questions-and-answers [5], crowdsourcing scientific
project code 213904.
knowledge production [6, 7] and scientific discovery games [8]. Similar phenomena can be
Competing Interests: The authors have declared observed in business/economics-associated online social networking [9–11]. On the other
that no competing interests exist. hand, the study of the collective knowledge creation opens new topics of research interests. In
particular, it provides ground to examine a novel type of collective dynamics in social systems
in which each actor possesses certain limited expertise. In the course of the collaborative social
efforts to solve a problem, such as communications through questions-and-answers that we
consider here, the tacit knowledge and the expertise of individual actors are externalised and
dynamically shared with other participants who take part in the process. When a systematic
tagging applies to the shared cognitive contents, the process leads to an explicit knowledge [3]
as the output value (the network of knowledge contents), from which others can learn. Further-
more, the dynamics underlying knowledge creation exemplifies multi-scale phenomena related
to the cognitive recognition, which may occur in a wider class of systems, social, biological and
physical [17].
By the nature of the underlying stochastic processes, the knowledge networks that emerge
through the collaborative social endeavours necessarily reflect the expertise and the activity
patterns of the involved participants. Furthermore, these networks tend to capture the logical
relationship among the used cognitive contents as it resides in the mind of each participating
individual. In this regard, these networks substantially differ from the commonly studied
knowledge networks, which are produced in ontological initiatives [12–14] such as those from
the online bibliographic data and Wikipedia, or the mapping citation relationships between
journal articles [15], to name a few. Also, the stochastic process of knowledge creation through
questions and answers are different from the spreading dynamics of scientific memes, whose
inheritance patterns are identified in citation networks [16].
In recent work [5], we have shown that the knowledge creation by questions-and-answers
involve two-scale dynamics, in which the constitutive social and cognitive elements (individual
experts or actors and the knowledge contents that they use) interact and influence each other
on the original scale. This complex system evolves in a self-organised manner leading to the
emergence of socio-technological structures where the involved actors share the accumulated
knowledge. These structures are visualised as communities on the related bipartite network of
actors and their artefacts [5]. Furthermore, the advance of innovation in this process, which
builds on the expertise of the involved participants, leads to the expansion of the knowledge
space by adding new cognitive contents. The central question for the research and applications
of the collective knowledge creation is how these stochastic processes work and potentially can
be controlled to converge towards the desired outcome. Furthermore, what is the structure of
the emergent knowledge that can be used by others?
A part of the answer relies on the structure of the networks, co-evolving with the knowl-
edge-sharing processes among the actors possessing the required expertise. In [5] the empirical
data from the Stack Exchange site Mathematics (http://math.stackexchange.com/) were down-
loaded and analysed, as a prototypal example. The sequence of events in the process of ques-
tions-and-answers (Q&A) suitably maps onto a growing bipartite network of actors, as one
partition, and their questions and answers, as another partition. The emergent communities on
these networks have been identified, consisting of the involved actors and the connected ques-
tions-and-answers. As a rule, in each community a dominant actor is found, representing an
active user with a broad expertise. The knowledge elements of each question are specified
according to the standard mathematical classification scheme by one to five tags (for instance,
“functional analysis”, “general topology”, “differential geometry”, “abstract algebra”, “algebraic
number theory”). Consequently, the expertise of the actor can be specified as a combination of
PLOS ONE | DOI:10.1371/journal.pone.0154655 May 12, 2016
, Knowledge Networks Topology and Innovation
tags that the actor had frequently used. Assuming that a minimal matching applies among the
actor’s expertise and the contents of the answered question, and using theoretical modelling
based on the empirical data, it was shown [5] that the emergent communities and the knowl-
edge that they share strongly depend on the population of the involved experts and their activ-
ity patterns.
In this work, using the same empirical dataset, our focus is on the networks of cognitive ele-
ments (tags) that emerge in these processes with questions-and-answers. Different from the
aforementioned bipartite networks, these emergent knowledge networks contain subelements
of both partitions, namely, knowledge contents of questions as well as a measure of the users’
expertise. Such networks, supported by the current information and computer technology
(ICT) systems, embody the collective knowledge that emerges via the cooperative social efforts
and can be used by others to learn. Moreover, the relevance and speed of knowledge acquisition
from these networks may be more efficient than from the networks generated through wide-
scale ontological plans and efforts. We apply the techniques of algebraic topology of graphs
[18–22] to investigate higher-order structures that characterise the connection complexity
between knowledge elements in the emergent networks. Specifically, we aim to determine
• the metrics to quantify the higher-order combinatorial structures which contain the logical
units of knowledge as the actors use them in communication;
• the role of innovative contents brought over time by the experts in building the network
architecture.
In addition to the standard graph-theoretic metrics and community detection in the emer-
gent networks of knowledge units, we describe their hierarchical organisation using several
algebraic topology measures. Further, we identify the appearance of new tags over time and
investigate the subgraphs (innovation channels) where these new cognitive elements attach to
the existing network. By tracking topology measures over the consecutive time periods for the
innovation channel together with the topology of the entire network, we quantify the impact of
the new-added contents. Our main findings indicate that the networks of cognitive elements
map to a nontrivial hierarchical architecture which contains aggregates of high-order cliques.
The increasing structural complexity of these networks over time, owing to the innovation
expansion, is consistent with the logical structure of knowledge that they contain and temporal
correlations in the appearance of new cognitive contents.
In the following, the networks of tags are built from the empirical data for four successive
one-year periods. At the initial stage, the networks are filtered to remove redundant links. At
the next stage, network measures are obtained at the graph level, and the community structure
is determined. At the final stage, the algebraic topology analysis of these networks for different
periods and filtering levels is performed. The analysis is focused on the subgraphs, which are
related to the appearance of new tags, representing the innovation channels of these networks.
Emergence of the tags networks
The Q&A process and structure of the empirical data
In this work, we have constructed knowledge networks from the empirical data, which are col-
lected and described in Ref. [5]. In the data, the knowledge contents are mathematical tags
used in the communications on Q&A system Mathematics Stack Exchange. In particular, the
content of each question is specified (tagged) by one or more (maximum five) tags according
to the standard mathematical classification scheme. While in Ref. [5] we investigated the role
of expertise in the social process taking part on the co-evolving bipartite network of users-and-
PLOS ONE | DOI:10.1371/journal.pone.0154655 May 12, 2016
