# Topological insulators and enhancers in networks under generic problem-
solving dynamics

J.F.Johannes Falk  E.E.Edwin Eichler  K.W.Katja Windt  M-T.H.Marc-Thorsten
Hütt  School of Science, Constructor University, Bremen, Germany  SMS Group
GmbH, Düsseldorf, Germany  EICHLER Consulting AG, Weggis, Switzerland  School
of Business, Social & Decision Sciences, Constructor University, Bremen,
Germany

###### Abstract

The collective coordination of distributed tasks in a complex system can be
represented as decision dynamics on a graph. This abstract representation
allows studying the performance of local decision heuristics as a function of
task complexity and network architecture. Here we identify hard-to-solve and
easy-to-solve networks in a social differentiation task within the basic model
of small-world graphs. We show that, depending on the details of the decision
heuristic as well as the length of the added links, shortcuts can serve as
topological enhancers, which speed up the finding of a solution, but also as
topological insulators, which make the network more difficult to solve. Our
findings have implications for situations where, in distributed decision
systems, regional solutions emerge, which are globally incompatible such as
e.g. known from the emergence of standards.

Graph Coloring Dynamics,

Distributed Decision Strategies,

Global Coordination,

###### keywords:

Research

##  1 Introduction

Dynamics on graphs are an important concept to analyze distributed decision-
making and task coordination. Beyond social sciences [1, 2, 3] also logistics
[4] and computer science [5, 6, 7] are interested in how distributed decisions
can efficiently lead to global coordination, e.g. to avoid queuing or to
minimize interference between WLANs. In the simplest coordination problems, a
node of the graph can select a decision (a ’color’) out of a list of allowed
decisions based on the observed decision states of their direct neighbors. The
local decision heuristics (i.e., the decision selection criteria at each node)
represent the goal of the systemic task. Such coordination tasks come in two
variants [8]: Either the task is related to some type of consensus across the
whole system. In this case, the graph is ’solved’, when no different colors
are linked. Alternatively, these coordination tasks can be related to social
differentiation, scheduling, or resource allocation. In this case, the graph
is ’solved’, when no same colors are linked. Here we focus on the second
scenario of social differentiation and scheduling. Its abstraction as color
dynamics on graphs, related to the graph coloring problem, has been made
popular by the seminal work of Kearns et al. [1]. This framework has led to
relevant insight into problem-solving dynamics and some ’stylized facts’ about
distributed decisions. Examples include the positive effect of random agents
in such a distributed decision system [3], the effect of a wave-like
organization of attention and strategic waiting on these decision dynamics
[9], and the effect of shortcuts in a small-world architecture on the
convergence toward a fully solved system. This is visible, both in experiments
with human subjects [1] and numerical simulations involving simple heuristics
[9].

The decision heuristics introduced in Hadzhiev et al. [9] furthermore provided
some insight into the interplay of centralized and autonomous, decentralized
control in manufacturing planning and control [10, 11].

However, a striking characteristic of graph coloring dynamics has not been
analyzed in the past: For a few or even a single shortcut (i.e., a small
rewiring probability in the Watts-Strogatz model [12]) one observes a dramatic
variability of runtimes. Here we show that – besides the random
initialization, as well as the general stochastic nature of these dynamics –
this high variability is due to the network topology: Depending on the exact
positions as well as the used heuristic, shortcuts in a ring graph can
generate easy-to-solve and difficult-to-solve graphs.

The problem addressed by our investigation is of relevance for many real-world
applications: Regional solutions emerge rapidly, but they are incompatible on
a global scale and the diffusing remaining conflicts, which are the boundaries
of incompatible solution regimes, require an excessive amount of local
reorganization, until one region switches to a solution compatible with
another region. This problem of different locally valid solutions that are
globally incompatible can especially be observed in the emergence of
compatibility standards [13]. Different technical devices may be locally
compatible based on one standard, but incompatible with functionally
equivalent standards from other areas, leading to competition between
alternatives [14] and ultimately resulting in a global standard. Examples of
such battles are BlueRay vs HD DVD or Wi-Fi vs HomeRF [15]. There already
exist some models to explain the success or failure of standards. But as
economic models, they are focused on the interplay of strategic factors,
business models, and business actors [16, 17]. Our investigation rather
contributes to understanding the spatial organization of standards and hence
the influence of the network topology on the time until a standard settles.

We show that – depending on the heuristic and the length of the links –
shortcuts can act as topological insulators and topological enhancers, i.e.
either delaying or accelerating the regional reorganization efforts towards a
trans-regionally compatible solution.

##  2 Methods

In this paper, we investigate heuristics that can solve the graph coloring
problem. In this problem from graph theory, the goal is to assign colors to
the vertices of a graph such that no two adjacent vertices have the same
color. The minimal number of colors that are needed to color a network in this
way is known as the chromatic number $\chi$ of the graph. In this section, we
explain how we generate graphs with a given chromatic number, introduce
different heuristics, and present a genetic algorithm that we use to generate
networks with specific properties.

###  2.1 Small-World Networks

In this analysis, we mainly focus on small-world networks with few inserted
links as a toy model for graphs with high clustering and small shortest path
length. The idea of the graph generation follows [12]. However, since the
networks are supposed to be solvable with a given number of $\chi$ colors (the
chromatic number), we generated them as follows: 40 (39 for $\chi=3$) nodes
are arranged as a circular graph, where every node $i$ is connected to its
$\chi-1$ closest neighbors in both directions. A given number of shortcuts is
then added such that every shortcut connects only nodes with a different value
of $mod(i,\chi)$, where $i$ is the node’s index, thus preserving the graph’s
chromatic number $\chi$. To compare how fast different network topologies can
be solved, we look at the number of color changes that have been performed
until the network is in a solved state. The color changes then set a time
scale where each time step is equal to one color change.

###  2.2 Other graph topologies with $\chi=2$

To extend our results to more general statements we generate three other types
of random networks (only for $\chi=2$):

  * •

BA: For this network, we start with a simple path graph with 4 numbered nodes.
Subsequently, we add nodes and links following preferential attachment as
described in [18] where every new vertex (labeled with a consecutive number)
is attached to existing nodes via two links. However, and in contrast to the
reference, to ensure that the graph has a chromatic number of $2$, for an even
(odd) number of already existing nodes, a newly added node can only connect to
nodes with an odd (even) label.

  * •

Random: The procedure to create this graph starts with a graph of $N$
unconnected nodes, labeled with an integer $i$. A given number of edges is
then sampled randomly from all edges that would connect two nodes with an even
and an odd label. This ensures a chromatic number of $\chi=2$. If the
resulting graph is not connected, the procedure is repeated with a different
set of randomly selected edges.

  * •

Modular: This graph consists of two separate graphs of type random that are
connected by a single edge connecting two randomly selected nodes.

###  2.3 Neighborhood assessment strategies

Agent-based models to solve graph coloring problems have already been analyzed
in various variations. Inspired by the results from [1], Hadzhiev et al. [9]
developed a family of heuristics that allow agent-based networks to be solved
in reasonably short times. Following the concepts from [9], a graph coloring
heuristic consists of two components: One strategy for the temporal
organization (which node acts next) and one for the neighborhood assessment
(which color does the active node select). To simulate the behavior of
independent distributed systems as closely as possible, we use random
sequential updates (R) for the temporal organization, which means that every
time step the next node is selected at random from all available nodes. For
the neighborhood assessment heuristic, we first refer to three strategies from
[9], namely $R$ (random), $M$ (color minimizing), and $W$ (strategic waiting).
Subsequently, we develop a new ($N$) heuristic whose behavior can be
continuously tuned by a parameter $r$ (reasoning): For large values of $r$ the
agents always select their color by reasoned considerations. The smaller $r$,
the more often the color choice happens randomly. In all strategies, the
active node first assesses the colors of its connected neighbors. If possible,
the node randomly selects one of the colors that do not appear in its
neighborhood (conflict-free color). Otherwise, the different strategies
proceed as follows:

  * •

R (random color): The node selects a color at random from all available colors

  * •

M (conflict minimizing color): The node selects randomly a color from the set
of colors that minimizes the number of conflicts. If the node has already the
unique conflict-minimizing color, a color is selected at random.

  * •

W (strategic waiting): Equal to the M scheme, however, if the node has already
the unique conflict-minimizing color, the present color is retained with
probability $p=0.9$.

  * •

N (reasoning): With a probability of $r$ the node randomly selects a color
that minimizes the conflicts (reasoned-acting). In the other case (with a
probability of $1-r$) it randomly selects a color from the list of all
available colors.

The $N$ heuristic can hence be understood as a generalization of the three
other heuristics. For small $r$ the $N$ heuristic is similar to the $R$
heuristic, for intermediate $r$ it is similar to the $M$, and for large $r$ to
the $W$ heuristic.

In order to name the full heuristics, we follow the naming scheme that was
also used in [9]: $XY$ means that we used $X$ as temporal organization
strategy and $Y$ as neighborhood assessment strategy.

###  2.4 Genetic Algorithm

To assess how strongly the topology of a network (with a fixed number of
shortcuts) affects the runtime, we use a genetic algorithm that evolves to
easy-to-solve or hard-to-solve networks (with respect to a given heuristic).
The algorithm starts with an ensemble of six randomly selected small-world
networks with the given number $S$ of shortcuts and proceeds as follows:

  * •

Each network of the ensemble is randomly colored and then solved by the
respective strategy. The time until solved (measured in activation steps) is
averaged over 500 runs.

  * •

The two fastest (slowest) solved networks are kept for the next run,
additionally, four networks are generated by mutations (re-connection of one
shortcut) and by recombination (take $n$ shortcuts from one network and $S-n$
shortcuts from the other network) of these two fastest (slowest) networks.

  * •

These six new networks are the new ensemble for the first step.

The process is terminated after 1000 evolution steps and the obtained
topologies are saved.

##  3 Results

In this investigation, we take the observed high variability of the
distributed graph-coloring problem as an opportunity to examine how the
network topology influences the runtime. To focus the analysis we limit
ourselves to networks with a chromatic number of $\chi=2$. In the last part of
the results section, we explain why networks with $\chi>2$ show a
significantly more complicated behavior, which results from the interaction of
different mechanisms and thus defies a simple mechanistic explanation.

We begin our investigation by looking at some results from [9]. The authors
analyzed how different graph coloring heuristics perform in small-world
networks if the number of shortcuts increases. In Fig. 1 we show the
performance of the three heuristics that use random neighborhood assessment
$R$ and $R$, $M$ or $W$ as neighborhood assessment (see 2.3 for details). With
the $RR$ and $RM$ heuristic, the more shortcuts the network has, the longer
(on average) the nodes need to organize and finally solve the network. In
contrast, using the $RW$ heuristic the solution is reached faster with more
added links, as it was also observed in human subject networks [1]. Looking at
Fig. 1, it is also noticeable that – for a fixed number of shortcuts – the
variance of the time steps required is strikingly high. Since the initial
conditions for each run are chosen randomly and the heuristic contains
stochastic components, a certain variance is to be expected. An open question,
however, is whether the topology, i.e. the location of the shortcuts, has an
impact on the solvability.

Figure 1: Mean number of time steps (color changes) until the network is
solved vs. the number of shortcuts for small-world networks using the $RR$,
$RM$, and $RW$ heuristic. The light area denotes the standard deviation.
(reproduced from [9])

To test and quantify the impact of the topology, we use a genetic algorithm
(see 2.4) that is designed to generate easy and hard-to-solve small-world
graphs with a small number of 5 added links. A strong difference between the
runtimes of the extreme graphs could indicate whether and how the topology
affects the runtime. Exemplary topologies for the $RR$, as well as the $RW$
heuristic, are presented in Fig. 2. The large difference between the fastest
and slowest networks ($120$ vs. $2531$ color changes for $RW$ heuristic, $406$
vs. $1206$ color changes for the $RR$ heuristic) indicates that – for a fixed
number of shortcuts – the runtimes depend strongly on the shortcuts’
positions. Additionally, the resulting topologies seem to have characteristic
features. However, these characteristic features are opposite: Long-range
links facilitate a fast solution finding for the $RW$ heuristic but create a
difficult-to-solve network for the $RR$ heuristic. Likewise, the easy-to-solve
network for the $RR$ heuristic is characterized by maximally short links,
whereas for the $RW$ heuristic the short links appear in the difficult graph.

Figure 2: Exemplary results of the genetic algorithm for (top) hard-to-solve
and (bottom) easy-to-solve network with 5 shortcuts optimizing for (left) $RW$
and (right) $RR$ heuristic. The numbers indicate how many time steps the
heuristics required to solve the respective graph (averaged over 500 random
initial conditions).

In what follows we will introduce a generalized heuristic and extract general
features that can explain the interdependence between topology and runtime.
Long-range links are often considered to be beneficial for a system-wide
organization because they allow transmitting information over a long distance.
Our analysis is based on the idea that the respective agent must be able to
process the additional information provided by a long link. When agents
evaluate the observations from their neighborhood reasoned, the remote
information helps them to adapt themselves to the global solution. If, on the
other hand, the agents do not operate reasoned, the additional source of
information creates confusion, which hinders the stabilization of local
solutions. To test this proposition, we introduce a new heuristic that can be
continuously adjusted from perfectly reasoned to random behavior. Our new
heuristic $N$ (details in Sec. 2.3) has one parameter $r$, the reasoning of
the respective agent. A large value of $r$ means that the nodes preferentially
select their color by minimizing the conflicts with their neighborhood
(similar to the $W$ heuristic). In contrast, for small values of $r$, the
nodes tend to select a color randomly from all available colors once they
observe a conflict within their neighborhood (similar to the $R$ heuristic).

We created a ring lattice with 40 nodes and added a single shortcut (with the
constraint that the chromatic number $\chi=2$ is conserved, see also Sec. 2.1)
For Fig. 3 we set $r$ to different values and analyzed how the runtime depends
on the relative length of the added shortcut (averaged over 10.000 runs each).
As expected, if the heuristic is very reasoned (large $r$) the time until
solved decreases for longer shortcuts. In contrast, if the heuristic contains
a lot of randomnesses (small $r$), long-range links deteriorate the
solvability of the graph. An additional observation is that the reasoned
strategies work poorly when the inserted link is very short (an increase of
the required time by about 30%).

Figure 3: Relative extra time until ring graphs with 40 nodes and a single
shortcut are solved vs. the relative length of the added shortcut for
different values of $r$. A relative length of $1$ refers to a shortcut of
maximal length, hence spanning 20 nodes. The time is measured relative to the
time that is needed if the ring graph does not have any shortcut.

###  3.1 Reasoned-acting Agents (large $r$)

For large $r$ the results are in line with the slow network obtained for the
$RW$ heuristic in Fig. 2. The slow network is characterized by comparably
short links that create two densely connected areas. These clusters foster a
fast emergence of local solutions. Additionally, the short shortcuts stabilize
the local solution against fluctuations from the outside. Figure 4a shows an
example of such stabilization of a local solution. The larger the parameter
$r$, the more stable the locally solved areas. However, in the likely case
that the local solution is not compatible with the currently prevailing global
solution domain, the system is in a hard-to-solve state: The reasoned-acting
agents cling to their local solution, the added link acts as a topological
insulator. Contrarily, evolving towards topologies that are easy to solve for
the $RW$ heuristic, the resulting network is characterized by a few nodes that
are connected to various areas of the network and that act as ordering nodes.
These ordering nodes synchronize the local solutions already during their
build-up. An example of the effect of a single long-range shortcut is shown in
Figure 4b. Without the shortcut, the node labeled with “A” could either stay
red or change its color to blue. In both cases, the result would be a single
conflict with one neighbor. However, due to the shortcut – that is by
definition inserted such that it does not alter the graph’s chromatic number
and, hence, a global solution is possible – a change to blue minimizes the
local color conflicts and acts as a local reference for the global solution
domain.

Figure 4: Comparison of the two effects a shortcut can have: (a) A short link
stabilizes a solution regime against perturbations from the outside. In the
example, there is a color conflict between the two red nodes (indicated by a
red link). The right red node has two blue neighbors (one direct and one via
the shortcut). If the node acts reasoned its color is stabilized since red
minimizes the conflicts. (b) The sketch shows two sections of a large ring
graph (indicated by the gray dashed line). The long shortcut organizes two
apart sections and orders them. Without the shortcut, the node with the label
“A” would have a 50% chance of keeping its color, compared to changing to
blue. Due to the shortcut, reasoned-acting nodes will change to blue, since
this is the conflict-minimizing color.

###  3.2 Irrational Agents (small $r$)

Things are different for the irrational agents with small $r$ (similar to the
$RR$ heuristic). Here, Fig. 1 tells us that shortcuts always create graphs
that are more difficult to solve than the pure ring graph, where the effect is
stronger the longer the added link. Consequently, the results from Fig. 2 show
that the fast networks are characterized by short links. For the $RR$
heuristic, the difficult-to-solve networks are characterized by long-range
links, very similar to the graphs that are easy to solve for the $RW$
heuristic. For irrational acting agents (as in the $RR$ heuristic), the long
links that connect a single node to various areas of the graph act like a
source of noise: A color-fluctuation of the highly connected node immediately
destabilizes the colors of all connected nodes, spread over the full network.

###  3.3 Complex Topologies

Having analyzed the interplay between the length of added links and the
reasoning of the acting agents in small-world graphs, it is now natural to ask
whether this behavior can also be observed in more complex networks. As
described in Sec. 2.2 we generated a random graph (40 nodes, 80 edges), a
modular graph (2 x 20 nodes with 40 edges each), and a BA graph (40 nodes).
All graphs are generated such that $\chi=2$. In Fig. 5 we show the time until
solved vs the reasoning of the heuristic. For both the random network and the
BA graphs, the more reasoned the agents act, the faster they are. Note,
however, that for $r=1.0$ dead-lock situations are possible that cannot be
solved (see e.g. Fig.2 in [9]). The results confirm the observations from the
small-world networks: Random networks as well as BA networks have small
modularity and high connectivity. It is therefore unlikely that globally
incompatible solutions can stabilize against the rest of the network. The
modular network is, however, specifically designed to have two almost separate
modules. Fig. 5 indicates that in this case heuristics that act too reasoned
have a disadvantage: If the two modules converge to different solution
domains, it is difficult for the heuristic to overturn one solution.

Figure 5: Mean number of timesteps (color changes) until the network is solved
vs. the reasoning of the heuristic for three different graph topologies (see
Sec. 2.2).

###  3.4 Extension to $\chi=3$

The natural extension of your investigation would be to increase the chromatic
number of the graphs. For Fig. 6 we performed a similar analysis as for Fig. 3
but with a ring graph with 39 nodes and a chromatic number of $\chi=3$.
Depending on the length of the added shortcut the system takes longer or is
faster than without a shortcut. The general behavior of the network is on
average also similar to the one with a chromatic number of two (short
shortcuts lead to longer times). However, there are also two drastic
differences: (1) The curve shows an alternating behavior that was not present
for the $\chi=2$ graph. The reason is a complicated interplay between the
shortcuts and the different possible solution regimes. For two colors there
are only two possible solutions: $abab$ or $baba$. However, for three colors
there are $3!=6$ possible solution domains that are facilitated or suppressed
depending on the position of the shortcut. (2) The relative effect of a single
shortcut is not as strong as for the $\chi=2$ graph. The main reason is that a
shortcut at each end excludes only one color at a time. If there are only two
colors a single disallowed color directly determines the correct color:
$\neg\text{red}\rightarrow\text{blue}$. However, the more colors we have the
less effect has the banning of a single color. To overcome this problem one
would need to generalize the definition of a shortcut. For $\chi=3$ such a
generalized shortcut would hence consist of four conventional shortcuts that
all-to-all connect two adjacent nodes with two other adjacent nodes.

##  4 Conclusion

Within small-world networks, shortcuts decrease the average path length and
facilitate the transport of local information through the system [19]. One
would therefore expect that distributed coordination problems on graphs always
benefit from shortcuts, albeit the effect size might depend on the respective
length of the shortcut. In this manuscript, we discussed the graph coloring
problem as a simple form of distributed coordination problem. We analyzed how
shortcuts affect the time a local heuristic needs to solve the coloring
problem. Depending on how reasoned the agents act, added shortcuts give rise
to different mechanisms: They synchronize the solution domains between distant
sections of the network, stabilize parts of the network against fluctuations,
or they create perturbations. For reasoned heuristics, short shortcuts tend to
insulate locally solved but globally incompatible solutions against each
other, finally leading to an increase in the overall time until a solution is
found. We call shortcuts that create such separated domains topological
insulators. In contrast, long shortcuts foster early synchronization of
otherwise distant areas of the network, which is why we call them topological
enhancers.

The graph coloring problem is the simplest model to analyze the conflicts
between and the solvability of distributed logical systems: The conflicts
encountered in graph coloring dynamics on a ring arise due to two (or more)
coloring domains that are structurally equal (they are correctly colored) but
locally different (they follow a different color permutation). From a
mathematical point of view, this inconsistency between local logical systems
relates to distributed logic. Our results can hence be interpreted from the
perspective of Gotthard Günther’s theory of polycontexturality (often also
termed transclassical logic) [20].

In this theory, different subjects, $S_{1}$ and $S_{2}$, observing the same
object, $O$, may arrive at different but equally correct conclusions about its
properties. In our model, this corresponds to different solutions emerging at
different locations on the ring. According to Günther’s theory, a third
subject, $S_{3}$, observing the situation, is able to reflect on the different
conclusions of $S_{1}$ and $S_{2}$ observing $O$. In our model, this
corresponds to a node that is enabled to compare different local solution
regimes via a long-ranging (enhancer) shortcut. This addition to the
neighborhood is enough to empower this node to modify the local decision
pattern and thus facilitate the emergence of a global solution.

In spite of the simplicity of the local decision heuristics and the stylized
nature of the task, the different roles of nodes (subjects) proposed by the
theory of polycontexturality establish themselves due to the structural
differences between nodes with and without a long-ranging shortcut.

In this view, it also becomes intuitive, why longer shortcuts serve as
enhancers and shorter shortcuts serve as insulators: For a node with a
shortcut, its role as the third subject requires a link to truly independent
information, transcending the local solution regime.

As a minimal model for the effects of links or information flow within
polycontextural systems, the analysis of the graph coloring problem can hence
contribute to heterarchical approaches in biology [21], consensus finding
[22], complex and reflexive relations in social systems [23, 24], or
transformations in physics [25].

We also believe that our findings have implications for the understanding of
the emergence of standards (here represented by globally compatible
solutions), as well as for the development of more robust scheduling schemes
in manufacturing and resource distribution [26].

Figure 6: Relative extra time until a ring graph with 39 nodes and a chromatic
number of 3 is solved vs. the length of the added shortcut for three different
values of $r$. A relative length of $1$ refers to a shortcut of maximal
length, hence spanning half of the system. The time is measured relative to
the time that is needed if the ring graph does not have any shortcut.

## Availability of data and materials

Not applicable

## Competing Interests

The authors declare that they have no competing interests

## Funding

Not applicable

## Authors’ contributions

MH and KW conceived this study. J.F. and M-T.H. developed the model, and J.F.
ran the simulations and analyzed the data. J.F. and M-T.H. wrote the
manuscript. E.E., K.W., and M-T.H. supervised the project. All authors
discussed the results and implications and commented on the manuscript at all
stages.

## Acknowledgments

Not applicable

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