Solitary states in complex networks: impact of topology [original]

Eur. Phys. J. Spec. Top. (2022) 231:4123–4130

https://doi.org/10.1140/epjs/s11734-022-00713-4

THE EUROPEAN

PHYSICAL JOURNAL

SPECIAL TOPICS

Regular Article

Solitary states in complex networks: impact of topology

Leonhard Sch¨ulen1,a, Maria Mikhailenko2, Everton S. Medeiros3, and Anna Zakharova1,4

1Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

2Charit´e-Universit¨atsmedizin Berlin, Einstein Center for Neurosciences Berlin, Charit´eplatz 1, 10117 Berlin, Germany

3Institute for Chemistry and Biology of the Marine Environment, Carl-von-Ossietzky-University Oldenburg, Carl-von-

Ossietzky-Straße 9 - 11, 26111 Oldenburg, Germany

4Bernstein Center for Computational Neuroscience, Humboldt-Universit¨at zu Berlin, Philippstraße 13, 10115 Berlin,

Germany

Received 31 August 2022 / Accepted 26 October 2022 / Published online 10 November 2022

©The Author(s) 2022

Abstract The dynamical behavior of networked systems is expected to reﬂect the properties of their cou-

pling structure. Yet, symmetry-broken solutions often occur in symmetrically coupled networks. An exam-

ple are so-called solitary states where the dynamics of one network node is diﬀerent from the synchronized

rest. Here, we investigate the structural constraints of networks for the appearance of solitary states. By

performing a large number of numerical simulations, we ﬁnd that such states occur with high probability

in asymmetric networks, among them scale-free ones. We analyze the structural properties of the networks

that support solitary states. We demonstrate that the minimum neighbor node degree of a solitary node is

crucial for the appearance of solitary states. Finally, we perform bifurcation analysis of dimension-reduced

systems, which conﬁrm the importance of the connectivity of the neighboring nodes.

1 Introduction

Understanding the interplay between network topology,

coupling scheme and emerging dynamics is one of the

central issues in the ﬁeld of nonlinear dynamics and

the theory of complex networks [1–4]. Often, perfectly

symmetric networked systems exhibit symmetry-broken

solutions [5–7]. For networks of oscillators, a particu-

larly interesting example is the extreme case of cluster

synchronization in which only one oscillator does not

synchronize with the rest of the network. This highly

unbalanced network conﬁguration is called “solitary”

state [8,9]. The occurrence of these states has been

observed in a variety of symmetrically coupled systems

such as globally coupled nonlinear oscillators [8], in

non-locally (and globally) coupled networks of phase

oscillators [10,11], adaptive networks [12], coupled

chaotic maps [13–15], coupled excitable systems [16],

multilayer networks [17–20], and time-delayed systems

[21]. Such ubiquity naturally drew attention to soli-

tary states and raised questions about the mechanism

behind their appearance [18,22]. Recently, Sch¨ulen

et al. have addressed this issue for coupled neural oscil-

lators and found that solitary states are created sub-

critically in a fold bifurcation [23]. Additionally, they

ae-mail: l.sch[email protected]erlin.de (corresponding

author)

have demonstrated that the dynamics of the solitary

oscillator can also be chaotic, while the synchronized

cluster stays periodic [23].

Despite the advanced understanding of ubiquity and

onset of the solitary states, some essential questions

remain unanswered. For instance, what are the topo-

logical constraints for the emergence of these asym-

metric solutions? Naturally, symmetry-broken solutions

are unexpected in the scope of symmetric systems with

uniform distribution of node degree. However, complex

networks with asymmetries in their structure of connec-

tions, such as scale free ones, which have a power-law

distribution of node degrees, do not oﬀer any imme-

diate advantage for the occurrence of solitary states.

In fact, it has been recently found that asymmetries

can actually favor synchronization [24,25] or regular

behavior [26,27]. Therefore, the onset of solitary states

in asymmetric networks is as equally important as the

symmetric case, especially for a better understanding

of how these states occur in general setups.

In this work, we address the onset, and the topologi-

cal dependencies of solitary states in complex networks

of FitzHugh–Nagumo oscillators. We begin our inves-

tigation by studying the appearance of solitary states

in three diﬀerent network topologies: a symmetric non-

locally coupled ring, a random, and a scale-free graph.

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4124 Eur. Phys. J. Spec. Top. (2022) 231:4123–4130

To establish the range of parameters in which the soli-

tary states would appear, we use a symmetric non-

locally coupled ring as a baseline for the parameter

search. We ﬁnd that random and scale-free networks

exhibit solitary states for similar parameter values as

the symmetric one. These results are veriﬁed over an

ensemble of realizations of such networks with diﬀerent

initial conditions. Further, we proceed to the topologi-

cal analysis of the networks, aiming to identify a com-

mon feature of solitary nodes. We analyze their node

degrees, neighbor node degrees, and eigenvector cen-

trality. Among these measures, we found that a high

neighbor node degree correlates with the onset of soli-

tary states. Our next step is to verify this observation

more rigorously by performing bifurcation analysis. We

rely on geometrical arguments to obtain a dimension-

reduced system, where the bifurcation analysis demon-

strates the onset of the solitary states in a fold bifurca-

tion and their dependencies on the minimum value of

the neighbors node degree.

2 Solitary states in complex networks

We consider diﬀerent network topologies, each network

consisting of identical FitzHugh–Nagumo (FHN) ele-

ments in the oscillatory regime. Such networks are

numerically simulated using the following dimension-

less equations:

εdui

dt =f(ui,v

i)+



j=1

Aij

[σu(uj−ui)+σv(vj−vi)],

dvi

dt =g(ui,v

i),(1)

where the pair (ui,vi) accounts for the activator and

inhibitor variables, respectively, of each FHN oscillator

iwith i=1,...,N. The parameter Nprescribes the

network size. The functions:

f(ui,v

i)=ui−u3

3−vi,g(ui,v

i)=g(ui)=ui+a

(2)

specify the local dynamics of the variables uiand

vi. The parameter εdeﬁnes the time scale separation

between the fast (ui) and slow (vi) variables. Through-

out this study, we ﬁx ε=0.1. For isolated oscilla-

tors, the threshold parameter aseparates the excitatory

(|a|>1) from the oscillatory (|a|<1) regime through a

supercritical Hopf bifurcation. We set each oscillator in

the oscillatory regime by keeping a=0.5. The diﬀerent

network topologies are characterized by their adjacency

matrix Aij. The FHN oscillators are diﬀusively coupled

via both variables uiand vi, similarly to [22,28–31].

The strength of the diﬀusive coupling is given by σuand

σvfor the activator and the inhibitor variable, respec-

tively. The coupling function of a given FHN oscillator

iis normalized by the number of oscillators coupled to

it, i.e., its node degree di=N

j=1 Aij.

In our study, we consider three diﬀerent network

topologies, namely, a symmetric non-locally coupled

ring, a random, and a scale-free network. The topology

of the symmetrically coupled ring is determined by a

coupling radius Rcommon to all nodes in the network.

The coupling radius Rdeﬁnes the number of neighbors

in each direction on a ring. This coupling scheme yields

a uniform node degree di=2R∀i∈[1,N], see the

corresponding graph in Fig. 1(a). This symmetric case

is used as a reference case for the comparison with the

asymmetric structures. To obtain the random topology,

we consider the well-known Erd˝os–R´enyi (ER) algo-

rithm [32]. Starting from N= 100 unconnected nodes,

we establish a connection between any given pair of

nodes with a probability p=0.18. This procedure

results in a normal distribution of node degree with

average value of di= 18, see the corresponding graph

in Fig. 1(b). To obtain a scale-free network, we use the

Barab´asi–Albert (BA) algorithm which is based upon

preferential attachment in the network growth process

[33,34]. Starting from a network with a star-like topol-

ogy, at each step of the BA algorithm anodewithmnew

links is added to the existing network. The newly added

links are distributed across the network as follows: a

given node ireceives a new link in accordance with its

current degree diwith the probability pi=di

idi. This

procedure yields a network with a power-law distribu-

tion of node degrees. In Fig. 1(c), we show the resulting

graph of the BA algorithm with m=5.

In all three network structures, we are able to ﬁnd

solitary states for various initial conditions. Each red

node in the network graph depicts an oscillator that

is not in sync with the rest colored in dark blue. The

fact that such states emerge for networks with diﬀerent

average node degrees already suggests that this mea-

sure alone does not determine their existence. Note that

dav =diis generally a poor measure to compare net-

works with vastly diﬀerent node distributions.

We now investigate the onset of solitary states in the

considered asymmetrically coupled networks. For that,

we assign random initial conditions (ICs) to the system

in Eq. (1) with the adjacency matrix Aij for a scale-free

topology as shown in Fig. 1(c). In Fig. 2(a), we show the

time evolution of the network, illustrating the presence

of one solitary node split oﬀ from the main synchronized

cluster. This behavior can be further visualized in a

snapshot of the variable uiat t= 2000 (arb. units)

as shown in Fig. 2(b). In general, the trajectories of

solitary nodes approach an attractor coexisting with the

one hosting the synchronized cluster. The attractor of a

solitary node can be periodic or chaotic [22,23]. For the

case depicted in Fig. 2(a) and (b), the dynamics of the

solitary node follows a period-3 limit cycle as shown in

the state-space projection (ui,vi) depicted in Fig. 2(c).

This ﬁgure demonstrates the existence of solitary states

in asymmetric networks.

To establish the coupling strengths for which soli-

tary states arise in complex networks, we obtain the

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Eur. Phys. J. Spec. Top. (2022) 231:4123–4130 4125

Fig. 1 Three examples of solitary states for various topologies, namely: aa non-locally coupled ring (for better visibility

we use N=30andR= 5, yielding di=d= 10); ban Erd˝os–R´enyi network with N= 100, p=0.18, yielding an average

node degree of di≈18; cBarab´asi–Albert network with N= 100 and m= 5 links added at each algorithm step, yielding

di≈9.5. The size of the nodes is proportional to the respective node degree

Fig. 2 Exploring the dynamics of a solitary state with a single solitary node in a scale-free network with N= 100, m=5

(as shown in Fig. 1c), σu=0.12, and σv=0.15: aSpace-time plot for the activator variable ui;bSnapshot of the activator

variable uitaken at t= 2000 (arb. units). Blue circles correspond to the synchronized cluster, while the red one shows the

solitary node; cState-space projections for the solitary node (red) and the synchronized cluster (blue)

Fig. 3 Regionsofexistenceofsolitarystatesinthe(σu,σ

v)-parameter plane. Color-coded is the probability of obtaining

a solitary state for the following topologies: aA non-locally coupled ring network with R=9;bA random (Erd˝os–R´enyi)

network with average degree p=0.18; cA scale-free (Barab´asi–Albert) network with m= 10. The probability is estimated

from 100 network realizations with diﬀerent ICs uniformly distributed over the intervals ui∈[−2.2,2.2], vi∈[−1.1,1.1].

The network size is N= 100 in all cases

map of regimes in the (σu,σ

v)-parameter plane for

all three topologies shown in Fig. 1. We consider an

ensemble of 100 network realizations with diﬀerent ICs.

For each simulation, we integrate a time interval of

Δt = 2000 (arb. units) before checking the existence

of solitary states. This approach provides statistical

results in which the fraction of simulations resulting

in solitary states is characterized by the probability

of their occurrence. With this, in the map of regimes

shown in Fig. 3(a)–(c) for the respective topologies,

the color-code stands for the probability of obtaining

a solitary state for a given parameter pair (σu,σ

v). We

observe that the parameter regions with a high prob-

ability of ﬁnding solitary states occur for very similar

values of the coupling intensities σuand σvfor all three

topologies. In addition, the shape of such parameter

regions is similar across the topologies. Therefore, the

analysis of the (σu,σ

v)-parameter plane does not cap-

ture the inﬂuence of topological asymmetries. To tackle

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4126 Eur. Phys. J. Spec. Top. (2022) 231:4123–4130

this issue, we analyze in more detail the topological fea-

tures of networks exhibiting solitary states in the sub-

sequent sections.

To complement our analysis of the coupling inten-

sities, we point out that in a recent work for a glob-

ally coupled system (symmetric), Sch¨ulen et al. have

obtained similar patterns in the (σu,σ

v)-parameter

plane [23]. For a class of such systems, the authors have

derived the bifurcation curves delimitating the regions

of high probability for the occurrence of solitary states

[23]. The similarity of the patterns observed for the

globally coupled system and the networks studied here

suggests that the same bifurcation scenario gives rise to

solitary states in the networks with complex topologies.

This fact is also investigated subsequently in this work.

3 The role of the network connectivity

and local topological features

The results conveyed in Sect. 2show that solitary states

arise in networks with complex topologies for the same

range of coupling strength as for simple regular net-

works. This raises a general question: What would be

the topological requirements for the onset of the soli-

tary states? To investigate the existence of possible con-

straints, we concentrate our eﬀorts on networks with

topologies represented by BA graphs as they consti-

tute the example with asymmetric distribution of node

degrees. As discussed in Sect. 2, the connectivity of BA

graphs is speciﬁed by the number of links madded at

every step of the growth algorithm. We investigate the

onset of solitary states in such networks for diﬀerent

levels of connectivity by varying m. For that, we ﬁrst

adopt a statistical approach by considering 1000 real-

izations of the system Eq. (1) with diﬀerent ICs chosen

randomly as discussed in Fig. 3. In the top panel of

Fig. 4(a), we show the share of ICs resulting in solitary

states with diﬀerent number of solitary nodes for vary-

ing mas follows: one (green), two (orange), three (blue),

four (orchid), and ﬁve or more (red). The absence of

solitary nodes (regime of synchronization) is marked in

gray. The colored region is the relative fraction that of

all ICs that goes to this speciﬁc state. For example, if

m= 10, about 35% of the simulations go to a state

with a single solitary oscillator, about 60% go to either

a single or two solitary nodes, so the relative share of

two solitaries is about 25%. Thus, they are given by

the diﬀerences between two curves. In eﬀect, we have a

100% stacked bar for each value of m. For low values

of m(low connectivity), we observe that solitary states

occur for a small share of the ICs and they contain only

one solitary node. By increasing m, the share of realiza-

tions corresponding to the completely synchronized net-

work (gray) diminishes giving room to a larger variety

of solitary states. For instance, for m≈8 solitary states

are prevalent among all the realizations being observed

with a diﬀerent number of solitary nodes. Despite the

fact that they are observed for all levels of network con-

nectivity, the ﬁndings in Fig. 4(a) indicate that high

connectivity favors the formation of such states.

Next, we reﬁne our knowledge by analyzing how

higher connectivity inﬂuences the formation of solitary

states. Speciﬁcally, we investigate local node properties

to ﬁnd out if the success of high network connectivity

can be captured by measures such as the node degree,

average neighbor node degree, and eigenvector central-

ity. As the node degree dihas been already deﬁned in

Sect. 2, the average neighbor node degree of node iis

given by dav,i =N

j=1 Aij dj

di. This measure accounts for

the average level of connectivity of adjacent nodes of

i. The eigenvector centrality of a node iis given by

ci=1

λN

j=1 Aijcj, where λis the largest eigenvalue cal-

culated via Ac =λc. This measure ranks the node ifol-

lowing the connectivity of its neighbors. We ﬁrst obtain

the average of these three measures over the entire net-

work, denoted by .all. We then obtain the average

only over the solitary nodes, which we denote by .sol.

We calculate the ratio .sol/.all as a function of the

parameter min the bottom panel of Fig. 4(a). Interest-

ingly, for m8, the average over solitary nodes diﬀers

from the network average. In particular, we point out

the fact that the average neighbor node degree is much

higher for the solitary nodes, i.e., dav,isol >dav,iall

[yellow curve in the bottom panel of Fig. 4(a)]. This

measure will be justiﬁed later on by the bifurcation

analysis (Sect. 4). In addition, by comparison with the

top panel of Fig. 4(a), we conclude that the asymme-

tries between the solitary nodes and the nodes in the

synchronized cluster favor the appearance of solitary

states with only one solitary node [green region in the

top panel of Fig. 4(a)]. In contrast, at m≈8, the ratio

.sol/.all of all measures approaches one, indicating

the establishment of homogeneity among the solitary

and the synchronized nodes. This conﬁguration holds

for m>8, giving rise to the regime dominated by soli-

tary states with a large number of solitary nodes.

To further analyze the dependencies observed in

Fig. 4(a), we now investigate the probability of a node

being a solitary given one of its characteristics dis-

cussed above such as node degree, average neighbor

node degree, eigenvector centrality. For that, we employ

the concept of conditional probability, given by

p(s|x)=p(s∩x)

p(x),(3)

where p(s—x) stands for the probability of a node being

a solitary sgiven a topological property x, e.g., the node

degree dor the average neighbor node degree dav.To

calculate this conditional probability, we consider 1000

realizations with diﬀerent ICs of BA networks possess-

ing N= 500 nodes (for better statistical quality). With

this, in the top panel of Fig. 4(b), we obtain the con-

ditional probability of a node being a solitary given its

node degree, i.e., p(s—d). For m= 4 (green circles), we

observe that only nodes with low d<10 degree have a

nonzero probability of being solitary [see the inset in the

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Eur. Phys. J. Spec. Top. (2022) 231:4123–4130 4127

Fig. 4 The impact of various topological measures on solitary states aTop panel: Share of network realizations yielding

a solitary state with one (green), two (orange), three (blue), four (light purple), and ﬁve or more (red) solitary nodes for

m∈[1,25]. The gray color corresponds to realizations with no solitary states. Bottom panel: Ratio .sol/.all between

average network measures, node degree (blue), centrality (red), and average neighbor degree (yellow). Parameters are:

σu=0.12, σv=0.12, and N= 100. bConditional probability p(s—d) to obtain a solitary state for a given node degree

din a Barab´asi–Albert network with m= 4 (green) and m= 8 (magenta). cConditional probability p(s|dav ) to obtain

a solitary state for a given average neighbor degree dav in a Barab´asi–Albert network with m= 4 (green) and m=8

(magenta). Parameters in band c:σu=0.12, σv=0.15, and N= 500

top panel of Fig. 4(b)]. The probability of occurrence

of solitary nodes with a degree above this threshold is

zero. This suggests that the basins of attraction of the

solitary states are rather small and does not necessar-

ily mean that solitary states are impossible for these

node degrees. On the other hand, for m= 8 (pink cir-

cles), we ﬁnd that both nodes at the lower and the

higher end of the degree distribution have a high prob-

ability of becoming a solitary node. This inconsistency

between the cases m=4andm= 8 suggests that the

probability of a node being solitary is not only deter-

mined by its respective degree. Therefore, we now look

into the conditional probability considering the average

neighbor degree as the underlying condition p(s|dav). In

Fig. 4(c) for both cases (m=4andm= 8), we observe

that a high average neighbor degree corresponds to a

higher probability of the respective node being solitary.

This observation is in line with the ﬁndings in the bot-

tom panel of Fig. 4(a). Note that for m8, the average

neighbor degree is the only measure that is on average

higher for the solitary nodes with respect to mean of

the network, i.e., dav,isol >dav,iall.Asmincreases,

the high connectivity in the network provides a higher

average neighbor degree to more nodes, creating the

abundance of solitary states observed in the top panel

of Fig. 4(a).

4 Bifurcation analysis

The statistical analysis performed in Sect. 3has shown

that the average neighbor node degree plays an impor-

tant role in the formation of solitary states. Now, we

deepen this knowledge by investigating the bifurcation

scenario giving rise to the solitary states in the consid-

ered complex network. To this end, we make use of the

fact that the entire synchronized cluster inﬂuences the

solitary node as a single input. That is, if node sis a

solitary and it is connected to dsoscillators that are all

in sync with each other, the resulting coupling term in

sis given Cs=σu(ub−us)+σv(vb−vs), where (ub,v

are the state variables of oscillators in the synchronized

cluster. In turn, an oscillator n, with node degree dn,

belonging to the synchronized cluster, and directly cou-

pled to the solitary node, has nonzero coupling input

only from the solitary node. The resulting coupling

term in nreads Cn=1/dn·[σu(us−un)+σv(vs−vn)].

With this, one can immediately infer that for large val-

ues of dn, the coupling term for the node nvanishes

and it decouples from the solitary node s. Therefore,

to ensure the coupling between the nodes sand n,we

assume that nhas the minimum node degree dnamong

all neighbors of s. In addition, we also assume that the

rest of the network stays synchronized at all times. With

these assumptions, we can reduce the dynamics of the

network to the following equation of two interacting

nodes [35]:

εdus

dt =f(us,v

s)+σu(un−us)+σv(vn−vs),

dvs

dt =g(us,v

s),

εdun

dt =f(un,v

n)+ 1

[σu(us−un)+σv(vs−vn)],

dvn

dt =g(un,v

n),(4)

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4128 Eur. Phys. J. Spec. Top. (2022) 231:4123–4130

Fig. 5 Bifurcations in the minimal neighbor degree of a solitary node. aBifurcation diagram of the reduced systems

varying dnas a bifurcation parameter. Red crosses indicate saddle-node bifurcations. The vertical dashed line in magenta

indicates the threshold value dc= 7. Top panel: Reduced system in Eq. (4). Bottom panel: Reduced system in Eq. (5)

for ds= 2 (yellow), ds=4(red)andds= 8 (purple). bThe minimal neighbor degree of all nodes in Barab´asi–Albert

network with m= 4 (top panel) and m= 8 (bottom). Red circles indicate nodes that are observed as solitaries. Blue circles

are the nodes that are always in sync. The size of the red circles is proportional to how often the corresponding node is

observed as a solitary. cConditional probability p(s|dmin) to observe a solitary for a given minimal neighbor node degree

for Barab´asi–Albert networks with m= 4 (green) and m= 8 (pink). Parameters: σu=0.12, σv=0.15, and N= 500

where the functions f(u,v)andg(u,v) are given in Eq.

(2). Following this dimension reduction, we are now able

to study the stability of solutions of the system Eq. (4).

Considering the node degree dnas a continuous param-

eter, we employ numerical continuation analysis to fol-

low a solitary solution as dnis varied. For this task,

we use the software auto-07p [36]. The resulting bifur-

cation diagram is shown in the top panel of Fig. 5(a).

Even though, in reality the minimum node degree dnis

an integer number, the bifurcation analysis in Fig. 5(a)

treating it as continuous provides deep insight into the

stability proﬁle of the solitary node s. Note that the

limit cycle hosting the trajectory of the solitary node

appears in a fold bifurcation at dn≈5.701, see the red

cross in the top panel of Fig. 5(a). This analysis indi-

cates that the neighbor nodes of a solitary node must

have a minimum node degree of dn≥6. Therefore, we

establish this value of dnas a threshold value dcfor the

occurrence of solitary states in the network. In addition,

it is also visible in the top panel of Fig. 5(a) that the

increase of the parameter dnleads to a period-doubling

bifurcation transforming the dynamics of the solitary

node, see a blue cross. This observation suggests that

the larger the neighbor node degree, the higher is the

period of the limit cycle hosting the solitary node.

The results shown in the top panel of Fig. 5(a) are

obtained by assuming the dynamics of two coupled

oscillators as a descriptor of two interacting subsets of

the network, i.e., the synchronized cluster and the soli-

tary node. This approach can be improved by splitting

the synchronized cluster into the dynamics of a mean-

ﬁeld component and the dynamics of the node directly

coupled to the solitary node. With this, the dynamics

of the solitary node sand the directly coupled neigh-

bor ndescribed in Eq. (4) receive an extra dynami-

cal input corresponding to the synchronized mean-ﬁeld.

The dynamics of the mean-ﬁeld is described by an extra

oscillator with state variables (ub,v

b), which are not

aﬀected by the nodes nand s. The resulting equations

for this version of the reduced system are given by:

εdub

dt =f(ub,v

b),dvb

dt =g(ub,v

b),

εdun

dt =f(un,v

n)+1−

dn[σu(ub−un)+σv(vb−vn)]

[σu(us−un)+σv(vs−vn)] dvn

dt =g(un,v

n),

εdus

dt =f(us,v

s)+1−

ds[σu(ub−us)+σv(vb−vs)]

[σu(un−us)+σv(vn−vs)],dvs

dt =g(us,v

s),

(5)

with f(u,v)andg(u,v)asinEq.(2). As before, the

bifurcation analysis is performed by varying the min-

imum degree dnof the neighbor node, see the bot-

tom panel of Fig. 5(a). We repeat the analysis for

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Eur. Phys. J. Spec. Top. (2022) 231:4123–4130 4129

three diﬀerent values of the parameter attributed to

the degree of the solitary node, namely ds∈[2,4,8].

Surprisingly, the choice of dsdoes not inﬂuence much

the threshold for the existence of solitary states. In all

three cases, we ﬁnd the fold bifurcation located between

6<d

n<7, slightly higher compared to the case of

the two-node reduction. This implies that the lower

threshold is dn≥7. We can therefore conﬁrm the claim

that the crucial parameter for the formation of solitary

states is the minimal neighbor node degree dn. Neither

the node degree of the solitary node nor the average

neighbor node degree by itself has a causal role in the

formation.

Finally, we statistically verify this result in the full

system, i.e., via an ensemble of realizations of the com-

plex network. We consider again realizations of BA net-

works with two diﬀerent values of the parameter mto

compute the quantity dmin,i, the minimum neighbor

node degree of each network node i.Form=4,inthe

top panel of Fig. 5(b), we obtain dmin,i for all nodes i.In

this ﬁgure, among the diﬀerent realizations, the nodes

that are solitary at least once are colored in red, while

the other nodes are marked in blue. In addition, the size

of the red circles is proportional to how often the corre-

sponding node is observed as a solitary. We observe that

the nodes with a higher value of dmin,i are more often

solitaries, indicating that indeed the higher value of the

minimum neighbor node degree yields a higher proba-

bility of being solitary. Moreover, for the m=4,theBA

network has low connectivity implying that most nodes

have a dmin below the threshold dc= 7 unraveled by

the bifurcation analysis in Fig. 5(a). The dashed hori-

zontal line in Fig. 5(b) marks this threshold and indeed

all nodes below this threshold do not become solitary

(blue circles). Conversely, in a BA network with m=8,

all nodes have a degree larger than the critical thresh-

old dc. In such a network, all nodes meet the criteria we

established via bifurcation analysis. The network simu-

lations shown in the bottom panel of Fig. 5(b) conﬁrm

this hypothesis. Note that dmin for all nodes falls above

the dashed horizontal line marking dcand most of the

nodes can be solitaries (red circles). Also, in this case,

higher dmin corresponds to a higher probability of being

solitary. Finally, in Fig. 5(c), we complement our anal-

ysis by estimating the conditional probability p(s|dmin)

of a node iwith a given dmin,i being solitary. For m=4

(green circles), the observed probability is nonzero only

for the nodes having dmin,i >d

c. In addition, the prob-

ability increases for nodes with higher values of dmin,i.

For m= 8 (pink circles), as all nodes already fall into

the criteria for solitary (dmin,i >d

c∀i∈[1,500]),

there is no node with zero probability of being a soli-

tary. Again, the probability is higher for nodes with

higher dmin,i. Both cases conﬁrm the existence of the

threshold dcfor the minimum node degree of adjacent

nodes provided by the bifurcation analysis.

5 Discussion

In summary, by approaching the onset of solitary states

in complex networks, we have demonstrated that the

connectivity of nodes adjacent to a solitary one is an

essential topological feature for the appearance of soli-

tary states. More speciﬁcally, we found a threshold for

the minimum value of the degree of nodes neighboring

the solitary ones in the network. This ﬁnding is sta-

tistically demonstrated by estimating the conditional

probability of its occurrence in ensembles of realiza-

tions of scale-free networks. Furthermore, a dimension-

ality reduction of the network dynamics made possible

a bifurcation analysis conﬁrming the existence of the

minimum threshold.

Due to the symmetry-broken character of solitary

states, their occurrence is undesirable in many con-

texts. Therefore, the knowledge of topological con-

straints for the onset of these states sheds light on a

perspective of their control, i.e., suppression or initi-

ation. Finally, we emphasize that the results reported

here can be extended to other classes of networked sys-

tems since there are no particular dynamical, or topo-

logical, restrictions on them.

Acknowledgements We thank Matthias Wolfrum and

Alexander Gerdes for fruitful discussions. E.S.M acknowl-

edges the support by the Deutsche Forschungsgemein-

schaft (DFG) via the project number 454054251. This work

was supported by the Deutsche Forschungsgemeinschaft

(DFG, German Research Foundation) - Projektnummer -

163436311 - SFB 910.

Funding Information Open Access funding enabled and

organized by Projekt DEAL.

Open Access This article is licensed under a Creative

Commons Attribution 4.0 International License, which per-

mits use, sharing, adaptation, distribution and reproduction

in any medium or format, as long as you give appropri-

ate credit to the original author(s) and the source, pro-

vide a link to the Creative Commons licence, and indi-

cate if changes were made. The images or other third

party material in this article are included in the arti-

cle’s Creative Commons licence, unless indicated other-

wise in a credit line to the material. If material is not

included in the article’s Creative Commons licence and

your intended use is not permitted by statutory regu-

lation or exceeds the permitted use, you will need to

obtain permission directly from the copyright holder. To

view a copy of this licence, visit http://creativecomm

ons.org/licenses/by/4.0/.

Data availability The datasets generated during and ana-

lyzed during the current study are available from the corre-

sponding author on reasonable request.

123

4130 Eur. Phys. J. Spec. Top. (2022) 231:4123–4130

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