# Identifying the structure patterns to govern the performance of localization
in regulating innovation diffusion

Leyang Xue1, Peng-Bi Cui1,∗, Zengru Di1 1International Academic Center of
Complex Systems, Beijing Normal University, Zhuhai, 519087, China
<EMAIL_ADDRESS>
###### Abstract

The macro social influence is recognized as a non-negligible ingredient in
innovation propagation: more adopters in the network lead to a higher adoption
tendency for the rest individuals. A recent study to incorporate such a
crucial mechanism shows that sufficient intensity of macro-level social
influence can cause a change from a continuous to discontinuous transition,
further indicating the existence of a tricritical point. Although network
localization strength determines the tricritical point, it remains unclear
what network quantities govern the performance of localization in regulating
innovation diffusion. To address this issue, we herein consider the model
incorporating both the micro- and macro-levels social influence. We present a
dynamic message-passing method to analytically treat both the outbreak
threshold and recovered population, and validate the predictions through
agent-based simulations. Extensive analysis on the classical synthetic
networks shows that sparsely available connections, and relatively
heterogeneous degree distribution, either assortative or extremely
disassortative configurations are favorable for continuous transition. In such
cases, the employed network can yield a strong localization effect so that the
innovation is trapped in the configurations composed of the hubs with high
non-backtracking centrality. We further explore the dependence of both
tricritical point and localization strength on three structural quantities:
network density, heterogeneity, and assortativity, which gives a clear
physical picture of the joint effects of the three structure quantities on the
localization strength. Finally, we conclude that the core-periphery structure,
being sensitive to the change of the three structure quantities, essentially
determines localization strength, and further regulates the phase transition.

Keywords: complex network, contagion dynamics, macro-level social influence,
phase transition, localization

##  1 Introduction

The propagation of innovation such as new ideas, technology, and products, is
regarded as a fundamental process to drive the development of socioeconomic
context [1]. However, it is quite difficult to exclude the effects of
confounding factors in empirical studies. Therefore, how to model innovation
diffusion becomes an increasing concern in understanding the underlying
mechanisms of adoption decisions and guiding marketing strategies. Extensive
models have been proposed and studied from empirical and theoretical aspects.
Some studies have suggested that the payoff gain from innovation entities
largely determines the final market penetration through shaping game profiles
among individuals during the diffusion process [4, 2, 3]. By means of a
reaction-diffusion framework, others have highlighted the crucial role of
local connectivity patterns of the target population in regulating the speed
and extent with which innovation prevails [7, 5, 8, 6]. For example, the hubs
of the networks can largely facilitate the product [8]. General mechanisms of
social contagion have been additionally incorporated in the models belonging
to this framework to enable a discontinuous transition, such as reinforcement
[10, 9] and mutual cooperation between multiple diffusion process [11].
However, these research efforts mainly treat innovation diffusion as the
context of social contagion [7, 12], calling for a fundamental understanding
of network properties that regulate the innovation diffusion process.

Recently, Ref. [13] has attempted to close this gap by proposing a model of
innovation diffusion based on the susceptible-adopted-recovered (SAR) model,
which incorporates social influence at micro and macro levels. The micro-level
influence is conveyed through pairwise transmission between neighbors [14],
representing the well-documented word-of-mouth effect [15]. While the macro-
level influence promotes individuals’ preference for adopting the new product
as market penetration increases. Such influence has at least three roots which
innovation propagation is deeply involved: (i) People become more willing to
follow the social norm with more adopters in the population, as a result of
psychological effect such as group conformity [17, 16, 4, 18, 19, 22, 20, 21,
23, 9]; (ii) The utility of the product increase due to network externalities
(e.g. telecommunication products like fax, email, SNS and smartphone, new
energy vehicles and artificial intelligence technology) [26, 24, 1, 25, 27,
29, 28]; (iii) The product price might decrease with the marginal cost decline
induced by economies of scale [32, 30, 31]. Both types of social influences
together are assumed to contribute to the transmission, such that the
transmissibility is a monotonic and increasing function of the recovered
population. In Ref. [13], the authors study the model on a collection of real-
world networks and observe that the macro-level influence, depending on the
proportion of the recovered population, can cause a change of the transition
from continuous to discontinuous. There exist a tricritical point that
separates the continuous and discontinuous phase transitions. They further
focused on how the type of transition depends on the contact network topology
by identifying the dependence of the tricritical points on the heterogeneity
of non-backtracking centrality distribution. More in detail they argue that a
discontinuous transition is more likely to occur if the network is dense and
the high-centrality nodes are well separated, i.e., the localization effect is
weak. From this, they developed a novel metric to measure localization
strength. Using the metric, one can accurately predict the position of a
tricritical point on a large collection of empirical networks. Moreover, the
metric can deeply reveal the relation between the critical and tricritical
points, which further indicates a trade-off: networks that allow less
attractive products to prevail instead tend to yield slower diffusion and
lower market penetration, and verse versa.

An open question left by the study in Ref. [13] has to do with what structure
patterns or quantities determine the performance of localization in regulating
innovation diffusion with macro-level social influence. Simulations performed
on a large collection of empirical networks seem to indicate that the
tricritical point positively depends on the heterogeneity of non-backtracking
centrality distribution. However, it is not clear what structure patterns
affect the distribution and what happens for innovation diffusion with macro-
level social influence if we tune these structure patterns. A clear physical
picture of why the localization strength of the network can largely determine
the magnitude of the tricritical point is still not integrated.

In this paper, to address these questions, we systematically explore how
classical synthetic networks govern the dynamics of innovation diffusion ruled
by the model of Ref [13]. We herein use the dynamic message-passing (DMP)
method to analytically track both the outbreak threshold and the final
recovered population of the system. Numerical simulations validate the
analytical predictions. We perform the study on the following classical
synthetic networks: two-dimensional (2D), Erdös-Rényi (ER), Watts–Strogatz
(WS), Barabasi-Albert (BA), and scale-free (SF) networks generated according
to the uncorrelated configuration model [33]. As expected, with a given
intensity of macro social influence, the innovation diffusion on the first
four networks gives rise to discontinuous transitions. Moreover, the
discontinuous ones become more apparent with denser connections. Weird
contrary phenomena to the above cases are obtained for SF networks, changing
degree exponent leads to a highly nonlinear change of phase transition. In
more detail, continuous phase transitions can still take place when the
network is sparse, assortative, or has a relative heterogeneous degree
distribution, reflecting the crucial roles of the spatial configurations of
nodes, especially hubs. Furthermore, the effects of node configurations on the
nature of phase transition are then specifically elucidated by varying the
assortativity of SF networks. The likelihood of discontinuous transition
decrease with network assortativity. Using the metric developed in Ref. [13],
we then show the dependence of both the tricritical point and localization
strength on the three structure quantities. The dependent behaviors
surprisingly coincide with each other. Therefore, it is claimed that network
density, heterogeneity, and assortativity are the structure quantities we are
seeking for. Finally, we point out that core-periphery structure, being
dependent on the three structure quantities, essentially determines
localization strength, and further regulates the phase transition.

The paper is organized as follows. In Sec. 2.1, we formally describe the
innovation diffusion model incorporating macro-level social influence,
followed by theoretical analyses in Sec. 2.2. In Sec. 3.1, we simulate our
model on various networks and identify the key network structure
characteristics that have a strong influence on the dynamical process. Then a
comprehensive analysis of the localization effect and its relation with the
tricritical point is provided in Sec. 3.2. We conclude with a summary of the
results and an outlook for future studies in Sec. 4.

##  2 Method

###  2.1 Model

We consider an innovation diffusion model characterized by a social contagion
process with a macro-level social influence. Specifically, we use a SIR-like
model to describe the contagion process, where an individual can be in one of
three different states, i.e susceptible (denoted by $S$), adopted (denoted by
$A$), or recovered (denoted by $R$). In a discrete time step, the adopted
individuals are enthusiastic about transmitting the contagion to their
susceptible neighbors. After interacting with an adopted neighbor, a
susceptible individual may become an adopter as well with a probability
$\beta$. The adopted individuals may lose interest in advertising and become
recovered (exhausted) with a probability $\mu$. Moreover, the contagion
process takes place on a network with $N$ nodes and $L$ edges.

To capture the individual’s tendency from social influence, the
transmissibility $\beta$ has two components, a constant part, and a time-
dependent part, which is defined as

$\displaystyle\beta(t)=\min(1,\beta_{0}+\alpha\frac{R(t-1)}{N}),\alpha\geq
0,\beta_{0}\in\left[0,1\right],$ (1)

where $\beta_{0}$ model the influence of direct and indirect biased
transmission, which is the basic diffusion force from intrinsic attractiveness
of the new entity, influenced by relative advantage, economic advantage,
perceived compatibility, complexity, observability and so on [34]. The second
term $\alpha\frac{R(t-1)}{N}$ represents the contribution from the macro-level
social influence, regarded as positive feedback from the macro-level back to
the micro-level. The parameter $\alpha$ measures the strength of influence,
and $R(t-1)$ is the number of recovered individuals at time $t-1$. By varying
the value of $\alpha$, we can control the intensity of macro-level social
influence.

Eq. 1 indicates our assumption that the size of recovered population are
transparent to the others. It is reasonable. Since we focus on innovation
diffusion, the market penetration of a product or a new technology is usually
widely reported in the mass media and on the Internet. Our assumption is also
supported by a wide array of existing literature on conformity effect [21, 4,
23, 9], network externalities [26, 24, 27], and decline of marginal cost due
to economies of scale [32, 30, 31].

With the model formally defined, we can simulate it on different network
structures. In each simulation, a node is randomly assigned the $A$ state to
initiate the diffusion, while the remaining nodes are in the $S$ state. The
contagion process proceeds following the rules described above until an
absorbing state is reached where no changes can be observed anymore.
Throughout this paper, the recovery probability $\mu$ is set to $1$.

###  2.2 Theoretical Analysis

In this section, we employ the DMP method to characterize the outbreak size
and threshold analytically. DMP prevents the contagion from backtracking to
the node that it comes from, avoiding the mutual transmission effect [35, 36,
37, 38, 39]. Regardless of the initial conditions, DMP can accurately
approximate the probability of each node being in a specific state at time
$t$, especially for tree-like networks.

####  2.2.1 Dynamic message-passing

First, let us derive the exact equations of DMP. The probabilities of node $i$
being in $S$, $A$, or $R$ states at time $t$ are represented by
$P^{i}_{S}(t)$, $P^{i}_{A}(t)$ or, $P^{i}_{R}(t)$, respectively, with the
following constraint:

$P_{A}^{i}(t)+P_{S}^{i}(t)+P_{R}^{i}(t)=1.$ (2)

The recovery process for an adopted node $i$ does not depend on its neighbors’
state, so $P^{i}_{R}(t)$ can be represented as

$P_{R}^{i}(t)=P_{R}^{i}(t-1)+\mu P_{A}^{i}(t-1).$ (3)

Subsequently, the global quantity $R(t)$ can be obtained using

$R(t)=\sum_{i}^{N}{P_{R}^{i}(t)}.$ (4)

Substituting Eq. (4) into Eq. (1), we have

$\beta(t)=min(1,\beta_{0}+\alpha\frac{\sum^{N}_{i=1}P^{i}_{R}(t-1)}{N}),$ (5)

which increases with $R(t-1)$ until it reaches the upper bound $\beta(t)=1$.

Next, we derive the expression for $P^{i}_{S}(t)$. If we use $\Phi_{i}(t)$ to
denote the probability that the contagion has not been successfully
transmitted to node $i$ from its adopted neighbors until time $t$,
$P^{i}_{S}(t)$ can be represented as

$P_{S}^{i}(t)=P_{S}^{i}(0)\Phi_{i}(t).$ (6)

DMP assumes that the underlying network is a tree and that the state evolution
of node $i$’s neighbors is independent. However, once node $i$ is successfully
persuaded by its neighbor $z$ and switches to $A$ state, it would attempt to
pass the contagion to another neighbor $z^{\prime}$ ($z^{\prime}\neq z$). It
gives rise to a dilemma because the state transitions of node $z^{\prime}$ and
$z$ are clearly correlated. To alleviate this issue, we assume that the focal
node $i$ is a cavity and define $\theta^{z\rightarrow i}(t)$ as the
probability that node $i$ has not been successfully persuaded by node $z$
until time $t$. $\Phi_{i}(t)$ can then be factorized as $\prod_{z\in\partial
i}\theta^{z\rightarrow i}(t)$ where $\partial i$ is the neighbor set of node
$i$. Substituting it into Eq. (6), we have

$P_{S}^{i}(t)=P_{S}^{i}(0)\prod_{z\in\partial i}\theta^{z\rightarrow i}(t).$
(7)

Message-passing is directional, meaning that $\theta^{i\rightarrow
z}(t)\neq\theta^{z\rightarrow i}(t)$ for undirected networks. In our analysis,
we treat each undirected edge as two directed edges pointing to opposite
directions. Initially, we have $\theta^{z\rightarrow i}(0)=1$ for all edges in
the network. Later, $\theta^{z\rightarrow i}(t-1)$ decreases as the contagion
transmits from node $z$ to $i$, which occurs with the probability
$\beta(t)\phi^{z\rightarrow i}(t-1)$ where $\phi^{z\rightarrow i}(t-1)$ is the
probability that the adjacent adopted node $z$ has not passed the contagion to
node $i$ until time $t-1$. Hence, $\theta^{z\rightarrow i}(t)$ follows the
updating rule:

$\theta^{z\rightarrow i}(t)=\theta^{z\rightarrow
i}(t-1)-\beta(t)\phi^{z\rightarrow i}(t-1).$ (8)

Now we need to find the expression for $\phi^{z\rightarrow i}(t)$. On the one
hand, $\phi^{z\rightarrow i}(t)$ decreases when node $z$ in $A$ state
recovers, or when the contagion is successfully transmitted from $z$ to $i$,
or when the two processes occur simultaneously. The corresponding
probabilities for these events are $\mu$, $\beta(t)$, and $\mu\beta(t)$,
respectively. On the other hand, $\phi^{z\rightarrow i}(t)$ increases when
node $z$ in $S$ state becomes $A$. The changing rate $\Delta P^{z\rightarrow
i}_{S}(t)$ can be calculated as $P_{S}^{z\rightarrow
i}(t-1)-P_{S}^{z\rightarrow i}(t)$ where $P_{S}^{z\rightarrow i}(t)$
represents the probability of node $z$ staying in $S$ state after interacting
with the cavity node $i$. By combining all terms we have

$\displaystyle\phi^{z\rightarrow i}(t)$ $\displaystyle=\phi^{z\rightarrow
i}(t-1)-\beta(t)\phi^{z\rightarrow i}(t-1)-\mu\phi^{z\rightarrow i}(t-1)$
$\displaystyle+\mu\beta(t)\phi^{z\rightarrow i}(t-1)+P_{S}^{z\rightarrow
i}(t-1)-P_{S}^{z\rightarrow i}(t)$
$\displaystyle=(1-\beta(t))(1-\mu)\phi^{z\rightarrow
i}(t-1)+P_{S}^{z\rightarrow i}(t-1)-P_{S}^{z\rightarrow i}(t).$

The next step is to express $P_{S}^{z\rightarrow i}(t)$ explicitly. Since node
$i$ is a cavity, node $z$ will stay susceptible as long as it does not get the
contagion from another neighbor $j$. Using Eq. (7), we can obtain the
probability that $z$ remains susceptible when its neighbor $i$ is a cavity:

$P_{S}^{z\rightarrow i}(t)=P_{S}^{z}(0)\prod_{j\in\partial z\backslash
i}\theta^{j\rightarrow z}(t),$ (10)

where $\partial z\backslash i$ represents the neighbors of $z$ except for $i$.
At $t=0$, we have $P_{S}^{z\rightarrow i}(0)=0$ if $z$ is the seed that
initiates the contagion process and $P_{S}^{z\rightarrow i}(0)=1$ otherwise.
This can be represented as $P_{S}^{z\rightarrow i}(0)=1-\delta_{q_{z}(0),A}$
where $\delta_{q_{z}(0),A}$ is the Kronecker function and $q_{z}(0)$
represents the state of $z$ at time $t=0$.

Finally, we can employ Eqs. (5), (8)-(10) to track the exact trajectory of
$\theta^{z\rightarrow i}(t)$, $\phi^{z\rightarrow i}(t)$, and
$P_{S}^{z\rightarrow i}(t)$ with the following initial conditions:

$\theta^{z\rightarrow i}(0)=1,$ (11) $\phi^{z\rightarrow
i}(0)=P^{z}_{A}(0)=\delta_{q_{z}(0),A},$ (12) $P^{z\rightarrow
i}_{S}(0)=P^{z}_{S}(0)=1-\delta_{q_{z}(0),A}.$ (13)

By further combining $P_{R}^{i}(0)=0$ and $\beta(0)=\beta_{0}$ with Eqs. (2),
(3), (5), and (7)-(10) we can calculate the trajectory of $P^{i}_{S}(t)$,
$P^{i}_{A}(t)$, $P^{i}_{R}(t)$, and obtain the order parameter $R(\infty)$.
The computation complexity of DMP is $O(L)$.

####  2.2.2 Outbreak Threshold

To obtain the outbreak threshold, we exert a small perturbation on the
adopter-free state of the system and analyze its linear stability [38, 39]. If
the transmissibility reaches the threshold, the perturbation will lead to a
global outbreak. At the initial stage, we introduce two perturbation
quantities $\epsilon_{i}\ll 1$ and $\delta^{z\rightarrow i}(t)\ll 1$ to Eqs.
(7) and (8) and express it as the following forms:

$P^{i}_{S}(0)=1-\epsilon_{i},$ (14) $\theta^{z\rightarrow
i}(t)=1-\delta^{z\rightarrow i}(t).$ (15)

Equation (15) and Eq. (8) together imply that

$\delta^{z\rightarrow i}(t)=\delta^{z\rightarrow
i}(t-1)+\beta(t)\phi^{z\rightarrow i}(t-1).$ (16)

By substituting Eqs. (14) and (15) into Eq. (10) we get

$P^{z\rightarrow i}_{S}(t)=(1-\epsilon_{z})\prod_{j\in\partial z\backslash
i}(1-\delta^{j\rightarrow z}(t)),$ (17)

and apply the approximation:

$P^{z\rightarrow i}_{S}(t)\approx 1-\epsilon_{z}-\sum_{j\in\partial
z\backslash i}\delta^{j\rightarrow z}(t).$ (18)

By substituting Eq. (16) into Eq. (18) we get

$\displaystyle P^{z\rightarrow i}_{S}(t)$
$\displaystyle=1-\epsilon_{z}-\sum_{j\in\partial z\backslash
i}\delta^{j\rightarrow z}(t-1)-\beta(t)\sum_{j\in\partial z\backslash
i}{\phi^{j\rightarrow z}(t-1)},$ $\displaystyle=P_{S}^{z\rightarrow
i}(t-1)-\beta(t)\sum_{j\in\partial z\backslash i}{\phi^{j\rightarrow
z}(t-1)},$

where we apply the approximation in Eq. (18) again for $P_{S}^{z\rightarrow
i}(t-1)$. We further incorporate Eqs. (2.2.2) back into Eq. (2.2.1) and obtain
the approximation of $\phi^{z\rightarrow i}(t)$:

$\phi^{z\rightarrow i}(t)\approx(1-\beta(t))(1-\mu)\phi^{z\rightarrow
i}(t-1)+\beta(t)\sum_{j\in\partial z\backslash i}{\phi^{j\rightarrow
z}(t-1)},$ (20)

which is a self-consistent equation.

To track the dynamical trajectory of the whole system, we express Eq. (20) in
matrix format as

$\bm{\Phi}(t)=[(1-\mu)(1-\beta(t))\bm{I}+\beta(t)\bm{B}]\bm{\Phi}(t-1),$ (21)

where $\bm{\Phi}(t)$ represents a vector with $\phi^{z\rightarrow i}(t)$ as
the elements, $\bm{I}$ is a $2L\times 2L$ identity matrix, and $\bm{B}$ is a
$2L\times 2L$ non-backtracking matrix with elements

$B_{z\rightarrow i,j\rightarrow
z^{\prime}}=\left\\{\begin{array}[]{cc}1&if\quad z^{\prime}=z,j\neq i,\\\
0&otherwise.\\\ \end{array}\right.$ (22)

For simplicity, we denote

$\bm{\Phi}(t)=\bm{C}(t)\bm{\Phi}(t-1),$ (23)

and use a time-dependent propagator $\bm{P}$ to represent the accumulated
effects of $\bm{C}(t)$:

$\bm{P}(T,\beta_{0},\mu,\alpha)=\prod_{t=1}^{T}\bm{C}(t),$ (24)

and further approximate Eq. (21) as

$\bm{\Phi}(T)=\bm{P}\bm{\Phi}(0).$ (25)

However, it is infeasible to qualitatively evaluate the asymptotic behavior of
the dynamics through the spectral radius of $\bm{P}$ since it depends on time
$t$.

To circumvent this obstacle, we define $\beta_{c}$ as the critical threshold.
If $\beta_{0}<\beta_{c}$, the contagion remains confined to a small set of
nodes and we have $\frac{R(t)}{N}\approx 0$ in the thermodynamic limit. As a
result, we have

${\lim_{N\to+\infty}}\beta(t\rightarrow\infty)=\beta_{0}+\frac{\alpha
R(t\rightarrow\infty)}{N}=\beta_{0},$ (26)

meaning that macro-level social influence is absent in this case. Now consider
the scenario of $\beta_{0}>\beta_{c}$ where nodes in $R$ state will accumulate
in the system and further amplify the positive feedback. It suggests macro-
level social influence only plays a critical role above $\beta_{c}$. The
position of phase transition does not change but is independent of $\alpha$,
and can thus replace $\beta(t)$ with $\beta_{0}$ in Eq. (21). As a result, we
have $\bm{C}=[(1-\mu)(1-\beta_{0})\bm{I}+\beta_{0}\bm{B}]$ and
$\bm{P}=\prod_{t=1}^{T}\bm{C}$ and it is possible to evaluate its asymptotic
behavior through the spectral radius of $\bm{C}$, i.e., $\rho(\bm{C})$. When
the phase transition occurs, we have

$\rho([(1-\mu)(1-\beta_{0})\bm{I}+\beta_{0}\bm{B}])=1.$ (27)

Given that $0\leq\beta_{0},\mu\leq 1$ and all nodes in the network belong to a
strongly giant connected component, $\bm{C}$ is non-negative and irreducible.
According to the Perron–Frobenius theorem [40, 41], we have
$\lambda_{C}^{max}=\rho(\bm{C})$, $\lambda_{C}^{max}>0$, and there exists an
positive eigenvector $v$ corresponding to $\lambda_{C}^{max}$. With these
relations, we can further simplify Eq. (27) as
$(1-\mu)(1-\beta_{0})+\beta_{0}\rho(\bm{B})=1$ and obtain the outbreak
threshold as

$\beta_{c}=\frac{\mu}{\rho(\bm{B})+\mu-1},$ (28)

where $\rho(\bm{B})$ is the the spectral radius of $\bm{B}$. The result
suggests that the outbreak threshold is only determined by the network
structure and recovery rate, regardless of the intensity of the macro-level
social influence. Specifically, we have $\beta_{c}=\frac{1}{\rho(\bm{B})}$ if
$\mu=1$.

##  3 Results

In this section, our study consists of two parts. Firstly, we in turn
implement the SAR model on classical networks, including 2D lattices, ER, WS,
BA, and SF networks with different heterogeneity and assortativity.
Subsequently, we refine a physical picture to understand why localization
strength determines the magnitude of the tricritical point.

###  3.1 Results on Network Models

Figure 1: Understanding the effects of network density on spreading dynamics.
The density of final adopted population $R(\infty)/N$ is plotted versus
intrinsic attractiveness of medium $\beta_{0}$ for four types of classical
networks of size $N=10^{5}$: (a) 2D lattices, (b) WS, (c) ER and (d) BA
network. In each subplot, the results with $\langle k\rangle=4,6,8$ are
plotted, where each marker denotes a single numerical realization of the
developed SAR model with $\alpha=1$ and colored solid lines are obtained by
the DMP-based approach.

Figure 1 shows the final adopted populations obtained from the developed SAR
model, on classical networks such as 2D lattices, WS, ER, BA, with different
mean degrees $\langle k\rangle=4,~{}6,~{}8$. Definitely differing from the
dynamic scenarios formulated by the SIR model, the macro-level social
influence can widely induce a discontinuous phase transition. More
specifically, $R(\infty)/N$ abruptly jumps to $1$ at the threshold on 2D
lattices, attributing to that the nodes have the same regular connections such
that PFDF can constantly push the contagion process forward. However, networks
with the same density present smaller jumps with the increase of the degree
heterogeneity from WS, ER, to BA [Figs. 1(b)-(d)], suggesting that the
intensity of feedback is gradually weakened. The effect of density on the
critical behavior of the system also depends on the heterogeneity of degree,
highlighting that both zero- and first-order statistics of networks determine
the critical behaviors. Overall, the results in Fig. 1 seem to indicate a
discontinuous transition as long as there is a macro-level social influence in
the process of adoption decision, regardless of the structure details of the
networks. Furthermore, the DMP-based approach agrees well with numerical
simulations except for 2D lattices and WS, because numerous short-loop
structures break the assumption of independence between nodal states.

Figure 2: Understanding the effects of network heterogeneity on spreading
dynamics. Numerically obtained (blue circles) and theoretically predicted
(solid lines) final fractions of the adopted population ($R(\infty)/N$) are
plotted versus $\beta_{0}$ for SF networks with different values of $\gamma$.
The network size, the parameter $\alpha$, and the numerical simulation are the
same as that in Fig. 1. For $\gamma=2.4,3.3$, the height of the jump is close
to zeros, causing the nature of phase transition is not clear-cut. According
to the change from $\gamma=2.2$ to $3.5$, we infer that $\gamma=2.4,3.3$ is at
the boundary between continuity and discontinuity.

More insights are obtained by investigating the effect of network
heterogeneity on the nature of phase transition. Here, we generate the SF
network with different degree exponents $\gamma$ by means of a configuration
model. More structure details of the SF network are listed in Table 1 of
Appendix A. Correspondingly, Fig. 2 illuminates the complex effects of network
heterogeneity in tuning the nature of phase transitions. When the network is
nearly star-like ($\gamma$ approaching $2$), $R(\infty)/N$ undergoes a
discontinuous transition [Figs. 2(a)-(c)]. A striking phenomenon then is
exhibited, that continuous transitions instead emerge as $\gamma$ increase to
the intermediate range of the scale-free regime ($2.6\lesssim\gamma\lesssim
3.2$) [Figs. 2(d)-(f)]. However, in the regime of random networks
($\gamma>3.2$), again, the continuous transition is replaced by discontinuous
class [Figs. 2(g)-(i)]. The above phenomena are contrary to our expectation
based on the results given by Fig. 1, that macro-level social influence is
destined for a discontinuous transition; and unlike the conclusions drawn from
previous threshold models concerning reinforcement, that increasing network
heterogeneity would lead to a continuous transition [42]. Note the good
agreement between numerical and theoretical results, which validates the DMP-
based approach. It is claimed that the unusual shift of phase transition with
heterogeneity is related to the spatial configurations of nodes, naturally,
indicating the necessity to further understand the effects of the second-order
statistics, i.e. network assortativity.

Figure 3: Illustrating the effects of network assortativity on phase
transitions. In (a)(e)(i), numerically obtained (markers) and theoretically
predicted (solid lines) final fractions of the adopted population
($r(\infty)$) are respectively plotted versus $\beta_{0}$, for neutral,
assortative, and disassortative SF networks with $\gamma=3.3$. The assortative
($r=0.170$) and disassortative networks ($r=-0.057$) are generated by the
Xulvi-Brunet-Sokolov algorithm, without changing the degrees of nodes. The
configurations of simulations are the same as that in Fig.1. Black vertical
lines mark the position of the outbreak thresholds estimated in terms of the
susceptibility $\chi=[\langle r(\infty)^{2}\rangle-\langle
r(\infty)\rangle^{2}]/\langle r(\infty)\rangle$), and colored vertical lines
the theoretical thresholds given by Eq. (28). Correspondingly, the evolution
of adopted nodes for different degree classes around the outbreak threshold is
illustrated for the neutral network (b)-(d), assortative network (f)-(h), and
disassortative network (j)-(l).

The effects of network assortativity on the nature of phase transition are
elucidated by varying the assortativity of SF networks, as shown in Fig. 3.
Here, we employ the Xulvi-Brunet and Sokolov algorithm to generate either
assortative or disassortative networks, by preserving the same degree sequence
as neutral SF networks [43]. Interestingly, in spite of following the
identical degree sequence, increasing assortativity alone can change the
transition from being abrupt to continuous. Figure 3(b)-(d),(f)-(h),(j)-(l)
provide a visual picture of the results in Figs. 3 (a), (e) and (i), which
give a clear understanding. In the assortative network, the adoption seldom
occurs in nodes with small degrees, it tends to localize near the hubs under
the macro-level social influence [Figs. 3(f)-(h)]. The degree correlation
coefficient of the neutral network is slightly positive, hubs do not have
sufficient feedback to drive non-hubs [Figs. 3(b)-(d)], exhibiting a
continuous transition. Figure 3(j)-(l) document a much more sustainable
process, i.e., an avalanche-like outbreak happens as the degree correlation
coefficient $r$ decreases to be negative. Notice that the threshold becomes
larger with decreasing assortativity, in accordance with the interpretation
put forward in Ref. [44].

Figure 4: The nature of phase transition on the SF networks of different
assortativity: (a) neutral network, (b) assortative network, and (c)
disassortative network. Mass distribution of the final adopted population at
the outbreak threshold is obtained from $10^{6}$ realizations. The size of the
employed networks ranges from $N=10^{4}$ to $5\times 10^{5}$. $\gamma=3.3$ and
$\alpha=1$. The solid straight lines represent the decay, where the exponent
is approximated through the maximum likelihood method. The peaks on the right
side, separated clearly from the left power-law distribution, corresponds to
giant adopted clusters. The emergence of a giant adopted cluster and linear
scaling relationship $m_{peak}\sim N$ indicates a discontinuous transition,
and a continuous transition otherwise. $m_{peak}$ denotes the size of the
giant adopted cluster.

Still, the DMP-based approach provides accurate predictions, with respect to
not only the final adopted fractions of the population but also the position
of the threshold [Fig. 3(a)(e)(i)]. We assume that the threshold does not
change with the macro-level social influence, which is proved to be rational
by the results illustrated in Fig. 8. The threshold does not depend on
$\alpha$ and is equal to the threshold of the classical SIR. One clearly
observed that the theoretical lines and numerical markers highly coincide with
each other, regardless of assortativity (See Appendix C for more details).
Moreover, the conclusion about the nature of the transitions in Fig. 3 is
strongly supported by the simulation results illustrated in Fig. 4. As the
networks change from assortative to disassortative, the nature of the
transition passes from continuous to discontinuous, along with a larger
threshold.

###  3.2 A comprehensive understanding of the joint effects

Figure 5: Understanding the joint effects of network structures and macro-
level social influence. (a)-(c) The tricritical point $\alpha_{c}$ as function
of mean degree $\langle k\rangle$, degree exponent $\gamma$, and degree
correlation coefficient $r$, respectively. $\alpha_{c}$ is identified through
the patterns of mass distribution of $r(\infty)$ at the theoretical
$\beta_{c}$, as shown in Fig. 4. The size of networks is $N=10^{5}$. In (a),
the SF networks with different degrees are generated by the configuration
model, by increasing the minimum degree $k_{min}$ from $2$ to $10$. The
positions of the four labeled parameter points are $\langle k\rangle=2.75$,
$\langle k\rangle=4.50$, $\langle k\rangle=9.84$ and $\langle k\rangle=15.06$,
with $\alpha=1.0$. In (b), we generate the SF network with different $\gamma$
by the Goh-Kahng-Kim algorithm put forward in Ref. [45]. The positions of the
four labeled parameter points are $\gamma=2.1$, $\gamma=2.6$, $\gamma=3.1$ and
$\gamma=3.9$, with $\alpha=2.16$. In (c), the networks with different $r$ are
generated by the Xalvi-Brunet-Sokolov algorithm, starting with a neutral
scale-free network with $\gamma=3.3$ and $\langle k\rangle=2.75$. The
positions of the four labeled parameter points are $r=-0.060$, $r=-0.048$,
$r=-0.016$, and $r=0.059$, with $\alpha=1.0$. For each subplot in (a)-(c), the
dynamic evolution of the order parameter $R(t)/N$ at theoretical $\beta_{c}$
are illustrated in (d)-(f), corresponding to the labeled points, i.e. I, II,
III, IV in (a)-(c). The simulation results are from 2000 realizations, where
each realization represents a curve. In (g)-(i), the structural patterns of
size $N^{{}^{\prime}}=50$ give a more intuitive picture, without leaving the
key structural properties of the employed networks in (d)-(f). Darker and
larger nodes indicate more connections they have.

The aforementioned results for classic networks highlight three structure
terms that can regulate the macro-level social influence on phase transitions:
density (or mean degree), heterogeneity, and assortativity, as 0-order,
1-order, and 2-order statistics of the network. A continuous transition occurs
if relatively heterogeneous networks are assortative and sparse enough.
However, a clear physical picture of the joint effects of network structure
and macro-level social influence is still lacking. In this section, we will
give a comprehensive understanding of the dependence of the nature of phase
transition on strength of macro-level social influence induced by structural
patterns of SF networks.

Figure 6: Localization effect on the networks corresponding to Fig. 5. (a)-(c)
Non-backtracking centrality $f_{i}(\Lambda_{1})$ of each node in the network
is exhibited in ascending order. The continuous transition is plotted with
solid lines if $\alpha_{c}>\alpha$, otherwise dashed. (a) $\alpha=1$. (b)
$\alpha=2.16$. (c) $\alpha=1$. (d)-(f) Localization strength ($\mathcal{L}$)
versus structural quantities. $\mathcal{L}=\frac{1}{\langle
k\rangle}(\frac{1}{N}\sum_{i=1}^{N}(f_{i}(\Lambda_{1})-\langle
f_{i}(\Lambda_{1})\rangle)^{2})^{1/2}$ where $f_{i}(\Lambda_{1})$ denotes the
non-backtracking centrality of node. A larger $\mathcal{L}$ corresponds to a
stronger localization effect in networks.

We vary individual structural parameters to generate the networks, with other
structure variables remaining fixed as much as possible. Figure 5 focuses on
how the critical strength of macro-level social influence $\alpha_{c}$, beyond
which the transition changes from continuous to discontinuous, behaves with
network density (mean degree), heterogeneity, and assortativity, respectively.
Figure 5(a) shows an overall decrease of $\alpha_{c}$ with mean degree
$\langle k\rangle$, but a more slow descent as the networks become rather
dense. This is in agreement with what is found in Fig. 1, that the denser the
network is [Fig. 5(g)], the more easily macro-level social influence can
facilitate an avalanche-like contagion [Fig. 5(d)]. Sparse networks have many
smaller-degree nodes and long-loops [Fig. 5(gI)], contributing to a weaker
PFDF and thus a continuous transition [Fig. 5(dI)]. A distinctive behavior
observed in the regions of discontinuous transitions is an additional
explosive growth [Fig. 5(dII)(dIII)(dIV)], since the contagion can reach the
hubs and becomes avalanche-like through sufficient available channels to those
small-degree nodes [Fig. 5(gII)(gIII)(gIV)].

Nevertheless nonlinear dependence of $\alpha_{c}$ on degree exponent $\gamma$
and degree correlation coefficient $r$ can be correspondingly observed in
Figs. 5(b) and (c). More in detail, in Fig. 5(b), $\alpha_{c}$ firstly
increases to about $3.8$ in the intermediate scale-free regime and then keep
decreasing to a flat region in the random network regime. It is evident that a
discontinuous transition is more likely to occur if connections are either
extremely heterogeneous or homogeneous, in line with the conclusions drawn
from Fig. 2. Discontinuous transition can be found at the parameter points I
and IV, however, the mechanisms lying behind are fundamentally different. With
small $\gamma$, star-like SF network owns more numerous small-degree nodes, as
well as super hubs, which together bring about larger mean degree [Table. 1
and Fig. 5(hI)]. Due to a considerable number of close contacts and shorter
length of the path, the super hubs can readily push the contagion to
penetratingly reach the patterns of small-degree nodes, to some extent which
weakens the macro-level social influence [see Fig. 5(eI) and Fig. 2(a) for
jumps of smaller size]. In cases of large $\gamma$ (e.g. parameter point IV),
hubs continue to be present, but small and less numerous, the difference in
the degree of individuals is thus insignificant, as shown in Fig. 5(hIV).
There are abundantly available transmission channels among hubs and non-hubs
for the contagion, rather than being simply impeded by hubs. Hence, the macro-
level social influence persistently reinforces individual adoption
probability, more easily leading to an avalanche-like outbreak [Fig. 5(eIV)].
The picture is obviously distinct when the networks are in an intermediate
range of $\gamma$ (e.g. parameter point II), where hubs are numerous and
cluster together, such that they have insufficient connections to make the
contagion immediately stretch over the ”periphery architecture” consisting of
small-degree nodes [Fig. 5(hII)]. This is confirmed by the temporal evolution
patterns that the growth rate is relatively low, along with a short duration
[Fig. 5(eII)]. It is consequently easy to have a continuous transition. In
addition, parameter point III is at the boundary between continuous and
discontinuous transition, at which the temporal evolution pattern has no
avalanche [Fig. 5(eIII)].

Figure 5(c) shows that increasing $r$ has a monotonous growth of $\alpha_{c}$,
except for a slight drop when the network is disassortative. The generated
assortative network contains a tight core of nearly all hubs through which
numerous long loops pass, surrounded by sparse periphery architectures of
small-degree nodes [Fig. 5(iI)]. Therefore, it is not easy for the contagion
to persist along these long loops for a long time [Figs. 3(f)-(h) and Fig.
5(fI)], especially after that it stretches into the periphery architectures
far away from the hubs. As the network approaches a disassortative state, hubs
are increasingly partitioned by small-degree nodes [Fig. 5(iII)(iIII)], and
hubs have more chances to connect with non-hubs. In such cases, the hubs play
the role of ”Destination Charger” with the assistance of the macro-level
social influence, which can drive the medium to diffuse through surrounding
circles of non-hubs to reach other hubs, triggering further contagion. The
above process repeats constantly and finally gives rise to an avalanche-like
outbreak [Fig. 5(fII)(fIII)]. However, a more drastic avalanche cannot be
expected as $r$ decreases to be rather negative [Fig. 5(cIV)], attributing to
a long distance between the hubs and that the non-hubs with a larger degree
are moving away. There are thus more long loops along which the contagion
frequently fails to persist [Fig. 5(iIV)].

We finally uncover a sufficient condition for a continuous transition:
simultaneously, being sufficiently sparse, heterogeneous, either assortative
or rather disassortative; otherwise, the transition is instead discontinuous.
However, one needs to know what mechanism ultimately dominates the nature of
phase transition in the macro-level social influence, whose effects can also
be regulated by network density, heterogeneity, or assortativity. Figure
6(a)-(c) plot the non-backtracking centrality of nodes in ascending order, for
the networks employed in Fig. 5(a)-(c), which provides key hints on the issue.
Combining with results in Fig. 6, we find that, if the divergence of the non-
backtracking centrality is sufficiently great, i.e. more apparent height
difference between the plain and the platform or peak, the transition is
continuous; and discontinuous otherwise. The nodes lying in the plain belong
to periphery structure patterns, in contrast to those hubs constructing a
tight core located in the platforms or peaks. While the core-periphery
structure indicates strong localized effects, which is the mechanism we are
seeking for. An avalanche happens on condition that the contagion is not
localized in the core pattern, and persistently preserved by the ”Destination
Chargers”-hubs to stretch over the periphery areas. In other words, more
likely, we will have a continuous transition if the localization effects
become pronounced, meaning that there is a tight core pattern to capture a
considerable fraction of connections, leaving fewer connections to the
periphery structure (this is why the whole network must be sparse).
Accordingly, we use the metric $\mathcal{L}$ to quantify the localization
strength of networks [13]. Figure 6(d)-(f) show that the changing trend of
$\mathcal{L}$ is qualitatively consistent with $\alpha_{c}$ by varying the
network density, heterogeneity, and assortativity, revealing that the
magnitude of the tricritical point is governed by the localization strength of
network structure. Therefore, it is easier to preserve a continuous transition
if $\mathcal{L}$ in the network is larger, which provides a basis to evaluate
the phase transition for real networks. While the measures reducing the
localization effect of the network, such as increasing the mean degree and
making the network less heterogeneous or assortative, can change the
transition from continuous to discontinuous.

##  4 Discussion

In this paper, we have performed analysis of the innovation diffusion model
incorporating both the micro and macro level social influence on classical
synthetic networks. In addition, we have applied the DMP method to track the
final fractions of recovered population, and to derive an expression for the
outbreak threshold which does not depend the intensity of the macro social
influence, in agreement with simulations. With a given intensity of macro
social influence of $\alpha=1$, a discontinuous transition is still exhibited
on classical networks such as 2D lattices, WS, ER, and BA networks, regardless
of the structure details. Moreover, network heterogeneity is the premise to
unfold the effects of network density in facilitating a discontinuous phase
transition. Then, an interesting and striking point regards what happens for
SF networks, i.e., the transition is continuous when the network is in the
intermediate range of scale-free regime ($2.6\lesssim\gamma\lesssim 3.2$). We
further tune the assortativity of the SF network with $\gamma=3.3$ by means of
the Xulvi-Brunet and Sokolov algorithm, uncovering that whether continuous
transition can be observed depends on that positive feedback capable of
promoting the contagion tends to be localized near those hubs, which requires
an assortative structure.

In the second part of this paper, we provided a comprehensive understanding of
the dependence of tricritical point $\alpha_{c}$ on three structure
quantities: network density, heterogeneity, and assortativity, giving a clear
physical picture of their joint effects on the localization strength. In more
detail, we find that, apart from a weak intensity of macro-level social
influence, sparsely available connections, and relatively heterogeneous
distribution, either assortative or extremely disassortative configurations
are simultaneously indispensable for a continuous transition. The core-
periphery structure is thus required. Otherwise, a discontinuous transition is
instead more easily achieved. The hubs persistently preserve the contagion to
have an avalanche, when there are sufficient available channels, the network
is either star-like or homogeneous, or there are appropriate intervals of non-
hubs between the hubs i.e., being appropriately assortative. We further report
a strong confirmation of the association between the tricritical point and
localization effects. The issue can be clarified by plotting the spectrum of
non-backtracking centrality of networks, and the localization strength
measured by the metric developed in Ref. [13]. More specifically, core-
periphery structures yield a strong localization effect, and thus a continuous
transition. Our analysis highlights that core-periphery structure, being
dependent on network density, heterogeneity, and assortativity, is the
structure pattern that essentially governs localization effects, and the
magnitude of the tricritical point in presence of macro-level social
influence.

We conclude our study by providing two general remarks. The macro-level social
influence turns out to be a non-negligible ingredient in innovation
propagation. It essentially acts on the diffusion process with a form of
positive feedback and is thus capable of inducing a discontinuous transition.
However, based on our results, we can say that strong positive feedback is not
a sufficient condition for the occurrence of discontinuous phase transition.
It is also up to localization strength governed by core-periphery structure.
Secondly, our study suggests that the positive feedback between the macro and
micro scales of complex systems is proved to be a useful tool to underline the
effects of the mesoscopic structure patterns such as core-periphery structure
on the local micro dynamics. It guides future research direction.

It should be noted that the network models employed in this manuscript cannot
capture all structural characteristics of natural or man-made networks,
especially higher-order statistics such as high clustering or community
structures found commonly in real-world networks. These factors are
potentially related to the change of core-periphery structure [53]. Therefore,
further attempts to explore the relationship of them are still required in the
future. In addition, the investigation of multiple feedback in general complex
contagion dynamics remains a very interesting avenue for future research
activity. We also hope to be able to extend our study to different contexts,
as well as to more diverse complex networks such as metapopulation networks
[46, 47], temporal networks [48], multilayer networks [49, 50], hypergraphy
[51, 52] and those have hierarchical core-periphery structures [53].

## Acknowledgments

This work was supported by the Key Program of the National Natural Science
Foundation of China (Grant No. 71731002), and by Guangdong Basic and Applied
Basic Research Foundation (Grant No. 2021A1515011975).

##  Appendix A Statistical characteristic of SF networks

Table 1: The basic statistical characteristic of SF networks generated by configuration model: degree exponent $\gamma$, the number of nodes $N$, the number of edges $L$, mean degree $\langle k\rangle$, assortativity coefficient $r$, where the minimum and maximum degree are respectively set as $k_{min}=1$ and $k_{max}\sim\sqrt{N}$ to avoid structural disassortativity. $\gamma$ | $N$ | $L$ | $\langle k\rangle$ | $r$
---|---|---|---|---
2.2 | 100000 | 296511 | 5.9302 | -0.0049
2.3 | 100000 | 262034 | 5.2407 | -0.0047
2.4 | 100000 | 235218 | 4.7044 | -0.0015
2.6 | 100000 | 197329 | 3.9466 | -0.0012
2.9 | 100000 | 162283 | 3.2457 | -0.0046
3.2 | 100000 | 143425 | 2.8685 | 0.0002
3.3 | 100000 | 137872 | 2.7574 | 0.0044
3.4 | 100000 | 135253 | 2.7051 | 0.0006
3.5 | 100000 | 131266 | 2.6253 | -0.0033

##  Appendix B Outbreak threshold

Figure 7: The convergence of DMP. The density of the final adopted population
at time $T$ is plotted as a function of the intrinsic attractiveness
$\beta_{0}$. The result is obtained on the neutral scale-free networks
($\gamma=3.3$) by averaging $20$ different initial conditions.

The density of the final adopted population is plotted as a function of
$\beta_{0}$ for different time steps [Fig.7]. For T $\geq$ 200, all points
agree well with each other, suggesting that the result obtained through DMP-
based approach has converged. Therefore, we obtain $r(\infty$) for $T=300$ in
this paper.

Figure 8: The outbreak threshold derived from the DMP-based approach and
numerical simulation. (a) Temporal evolution of $\beta(t)$ for the network of
different sizes ($N$) at theoretical outbreak threshold $\beta_{c}$. (b)
Temporal evolution of $\beta(t)$ for different intrinsic attractiveness
($\beta_{0}=\beta_{c}$, $\beta_{c}^{up}$, $\beta_{c}^{down}$). (c) The
adoption probability at $T=300$ is plotted as a function of $\beta_{0}$. The
gray line denotes $\beta_{0}=\beta_{c}$, and a black line with a slope of 1 is
used to guide the eye such that one observes clearly $\beta(t=300)=\beta_{0}$
when $\beta_{0}\leq\beta_{c}$. (d)-(f) The outbreak thresholds under different
$\alpha$ are identified by the DMP-based approach (solid line) and the
susceptibility (markers). The employed SF networks have different
assortativity: (d) neutral, (e) assortative, and (f) disassortative networks.
The network size is $N=10^{5}$, and $\gamma=3.3$.

The phase transition derived by the DMP-based approach does not change with
the macro-level social influence as the system approaches the thermodynamic
limit ($N\rightarrow\infty$). Here, we verify that the argument still holds in
a large-scale system by the temporal evolution of adoption probability. Fig.
8(a) shows that $\beta(t)$ does not change with the evolution of $t$ when the
size of the network reaches the $10^{5}$, suggesting that the macro-level
social influence does not work at theoretical $\beta_{c}$. However, $\beta(t)$
changes dramatically with $t$ once a small perturbation occurs above
$\beta_{c}$ [Fig. 8(b)]. A clear result is shown in Fig. 8(c), $\beta(t)$
starts to deviate from the diagonal when $\beta_{0}>\beta_{c}$. In addition,
outbreak thresholds identified by susceptibility on networks with different
degree correlation coefficients do not depend on $\alpha$ [Fig. 8(d)-(f)],
further supporting the conclusion that the position of phase transition does
not change with the macro-level social influence.

##  Appendix C The effect of multiple seeds

To test the effect of multiple initial seeds on the diffusion process, we
consider scale-free networks with different assortativeness levels and vary
the number of seeds. We find that adding more seeds can greatly increase the
probability of successfully initiating diffusion (i.e., $R(\infty)/N>5\%$, see
Fig. 9) and result into shrinkage of zero-value markers (see Fig. 10), while
leaving the dynamical characteristics of the diffusion process unchanged (see
Fig. 10). Therefore, we can say the number of seeds does not affect our
conclusion from the main text.

Figure 9:  The effect of seed number on the proportion of successful outbreaks
(i.e., $R(\infty)/N>5\%$). We show the results on (a1)-(a4) neutral, (b1)-(b4)
assortative and (c1)-(c4) disassortative SF networks, which are the same as
those used in Fig. 3 in the main text. The value of $p$ indicates the
percentage of seed nodes in each case. The results are obtained from $100$
independent realizations. The zero-value markers are excluded for clarity.
Figure 10:  The effect of seed number on the diffusion dynamics. We show the
results on (a1)-(a4) neutral, (b1)-(b4) assortative, and (c1)-(c4)
disassortative SF networks, which are the same as those used in Fig. 3 in the
main text. The value of $p$ indicates the percentage of seed nodes in each
case. The results are obtained from $100$ independent realizations.

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