Inertia mechanisms in networks and knowledge diffusion in team inertia networks

Abstract

Knowledge diffusion is a critical driver of innovation, and team-based diffusion has increasingly become the dominant mode of knowledge diffusion. However, its efficiency is strongly shaped by inertia mechanisms embedded within teams. To capture this phenomenon, this paper first formalizes inertia mechanisms in networks, distinguishing between temporal inertia and spatial inertia. Building on this foundation, we propose a team inertia network model that integrates both forms of inertia and specify its evolutionary dynamics. Through extensive simulation experiments, we examine how temporal and spatial inertia jointly affect knowledge diffusion. The results indicate that moderate spatial inertia can enhance the average knowledge level of a team, whereas excessive or insufficient spatial inertia reduces diffusion efficiency. The impact of temporal inertia depends on communication density and knowledge dimensionality, where tighter connectivity enables members to gain more balanced benefits, reflected in a more stable Gini coefficient and more stable negative spatial autocorrelation. This work contributes a novel modeling framework for analyzing team knowledge evolution and provides practical insights for designing team structures and communication strategies that leverage inertia mechanisms to improve knowledge sharing, innovation capacity, and collaborative efficiency.

Introduction

Knowledge diffusion is a crucial part of the knowledge production¹ and is also an inevitable result of the imbalance in knowledge production². With the exponential growth of knowledge in human society, the specialization and refinement of knowledge production have continuously advanced. The crucial role of knowledge diffusion in the production is increasingly emphasized³, becoming a significant factor influencing the efficiency of knowledge innovation⁴. It also promotes team collaboration as the main form of knowledge production and the key to major scientific discoveries^5,6,7.

Scientific research collaboration networks are a structured form for heterogeneous team members to share, absorb, and exchange knowledge, achieving the internal knowledge diffusion within research teams⁸. Related studies have shown that individual knowledge diffusion behavior is relatively stable over time^9,10, which implies that there exists a mechanism of temporal inertia in knowledge diffusion within a team, which makes the current knowledge diffusion behavior of team members influenced by past behavior. Due to the existence of decision preferences, team members are not entirely influenced by history, individual behavioral preference decisions still have a significant impact on cooperation¹¹. Since decisions are made by members occupying specific positions within the network, we refer to this as spatial inertia in this paper. In this way, the temporal and spatial inertia of a team will jointly shape its knowledge diffusion dynamics.

Currently, most of the research on knowledge diffusion is based on complex networks¹², and many scholars have fully researched knowledge diffusion from a variety of complex network models^13,14,15,16. Nevertheless, the team inertia mechanism and its impact on knowledge diffusion have received scant attention from scholars. In view of this, this paper lucidly expounds the inertia mechanism of the network. Based on this, the team inertia network model and its evolution mechanism are abstracted, and the knowledge diffusion effect of this network is simulated and analyzed. This work aims to more accurately capture the dynamics of team knowledge diffusion and is anticipated to provide meaningful implications for enhancing the efficiency of knowledge diffusion in team settings.

Related work

Academic research on knowledge emerged in the 1980s¹⁷. At that point, the research emphasis was primarily centered on knowledge management within enterprises and the long-term growth of the national economy^18,19. At the end of the 20th century, the statistical mechanics of complex networks attracted great attention from the scientific community. The study of the topological properties of large networks led to the emergence of small-world networks and scale-free networks²⁰, which provided a completely new methodological approach for the study of knowledge diffusion.

Knowledge diffusion takes place on the social network established by specific social relationships²¹, and it is a process wherein network members are impacted by the experiences of others²². Its outcome is frequently manifested through the performance of the knowledge recipient²³, Phelps et al.²⁴ refer to such networks as “knowledge networks”. Academic research on knowledge diffusion follows two paths: one is to construct knowledge networks based on real data and reveal the nature of knowledge diffusion, and the other is to construct knowledge networks through modeling and discussing the efficiency of knowledge diffusion. The former has a relatively small amount of literature, while the latter is more abundant.

Scale-free or small-world properties are prevalent in large real networks^25,26, and similar properties on real knowledge networks have received support from relevant studies. Chen & Hicks²⁰ analyzed the patent citation data of the United States from 1975 to 1999, confirmed that the patent citation network is a scale-free network, the citation degree distribution follows a power law, and on this basis proposed a visualization method for the knowledge diffusion path. Choe et al.²⁷ undertook topological analyses of multinational patent citation networks from the viewpoints of countries, institutions, and technical fields. The findings revealed that all of them displayed scale-free characteristics with a power-law degree distribution. Existing studies have affirmed that knowledge networks with high clustering have a more rapid rate of knowledge diffusion²⁸, Xiang & Cai²⁹ carried out a research on the invisible knowledge diffusion network established based on the patent data of the Chinese power system, and found that it has the characteristics of an obvious small-world network, and higher clustering coefficient shorten the overall distance of the network, which facilitates the rapid diffusion of knowledge. Sadatmoosavi et al.³⁰ studied a co-author network consisting of countries in the field of nuclear science and technology, and the results showed that the network has high clustering coefficient and short mean paths, and exhibits small-world network characteristics.

Differences in the structure of knowledge networks usually show the heterogeneity of knowledge diffusion effects^31,32,33,34, which is more reflected in the knowledge diffusion modeling based on classical complex networks. Lin & Li¹⁵ studied the knowledge diffusion process on regular network, small-world network, random network, and scale-free network, and found that there are large differences in the knowledge diffusion effects of different network structures, and scale-free networks have the best knowledge diffusion effects. Zhang et al.¹⁴ also simulated the knowledge diffusion efficiency of these four representative network models, and showed that the small-world network has excellent performance in terms of total knowledge and growth rate, and the most uniform knowledge distribution. Representative scholars have also introduced different types of hypernetworks into knowledge diffusion modeling. Zhao et al.³⁵ constructed a dynamic knowledge hypernetwork model that includes both knowledge networks and enterprise networks, studying the process of enterprise knowledge creation and diffusion in uncertain network environments. The results show that the stock of network knowledge grows exponentially, and cooperation strategies based on knowledge are superior to those based on networks. Li & Ba¹² introduced hypergraph mathematical theory to construct an evolution model of knowledge diffusion based on a scientific research cooperation hypernetwork. They conducted simulation studies on the structural characteristics of the scientific research cooperation network, node selective preference, knowledge growth and aging, knowledge diffusion pathways, and the knowledge diffusion process. The relevant conclusions provide foresight for this study. Some scholars have constructed knowledge diffusion models from different perspectives. Cowan & Jonard³⁶ modeled knowledge diffusion as a barter process, and research shows that the performance of knowledge diffusion is influenced by the network structure and exhibits clear small-world characteristics. Li & Sun¹⁸ regard the knowledge growth caused by knowledge diffusion as a cooperative production of knowledge products. Based on the WS model, they introduced the Cobb-Douglas production function to construct a knowledge diffusion model. The study indicates that under the same conditions, the degree of network randomness is closely related to the speed of knowledge diffusion and the uniformity of knowledge distribution.

In reality, knowledge diffusion frequently takes place at the team level³⁷. Knowledge is transferred, shared, or integrated among team members³⁸, and during this process, it demonstrates inertial characteristics. From the perspectives of organizational behavior and behavioral economics, knowledge production is a highly complex and decentralized cognitive process that is difficult for an individual to independently generate high-quality knowledge³⁹. Teams modularize complex tasks through functional division and role complementarity⁴⁰, allowing members to focus on their respective areas of expertise, thereby achieving collaborative knowledge creation and efficiency optimization. At the same time, institutionalized communication norms, collaboration processes⁴¹, and sharing mechanisms within the team not only ensure the smooth flow of knowledge among members but also facilitate the accumulation and reuse of knowledge⁴², making the team itself a core unit for knowledge production and dissemination. On this basis, the historical experiences of team members will generate path dependence on a temporal scale⁴³, while cooperative preferences prompt knowledge to flow selectively along existing channels. These behavioral inertias further shape the efficiency and direction of knowledge diffusion. Evidence also exists that path dependence and cooperation preferences manifest at the level of network formation and evolution, with cooperation preferences often expressed through homophily and preferential attachment mechanisms⁴⁴. Guimera et al.⁴⁵ earlier proposed a self-assembly model for creative teams, incorporating the tendency of incumbents to repeat previous collaborations; Palla et al.⁴⁶ investigated the time-dependence of overlapping communities for the first time on a large-scale dataset, revealing the fundamental relational features of community evolution; Uzzi et al.⁴⁷ analyzed 17.9 million papers in the scientific field and found that influential scientific discoveries are primarily based on previous work. Wu et al.⁴⁸ based on data from more than 65 million papers, patents, and software products, found that large teams are more path-dependent and tend to develop existing ideas and opportunities than small teams. Furthermore, Ye et al.⁴⁹ explicitly identified “individual inertia” and “trend-seeking” as individual decision-making mechanisms, revealing how these two behavioral mechanisms shape the collective patterns of social diffusion at the group level and highlighting the significant role of inertia in social transmission. However, despite these insights, the study of knowledge diffusion at the team level, and the behavioral inertias manifested through intra-team path dependence and cooperation preferences, has not yet received systematic attention in the academic literature. While path dependence and cooperation preferences capture behavioral inertias at the individual and intra-team level, a systematic analysis of network-level knowledge diffusion requires an abstracted framework that can formalize these tendencies. To this end, we conceptualize behavioral inertia along two complementary dimensions: temporal inertia, reflecting the persistence of actions over time, and spatial inertia, capturing structural preferences in network interactions.

Inspired by these underexplored aspects, this paper clearly conceptualizes behavioral inertia in networks, abstracting it as temporal inertia, which captures the dependence of the current network configuration on previous states and reflects the persistence of existing connections over time, and spatial inertia, which supplements node-level connections through a combination of preference-driven selection and random mechanisms, and formally models these mechanisms. On this basis, the team is regarded as an integrated unit, combining the two types of inertia to construct the team inertia network and its evolutionary dynamics, which then serves as the foundation for examining patterns of knowledge diffusion.

Methods

Temporal-spatial inertia in networks

As mentioned above, the inertia mechanism in this paper includes temporal inertia and spatial inertia. The former captures the dependence of the initial network configuration on previous time points, while the latter supplements node-level connections through preference-driven selection combined with a random mechanism. This inertia mechanisms jointly governs the network evolution, influencing both the persistence of existing connections and the formation of new ones, thereby shaping the structural characteristics of the network at any given time.

The network at time t is defined as a triplet ${G_t} = \left( V(G_t),E(G_t),W_{G_t} \right)$, where $V(G_t)$ is the set of nonempty vertices and $E(G_t)$ is the set of directed edges; if there exists a directed edge between $u \in V(G_t), v \in V(G_t)$, then $(u, v) \in E(G_t)$; $W_{G_t}: E(G_t) \rightarrow \mathbb {R}$ is the weighting function.

Due to temporal inertia, the initial network structure $G_t^0$ at time t depends on the structures at several previous moments. Let $\Gamma _t: T \rightarrow \mathbb {R}$, with $T = [0, t)$, denote the function used to scale the network weights at each time step. Then, the discrete form of $G_t^0$ is given by:

$$\begin{aligned} G_t^0 = \sum _{i = s}^{t - 1} \Gamma _t(i) \times G_i, \end{aligned}$$

(1)

where the summation merges networks over $[s, t-1]$, i.e., $G_t^0 = (V(G_t^0), E(G_t^0), W_{G_t^0})$ with $V(G_t^0) = V(G_{t-1})$, $E(G_t^0) = \{(u,v) \in \bigcup _{i=s}^{t-1} E(G_i) \mid u,v \in V(G_t^0)\}$, and $W_{G_t^0}(u,v) = \gamma (\{w_{uv}^{(i)}\}_{i=s}^{t-1})$, where $\gamma : \{w_{uv}^{(i)}\}_{i=s}^{t-1} \mapsto w_{uv}^{(t)}$ aggregates historical edge weights over $[s, t-1]$; s determines the starting time affecting $G_t^0$, and the temporal inertia index $i_t$ defines the backtracking window width $W_{mw} = \lceil (t+1) i_t \rceil$, thereby determining s:

$$\begin{aligned} s = {\left\{ \begin{array}{ll} t - {W_{mw}},t - {W_{mw}} \ge 0,\\ 0,t - {W_{mw}} < 0. \end{array}\right. } \end{aligned}$$

(2)

The evolution of the network structure over time is influenced by spatial inertia, so the final network $G_t$ is not determined directly by the initial network $G_t^0$, but is formed based on node selection preferences. First, we define the preference function:

$$\begin{aligned} \Phi _t: V(G_t^0) \times V(G_t^0) \rightarrow \mathbb {R}, \end{aligned}$$

(3)

which characterizes the preference strength of nodes at time t toward their potential connection targets. Based on this function, for any node $r \in V(G_t^0)$, the set of candidate connections to its neighbors in $G_t^0$ is given by:

$$\begin{aligned} S_{t,r} = \{(u,p) \mid u \in V(G_t^0), (r,u) \in E(G_t^0), p = \Phi _t(r,u)\}. \end{aligned}$$

(4)

From this set, assuming $k \le |S_{t,r}|$, we extract the k nodes with the highest preference values, forming the connection set (if $k > |S_{t,r}|$, all u in $S_{t,r}$ are used, and in this case k is set to $|S_{t,r}|$):

$$\begin{aligned} C_{t,r} = \{(u_1,w_1),(u_2,w_2),\ldots ,(u_k,w_k)\}, \end{aligned}$$

(5)

where $(u_i,w_i)$ denotes the i-th node selected by r at time t along with its associated weight. To incorporate spatial inertia, we introduce the spatial inertia index $i_s$, with $0 \le i_s \le 1$. The final number of retained connections is $\lfloor i_s \times k \rfloor$, which are chosen according to a specific criterion (in this paper, the nodes with the highest knowledge stock in $C_{t,r}$), while the remaining $k - \lfloor i_s \times k \rfloor$ connections are supplemented through a selection mechanism such as random choice. Thus, the final network $G_t$ is obtained from $G_t^0$ by integrating preference-driven selection with complementary selection mechanisms (this paper uses random supplementary selection).

From the above introduction, it can be found that inertia stems from the combination of temporal and spatial effects. Temporal inertia reflects the characteristics of the network formed over time, that is, the initial structure at any moment is shaped by previous configurations. Spatial inertia, based on this time-affected structure, controls the connection choices at the node level through a preference-based selection mechanism and supplements the remaining connections through random selection. These two complementary mechanisms jointly determine the evolution of the network, influencing both the persistence of existing connections and the formation of new ones.

Fig. 1 depicts the influence of temporal inertia and spatial inertia mechanisms in the network on its formation. This figure encompasses two major components. The direct process on the left indicates the formation outcome of the network at discrete time points, while the intermediate process on the right represents the formation details of the network under the effect of inertia. For the sake of convenience in expression, it is assumed that $s = t - 2$ , that is, the network at time t relies on the network structure of the previous two time points. Due to the relatively small scale of this demonstration network, no restrictions are imposed on k (which also means that the actual value of k is the number of nodes actively connected to each node). Under the influence of temporal inertia, the initial structure of the network at time t, $G_t^0$, is augmented on the basis of $G_{t-1}$ by several edges of $G_{t-2}$, which are marked with blue directed edges in the figure. On the basis of $G_t^0$ , the spatial inertia index $i_s$ is set to 0.7; therefore, the directed edges of some nodes need to be adjusted. The deleted edges in the figure are marked in red, and the newly added edges are marked in blue as well. Under the effect of spatial inertia, the final network structure $G_t$ at time t is formed.

Fig. 1

Network formation under the influence of temporal inertia and spatial inertia mechanisms.

In addition to Fig. 1, we provide pseudocode for temporal inertia and spatial inertia, along with necessary annotations, presented as Algorithms 1 and  2, respectively. Since the details of certain functions are very cumbersome and may obscure the core algorithmic logic, for clarity, we represent such functions directly using uppercase text or natural language. The same approach applies to the subsequent algorithms.

Algorithm 1

The initial network structure determined by temporal inertia.

Algorithm 2

The final network structure determined by spatial inertia.

To thoroughly demonstrate the existence of the inertia mechanisms in the real world, we measured temporal inertia and spatial inertia on a large-scale dynamic social network. The results strongly support our perspective. For details, please refer to Supplementary Material 1.

Team inertia network

To discuss the impact of the inertia mechanism in the network on knowledge diffusion, this paper constructs a team inertia network based on the above-mentioned temporal-spatial inertia mechanisms. First, the definition of a team in the network must be given: A team is a highly connected directed network composed of n nodes with d types of heterogeneous knowledge, and it has the following characteristics:

• (1) The team is defined based on a directed graph. Knowledge diffusion has directionality⁵⁰, and Yue et al.⁵¹ summarize the knowledge diffusion behavior into the barter trade-type Cowan model, the knowledge promotion-type Push model, and the knowledge search-type Pull model. Their commonality lies in the fact that knowledge diffusion is directional. Therefore, this paper defines the team on a directed graph to depict the unevenness of knowledge stock between nodes within the team and the directionality of knowledge diffusion.

• (2) The team is highly interconnected internally. Since team members are often familiar with one another and frequently collaborate, knowledge diffusion within the team tends to be unimpeded, necessitating a high level of connectivity among team members. In the process of team evolution, due to the existence of node elimination mechanisms, team connectivity often weakens. This paper proposes bridging operations to ensure high connectivity within the team within a certain time frame.

Within a team, there exists not only a tendency to maintain existing cooperative relationships but also selective interactions among members based on preferences^52,53. This precisely reflects the behavioral persistence induced by temporal inertia and the structural selection constraints brought by spatial inertia, as discussed above. Thus, at the team level, this inertia manifests as the continuity of behavior and preference-driven interactions during knowledge diffusion, which constitutes the core of team inertia. To capture this feature, we introduce the team inertia network, a directed network composed of multiple teams. Within the network, members of each team are highly interconnected, and directed edges between teams reflect knowledge diffusion across teams. However, this is not the focus of this paper.

To fully define the team inertia network, it is necessary to specify the function ${\Gamma _t}$, which governs dynamics on the temporal scale, and the node selection preference ${\Phi _t}$, which operates on the spatial scale.

The function ${\Gamma _t}$ essentially reflects the network’s memory, recording significant moments in the network’s evolutionary process that will affect the current network structure. For simplicity, this paper assumes that the temporal function ${\Gamma _t}$ for all nodes in the network at all times has a consistent power decay function form, taking into account the decay patterns observed in previous studies^54,55 on temporal and scale-free networks, that is:

$$\begin{aligned} \Gamma (i) = a(t - i)^{-b},0< i < t, \end{aligned}$$

(6)

where $t - i$ denotes the relative time distance.

The weight reductive function $\gamma$ is defined as:

$$\begin{aligned} \gamma (\{w_{uv}^{(i)}\}_{i=s}^{t-1}) = \sum _{i=s}^{t-1} w_{uv}^{(i)}, \end{aligned}$$

(7)

Intuitively, a node is more likely to link to another node with a similar knowledge structure to acquire knowledge, which is referred to as homology diffusion⁵⁶. Therefore, this paper considers using cosine similarity to measure the similarity of the knowledge stock $\textbf{x},\textbf{y}$ between two nodes, to express $\Phi _t$:

$$\begin{aligned} \Phi (\textbf{x},\textbf{y}) = \frac{\textbf{x} \cdot \textbf{y}}{\left\| \textbf{x} \right\| \times \left\| \textbf{y} \right\| }. \end{aligned}$$

(8)

Here, for simplicity, weights are not introduced into the preference function, aligning with the methodology of spatial econometrics⁵⁷ where the use of unweighted adjacency measures is often preferred as a robust baseline to focus analysis on structural effects and enhance model interpretability.

Adaptive evolution of team inertia network

Next, we consider the evolutionary dynamics of the team inertia network, and the diffusion of knowledge based on it. The evolution of the team inertia network is team-based, assuming that there are m teams in the network, which have the same upper size limit L, this paper divides the team evolution into the following three phases:

• (1) Initialization phase. The initialization phase is the starting point of team evolution. At $t = 0$, $n = n_0(n_0 \le L)$ fully connected nodes are added to the team, and the weights of the directed edges are all initialized to 1 for the initialization operation of the team. This implies that right at the inception of the team’s establishment, it possesses $n_0$ team members and is strongly interlinked within the team. Additionally, for each of the m teams, it is paired with the other $m-1$ teams. In each pair, one node is randomly selected from each team, and a cross-team edge is established between these two nodes. By repeating this procedure for all teams, a total of $m \times (m-1)$ cross-team edges are formed in the network.

• (2) Growth phase. After the team is initialized ($t > 0$) and before it reaches its size limit ($n < L$), at each time step the current network is first determined through the inertia mechanisms, and then one node is uniformly added to the team. With probability $p_i = d_i / \sum _j d_j$, it establishes $2 \times n_0$ mutually connected directed edges with existing members of the team, and all edge weights are initialized to 1. This means that team growth does not affect connectivity within the team network.

• (3) Refining phase. After the team reaches its size limit ($n = L$), if new nodes still need to be added (i.e., $t > L - n_0$), then at each time step the current network is first determined through the inertia mechanisms, and subsequently the node with the lowest indegree in the team is eliminated to free capacity, allowing new nodes to be accommodated seamlessly. Node elimination may cause a team to become temporarily disconnected. To address this, a bridging mechanism is applied to connect multiple strongly connected components within the team, ensuring that the refined team remains connected over a short time window. Although connectivity is not guaranteed at every time step, the team consistently achieves full connectivity within a relatively short period, which aligns with real-world observations where teams are not perfectly cohesive at all times.

For the three-stage evolution process of the team inertia network described above, we provide pseudocode as shown in Algorithms 3, 4, and 5. Algorithm 3 presents the logic of the network initialization stage, Algorithm 4 details the bridging mechanism, and Algorithm 5 illustrates the combined logic of the growth and refinement stages.

Algorithm 3

The initial stage of the evolution of the team inertia network.

Algorithm 4

Bridging mechanism for incremental team connectivity.

Algorithm 5

The growth and refinement stage of the team inertia network.

It is easy to find that the team has different evolution characteristics at different phases: before the team size reaches the upper limit, the team’s evolution is only growth without node elimination; after the team size reaches the upper limit, the team enters into the refining phase, at this time, node elimination will be carried out first and then node growth, and thereafter the elimination and growth of the nodes will be overlapped. In particular, the network evolution algorithm proposed in this paper fully draws on the work of Barabási & Albert⁵⁸, so that there is preferential attachment to the growth process of the team inertial network. In addition, we analyzed the network characteristics of the team inertia network at the final moment (including in-degree, out-degree, density, clustering coefficient, shortest path, small-world index, etc), for details, please refer to Supplementary Material 2.

The knowledge diffusion in the team inertia network is carried out in synchronization with the evolution of the team inertia network. This paper mainly refers to the knowledge diffusion mechanism by Zhang et al.¹⁴. Specifically, knowledge diffusion in the team inertia network is mainly driven by the knowledge innovation ability and knowledge absorption ability of the network nodes, under the influence of knowledge innovation ability, the increment of the node’s knowledge stock is given by the following equation:

$$\begin{aligned}&\Delta I_{ic}(t + 1) = \beta _i v_{ic}(t), \end{aligned}$$

(9)

$$\begin{aligned}&\beta _i = b_i\overline{\beta }, \end{aligned}$$

(10)

where ${v_{ic}}\left( t \right)$ denotes the stock level of knowledge of node i in category c, ${\beta _i}$ is the knowledge innovation capability of the node, $\bar{\beta }$ is the upper limit of knowledge innovation capability of all nodes, and ${b_i}$ is a random number following a uniform distribution on (0, 1]. The above equation shows that the incremental knowledge stock $\Delta I$ of node i of type c at the time $t + 1$ is a linear function of the knowledge stock of the same type at that node at the time t.

The increment in the knowledge stock of the node determined by the knowledge absorptive capacity is given by the following equation:

$$\begin{aligned}&\Delta A_{ic}(t + 1) = \sum _{j \in V_i} \Delta A_{ijc}(t), \end{aligned}$$

(11)

$$\begin{aligned}&\Delta A_{ijc}(t) = {\left\{ \begin{array}{ll} (v_{jc}(t) - v_{ic}(t))\alpha _i,s.t.v_{jc}(t)> v_{ic}(t),c = ic(i),\\ (v_{jc}(t) - v_{ic}(t))\alpha _iw,s.t.v_{jc}(t) > v_{ic}(t),c \ne ic(i),\\ 0,s.t.v_{jc}(t) \le v_{ic}(t), \end{array}\right. } \end{aligned}$$

(12)

$$\begin{aligned}&\alpha _i = \underline{\alpha } + a_i r. \end{aligned}$$

(13)

That is, the incremental knowledge stock $\Delta A$ of node i of type c at time $t + 1$ comes from the sum of knowledge absorbed by that node from all the nodes it points to. In Eq. (12), $\Delta {A_{ijc}}$ is expressed as a linear function of the difference in knowledge stock between two nodes. If the knowledge stock of type c at node i is less than that at node j, it is absorbed directly through its knowledge absorption ability ${\alpha _i}$, otherwise, there is no increment. In particular, nodes are assumed to be better absorbers in the domains in which they specialize (specialization is specified by the function ic), and if not, they are decayed appropriately by the parameter $w, 0< w < 1$. In Eq. (13), $\underline{\alpha }$ is the lower bound of knowledge absorption capacity, $a_i$ is a random number following a uniform distribution on (0, 1], and r is a global parameter.

Summarizing the above, the total knowledge stock of node i of type c at the time $t + 1$ is defined as:

$$\begin{aligned}&v = v_{ic}(t)(1 - df) + \Delta I_{ic}(t + 1) + \Delta A_{ic}(t + 1), \end{aligned}$$

(14)

$$\begin{aligned}&v_{ic}(t + 1) = {\left\{ \begin{array}{ll} v, v < ub, \\ ub,v \ge ub. \end{array}\right. } \end{aligned}$$

(15)

That is, the knowledge stock of type c for node i at time $t + 1$ is constituted by three components: the residual knowledge stock from time t, the increment of knowledge stock created at time $t + 1$, and the increment of knowledge stock absorbed at time $t + 1$. Particularly, there exists a theoretical upper bound ub for the total amount of each type of knowledge stock. Once the upper bound is reached, the knowledge stock of this type will remain at ub and no longer increase. However, this paper does not set a limit on ub.

For the initial knowledge stock of team nodes, this paper assumes that each node has an expertise knowledge category in the d types of heterogeneous knowledge. Based on this, team nodes are divided into two categories. One category is expert nodes, all of which enter the team at the initialization stage (all nodes at the initialization stage of the network are expert nodes). The knowledge stock of the expertise category of expert nodes is defaulted to 30, and the knowledge stock of the remaining categories is defaulted to 0. The other category is ordinary nodes, all of which enter the network successively after the initialization of the team. The knowledge stock of the expertise field of ordinary nodes is the sum of the average value of this type of knowledge of the team and a random number that follows the uniform distribution on [0, 1]. The knowledge stock of non-expertise fields is defaulted to the average value of this type of knowledge of the team.

Knowledge diffusion metrics for team inertia networks

This paper primarily focuses on revealing the impact of inertia mechanisms on knowledge diffusion within team inertia networks from the perspective of the team as a whole. In this section, the paper measures the overall knowledge level of the team, the disparity in knowledge stock among internal nodes, and the spatial distribution of team knowledge during the evolution of the team inertia network.

Diffusion and growth of overall team knowledge

Referring to the average knowledge stock indicator proposed by Lin & Li¹⁵, we define the average knowledge stock level of type c for team j at time t as:

$$\begin{aligned} \overline{v}_{jc}(t) = \frac{1}{n}\sum _{i = 1}^n v_{ic}(t). \end{aligned}$$

(16)

If we do not consider specific types of knowledge, then the overall average knowledge stock of team j at time t is:

$$\begin{aligned} \overline{v}_j(t) = \frac{1}{d}\sum _{c = 1}^d \overline{v}_{jc}(t) = \frac{1}{dn}\sum _{c = 1}^d\sum _{i = 1}^n v_{ic}(t). \end{aligned}$$

(17)

Based on Eqs. (16) and (17), we can temporally assess the convergence rate of the team’s total knowledge stock, that is:

$$\begin{aligned}&\overline{v}_{jc}(t = \tau ) = C,\end{aligned}$$

(18)

$$\begin{aligned}&\overline{v}_j(t = \tau ) = C. \end{aligned}$$

(19)

It means that for a given average knowledge stock C of a team, a smaller $\tau$ often corresponds to faster knowledge growth.

Additionally, based on Eqs. (16) and (17), the growth rate of the team’s knowledge stock can also be defined:

$$\begin{aligned}&g_{jc}(t) = \frac{\overline{v}_{jc}(t)}{\overline{v}_{jc}(t - 1)} - 1,\end{aligned}$$

(20)

$$\begin{aligned}&g_j(t) = \frac{\overline{v}_j(t)}{\overline{v}_j(t - 1)} - 1. \end{aligned}$$

(21)

The above formulas directly indicate the growth particulars of a certain type or the overall knowledge of the team.

Differences in knowledge stocks of team nodes

This paper uses the Gini coefficient to measure the disparity in knowledge stock among nodes within the team. According to Kakwani’s definition of the Gini coefficient⁵⁹, the Gini coefficient in the team inertia network can be defined as:

$$\begin{aligned}&G_{jc}(t) = \frac{\Delta _c}{2\mu _c}, \end{aligned}$$

(22)

$$\begin{aligned}&\Delta _c = \frac{1}{n(n - 1)}\sum _{i = 1}^n \sum _{j = 1, j \ne i}^n \left| v_{ic}(t) - v_{jc}(t)\right| , \end{aligned}$$

(23)

where ${\mu _c}$ represents the mean of the c-th type of knowledge for all members in the team at time t, and n is the number of members in team j at time t. If we do not consider the type of knowledge c, then the overall average Gini coefficient for team j at time t is:

$$\begin{aligned} G_{j}(t) = \frac{1}{d}\sum _{c = 1}^d G_{jc}(t). \end{aligned}$$

(24)

It is easy to show that when all members of a team have the same stock of knowledge of type c, then $\Delta _c = 0$, i.e., $G_{jc} = 0$, which implies that there is no difference in the stock of knowledge of type c among all members of the team. When all the c-th type of knowledge in the team is possessed by a single member, at this point $\Delta _c = 2\mu _c$, thus $G_{jc} = 1$, which means that the distribution of the c-th type of knowledge stock in the team is absolutely unequal. Therefore, $0 \le G_{jc} \le 1$ holds.

Spatial distribution of team knowledge

Existing studies have confirmed the phenomenon of local spillover in knowledge diffusion^60,61, which may lead to the spatial agglomeration of knowledge diffusion⁶². To measure whether there is a spatial agglomeration phenomenon in the dynamics of knowledge diffusion in the team inertia network, this paper introduces the following Global Moran’s I to measure the degree of spatial agglomeration of the c-th type of knowledge stock in team j at time t:

$$\begin{aligned} M_{jc}(t) = \frac{n}{W}\frac{\sum _i \sum _{j \ne i} X(i,j) (v_{ic}(t) - \overline{v}_c)(v_{jc}(t) - \overline{v}_c)}{\sum _i(v_{ic}(t) - \overline{v}_c)^2}, \end{aligned}$$

(25)

where X(i, j) is used to indicate whether there is an edge between node i and node j (without considering the direction of the edge or its weight), where it is 1 if there is an edge, and 0 otherwise; $W = \sum _i \sum _{j \ne i} X(i,j)$.

Notice that in Eq. (25), $\sum _i(v_{ic}(t) - \overline{v}_c)^2$ is n times the variance of $v_{ic}(t)$. Thus $\sum _i(v_{ic}(t) - \overline{v}_c)^2 = n\sigma ^2(t)$. In addition, W can also be moved into the summation of numerator and expressed together with X(i, j), thus Eq. (25) can be further simplified to:

$$\begin{aligned} M_{jc}(t) = \frac{\sum _i \sum _{j \ne i} w_{i,j} (v_{ic}(t) - \overline{v}_c)(v_{jc}(t) - \overline{v}_c)}{\sigma ^2(t)}, \end{aligned}$$

(26)

where $w_{i,j} = \frac{X(i,j)}{W} = \frac{X(i,j)}{\sum _i \sum _{j \ne i} X({i,j})}$.

Based on the aforementioned equations, the Moran’s I for the overall knowledge level of team j at any given time can be further defined:

$$\begin{aligned} M_{j}(t) = \frac{\sum _i \sum _{j \ne i} w_{i,j} (v_{i}(t) - \overline{v})(v_{j}(t) - \overline{v})}{\sigma ^2(t)}, \end{aligned}$$

(27)

where $v_i(t)=\frac{1}{d}\sum _{c=1}^d v_{ic}(t)$, $\sigma ^2(t)$ is the variance of $v_i(t)$, $w_{i,j}$ is defined as in Eq. (26).

The values of the Moran’s I defined above range between -1 and 1. Positive Moran’s I implies that there is positive spatial agglomeration within the team, i.e., nodes with more similar knowledge stocks are closer together; negative Moran’s I implies that there is negative spatial agglomeration within the team, i.e., nodes with less similar knowledge stocks are closer together; and if the Moran’s I is 0, it implies that there is no spatial agglomeration phenomenon.

Results

Parameter settings

Considering the parameter settings of existing research¹⁴, this paper sets the number of teams $m = 5$, the upper limit of team members $L = 25$, the initial network scale $n_0 = 10$; $k = 24$ in Eq. (5); $a = 5$, $b = 0.5$ in Eq. (6); $\bar{\beta }= 0.003$ in Eq. (10); $w = 0.01$ in Eq. (12); $\underline{\alpha }= 0.001$, $r = 0.02$ in Eq. (13); $df = 0.0005$ in Eq.(14), ub is not limited, meaning there is no boundary for the growth of the team’s knowledge. In particular, this paper generally sets the dimension of the knowledge vector $d = 1$, so that the aggregate metrics in the Section “Knowledge diffusion metrics for team inertia networks” are equivalent to the metrics for the specific knowledge types, thus eliminating the need to delve into the details of the different knowledge types, and instead focusing on revealing the diffusion of knowledge across the inertial network of the team. The meanings of related parameters are shown in Table 1. Since k and d need to be adjusted later on, they are not listed in the table.

Table 1 Model simulation parameters setting. The parameters $\overline{\beta }$, w, $\underline{\alpha }$, r, and df are referenced from Zhang et al. (2018), while the rest are original.

For the parameter settings presented in Table 1, the following guidelines are provided. The number of teams m and the team size L can be flexibly determined according to the research context, while the initial team scale $n_0$ must satisfy $n_0 \le L$. In the temporal inertia function, both parameters $a>0$ and $b>0$ control the intensity and decay rate of temporal memory, respectively. The upper limit of knowledge innovation ability $\overline{\beta }$ is typically assigned a small positive value to ensure system stability, but can be adjusted depending on the innovation environment. The decay coefficient for non-expert knowledge categories w usually falls within $0< w < 1$, representing the attenuation of non-core knowledge. The lower bound of knowledge absorption ability $\underline{\alpha }$ and the global regulation parameter $r>0$ jointly determine the efficiency of knowledge absorption, and their relative magnitudes may vary across scenarios. The knowledge depreciation rate df generally lies within $0< df < 1$, reflecting the natural decay of knowledge over time. Finally, the upper bound of maximum knowledge stock $ub>0$ represents the theoretical limit of an individual’s knowledge capacity and can be set as finite or unbounded depending on model design.

In the specific simulation, the time step of each experiment is 1000 steps and is independently repeated 30 times, and then the average value is taken as the final result to reduce the random error.

The effect of inertia on knowledge diffusion

In theory, all parameters in this paper will affect the knowledge diffusion of the team inertia network model, but parameters of different natures will produce different results. To grasp the key issues and reduce unnecessary analysis, this paper categorizes the parameters involved in the model into three types:

• (1) Core parameters: the temporal inertia index $i_t$ and the spatial inertia index $i_s$ are the most critical parameters affecting the diffusion of knowledge in team inertia networks, both will have a substantial structural impact on the diffusion of knowledge.

• (2) Control parameters: the maximum successor k of the node and the dimension d of the knowledge vector can be adjusted according to different scenarios, and both of them have mainly non-structural effects.

• (3) Flexible parameters: The rest of the parameters in the section “Parameter settings” are flexible and their adjustment does not have a substantial impact on the model. Because their effects are mostly linear, so there is no need to pay attention to them (we still provided a sensitivity analysis of the parameters related to knowledge diffusion. Please refer to Supplementary Material 3).

To investigate the impact of inertia on knowledge diffusion in the team inertia network, this paper will focus on the role of core parameters while adjusting and controlling the control parameters. Specifically, based on the experiment with $k = 24,d = 1$, two additional experiments were designed by replacing the control parameters, totaling three experiments, as follows:

• (1) $k = 24,d = 1$. The impact of $i_t$ and $i_s$ on knowledge diffusion when the number of successor nodes is 24 and the knowledge vector dimension is 1.

• (2) $k = 10,d = 1$. Adjusting k to 10 while controlling for d, and study the effects of ${i_t}$ and ${i_s}$ on knowledge diffusion.

• (3) $k = 24,d = 5$. Adjusting d to 5 while controlling for k, we study the effects of ${i_t}$ and ${i_s}$ on knowledge diffusion.

In terms of specific indicators for knowledge diffusion, since all teams have the same parameters, the focus is directly on the average knowledge level of the network at $t = 999$ (i.e., the 1000th step) under different levels of inertia, which is:

$$\begin{aligned} V(999) = \frac{1}{m}\sum _{j = 1}^m \overline{v}_j(999). \end{aligned}$$

(28)

Based on this equation, we can plot the heatmap of the mean knowledge level $\overline{V}(999)$, which represents the average of the 30 repeated experiments for each parameter combination of temporal and spatial inertia. To further illustrate the effect of each inertia dimension, we include two dedicated distribution plots. These plots are constructed by fixing one inertia dimension and showing the distribution of the 10 mean knowledge values corresponding to the other dimension at that fixed value. This approach provides a clear interpretation of how the distribution of average performance shifts as each dimension of inertia changes.

While these averaged results offer a clear overall picture, they may mask important variability inherent in the 30 individual experiments at each grid point. Therefore, to provide a rigorous measure of the practical statistical effect, we further introduce the extended Common Language Effect Size (CLES)⁶³:

$$\begin{aligned} \text {CLES}_{x,y} = P(X_{x,y} > \mu _0), \end{aligned}$$

(29)

where $X_{x,y}$ denotes the set of average knowledge levels across all repeated experiments for the parameter combination (x, y), and $\mu _0$ is the global mean across all experiments. A higher CLES value indicates that the network knowledge level under the given parameter combination tends to exceed the overall mean. As an effect size metric, CLES provides a relative interpretation of the results and can be directly compared with the mean value.

To assess the precision of the effect size estimates, we compute 95% confidence intervals $\text {CI}_{x,y} = [\text {Lower}_{x,y}, \text {Upper}_{x,y}]$ for each grid point using the Bootstrap with bias correction and acceleration (BCa) method. Then, the confidence interval width, $\text {CI Width}_{x,y} = \text {Upper}_{x,y} - \text {Lower}_{x,y}$, measures the uncertainty of the estimate: narrower intervals indicate more precise effect size estimates, whereas wider intervals suggest greater uncertainty for that parameter combination. Notably, the CI width is related to the effect size value: when the effect size approaches its extremes (close to 0 or 1), the confidence interval tends to be narrower, while values near the midpoint often exhibit wider intervals. This approach allows us to present both the network’s knowledge levels under different conditions and a quantitative measure of the associated uncertainty, providing a rigorous basis for subsequent analyses and interpretations. Additionally, we assess the statistical significance of $\text {CLES}_{x,y}$ relative to the global mean $\mu _0$ by comparing the 95% BCa confidence interval $\text {CI}_{x,y}$ to the null threshold of 0.5: if $\text {Lower}_{x,y} > 0.5$, the effect is significantly positive; if $\text {Upper}_{x,y} < 0.5$, it is significantly negative; otherwise, it is not significant. To control the false-discovery rate across all grid-wise comparisons, we further applied the Benjamini–Hochberg FDR correction ($\alpha = 0.05$) to the raw p-values derived from the BCa confidence intervals. Significant cells were highlighted with black borders, whereas non-significant regions should not be interpreted as irrelevant but as transitional zones between high- and low-CLES areas.

Fig. 2 displays the experimental results for $k = 24,d = 1$. As can be observed from Fig. 2a, at any level of temporal inertia, a moderate increase in spatial inertia can enhance the average knowledge level of the network. When spatial inertia reaches approximately 0.7 to 0.8, the average knowledge level of the network peaks, as more intuitively demonstrated in Fig. 2c. If spatial inertia continues to increase beyond this point, the average knowledge level of the network will sharply decline. This indicates that moderate spatial inertia is beneficial to team growth, while excessive spatial inertia is detrimental to team development. Temporal inertia has no direct impact on the diffusion of knowledge within the team inertia network, as can be easily observed in Fig. 2b, where the distribution of average knowledge levels is basically consistent at any level of temporal inertia. The remaining three heatmaps provide a rigorous statistical assessment of the results. Fig. 2d presents the Common Language Effect Size heatmap. Higher $\text {CLES}$ values (closer to 1) indicate a stronger positive effect, meaning the network consistently performs above the average under those specific inertia conditions. Evidently, the pattern in Fig. 2d aligns closely with the mean values shown in Fig. 2a, confirming that the observed average performance is statistically robust rather than driven by random fluctuation. Fig. 2e further displays the 95% confidence interval (CI) width of the $\text {CLES}$ estimates, which demonstrates that the effect size estimates in Fig. 2d are highly precise across most parameter combinations. In addition, the significance map in Fig. 2f shows that the estimated effects are strongly significant overall. The non-significant regions (where spatial inertia is 0.3 and 0.9) directly correspond to the transition zones in the $\text {CLES}$ values. In these zones, the practical effect size is naturally close to the null effect threshold of 0.5, which inevitably causes the 95% CI to encompass 0.5. Consequently, the performance achieved in these transition regions cannot be statistically separated from the global mean. This finding offers clear guidance for parameter selection: to ensure a robust and verifiable performance advantage, parameter selection must focus on the high-CLES regions where the effect is strongly significant, and extreme caution should be exercised when choosing parameters within the transitional spatial inertia ranges. The simulation we conducted later on the Gini coefficient and the Global Moran’s I was carried out under this guidance.

Fig. 2

The impact of inertia on knowledge diffusion with $k = 24,d = 1$.

The reason why the temporal inertia effect was not shown under the parameters of $k = 24$ and $d = 1$ is analyzed as follows: First, in this experiment, there are a larger number of subsequent nodes, which means that existing members of the team have a sufficiently high probability of being connected to any node within the team at any given time, thus exhibiting a uniform characteristic over time; Second, this study assumes that the temporal function is consistent in form at any time and is in the form of a power decay function. Due to its sensitivity to changes in time, the network structure at any given moment depends on the network from more recent times, making the effect of temporal inertia relatively uniform; Thirdly, spatial inertia shares some of the effects of temporal inertia. Since spatial inertia is considered after temporal inertia, it inevitably burdens some of the effects of temporal inertia. Due to the second and third reasons being related to the model’s settings, this paper focuses on discussing the potential implications of the first reason. By reducing the number of subsequent nodes for team members, setting $k = 10$, while keeping $d = 1$, we experimented again, and the results confirmed the above conjecture: As shown in Fig. 3a, when the number of subsequent nodes of team members decreases, the influence of temporal inertia becomes apparent. Fig. 3b visually represents from the perspective of temporal inertia that when the temporal inertia is 0.1, the average knowledge level distribution is more prominent, and the upper quartile is larger. As the temporal inertia increases, the upper quartile decreases until it stabilizes. Additionally, this experiment also confirmed the conclusion of spatial inertia when $k = 24, d = 1$, as shown in Fig. 3c, which indicates the positive effect of maintaining a moderate spatial inertia on the growth of the average knowledge level. The $\text {CLES}$ pattern in Fig. 3d remains largely consistent with the mean pattern in Fig. 3a, which further confirms the statistical robustness of the observed average performance. The confidence interval widths displayed in Fig. 3e validate the high estimation precision achieved in the high-$\text {CLES}$ regions. Furthermore, the significance test results in Fig. 3f reinforce our interpretation regarding parameter reliability. Crucially, our previous observation–that the non-significant areas correspond precisely to the transitional zones between high and low $\text {CLES}$ values–is reaffirmed. This pattern demonstrates that non-significance is a statistically expected consequence of the effect size approaching the null threshold of 0.5 in areas where the performance trend shifts.

Fig. 3

The impact of inertia on knowledge diffusion with $k = 10,d = 1$.

This paper further adjusted the dimension of the knowledge vector d on the basis of $k = 24,d = 1$, setting it to 5, and discussed the effect of knowledge diffusion in the team inertia network under the condition of interdisciplinary integration, as shown in Fig. 4. Since $k = 24$, the effect of temporal inertia does not manifest. However, the effect of spatial inertia is still strongly confirmed. It can be observed that an increase in the knowledge dimension reduces the optimal spatial inertia index, which can reach the peak of the network’s average knowledge level when it is between 0.6 and 0.7. The experimental outcomes here consistently validate that $\text {CLES}$ aligns closely with the mean value. Estimation precision is high in both high-$\text {CLES}$ and low-$\text {CLES}$ zones, while the transitional areas between these two regions predictably exhibit non-significant characteristics.

Fig. 4

The impact of inertia on knowledge diffusion with $k = 24,d = 5$.

In addition to the aforementioned three experiments, there is another possible combination, that is $k = 10,d = 5$, but this is often an exception. Since the k value is usually positively correlated with team size, a smaller k value indicates a smaller team size, and the possibility of interdisciplinary knowledge crossover within small teams is relatively low. However, this paper still conducted experimental analysis, and the results are shown in Fig. 5. It is not difficult to observe that when the number of subsequent nodes for team members is relatively small, but the dimension of the knowledge vector is relatively large, the optimal level of spatial inertia is around 0.8 to 0.9. Moreover, at the optimal level of spatial inertia, a moderately high level of temporal inertia can yield the highest average knowledge level. The conclusion about temporal inertia seems to differ from Fig. 3. The reason is that when k is small and d is large, the knowledge vector of team members only has a single dominant component, which leads to a fully averaged average knowledge level in the team inertia network. In this situation, enhancing the average knowledge level can only be achieved through a more secure approach – the “selecting the best from the familiar” mechanism of spatial inertia – for node selection. This is only possible when temporal inertia is maintained at a moderately high level. The experimental results for this parameter combination confirm that the $\text {CLES}$ consistently aligns with the mean results shown in Fig. 5a. However, the statistical analysis reveals a unique pattern: the Confidence Interval width is minimal only when $\text {CLES}$ approaches 0, and while narrow, the $\text {CI}$ width is not similarly small when approaching 1 (though not significantly wider). This phenomenon is rooted in the contradiction between the limited smoothing capability of the k value and the dimensionality of the knowledge space. For a team of $L=25$ members, the limited exploration capacity of $k=10$ is sufficient to dilute noise and maintain system stability in the low-dimensional knowledge space of $d=1$. However, in the high-dimensional and complex knowledge space of $d=5$, the smoothing capability of the k value is dispersed across multiple dimensions. This dispersion prevents the effective suppression of noise and fluctuation in each dimension, thus triggering systemic high variability (i.e., widened confidence intervals). This high variability makes it statistically impossible to reliably separate the effect size in the $\text {CLES}$ transition zones, consequently leading to the expansion of non-significant areas. This unique statistical pattern carries a significant implication for real-world teams operating with sparse learning sources (k small) and high-dimensional complex knowledge (d large): The average knowledge stock exhibits significant inherent randomness, making it difficult to establish a sustained, stable, and reliable knowledge advantage.

Fig. 5

The impact of inertia on knowledge diffusion with $k = 10,d = 5$.

The above experiments show that: First, a moderate level of spatial inertia can enhance the average knowledge level of a team, insufficient or excessive inertia is not conducive to the overall development of the team; Second, when the number of succeeding nodes of the team members is high, the effect of temporal inertia on the knowledge level of the team is more uniform, which means that the closer the communication among the team members is, the more favorable it is to the team’s development; Third, when the number of successor nodes of the team members is small and the dimension of the knowledge vector is small, the smaller temporal inertia is favorable to the team’s development because the team is less influenced by the history, but when the dimension of the knowledge vector is large, the characteristics of temporal inertia should be fully exploited (even if there is variability in the knowledge stock performance). In addition, it should be noted that when the upper limit of the team is large and the number of successor nodes of the team members is small, the connectivity of the team in the evolution process will be affected to a large extent, which is obvious.

The equality and spatial distribution of knowledge diffusion

Based on the experimental results from the previous section and the guiding principles derived therefrom, this paper chooses a set of core parameters $i_t = 0.1, i_s = 0.7$, and two sets of control parameters $k = 24,d = 1$ and $k = 10,d = 1$ to further discuss the equality and spatial distribution in the knowledge diffusion process of team inertia network.

In this paper, we are still concerned with the team inertia network as a whole, and the average Gini coefficient and the average Global Moran’s I of the network at any time are given by the following equations:

$$\begin{aligned}&G(t) = \frac{1}{m}\sum _{j = 1}^m G_j(t), \end{aligned}$$

(30)

$$\begin{aligned}&M(t) = \frac{1}{m}\sum _{j = 1}^m M_j(t). \end{aligned}$$

(31)

In our specific analysis, we visualized all experimental trajectories of G(t) and M(t), and combined these with Kernel Density Estimation (KDE) plots to illustrate the aggregation trends of the results across both the temporal and metric-value dimensions. Unlike the statistical analysis conducted for the heatmaps, the key objective in analyzing the Gini coefficient and Global Moran’s I lies in revealing their convergence behaviors–that is, to track the average evolutionary processes of G(t) and M(t), and rigorously determine the specific time points at which they have statistically converged to their final states.

To achieve this, we defined a unified statistical baseline for both indicators, represented by the average value across all experiments at the final time step:

$$\begin{aligned} \mu _0^{V} = \frac{1}{N} \sum _{i=1}^N V_i(T-1), \end{aligned}$$

(32)

where V denotes either G or M.

We then employed four statistical tools to assess the mean evolutionary trajectories of $\overline{G}(t)$ and $\overline{M}(t)$: First, we applied the Bootstrap BCa method to compute the 95% confidence intervals for the mean trajectory at each time step, thereby evaluating the estimation precision of the mean trajectory itself, where the CI width directly reflects the degree of variability in the results. Second, we used Cohen’s d⁶⁴ to quantify the effect size–that is, the standardized magnitude of deviation–of $\overline{V}(t)$ relative to the baseline $\mu _0^V$:

$$\begin{aligned} d(t) = \frac{\overline{V}(t) - \mu _0^V}{\sigma _{V,\text {pooled}}}. \end{aligned}$$

(33)

This standardized measure allows us to clearly identify when the team’s Gini coefficient or Global Moran’s I undergoes substantial change. Most critically, we applied the Wilcoxon signed-rank test in combination with the Benjamini–Hochberg (BH) procedure for False Discovery Rate (FDR) correction, to determine at which time steps G(t) and M(t) differ significantly from the baseline, thereby avoiding Type I errors caused by multiple comparisons.

Importantly, the baseline $\mu _0^V$ is set to the empirical mean at the final time step $T-1$, so our test evaluates whether the distribution at time t deviates from that final distribution, rather than from a fixed theoretical value. The Wilcoxon test is a one-sample signed-rank test on the paired differences $V_i(t)-\mu _0^V$ across experiments, with the null hypothesis $H_0: \text {median}[V(t)-\mu _0^V]=0$. Crucially, Cohen’s d is standardized using the pooled standard deviation ($\sigma _{V,\text {pooled}}$) across the entire time series V, instead of the time-varying standard deviation $\sigma _V(t)$. This ensures that the effect size d(t) remains comparable across all time steps and accurately reflects the magnitude of the mean deviation from the baseline.

Through this unified and rigorous analytical framework, we are able to evaluate the efficiency of knowledge diffusion and its spatial structuring under a common statistical standard, ensuring the scientific robustness and reliability of our conclusions.

Figs. 6 and 7 illustrate the evolution of the Gini coefficient during knowledge diffusion in the team inertia network, with experiments conducted over 1000 time steps and independently repeated 30 times. The evolution trajectory of the graph in subfigure (a) or the average trajectory of the graph in subfigure (b) indicates that the Gini coefficient rises rapidly during the growth phase, before the team reaches its member limit; once the network enters the refinement phase, where new nodes are added through the elimination of low in-degree nodes, the growth of the Gini coefficient slows considerably and eventually stabilizes. This occurs because the refinement phase preserves high in-degree nodes, reducing the pace of knowledge concentration. Kernel density estimation further shows that the trajectories overlap considerably in the early stage but diverge progressively over time, reflecting patterns consistent with real-world team or industry development: in the early stage, competing teams may perform similarly, but over time, differences emerge, with some teams outperforming others. From this perspective, the proposed team inertia network effectively captures the dynamics of knowledge accumulation and development. From a statistical standpoint, under both parameter sets the Gini coefficient of the team’s inertia network consistently converges within a finite number of steps (approximately between 400 and 600). This indicates that, as the network evolves, the distribution of knowledge among individuals rapidly settles into and is maintained in a structurally unequal steady state–that is, the network completes an early transition from a highly dynamic phase of redistribution to a relatively fixed “high–low” stratified pattern, and subsequent evolution only induces small fluctuations rather than systematic reallocation.

Beyond the statistical evidence, the dynamic changes in the Gini coefficient not only reflect inequality in knowledge diffusion but also potentially influence internal team innovation, collaboration quality, and decision-making efficiency. Higher Gini coefficients indicate that knowledge is concentrated among a few core nodes, which may enhance the innovation capacity of these key members but could also create knowledge barriers that hinder collaboration and diverse decision-making. Conversely, lower Gini coefficients suggest a more balanced knowledge distribution, promoting broad participation and collaboration, though possibly lacking strong driving forces. The results further show that the intensity of intra-team communication, represented by the parameter k, significantly affects the Gini coefficient fluctuations: when k is small, limited communication amplifies disparities in knowledge distribution; when k is large, communication is sufficient, leading to more stable Gini coefficients and facilitating knowledge sharing, innovation, and collaboration across the team. Overall, the evolution of the Gini coefficient reveals the complex relationship between internal knowledge distribution and team efficiency, innovation potential, and collaborative dynamics.

Fig. 6

Equality in the process of knowledge diffusion in team inertia networks($k = 24,d = 1$).

Fig. 7

Equality in the process of knowledge diffusion in team inertia networks ($k = 10,d = 1$).

Figs. 8 and 9 illustrate the dynamic evolution of the Global Moran’s I during the process of knowledge diffusion in the team inertia network. The experimental results indicate that the process of knowledge diffusion in the team inertia network exhibits an overall tendency of negative spatial autocorrelation. In other words, nodes with different levels of knowledge stock are more likely to cluster together in the network space. The underlying mechanism of this phenomenon lies in the inherent driving force of diffusion: the uneven distribution of knowledge across nodes. Knowledge typically flows from nodes with higher stocks to those with lower stocks, thereby leading to a negatively clustered spatial pattern. From a more rigorous statistical perspective, the Global Moran’s I of the team inertia network still converges within a finite number of steps, and the smaller the parameter k the slower the convergence–which indicates that k is a key limiting factor for the speed at which the team’s knowledge-space structure forms. Specifically, when k is relatively large, the team can sustain ample internal communication, which keeps Moran’s I relatively stable and limits fluctuations in spatial autocorrelation. By contrast, when k is small, local information exchange between individuals becomes sparse: the constrained communication opportunities substantially reduce the efficiency of knowledge transfer, thereby exacerbating negative spatial autocorrelation and causing a pronounced early-stage decline in Moran’s I during the diffusion process. However, as knowledge gradually disperses and the network structure rebalances, this negative autocorrelation eventually stabilizes, albeit with a longer overall convergence time. Overall, the experiments confirm that knowledge diffusion is driven not only by heterogeneity in nodes’ knowledge endowments but is also significantly modulated by the team’s internal communication frequency (i.e., the parameter k); together, these factors shape the dynamic evolutionary trajectory of the macro-level spatial distribution of knowledge.

Fig. 8

The spatial distribution of knowledge diffusion in team inertia network ($k = 24,d = 1$).

Fig. 9

The spatial distribution of knowledge diffusion in team inertia network ($k = 10,d = 1$).

Discussion

Knowledge diffusion is a key channel for narrowing the gap in knowledge stock between different regions, groups, and fields. It is also an important means to promote the dissemination of science and culture and to accelerate the development of social productivity. This paper clearly expresses the inertia mechanism in the network, proposes a team inertia network model based on the real knowledge diffusion process, gives the dynamic evolution process and knowledge diffusion mechanism in the model, and proposes the metrics centered on the average knowledge stock, the difference of knowledge stock, and the spatial distribution.

Simulation experiments and result analysis show that spatial inertia and temporal inertia in team inertia networks have complex and crucial impacts on team knowledge diffusion and overall development. Firstly, moderate spatial inertia can promote internal knowledge flow within the team, thereby enhancing the average knowledge level of the team; however, overly strong or weak spatial inertia will inhibit knowledge diffusion, restricting the overall knowledge accumulation of the team. This phenomenon theoretically reflects the balance mechanism that teams need to maintain a certain structural stability to preserve core knowledge accumulation while also possessing flexibility to allow new knowledge to enter and spread. Specifically, moderate spatial inertia acts as a structural stabilizer. By fostering strong and durable social ties, it provides the necessary redundancy, trust, and continuity for effective internal knowledge circulation. This stability enables repeated interactions among familiar partners, facilitating the transfer and internalization of tacit knowledge, reinforcing existing competencies, and thereby enhancing the team’s average knowledge level through knowledge exploitation. Conversely, overly strong spatial inertia generates network closure and structural rigidity, which restricts the inflow of heterogeneous external knowledge needed for exploration, limiting the team’s capacity to broaden knowledge and innovate. On the other hand, overly weak spatial inertia produces a fragmented, low-density network where interactions are too transient to allow knowledge sedimentation and collective memory formation. While this may temporarily facilitate superficial exposure to new knowledge, it undermines knowledge exploitation and results in diluted core competencies. Thus, moderate spatial inertia represents the optimal configuration through which teams balance the dual demands of retaining and leveraging existing knowledge while maintaining sufficient flexibility and openness to acquire and diffuse novel knowledge. This micro-level preference mechanism naturally gives rise to the observed macroscopic patterns of knowledge accumulation, internal distribution, and diffusion efficiency within the team.

Secondly, the closeness of internal team connections and communication frequency play a decisive role in knowledge diffusion: when team members have a large number of subsequent nodes, temporal inertia can balance knowledge growth within the team, indicating that frequent interaction helps reduce local imbalance in knowledge accumulation and enhances the team’s overall innovation potential; while when the number of subsequent nodes is limited and the dimension of knowledge vectors is small, a smaller temporal inertia is more conducive to team development because historical dependence is lower, allowing the team to adapt to new knowledge input more quickly; conversely, in high-dimensional knowledge environments, temporal inertia should be fully utilized to maintain knowledge continuity. This pattern can be interpreted through the lens of the team’s relational social capital and the micro-level deployment of temporal inertia. When the network is dense with many subsequent nodes, high redundancy supports reliable knowledge circulation. Temporal inertia stabilizes these interactions, reinforcing relational ties, facilitating codification and internalization of knowledge, and ensuring diffusion reaches all nodes–thus enhancing the team’s innovation potential. In contrast, when network connectivity is sparse and knowledge dimensions are low, smaller temporal inertia allows the team to remain agile, rapidly integrating new inputs and discarding outdated practices without being constrained by historical path dependence. This flexibility supports adaptive learning but limits long-term knowledge consolidation. For high-dimensional or complex knowledge, temporal inertia should be adjusted to preserve systemic continuity and collective memory. Sustained interactions prevent fragmentation of interdependent knowledge, maintain shared mental models, and facilitate the effective long-term utilization of complex knowledge, supporting both organizational sense-making and sustained innovation.

Thirdly, the research finds that in teams with moderate inertia, the inequality of internal knowledge distribution (Gini coefficient) will gradually stabilize over time, suggesting that the team inertia mechanism can self-regulate knowledge imbalance to a certain extent, thereby supporting sustainable development and innovation. This stabilization theoretically reflects the mechanism of systemic homeostasis and the maturation of collective absorptive capacity. It is not a state of static equality, but a dynamic equilibrium achieved through the team’s self-regulating capacity. Moderate inertia–arising from stable interaction patterns and structural coherence–acts as a structural dampener: when knowledge disparities become excessive, it ensures sufficient and sustained interactions, generating feedback loops that facilitate the sharing of core competencies and the transfer of knowledge from high-endowment members to others. This dynamic process effectively limits extreme divergence, preventing destructive knowledge silos or resource-hoarding behaviors. Moreover, the steady Gini coefficient indicates that the team has developed an effective collective absorptive capacity. The structural consistency provided by moderate inertia enables the team to consistently assimilate, transform, and exploit both internal and external knowledge. Newly acquired knowledge is diffused and integrated across the team rather than concentrating in a few individuals. This controlled heterogeneity, maintained within a productive range, preserves cognitive diversity necessary for innovation while guaranteeing a sufficient shared knowledge base for collective competence and sustainable development. Consequently, the team structure is not only stable but also resilient, capable of dynamically correcting internal imbalances and sustaining its learning trajectory over time.

Fourthly, due to the imbalance in knowledge sharing within the team, a negative autocorrelation phenomenon (Moran’s I is negative) occurs in the team inertia network, indicating that nodes with high knowledge stock tend to form complementary interactions with nodes with low knowledge stock. This discovery provides a precise theoretical lens for understanding the micro-mechanism of knowledge flow, highlighting the essential role of resource complementarity in driving effective knowledge exchange. A negative Moran’s I indicates that high-value nodes (high knowledge stock) are structurally adjacent to low-value nodes, implying that knowledge diffusion within the team emerges from the dynamics of mutual complementarity, where diverse knowledge endowments stimulate interactive learning and adaptive exchange. In practice, high-knowledge nodes engage with low-knowledge nodes to fill knowledge gaps, validate existing knowledge, and enable asymmetric learning. This purposeful interaction suggests that knowledge diffusion is not merely a passive spillover but an active, functional process. The inertia network thereby facilitates heterogeneous interactions, which empirical studies suggest are more effective at generating novel combinations and accelerating overall knowledge accumulation than interactions between similar nodes.

Although the “inertia mechanisms” has no clear precedent as an independent modeling object in the research of knowledge diffusion, the inertia-related effects have already received attention in related fields such as social diffusion. At the network research level, phenomena related to the semantics of inertia, such as repetitive cooperation, time dependence, and path dependence in scientific research or innovation activities, have an inherent continuity with the mechanisms proposed in this paper. This study explicitly presents temporal inertia and spatial inertia within the framework of complex networks and introduces them into the modeling and simulation of team knowledge diffusion. This formal treatment not only elevates previous qualitative observations to operational mechanisms but also reveals how the two types of inertia jointly shape the macroscopic pattern of knowledge diffusion through the interaction of micro-nodes, providing a new theoretical perspective for understanding the flow of knowledge and innovation patterns within teams.

The core contribution of this paper lies in systematically proposing an inertial mechanism in networks that encompasses both temporal and spatial inertia, and further introducing this mechanism into team contexts to construct a team inertial network with clear evolutionary rules. This network not only characterizes the patterns of knowledge interaction among team members based on historical behaviors and structural preferences but also provides a systematic tool for analyzing team knowledge diffusion, innovation, and collaboration. On this basis, we reveal the dynamic characteristics of knowledge diffusion in team inertial networks through simulation analysis, including the impacts of spatial and temporal inertia on the inequality of team knowledge accumulation (Gini coefficient) and spatial distribution (Global Moran’s I). The research results indicate that moderate spatial inertia and reasonable team communication frequency can enhance the average knowledge level of the team, alleviate the inequality of knowledge distribution, and boost team innovation capabilities, while insufficient or excessive inertia is detrimental to the overall development of the team.

Theoretically, this paper offers a new mathematical framework for explaining team evolution and knowledge diffusion, clarifying the role of inertial mechanisms in the internal knowledge flow, distribution patterns, and innovation potential formation within teams. It fills the gap in traditional network research and team theories regarding the lack of quantitative analysis of historical behavior dependence and structural inertia. Practically, this research provides clear implications for team design and management: by rationally regulating the communication intensity among team members, optimizing team size and structure, and moderately leveraging historical behavior experience, the overall knowledge level and innovation efficiency of the team can be enhanced, and the potential barriers caused by knowledge concentration can be reduced. Future research can further explore the optimal design of team inertial parameters under different types of knowledge, cross-team collaboration, external intervention, and high-dimensional knowledge scenarios to achieve a balanced optimization of knowledge diffusion and innovation efficiency.

Of course, this paper has several limitations. First, in order to reduce the number of model parameters and simplify the analysis, the effects of network weights were temporarily ignored; however, this factor should ideally be incorporated into node selection preferences. Second, aside from the impact of inertia on knowledge diffusion, this study only examined team knowledge diffusion when $d = 1$ and did not fully explore the complex effects of varying the parameter d on the diffusion process. Third, due to the current complexity of the model–which includes both structural information and detailed node-level knowledge, as well as numerous parameters–analyzing the network’s complex statistical properties remains challenging, and thus we did not investigate the full dynamic complexity of the team inertia network. Fourth, although we observed the inertia mechanism in real networks, limitations in data availability prevented us from validating the performance of the team inertia network using empirical team knowledge diffusion data. Fifth, the specific functional forms chosen for the temporal memory kernel and the spatial similarity measure were guided by existing studies, ensuring theoretical consistency. Conducting formal empirical validation or systematic robustness testing of these functions remains a valuable direction for future research; however, such tests are practically challenging due to the scarcity of suitable real-world data and the substantial computational resources required for extensive model evaluation. These limitations represent key directions for future research, which will be systematically addressed in subsequent studies.