A network model of social contacts with small-world and scale-free features, tunable connectivity, and geographic restrictions

1.   Introduction

The study of complex networks experienced significant advances, over two decades ago, due to the expansion of the Internet ^[1,2] and, more recently, due to the outbreak of the COVID-19 pandemic ^[3,4]. The characterization of the connection patterns among computers and among humans has been crucial to understand, for instance, the dissemination of factual and false information ^[5], the propagation of computer viruses ^[6], and the spreading of contagious diseases ^[7].

In addition to the network model based on purely random connections, first theoretically analyzed by Gilbert ^[8] and Erdös and Rényi ^[9,10], two other network models have garnered widespread attention and utilization in the academic research. These models are commonly referred to as "small world" and "scale free". Such models present crucial differences in their degree distributions $P(k)$ and average clustering coefficients $\langle c \rangle$. Recall that, in undirected graphs, the degree $k$ of a node is the number of links connected to this node, $P(k)$ denotes the percentage of nodes with degree $k$, the clustering coefficient $c$ of a node is defined as the ratio of the number of links among its neighbors to the maximum number of possible links among these neighbors, and the average degree $\langle k \rangle$ and the average clustering $\langle c \rangle$ are the average values of $k$ and $c$, respectively, by taking into account all $N$ nodes composing the network ^{[11,12,13,14]}. Small-world graphs, originally proposed by Watts and Strogatz ^[15,16], emerge from a rewiring process applied to a regular lattice with a constant $N$. In this transformative process, a portion of the regular links are probabilistically replaced by random links. Such graphs exhibit "high" average clustering coefficient and bell-shaped degree distribution (here, "high" $\langle c \rangle$ means $\langle c \rangle \gg \langle c \rangle_{random} = \langle k \rangle/N$, in which $\langle c \rangle_{random}$ is the average clustering obtained for a purely random network with the same values of $\langle k \rangle$ and $N$). Scale-free networks, employed by Barabási and colleagues ^[11,17] to investigate the topology of various complex systems, are built from a preferential attachment rule (previously introduced by other authors ^[18,19,20]), wherein the probability of a new node forming a connection with a pre-existing node is directly proportional to the degree of this pre-existing node. Such graphs exhibit "low" $\langle c \rangle$ and power-law degree distribution. Despite being widely employed, both these approaches have their shortcomings ^[21,22,23]: for instance, the rewiring process may be perceived as somewhat contrived and the preferential attachment demands that the new node possesses awareness of the degrees of all nodes composing the growing network.

Social interactions among individuals, groups, and even corporations have been represented by graphs ^{[24,25,26,27]}. In real-world scenarios, these interactions are simultaneously characterized by "high" $\langle c \rangle$ and $P(k) \sim k^{-\gamma}$; that is, a degree distribution following a power law. Therefore, actual social networks concurrently exhibit topological properties found in small-world and scale-free graphs ^{[11,12,13,14]}. For instance, for a network in which the nodes represent actors and actresses and a link between two nodes implies that they have acted in the same movie together, $\langle c \rangle = 0.79$ and $\gamma = 2.3$ ^[11,15,17]. The model investigated in this article embodies these characteristics in a network with fixed number of nodes. Furthermore, it presents "low" average shortest-path length $\langle \ell \rangle$, as observed in real-world networks and also in purely random, small-world, and scale-free graphs ^{[11,12,13,14]} (here, "low" $\langle \ell \rangle$ means $\langle \ell \rangle \approx \langle \ell \rangle_{random} \simeq \log N/\log \langle k \rangle$, in which $\langle \ell \rangle_{random}$ is the average shortest-path length obtained for a purely random network with the same values of $\langle k \rangle$ and $N$).

The network model numerically investigated here was originally proposed by us ^[28] for analyzing the impact of socioeconomic stratification on word-of-mouth dissemination of information about COVID-19. In that study ^[28], however, the influence of the model parameters on the network structure was not explored. This model is based on the assumption that each individual typically interacts with a local community, in close physical proximity, composed of nearby family members, friends, neighbors, coworkers. In the model, the individuals can establish a maximum number of links within a limited area representing the geographic region where they live. Thus, the individuals are primarily locally connected, leading to "high" clustering. The number of links per individual is randomly taken from a power-law distribution. Therefore, the network topology becomes scale free, since $P(k)$ obeys a power law. Simpler versions of this model were already employed in studies on neurophysiology ^[29], game theory ^[30], and epidemiology ^[31,32,33].

There have been proposed computational methods for growing scale-free networks with tunable clustering coefficient and/or degree distribution ^{[34,35,36,37,38,39]}. Also, there have been proposed computational methods for creating networks that take into account the distances between nodes ^{[29,30,40,41,42]}. The algorithm numerically explored here enables adjusting the power-law exponent $\gamma$ of $P(k) \sim k^{-\gamma}$ and the average degree $\langle k \rangle$ in a fixed-size network with geographic constraints. Realistic values of $\gamma$ and $\langle k \rangle$ have been determined for several social groups ^[11,12,28] and they can be used in this algorithm.

In short, this article is about a network model with $\langle \ell \rangle \approx \langle \ell \rangle_{random}$, $\langle c \rangle \gg \langle c \rangle_{random}$ and $P(k) \sim k^{-\gamma}$, with tunable values of $\langle k \rangle$ and $\gamma$, in which geographic constraints are considered in the probabilistic creation of links. A network model with all these features was not found in the literature. The remainder of this article is organized as follows. In Section 2, the developed algorithm is explained. In Section 3, network metrics usually employed to characterize the topological structure of graphs are listed. In Section 4, the results obtained from numerical simulations are presented. In Section 5, these results are discussed and the possible relevance of the network model is stressed.

2.   An algorithm for generating complex social networks

Consider a square grid composed of $\eta \times \eta$ cells, in which each cell represents an individual. Therefore, there are $N = \eta^2$ individuals in this social group. To mitigate edge effects, the left and right edges are connected, and similarly, the top and bottom edges are also connected. Thus, all individuals residing in this grid are geographically equivalent. In this network model, social contacts among individuals can occur within a neighborhood of radius $r$. These social contacts are represented by undirected links between individuals. These links are established through a random wiring process, in which an individual is connected to $k$ others positioned within the square matrix of size $(2r + 1) \times (2r+1)$ centered around such an individual (self-connections and multiple links are not allowed) ^[28,29,30]. In cellular automata literature, $r$ is called Moore's radius ^[43]. Figure 1 illustrates an individual (black cell) with $k = 5$ neighbors (dark gray cells) in a Moore neighborhood with $r = 2$ (light gray cells).

Figure 1.  A block $7 \times 7$ of a grid showing the (dark gray) neighbors of the (black) central cell. In this example, $r = 2$; hence, the neighbors of the central cell live in a (light gray) neighborhood $5 \times 5$. The layer $i = 1$ is composed of 8 cells and the layer $i = 2$ of 16 cells. Notice that the central cell has 3 neighbors in the layer $i = 1$ and 2 neighbors in the layer $i = 2$; therefore, for this cell, $k = 5$. In the model, the probability of two cells being connected (being neighbors) is given by Eq (2.1). The (white) cells in the layer $i = 3$ are beyond the neighborhood radius; hence, such cells cannot be connected to the central cell.

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In the model, the probability $q_i$ of connecting an individual to another individual belonging to the neighborhood layer $i$ ($i = 1, 2, ..., r$) is calculated from ^[28,29,30]:

Notice that $\sum_{i = 1}^r q_i = 1$ and $q_i$ decreases with $i$. Therefore, the greater the distance between two individuals is, the smaller the probability of them being connected. For instance, for $r = 2$, then $q_1 = 2/3$ and $q_2 = 1/3$. Since the links are primarily local, this model can be suitable for representing face-to-face encounters in a social group. Such encounters can impact, for instance, information diffusion, infection spreading, and goods exchange.

The degree $k$ of each individual is obtained as follows. First, a number $x$ is randomly picked from the standard uniform distribution $U(0, 1)$. Then, the value of $k$ is determined from $P(k) = x$; thus, $k = P^{-1}(x)$ (as in an inverse transform sampling) with $P(k) = A k^{-\gamma}$ ^[28], in which $A$ is the normalization constant and $\gamma$ is a positive exponent. As the degree distribution $P(k)$ follows a power law, the resulting graph will exhibit scale-free characteristics ^{[11,12,13,14,17]}. The normalization condition $\sum_{k = k_{min}}^{k_{max}} P(k) = 1$ imposes that the constant $A$ must be equal to:

By definition ^{[11,12,13,14]}, $P(k) = n_k/N$, in which $n_k$ is the number of individuals with degree $k$; hence, $\sum_{k = k_{min}}^{k_{max}} n_k = N$. As a consequence:

The average degree $\langle k \rangle$ is a relevant metric of realistic networks. In order to adjust the value of $\langle k \rangle$ of the resulting graph to the target value $k_{tar}$ found in a real-world society, the minimum degree $k_{min}$ and the maximum degree $k_{max}$ of $P(k)$ must be conveniently chosen. Recall that $\langle k \rangle$ can be computed from ^{[11,12,13,14]}:

By taking into consideration Eqs (2.2) and (2.3), then:

Notice that the condition $n_{k_{max}} \geq 1$ (that is, there is at least one individual with $k = k_{max}$) implies:

Equations (2.5) and (2.6) can be used to computationally construct a graph with $\langle k \rangle = k_{tar}$. Notice that $(2r+1)^2-1 \geq k_{max}$; that is, the maximum number of potential neighbors must be greater than or equal to $k_{max}$ (for instance, if $r = 1$, it is impossible to build a graph with $k_{max} = 9$, because each individual will have at most 8 neighbors).

To better explain how the values of $k_{min}$ and $k_{max}$ are chosen, Figure 2 shows $\langle k \rangle$ (the blue line) and $k_{max}$ (the green line) as functions of $k_{min}$ for $N = 4900$, $\gamma = 2.5$ and $\langle k \rangle_{tar} = 10$ (the red line). The vertical axis is logarithmically scaled. The plot of $\langle k \rangle$ was obtained from Eq (2.5) and the plot of $k_{max}$ from Eq (2.6) (by considering the equals sign). Notice that for $k_{min} = 3$, then $\langle k \rangle \simeq \langle k \rangle_{tar}$ and, consequently, $k_{max} = 113$. Therefore, for $\gamma = 2.5$ and $\langle k \rangle_{tar} = 10$, the algorithm takes $k_{min} = 3$ and $k_{max} = 113$.

Figure 2.  The plots of $\langle k \rangle$ (the blue line) and $k_{max}$ (the green line) as functions of $k_{min}$ for $N = 4900$, $\gamma = 2.5$ and $\langle k \rangle_{tar} = 10$ (the red line, which was included just for reference).

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A pseudocode of the algorithm is given below. The input parameter values are: $\eta$ (which determines the total number of individuals $N$), $r$ (the neighborhood radius), $\gamma$ (the exponent of the power law), and $k_{tar}$ (the average degree of the actual social group). The aim is to build a primarily locally connected graph with $\langle k \rangle = k_{tar}$ and $P(k)$ obeying a power law. In the algorithm, $n_k$ is taken as the integer part of the real number calculated from Eq (2.3); hence, $\langle k \rangle$ in the graph is usually smaller than $k_{tar}$ in the early stages of a simulation. If $\langle k \rangle > k_{tar}$ after building an initial graph (which is very rare), the algorithm stops because a graph with the specified values of $k_{tar}$ and $\gamma$ cannot be built with the proposed computational method.

The developed algorithm sequentially creates $M \simeq N k_{tar}/2$ links in order to obtain $\langle k \rangle = \langle k \rangle_{tar}$ and $P(k) = A k^{-\gamma}$ (recall that $\langle k \rangle = 2M/N$ ^{[11,12,13,14]}, in which $M$ is the number of links). Some observations regarding the pseudocode are as follows:

● About step 4, the algorithm takes into account the integer part of $n_k$ determined from Eq (2.3) (for instance, if $n_5 = 67.12$, then the algorithm will create a graph with 67 nodes with $k = 5$).

● About step 5, if there is already a link with the randomly selected node, a new random selection must be made (recall that self-connections and multiple links are prohibited).

● About step 7, in dozens of simulations for networks with $1000 \leq N \leq 10000$, isolated subgraphs were never found. The idea of step 7 is to connect a randomly chosen node from an isolated subgraph to another randomly chosen node from the largest subgraph.

● Algorithm convergence problems can occur only for $\gamma \leq 1$, which is a value of $\gamma$ not found in realistic social networks.

The topological structure of graphs is usually characterized by computing the metrics presented in the next section.

3.   Network metrics

The connection pattern of graphs can be quantified by calculating $P(k)$, $\langle k \rangle$, $\langle l \rangle$, $\langle c \rangle$, $\langle C_c \rangle$, and $\langle C_b \rangle$. These statistical measures are defined below.

As mentioned in the previous section, $P(k) = n_k/N$ and $\langle k \rangle = \sum_{k = {k_{min}}}^{k_{max}} k P(k)$, in which $n_k$ is the number of nodes with degree $k$.

For undirected graphs, the average shortest-path length $\langle \ell \rangle$ is obtained from ^[12,13,28]:

in which $\ell_{ij}$ is the shortest distance (the minimum number of links) between the nodes $i$ and $j$. This metric influences the velocity that information can travel in the network. The smaller $\langle \ell \rangle$, the quicker the communication among the nodes.

For the node $i$, the clustering coefficient $c_i$ is defined as ^[12,13,28]:

in which $b_i$ is the number of connections among its $k_i$ neighbors. Since this metric reflects the local connectivity, it is commonly used to identify communities.

Centrality measures are calculated to evaluate the relevance of the nodes composing the network. The closeness centrality $C_c(i)$ of the node $i$ is defined as ^[13,28,44]:

This metric reveals how quickly the node $i$ can communicate with others in the network.

The betweenness centrality $C_b(i)$ of the node $i$ is defined as ^[13,28,44]:

in which $g_{js}$ is the number of the shortest paths between the nodes $j$ and $s$, and $g_{j s}(i)$ is the number of the shortest paths between the nodes $j$ and $s$ passing through the node $i$. This metric quantifies the extent to which the node $i$ facilitates communication among other nodes.

For the whole network, average values of the metrics defined by Eqs (3.2)–(3.4) are computed from $\langle c \rangle = \sum_{i = 1}^N c_i/N$, $\langle C_c \rangle = \sum_{i = 1}^N C_c(i)/N$, and $\langle C_b \rangle = \sum_{i = 1}^N C_b(i)/N$.

In the next section, the numerical results obtained in simulations are presented. Dozens of graphs were computationally created and analyzed. Then, nine graphs were selected, because they serve as representative examples.

4.   Results of computer experiments

Numerical simulations were performed to investigate the influence of $r$, $\gamma$, and $k_{tar}$ on the metrics described in the previous section. In these simulations, $\eta = 70$ (thus, the graphs are composed of $N = 4900$ nodes). Three sets of simulations were run with the following parameter values:

● set 1: $r = 10$, $\gamma = 3$, and $k_{tar} \in \{20, 45,100 \}$;

● set 2: $k_{tar} = 20$, $\gamma = 3$, and $r \in \{5, 20, 60 \}$;

● set 3: $r = 10$, $k_{tar} = 15$, and $\gamma \in \{0.3, 2.5, 8\}$.

Figure 3 shows how $P(k)$ varies with $k$ for these three sets of parameter values. Set 1 corresponds to the first column of Figure 3, set 2 to the second column, and set 3 to the third column. Tables 1, 2, and 3 present the metrics computed for sets 1, 2, and 3, respectively. The averages and the standard deviations of these metrics were obtained in 5 simulations with the same parameter values. Notice that the standard deviations are less than 0.1%.

Figure 3.  Log-log plots (log base 10) of the degree distribution $P(k)$ of the computer-generated graphs with $N = 4900$ nodes. In the first column, $r = 10$, $\gamma = 3$, and $k_{tar} \in \{20, 45,100 \}$. In the second column, $k_{tar} = 20$, $\gamma = 3$, and $r \in \{5, 20, 60 \}$. In the third column, $r = 10$, $k_{tar} = 15$, and $\gamma \in \{0.3, 2.5, 8\}$. The red lines correspond to linear regressions for $k \ll k_{cut}$, with the value of $\gamma$ shown in each plot.

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Table 1.  Values of $100 \times \langle C_c \rangle$, $100 \times \langle C_b \rangle$, $\langle \ell \rangle$, $100 \times \langle c \rangle$, $\langle k \rangle$, and $\Delta k$ for set 1.

$k_{tar}$	$100 \times \langle C_c \rangle$	$100 \times \langle C_b \rangle$	$\langle \ell \rangle$	$100 \times \langle c \rangle$	$\langle k \rangle$	$\Delta k$
20	$23.90 \pm 0.03$	$0.06511 \pm 0.00003$	$4.189 \pm 0.008$	$6.513 \pm 0.003$	$20.1 \pm 0.3$	$29.20 \pm 0.03$
45	$28.59 \pm 0.02$	$0.05103 \pm 0.00002$	$3.499 \pm 0.005$	$11.687 \pm 0.004$	$45.2 \pm 0.2$	$56.4 \pm 0.1$
100	$31.72 \pm 0.03$	$0.04397 \pm 0.00002$	$3.153 \pm 0.006$	$22.352 \pm 0.008$	$100.3 \pm 0.1$	$83.80 \pm 0.05$

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Table 2.  Values of $100 \times \langle C_c \rangle$, $100 \times \langle C_b \rangle$, $\langle \ell \rangle$, $100 \times \langle c \rangle$, $\langle k \rangle$, and $\Delta k$ for set 2.

$r$	$100 \times \langle C_c \rangle$	$100 \times \langle C_b \rangle$	$\langle \ell \rangle$	$100 \times \langle c \rangle$	$\langle k \rangle$	$\Delta k$
5	$16.50 \pm 0.02$	$0.10339 \pm 0.00004$	$6.064 \pm 0.004$	$16.937 \pm 0.004$	$20.0 \pm 0.1$	$28.20 \pm 0.03$
20	$29.24 \pm 0.03$	$0.04956 \pm 0.00001$	$3.427 \pm 0.003$	$2.826 \pm 0.002$	$20.2 \pm 0.2$	$34.10 \pm 0.06$
60	$31.74 \pm 0.02$	$0.04406 \pm 0.00002$	$3.158 \pm 0.004$	$1.278 \pm 0.001$	$20.0 \pm 0.1$	$89.40 \pm 0.08$

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Table 3.  Values of $100 \times \langle C_c \rangle$, $100 \times \langle C_b \rangle$, $\langle \ell \rangle$, $100 \times \langle c \rangle$, $\langle k \rangle$, and $\Delta k$ for set 3.

$\gamma$	$100 \times \langle C_c \rangle$	$100 \times \langle C_b \rangle$	$\langle \ell \rangle$	$100 \times \langle c \rangle$	$\langle k \rangle$	$\Delta k$
0.3	$24.63 \pm 0.02$	$0.06286 \pm 0.00003$	$4.079 \pm 0.006$	$7.084 \pm 0.003$	$21.5 \pm 0.8$	$14.20 \pm 0.05$
2.5	$22.51 \pm 0.02$	$0.07046 \pm 0.00004$	$4.451 \pm 0.008$	$5.911 \pm 0.002$	$15.8 \pm 0.1$	$28.40 \pm 0.01$
8.0	$21.48 \pm 0.03$	$0.07469 \pm 0.00004$	$4.658 \pm 0.009$	$4.413 \pm 0.003$	$15.3 \pm 0.2$	$5.080 \pm 0.003$

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These tables also present $\Delta k = k_{cut}- k_{min}$, in which $k_{cut}$ is the degree above which the data deviate more significantly from the straight lines shown in Figure 3. These lines were obtained from the least squares fitting method ^[45] with the value of $\gamma$ given in each plot. In fact, various real-world degree distributions are usually better described by ^{[11,12,23,28,46]}:

that is, by a power law with exponential cutoff. Equation (4.1) implies:

Therefore, the linear regression yields a satisfactory fit in the log-log plot for $k \ll k_{cut}$.

For $r = 10$ and $k_{tar} = 15$ (as adopted in set 3), Figure 4 exhibits how $\Delta k$ depends on the exponent $\gamma$ for $0.3 \leq \gamma \leq 10$.

Figure 4.  The interval $\Delta k$ as a function of the exponent $\gamma$ for $r = 10$ and $k_{tar} = 15$.

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5.   Discussion and conclusion

Here, the algorithm developed to represent face-to-face social interactions through graphs was numerically explored, by investigating the impact of $r$, $\gamma$, and $\langle k \rangle_{tar}$ on the graph structure. As shown in Figure 3, the algorithm can generate graphs with $P(k) \sim k^{-\gamma}$, despite the exponential cutoff. Tables 1–3 indicate that $\langle k \rangle \simeq k_{tar}$ for most graphs. An evident exception is the plot with $r = 10$, $\gamma = 0.3$, and $k_{tar} = 15$. In this case, $P(k)$ exhibits a bell-shaped curve, which is typical of purely random and small-world networks ^[11]. This result was unexpected, but it is a consequence of taking $\gamma < 1$ (which is not found in actual networks ^{[11,12,23,46]}). For $\gamma \to 0$, then $P(k) \to A$, $k_{max} \to \infty$ and $\Delta k \to 0$. In this limit case, due to the geographical restrictions imposed by Eq (2.1), $P(k)$ becomes a Poisson-like distribution in the log-log plot. The simulations also show that, as $\gamma$ increases, the bell-shaped curve in $P(k)$ is stretched along the $k$-axis and its maximum shifts to the right. Thus, the typical tail usually found in $P(k)$ for scale-free networks ^[11,17] appears here as a vestige of the stretched Poisson-like distribution.

For purely random networks, $\langle \ell \rangle_{random} \simeq \log N / \log \langle k \rangle$ ^[11], which is a convenient formula to estimate $\langle \ell \rangle$ for the graphs analyzed here. For instance, for $r = 60$, $\gamma = 3$, and $k_{tar} = 20$ (see Table 2), then $\langle \ell \rangle = 3.16 \approx \langle \ell \rangle_{random} \simeq \log 4899/\log 20.04 = 2.83$. Also, for purely random networks, $\langle c \rangle_{random} = \langle k \rangle/N$ ^[11]. For the graphs analyzed here, $\langle c \rangle \gg \langle c \rangle_{random}$. For instance, for $r = 10$, $\gamma = 2.5$, and $k_{tar} = 15$ (see Table 3), then $\langle c \rangle = 5.9\% \gg \langle c \rangle_{random} = 15.776/4899 = 0.3\%$. Therefore, for the graphs exhibited in Figure 2, $\langle \ell \rangle \approx \langle \ell \rangle_{random}$ and $\langle c \rangle \gg \langle c \rangle_{random}$, which are typical topological properties of small-world networks ^[15].

Table 1 shows that, by increasing $k_{tar}$, $\langle \ell \rangle$ decreases and $\langle c \rangle$ increases. Thus, by raising the average number of links per individual, the distances become shorter and the neighbors become more connected, as expected. Also, $\langle C_c \rangle$ increases and $\langle C_b \rangle$ decreases, because the addition of links improves communication and reduces the relative importance of single nodes. The interval $\Delta k$ also increases with $k_{tar}$. Hence, the higher $k_{tar}$ is, the larger the interval within a pure power law remains accurate to describe $P(k)$.

Table 2 shows that, by increasing $r$, $\langle \ell \rangle$ and $\langle c \rangle$ decrease. Therefore, by enlarging the area where the connections are made, the neighbors can become geographically further apart, which can reduce the path (the minimum number of links) between individuals. However, the neighborhood connectivity is also reduced. Consistently, $\langle C_c \rangle$ increases and $\langle C_b \rangle$ decreases. In addition, $\Delta k$ increases with $r$.

Table 3 shows that $\langle \ell \rangle$ and $\langle C_b \rangle$ increase with $\gamma$ and $\langle c \rangle$ and $\langle C_c \rangle$ decrease with $\gamma$. Also, Figure 4 reveals that $\Delta k$ presents a maximum for $2 {\mathit{\lesssim }} \gamma {\mathit{\lesssim }} 3$, which contains the values of $\gamma$ usually identified in real-world networks ^{[11,12,13,14]}. This is a striking result, since the graphs analyzed here were not created from a preferential attachment rule, which naturally leads to a power law with $\gamma \approx 3$ ^{[11,12,17,18,19,20]}. In fact, it is surprising to find out that, in the primarily locally connected graphs created from Eqs (2.1) and (2.3), the interval $\Delta k$ in which $P(k)$ is a pure power law is greater for $2 {\mathit{\lesssim }} \gamma {\mathit{\lesssim }} 3$. In Figure 4, $r = 10$ and $k_{tar} = 15$; however, the simulations show that this result holds for other values of $r$ and $k_{tar}$.

In short, the developed algorithm can generate graphs with scale-free and small-world features with the desired values of $\gamma$ and $\langle k \rangle$. Such graphs can be used to represent the social contacts in real-world societies. The relation between $\Delta k$ and $\gamma$ was an unexpected outcome. This relation suggests the following conjecture: social networks present $2 {\mathit{\lesssim }} \gamma {\mathit{\lesssim }} 3$ because the connectivity in these networks is influenced by the spatial location of their nodes. Usually, the spatial coordinates of the nodes are neglected in network models. For instance, consider the network of actors and actresses mentioned in Section 1. In general, most of the artists involved in a movie are from the same country, but this fact is ignored in the process of graph construction. Evidently, this geographical issue can affect the formation of links, as shown in this study. Another well-known example in which location clearly matters is the network of human sexual contacts ^[47]. Therefore, the study presented here suggests that the spatial location of the constituent nodes should really be taken into account in social network models based on face-to-face encounters.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Data Availability

The data used to support the findings of this study are available from the first author (newton. licciardijr@gmail.com) upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

LHAM is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under the grant #302946/2022-5. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) - finance code 001.