# Strategies of Success for Social Networks: Mermaids and Temporal Evolution

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Network Growth Models

- Random order. Every edge will have the same probability $p=\frac{1}{\left|E\right|-|{E}_{i}|}$ to be selected during the growth process. This rule, as many studies showed, is far from real. However, it is a good candidate for a baseline.
- Aristocratic order. This rule is based on the preferential attachment process (cf. [23,24]) where older nodes have a higher probability of attracting new links. The process selects edges by choosing a source node, according to the degree, and a target node, randomly chosen on the available neighbors’ list of nodes. By randomly choosing target nodes, low degree nodes can acquire new links as well. More formally, the probability of selecting the source node is the following:$${p}_{u}=\frac{1+deg\left(u\right)\xb7\alpha}{{\sum}_{j\in V}(1+deg\left(j\right)\xb7\alpha )}$$
- Social order. This rule is inspired by the local clustering of small world networks (also known as triadic closure), and in particular from the observation that two friends of a person are likely to know each other (see [25]). This rule considers it more likely that the edges that close triangles will be selected. Edges that make more than one triadic closure are inserted sooner into the network than others. More formally, the probability of edge $(u,v)$ of being selected is the following:$${p}_{u,v}=\frac{1+soc(u,v)\xb7\alpha}{{\sum}_{j\ne k\in V}(1+soc(j,k)\xb7\alpha )}$$

#### Evolutionary Models: Serial and Parallel

Algorithm 1: Parallel networks simulation |

## 4. Mermaids

#### 4.1. Handling Mermaids

- m specifies the number of mermaids,
- a is the mermaids’ ability of attracting new edges (i.e., to generate interest in the community),
- d is the operational timespan of mermaids.

- during the network evolution, edges between mermaids $\left\{({s}_{i},{s}_{j})\right|{s}_{i}\in {V}^{s},{s}_{j}\in {V}^{s}\}$ are not allowed,
- mermaids $\{{s}_{1},{s}_{2},\dots ,{s}_{m}\}$ are active at the beginning of simulation only, i.e., from time ${t}_{0}$ to ${t}_{d}$.

Algorithm 2: Accelerated networks simulation with mermaids |

#### 4.2. Managing Cost

## 5. Experimental Results

#### 5.1. Datasets

#### 5.2. Results

- Q0
- Does each rule behave equally in the inertial (serial) context? What happens in the accelerated context?
- Q1
- How do the same cost configurations influence efficiency?
- Q2
- How do parameter variations influence global efficiency?
- Q3
- How much do we have to invest in special nodes?

**Q0: Unfolded serial setting**. Figure 7 shows the unfolded behavior of the systems for the three proposed rules, namely: random, aristocratic, and social. Each curve represents global efficiency ${E}_{glob}$ of the temporal networks that have been created by adding one edge at time. The plots allow for interesting observations. First, we note that until one sixth of the complete spectrum, each rule produces an indistinguishable behavior probably due to weak network structure. After that point, the cumulative effect of drawing edges in different ways starts to appear. The behavior detected is super-linear for the aristocratic rule meaning that preferential attachment is an effective way to boost network efficiency in networks. Conversely, with the social rule we observe a weak sub-linear increase ideally meaning that triadic closure is not the only key ingredient for network evolution. Linear increase is then detected for random rule. To avoid the bias of randomness, we made 100 simulations and then the averaged results are considered. Standard deviations are small and, therefore, are not plotted in favor of clearer plots.

**Q1: Same cost configurations**. The following set of figures (from Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16) represents how the same cost configurations affect the global efficiency. In particular, we consider the cost levels ${C}_{s}$ that have at least two configurations $\mu $, namely 1200 and 2400. Table 3 collects all the possible configurations with a specific cost.

**Q2: Parameters variation**. In the following experiments, we investigated the effects of parameters’ variation in configurations. In particular, we fixed the number of mermaids ($m=6$ and $m=12$) and checked the performance of other configurations compared to the baseline (that are $(6,10,10)$ and $(12,10,10)$ respectively).

**Q3: Trade-off between the benefit of investing on mermaids and the cost**. In the plots presented previously, we described how the timespan needed to get the reference efficiency varies according to ${C}_{s}$. Figure 23 and Figure 24 (rightmost panels) show ${T}_{min}$ as a function of ${C}_{s}$ and this allows to describe more quantitatively the benefit of investing on mermaids. In fact, plots clearly show that this is not linear as one might guess, instead it is inversely proportional as ${C}_{s}$. We think this is probably due to the system saturation. In other words, the network is not able to respond to high level of exogenous stimuli from mermaids resulting in performances that are comparatively similar to those obtained with lower cost configurations.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Example of social rule. The figure represents a hypothetical snapshot of graph G at time t during network evolution. Straight links indicate already existing edges whereas the dashed lines indicate the ones that will be added in the following steps. Edge $(a,c)$ closes three triads $(a,d)(d,c)$, $(a,b)(b,c)$, and $(a,f)(f,c)$, whereas $(d,f)$ and $(b,e)$ only close two and one triangle respectively. Therefore, the probability of been selected at time $t+1$ is $0.5$, $0.33$ and $0.16$ respectively.

**Figure 2.**How the scale parameter C influences the global efficiency curve. It directly impacts the time span needed to get the referring network ${E}_{glob}$.

**Figure 3.**Example of online social network with normal users and mermaids ${V}^{s}=\{{s}_{1},{s}_{2},{s}_{3}\}$. Nodes inside the rounded rectangle belong to the users’ graph. Mermaids aim to connect to nodes of the users’ network in order to increase overall utilization (for instance edge $({s}_{1},c)$).

**Figure 4.**Examples of the word of mouth model (

**a**,

**b**). ${s}_{i}\in {V}^{s}$ and $\{b,c,e,m\}\in V$. Highlighted links indicate the edges that have just been added to the network, straight lines are edges inserted in the previous steps, and dashed ones represent the possible options for new links (and that have a higher probability to be chosen).

**Figure 5.**Cumulative degree distributions ${P}_{cum}\left(k\right)$ of Communities (

**left panel**) and VirtualTourist (

**right panel**). k is the degree, and $\alpha $ is the coefficient of the fitting (dashed) line ${k}^{-\alpha}$. The Figure clearly shows a power law behavior in the degree and $\alpha $ is approximately equal to $2.5$ and $2.7$, respectively.

**Figure 6.**Node degree correlations in Communities (

**left panel**) and VirtualTourist (

**right panel**) online social networks. $\langle {k}_{nn}\rangle $ is the average degree of first neighbors. Figures show a negative correlation. In fact, Pearson correlation is equal to $-0.59$ and $-0.30$ respectively. The inset graphs contain the same data but plotted in linear axes.

**Figure 7.**Effect of serial network simulation for Communities (

**left panel**) and VirtualTourist (

**right panel**). After an initial time span (approximately one sixth of the entire simulation time), the preferential attachment rule (aristocratic) outperforms the others.

**Figure 8.**Effect of parallel network simulation for Communities (

**left panel**) and VirtualTourist (

**right panel**). These plots help us to understand how parallel links’ creation modifies the dynamics of online social systems. Surprisingly, preferential attachment (that outperforms other rules in inertial setting) is the slowest, obtaining bad performance in terms of time needed to reach the target efficiency. Curves start at simulation time ${t}_{0}$, but we cropped the points for low values of ${E}_{glob}$ for graphical clarity. Standard deviations are very small and are not plotted for graphical reasons.

**Figure 9.**Effect of simultaneous network simulation in randomized version of Communities (

**left panel**) and VirtualTourist (

**right panel**). Curve starts at simulation time ${t}_{0}$, but we cropped the points for low values of ${E}_{glob}$ for graphical clarity. Standard deviation is very small and therefore is not plotted.

**Figure 10.**The figures describe the benefit of higher attractiveness for the same cost configurations of the Communities online social network. In particular, we selected broadcast model, random (

**top panels**), and aristocratic (

**bottom panels**) rules. Two cost levels have been considered: ${C}_{s}=1200$ (

**left panels**) and ${C}_{s}=2400$ (

**right panels**). Configurations $(6,20,10)$ and $(12,20,10)$ outperform the others and in this case network efficiency will start to increase earlier, regardless of the growing rule of the users’ network.

**Figure 11.**Comparison between the same cost of configurations of the Communities online social network. We consider two cost level ${C}_{s}=1200$ (

**left panels**), ${C}_{s}=2400$ (

**right panels**), and random, aristocratic, and social rules. All plots refer to the word of mouth model. We clearly see that network efficiency increases faster in configurations that have a higher value of attractiveness, no matter what cost level or rule has been selected.

**Figure 12.**Accelerated analysis with mermaids, random, and aristocratic, social rules, preferential model, for the Communities social network with cost ${C}_{s}=1200$ (

**left panels**) and ${C}_{s}=2400$ (

**right panels**).

**Figure 13.**Spread of ${E}_{glob}$ curves over multiple runs of simulations (aristocratic rule) on the same VT dataset, with same cost configurations: $\mu =(6,10,20)$ in left panel (

**a**), $\mu =(6,20,10)$ in center panel (

**b**), $\mu =(12,10,10)$ in right panel (

**c**).

**Figure 14.**Behavior of the network’s ${E}_{glob}$ with two different cost levels: ${C}_{s}=1200$ (

**left panels**) and ${C}_{s}=2400$ (

**right panels**) for the VirtualTourist social network, broadcast model. In total, six configurations are considered. The one that has higher attractiveness is the favored one because can reach the efficiency of the original network faster than the others.

**Figure 15.**Same cost configurations for VirtualTourist, C

_{s}= 1200 (

**left panels**) and C

_{s}= 2400 (

**right panels**), word of mouth model. For each cost C

_{s}, three configurations are then considered. In all experiments, the configuration that performs better is the one that has fewer mermaids and higher attractiveness (or equivalently that last more). In accordance with the results of accelerated analysis with no mermaids (Figure 7), random and social rules attain the target efficiency in fewer steps than the aristocratic rule.

**Figure 16.**Accelerated analysis with mermaids, random, aristocratic, and social rules, preferential model, in the VirtualTourist social network with cost equal to ${C}_{s}=1200$ (

**left panels**) and ${C}_{s}=2400$ (

**right panels**). We clearly see that the configurations with higher attractiveness reach faster the target efficiency, regardless of the users’ growing rules.

**Figure 17.**Accelerated analysis with mermaids, broadcast model (for mermaids’ dynamics), Communities online social network, fixing $m=6$ (

**top panels**), and $m=12$ (

**bottom panels**). Three cost levels are then considered for each plot.

**Figure 18.**Accelerated analysis with mermaids fixing $m=6$ (

**top plots**) and $m=12$ (

**bottom plots**), random, aristocratic, and social rules, word of mouth model (for mermaids’ dynamics). Communities online social network.

**Figure 19.**Accelerated analysis with mermaids fixing the number of mermaids to $m=6$ and $m=12$, random, aristocratic, and social rules. The mermaids’ dynamics evolve according to the preferential model. Communities online social network.

**Figure 20.**Effect on parameters’ variation on the configurations fixing the number of mermaids to $m=6$ (

**top panels**) and $m=12$ (

**bottom panels**). Four cost levels are then considered in each plot, from 600 to 4800. Broadcast model, VirtualTourist online social network.

**Figure 21.**Accelerated analysis with mermaids fixing $m=6$ and $m=12$, random, aristocratic, and social rules, word of mouth model (for mermaids’ dynamics), VirtualTourist online social network.

**Figure 22.**Accelerated analysis with mermaids fixing $m=6$ and $m=12$, with random, aristocratic, and social rules, preferential model (for mermaids’ dynamics), VirtualTourist online social network.

**Figure 23.**Scatter plots between cost ${C}_{s}$ and ${T}_{min}$ in Communities. ${T}_{min}$ represents the minimum number of steps (in simulated time units) necessary to get the target efficiency (${E}_{glob}$). We consider three thresholds: half efficiency (leftmost column), one third (centermost column), and no threshold (rightmost column). Every row represents a different mermaids’ model namely broadcast, word of mouth, and preferential.

**Figure 24.**Scatter plots between cost ${C}_{s}$ and ${T}_{min}$ in VirtualTourist. ${T}_{min}$ represent the minimum number of steps (in simulated time units) necessary to get the target efficiency (${E}_{glob}$). We consider three thresholds: half (leftmost column), one third (centermost column) and full (rightmost column) efficiency. Every row represents a different mermaids’ model, namely broadcast, word of mouth, and preferential.

**Table 1.**Statistical features of the Communities and Virtualtourist online social networks, together with randomized versions of the same networks: number of nodes $\left|V\right|$, number of edges $\left|E\right|$, average node degree $\langle k\rangle $, maximum degree ${k}_{max}$, average shortest path L and average clustering coefficient C (for the largest connected component), global efficiency ${E}_{glob}$, local efficiency ${E}_{loc}$, cost, cost over efficiency, exponent of the cumulative degree distribution $\gamma $, number of connected clusters $\#CC$, and the correlation pattern $\rho $.

Feature | Communities | Virtual Tourist | Randomized CM | Randomized VT |
---|---|---|---|---|

$\left|V\right|$ | 12,479 | 57,639 | 12,479 | 57,639 |

$\left|E\right|$ | 60,209 | 211,415 | 60,209 | 211,415 |

$\langle k\rangle $ | 9.64 | 7.34 | 9.64 | 7.34 |

${k}_{max}$ | 656 | 963 | 24 | 21 |

L | 4.18 | 4.95 | 4.42 | 5.72 |

C | 0.1067 | 0.04425 | 0.0006 | 0.0001 |

${E}_{glob}$ | 0.238683 | 0.201822 | 0.23296 | 0.17817 |

${E}_{loc}$ | 0.131466 | 0.056248 | 0.00074 | 0.00013 |

$Cost$ (density) | 0.00077 | 0.00013 | 0.00077 | 0.00013 |

$Cost/{E}_{glob}$ | 0.00324 | 0.00063 | 0.00332 | 0.00073 |

$\gamma $ | 2.5 | 2.7 | ∼0 | ∼0 |

$\#CC$ | 161 | 2078 | 3 | 43 |

$\rho $ | −0.027 | −0.027 | −0.002 | 0.00082 |

**Table 2.**Summary of attractiveness values used during network simulations in Communities (first four rows) and VirtualTourist (last four rows). $\left|V\right|$ is the number of nodes, m the number of mermaids, $q\left(s\right)$ is the weight assigned to mermaids, ${a}_{n}$ is the attractiveness of normal nodes, and ${a}_{s}$ is the attractiveness of mermaids.

$\left|\mathit{V}\right|$ | m | $\mathit{q}\left(\mathit{s}\right)$ | ${\mathit{a}}_{\mathit{n}}$ | ${\mathit{a}}_{\mathit{s}}$ |
---|---|---|---|---|

12,479 | 6 | 10 | 0.000079751 | 0.000797512 |

12,479 | 6 | 20 | 0.000079371 | 0.001587428 |

12,479 | 12 | 10 | 0.000079371 | 0.000793714 |

12,479 | 12 | 20 | 0.000078623 | 0.001572451 |

57,639 | 6 | 10 | 0.000017331 | 0.000173313 |

57,639 | 6 | 20 | 0.000017313 | 0.000346266 |

57,639 | 12 | 10 | 0.000017313 | 0.000173133 |

57,639 | 12 | 20 | 0.000017277 | 0.000345548 |

**Table 3.**List of the all possible configurations available with $m=6,12$, $a=10,20$, and $d=10,20$ with the corresponding costs.

${\mathit{C}}_{\mathit{s}}$ | Configurations $\mathit{\mu}$ | ||
---|---|---|---|

600 | (6,10,10) | ||

1200 | (6,10,20) | (6,20,10) | (12,20,20) |

2400 | (12,10,20) | (12,20,10) | (6,20,20) |

4800 | (12,20,20) |

**Table 4.**Summary of ${T}_{min}$, i.e., the minimum number of simulated steps to get the original ${E}_{glob}$, for all accelerated simulations in Communities (CM), VirtualTourist (VT), and randomized version of both networks. Random (rnd), aristocratic (ari), and social (soc) rules are considered.

CM | VT | |||||
---|---|---|---|---|---|---|

rnd | soc | ari | rnd | soc | ari | |

Normal | 1384 | 1333 | 1931 | 3130 | 2996 | 7505 |

Randomized | 2585 | 2571 | 2294 | 13,718 | 13,704 | 12,003 |

**Table 5.**Summary of the average ${T}_{min}$ in all configurations of the number of mermaids (m), attractiveness (a), and length of time (d) of accelerated analysis with and without mermaids. Mermaid’s dynamics are broadcast (bro), word of mouth (word) and preferential (pref), combined with users’ dynamics random (rnd), aristocratic (ari) and social (soc). Communities online social network.

CM | bro rnd | bro ari | word rnd | word ari | word soc | pref rnd | pref ari | pref soc |
---|---|---|---|---|---|---|---|---|

(no mermaids) | 1381 | 1930 | 1381 | 1930 | 1328 | 1381 | 1930 | 1328 |

(6,10,10) | 112.92 | 128.16 | 111.66 | 126.95 | 113.01 | 106.82 | 118.48 | 108.16 |

(6,10,20) | 73.10 | 73.67 | 73.80 | 73.80 | 74.93 | 72.45 | 71.28 | 74.20 |

(6,20,10) | 68.28 | 68.29 | 68.61 | 68.61 | 69.90 | 67.38 | 66.88 | 69.08 |

(6,20,20) | 58.14 | 56.75 | 58.52 | 56.65 | 59.09 | 57.61 | 55.02 | 59.22 |

(12,10,10) | 72.35 | 73.32 | 72.07 | 74.23 | 73.34 | 70.51 | 70.39 | 72.34 |

(12,10,20) | 60.29 | 58.69 | 60.44 | 58.09 | 62.01 | 59.25 | 56.70 | 60.70 |

(12,20,10) | 55.08 | 52.73 | 55.21 | 52.92 | 56.12 | 53.90 | 51.90 | 55.60 |

(12,20,20) | 49.44 | 47.90 | 50.05 | 48.60 | 50.81 | 49.23 | 47.11 | 50.66 |

**Table 6.**Summary of the average ${T}_{min}$ in all configurations $(m,a,d)$ of accelerated analysis with and without mermaids. Mermaid’s dynamics are broadcast (bro), word of mouth (word) and preferential (pref), combined with users’ dynamics random (rnd), aristocratic (ari) and social (soc). VirtualTourist online social network.

VT | bro rnd | bro ari | word rnd | word ari | word soc | pref rnd | pref ari | pref soc |
---|---|---|---|---|---|---|---|---|

(no mermaids) | 3120 | 7496 | 3120 | 7496 | 2987 | 3120 | 7496 | 2987 |

(6,10,10) | 1051.37 | 2272.84 | 1041.35 | 2215.34 | 999.88 | 752.16 | 1607.41 | 771.30 |

(6,10,20) | 283.12 | 461.48 | 262.40 | 428.12 | 264.58 | 243.19 | 375.16 | 242.17 |

(6,20,10) | 293.92 | 512.32 | 276.52 | 454.18 | 276.97 | 253.02 | 392.34 | 245.84 |

(6,20,20) | 137.42 | 163.84 | 134.42 | 156.91 | 136.71 | 130.49 | 148.74 | 132.45 |

(12,10,10) | 431.78 | 807.28 | 432.89 | 725.23 | 415.61 | 311.19 | 521.40 | 333.13 |

(12,10,20) | 147.04 | 180.44 | 144.94 | 174.75 | 146.64 | 138.53 | 163.60 | 140.48 |

(12,20,10) | 145.96 | 184.40 | 143.78 | 178.64 | 146.17 | 137.33 | 168.59 | 138.82 |

(12,20,20) | 98.46 | 98.70 | 96.63 | 95.89 | 98.95 | 94.74 | 93.39 | 96.72 |

**Table 7.**Summary of ${T}_{min}^{\prime}\left({C}_{s}\right)$ and $\beta $ calculated for different threshold values of $E\left(G\right)$ and ${C}_{s}$.

${\mathit{T}}_{\mathbf{min}}^{\prime}\left(1200\right)$ | $\mathit{\beta}$ | ${\mathit{T}}_{\mathbf{min}}^{\prime}\left(2400\right)$ | $\mathit{\beta}$ | ${\mathit{T}}_{\mathbf{min}}^{\prime}\left(4800\right)$ | $\mathit{\beta}$ | |
---|---|---|---|---|---|---|

$1/1\xb7E\left(G\right)$ | $-0.05875$ | $17.02$ | $-0.01104$ | $90.56$ | $-0.001145$ | $872.72$ |

$1/2\xb7E\left(G\right)$ | $-0.055$ | $18.12$ | $-0.0094$ | $106.38$ | $-0.00041\overline{6}$ | 2400 |

$1/3\xb7E\left(G\right)$ | $-0.054$ | $18.23$ | $-0.0092$ | $108.10$ | $-0.0003$ | 3333 |

**Table 8.**Total cost ${C}_{t}={C}_{s}+\beta \xb7{T}_{min}\left({C}_{s}\right)$ as a function of ${C}_{s}$ and different values of $\beta $. First row, from left to right: $\beta =18.12$ (

**a**), $106.38$ (

**b**), and 2400 (

**c**) for half ${E}_{glob}$. Second row: $\beta =18.23$ (

**d**), $108.10$ (

**e**), and 3333 (

**f**) for one third of the efficiency. Third row: $\beta =17.02$ (

**g**), $90.56$ (

**h**), and $872.72$ (

**i**) with no threshold at all. Once $\beta $ is known, our method estimates the best ${C}_{s}$ to obtain the minimum cost. For instance, suppose that the cost per unit time $\beta $ is approximately equal to 90 (with no threshold on ${E}_{glob}$), the configurations that achieve the minimum cost are those with ${C}_{s}\in [1200,2400]$. Indeed, since there are many configurations with the same cost level, the ones that perform better (outlined in bold) are those with the higher value of attractiveness.

(a) | (b) | (c) | |||
---|---|---|---|---|---|

${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ |

600 | 1732 | 600 | 7248 | 600 | 150,600 |

1200 | 1732 | 1200 | 4327 | 1200 | 71,760 |

2400 | 2728 | 2400 | 4327 | 2400 | 45,888 |

4800 | 5110 | 4800 | 6621 | 4800 | 45,888 |

(d) | (e) | (f) | |||

${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ |

600 | 1700 | 600 | 7126 | 600 | 201,833 |

1200 | 1700 | 1200 | 4168 | 1200 | 92,733 |

2400 | 2698 | 2400 | 4168 | 2400 | 56,933 |

4800 | 5085 | 4800 | 6490 | 4800 | 56,933 |

(g) | (h) | (i) | |||

${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{s}}$ | ${\mathit{C}}_{\mathit{t}}$ |

600 | 2314 | 600 | 9724 | 600 | 88,527 |

1200 | 2314 | 1200 | 7132 | 1200 | 58,363 |

2400 | 3289 | 2400 | 7132 | 2400 | 48,000 |

4800 | 5642 | 4800 | 9283 | 4800 | 48,000 |

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## Share and Cite

**MDPI and ACS Style**

Marchiori, M.; Possamai, L. Strategies of Success for Social Networks: Mermaids and Temporal Evolution. *Future Internet* **2020**, *12*, 25.
https://doi.org/10.3390/fi12020025

**AMA Style**

Marchiori M, Possamai L. Strategies of Success for Social Networks: Mermaids and Temporal Evolution. *Future Internet*. 2020; 12(2):25.
https://doi.org/10.3390/fi12020025

**Chicago/Turabian Style**

Marchiori, Massimo, and Lino Possamai. 2020. "Strategies of Success for Social Networks: Mermaids and Temporal Evolution" *Future Internet* 12, no. 2: 25.
https://doi.org/10.3390/fi12020025