Two models of sparse and clustered dynamic networks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909394731532288 |
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| author | Bloznelis, Mindaugas Marma, Dominykas |
| author_facet | Bloznelis, Mindaugas Marma, Dominykas |
| contents | We present two models of sparse dynamic networks that display transitivity - the tendency for vertices sharing a common neighbour to be neighbours of one another. Our first network is a continuous time Markov chain $G=\{G_t=(V,E_t), t\ge 0\}$ whose states are graphs with the common vertex set $V=\{1,\dots, n\}$. The transitions are defined as follows. Given $t$, the vertex pairs $\{i,j\}\subset V$ are assigned independent exponential waiting times $A_{ij}$. At time $t+\min_{ij} A_{ij}$ the pair $\{i_0,j_0\}$ with $A_{i_0j_0}=\min_{ij} A_{ij}$ toggles its adjacency status. To mimic clustering patterns of sparse real networks we set intensities $a_{ij}$ of exponential times $A_{ij}$ to be negatively correlated with the degrees of the common neighbours of vertices $i$ and $j$ in $G_t$. Another dynamic network is based on a latent Markov chain $H=\{H_t=(V\cup W, E_t), t\ge 0\}$ whose states are bipartite graphs with the bipartition $V\cup W$, where $W=\{1,\dots,m\}$ is an auxiliary set of attributes/affiliations. Our second network $G'=\{G'_t =(E'_t,V), t\ge 0\}$ is the affiliation network defined by $H$: vertices $i_1,i_2\in V$ are adjacent in $G'_t$ whenever $i_1$ and $i_2$ have a common neighbour in $H_t$. We analyze geometric properties of both dynamic networks at stationarity and show that networks possess high clustering. They admit tunable degree distribution and clustering coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two models of sparse and clustered dynamic networks Bloznelis, Mindaugas Marma, Dominykas Probability Social and Information Networks 05C82, 91D30, 05C80 We present two models of sparse dynamic networks that display transitivity - the tendency for vertices sharing a common neighbour to be neighbours of one another. Our first network is a continuous time Markov chain $G=\{G_t=(V,E_t), t\ge 0\}$ whose states are graphs with the common vertex set $V=\{1,\dots, n\}$. The transitions are defined as follows. Given $t$, the vertex pairs $\{i,j\}\subset V$ are assigned independent exponential waiting times $A_{ij}$. At time $t+\min_{ij} A_{ij}$ the pair $\{i_0,j_0\}$ with $A_{i_0j_0}=\min_{ij} A_{ij}$ toggles its adjacency status. To mimic clustering patterns of sparse real networks we set intensities $a_{ij}$ of exponential times $A_{ij}$ to be negatively correlated with the degrees of the common neighbours of vertices $i$ and $j$ in $G_t$. Another dynamic network is based on a latent Markov chain $H=\{H_t=(V\cup W, E_t), t\ge 0\}$ whose states are bipartite graphs with the bipartition $V\cup W$, where $W=\{1,\dots,m\}$ is an auxiliary set of attributes/affiliations. Our second network $G'=\{G'_t =(E'_t,V), t\ge 0\}$ is the affiliation network defined by $H$: vertices $i_1,i_2\in V$ are adjacent in $G'_t$ whenever $i_1$ and $i_2$ have a common neighbour in $H_t$. We analyze geometric properties of both dynamic networks at stationarity and show that networks possess high clustering. They admit tunable degree distribution and clustering coefficients. |
| title | Two models of sparse and clustered dynamic networks |
| topic | Probability Social and Information Networks 05C82, 91D30, 05C80 |
| url | https://arxiv.org/abs/2411.12055 |