Analyzing the Impact of Storage Shortage on Data Availability in Decentralized Online Social Networks

Table 1

The notations that are used in the derivation.

Notations	Descriptions
𝑣𝑡	The user
𝑁	The number of the user’s friends
𝑡	Current time point
𝑡^′	Target time point in near future, 𝑡^′=𝑡+Δ𝑡, where Δ𝑡 is a time duration after 𝑡. We want to predict the state of the DOSN at the time point 𝑡^′
𝑡^𝑢out	The time point at which the user 𝑣𝑡 goes offline
𝑉on	The set of all online users in the friend circle of the user 𝑣𝑡 at current time 𝑡
𝑁on	The number of online users in set 𝑉on
𝑉off	The set of all offline users in the friend circle of the user 𝑣𝑡 at current time 𝑡
𝑁off	The number of offline users in set 𝑉on
𝑡^onin_𝑖	The latest login time of the online user 𝑣𝑖 in 𝑉on before current time 𝑡
𝑡^onout_𝑖	The first logout time of the online user 𝑣𝑖 in 𝑉on after current time 𝑡
𝑡^offout_𝑗	The latest logout time of the offline user 𝑣𝑗 in 𝑉off before current time 𝑡
𝑡^offin_𝑗	The first login time of the offline user 𝑣𝑗 in 𝑉off after current time 𝑡
𝐸login; 𝐸logout	The login and logout events, respectively. When any of these two events occurs, the state of a user changes from OFFLINE to ONLINE or from ONLINE to OFFLINE
𝑡on; 𝑓on𝑡on; 𝐹on𝑡on	The time duration of a user being online continuously (i.e., the time duration from an 𝐸login event to the following 𝐸logout event), which is a random variable and whose probability density function and probability distribution function are denoted by 𝑓on𝑡on and 𝐹on𝑡on, respectively
𝑡off; 𝑓off𝑡off; 𝐹off𝑡off	The time duration of a user being offline, which is also a random variable and whose probability density function and probability distribution function are denoted by 𝑓off𝑡off and 𝐹off𝑡off, respectively
𝑥; 𝑃pu(𝑥,𝑡)	The number of times that the user publishes the data, which is a discrete random variable and whose probability density function in a duration t is denoted by 𝑃pu(𝑥,𝑡)
𝑎	The statistical average size of the data published by the user each time. 𝑎 is a constant
𝑘	The replication degree, that is, the number of replicas created for each data item
𝑡𝑡𝑙	The publishing time of the oldest data stored in the storage pool
SS	The total storage capacity contributed by all online friends
S	The maximum storage capacity that each friend is able to contribute