Associative nature of event participation dynamics: A network theory approach

The affiliation with various social groups can be a critical factor when it comes to quality of life of each individual, making such groups an essential element of every society. The group dynamics, longevity and effectiveness strongly depend on group’s ability to attract new members and keep them engaged in group activities. It was shown that high heterogeneity of scientist’s engagement in conference activities of the specific scientific community depends on the balance between the numbers of previous attendances and non-attendances and is directly related to scientist’s association with that community. Here we show that the same holds for leisure groups of the Meetup website and further quantify individual members’ association with the group. We examine how structure of personal social networks is evolving with the event attendance. Our results show that member’s increasing engagement in the group activities is primarily associated with the strengthening of already existing ties and increase in the bonding social capital. We also show that Meetup social networks mostly grow trough big events, while small events contribute to the groups cohesiveness.


Network filtering
The Meetup dataset, containing information on organised events by certain Meetup group and members of that group that confirmed attendance at an event, allows us to construct member-event bipartite network with adjacency matrix B. For each member i ∈ {1, . . . , N } and event l ∈ {1, . . . , M }, matrix element B il = 1 if member i participated event l, or B il = 0, otherwise. The degree of member i is defined as the total number of events member i participated in, k i = l B il , and similarly, the degree of event l is defined as the total number of members attended the event, d l = i B il . Given the matrix B, social relations between Meetup members can be analysed using the projected unipartite member-member weighted network, where the weight of the link between two members is equal to the number of events they both attended. The observed weighted network is the dense network where some of the non-zero edges can be a matter of coincidence. For instance, two frequent attendees can meet several times due to a chance not due to the fact that there is some relation between them, which means that the connection between them is not significant for our analysis. Also, the connections between members that meet at big events and never again can not be regarded as social relations and thus they need to be excluded from our analysis. To make the distinction between significant and non-significant edges is nontrivial task [1][2][3]. Here we use the method which enables us to calculate the significance of the link between two members based on the probability for that link to occur in random network. As a null model we use configuration model of bipartite network [2][3][4][5].
First we describe general framework for constructing randomized network ensemble G with given structural constraints {x i }. The maximum-entropy probability of the graph in the ensemble, P (G), is given by where the λ i are Lagrangian multipliers and the partition function of these network ensembles is defined as The ensemble average of a graph property x i can be expressed as Then the constants λ i could be determined from (3).
Let us now consider configuration model of the member-event bipartite network with given degree sequence k i and d l . In this case the partition function can be written as The Lagrangian multipliers α i and β l are determined from Finally, we can calculate the probability p il that a member i attended event l. If we define coupling parameter λ il = α i +β l and write partition function in the form then, it holds Now, when the probability p il is given, the probability that members i and j both participated in event l is p ij (l) = p il p jl . The probability P ij (w) of having an edge of the weight w between the nodes i and j is given by Poisson binomial distribution where M w is the subset of w events that can be chosen from given M events [2,3,6]. We use DFT-CF method (Discrete Fourier Transform of characteristic function), proposed in [7], to compute Poisson binomial distribution. On the basis of P ij (w), we define p-value as the probability that edge (i, j) has weight higher or equal than w ij The edge (i, j) will be considered statistically significant if p-value(w ij ) ≤ α. In our case, threshold α = 0.05. If p-value(w ij ) > α, the edge (i, j) should be removed as spurious statistical connection between members (set w ij = 0).
We fit exponential function e −λx , power law function x −γ and truncated power law x −α e −Bx to the probability distribution of the total number of participations in group events using the maximum-likelihood fitting method [8]. It is evident from Fig A that the distribution does not follow exponential fit. We compare how the power law and the truncated power law distribution, which are the nested versions of each other, fit the data by calculating the log likelihood ratio R and π-value (see Ref. [8]). Here, the negative value of R indicates that the truncated power law is a superior fit to the power law. Additionally, when the value of R tends to 0, one can use π-value. The small π-value indicates that the power law distribution can be excluded. Table A shows that the truncated power law is a superior fit compared to power law for all four empirical distributions.

Data randomization
We randomize event participation patterns preserving the total number of participation for each member and the number of participants per event [9]. Firstly, we choose at random two members, i and j, and for each of them we choose randomly an event they participated, l i and l j . If l i = l j , and i didn't participate at event l j and j didn't participate at event l i , they are swapped. We perform 10 × (number of participants) × (number of events) swaps. The randomization of the event participation times induces transformations of associated weighted network. The number of participants at some event will stay the same, but participants will differ, resulting in weight increase of certain edges and likewise in weight decrease of some other edges in weighted network. The total weight of the network will be preserved.
For each Meetup group we generate 100 randomized weighted networks and filter out non-significant edges.