DANCer: dynamic attributed networks with community structure generation

Abstract

Most networks, such as those generated from social media, tend to evolve gradually with frequent changes in the activity and the interactions of their participants. Furthermore, the communities inside the network can grow, shrink, merge, or split, and the entities can move from one community to another. The aim of community detection methods is precisely to detect the evolution of these communities. However, evaluating these algorithms requires tests on real or artificial networks with verifiable ground truth. Dynamic networks generators have been recently proposed for this task, but most of them consider only the structure of the network, disregarding the characteristics of the nodes. In this paper, we propose a new generator for dynamic attributed networks with community structure that follow the properties of real-world networks. The evolution of the network is performed using two kinds of operations: Micro-operations are applied on the edges and vertices, while macro-operations on the communities. Moreover, the properties of real-world networks such as preferential attachment or homophily are preserved during the evolution of the network, as confirmed by our experiments.

Appendices

Appendix 1: Additional functions

See Table 4.

Table 4 Additional functions used in the algorithms

Appendix 2: User manual

The software DANCer as well as a detailed user manual is available at http://perso.univ-st-etienne.fr/largeron/DANC_Generator/. The user interface of DANCer generator is formed by three views as shown in Fig. 9.

1.1 Graph parameters

The parameters are on the left panel. They correspond to the parameters of algorithms given in Table 2. They are detailed below.

1.1.1 Communities

• K : Number of communities in the first graph;

• n : Number of vertices in the first graph;

• Nb. Rep. : Number of representatives in each community. The higher is the value, the slower is the computation;

• Theta : Percentage of vertices assigned to a random community. The higher is this value, the less likely the community will be homogeneous w.r.t. the attributes.

1.1.2 Attributes

• Nb. Attr. : Number of real attributes associated with the vertices. Each attribute is distributed according to centered normal distribution with mean equals to 0;

• Dev. i : Standard deviation of the ith attribute.

1.1.3 Edges

• Edges Within : Maximum number of within community edges added to a newly inserted vertex;

• Edges Between : Maximum number of between community edges added to a newly inserted vertex

• MTE : Minimum number of edges in the resulting graph (up to a graph where communities are cliques).

1.1.4 Micro-dynamic

• Proba Micro : The probability to perform a micro-update operation;

• Add Vertex : The ratio of vertices created at each timestamp. When set to 1, the number of vertices is doubled at each timestamp;

• Remove Vertex : The ratio of vertices removed at each timestamp;

• Update Attr. : The ratio of vertices having their attribute values updated;

• Add Btw. Edges : The ratio of edges inserted connecting two vertices in different communities;

• Remove Btw. Edges : The ratio of edges removed connecting two vertices in different communities;

• Add Wth. Edges : The ratio of edges inserted connecting two vertices in the same communities;

• Remove Wth. Edges : The ratio of edges removed connecting two vertices in the same communities;

1.1.5 Macro-dynamic

• Timestamps : The number of timestamp (i.e., the number of single graphs generated to form the dynamic network);

• Proba Merge : The probability to perform a merge operation at a single timestamp (i.e., merging two communities into a single one);

• Proba Split : The probability to perform a split operation at a single timestamp (i.e., split one community into two)

• Proba Migrate : The probability to perform a migrate operation at a single timestamp (i.e., migrate vertices from a community to either a new or an existing community).

1.1.6 Network reproduction

• Seed parameter : A seed is used for the random number generator. It allows to reproduce exactly the same network.

1.2 Graph visualization and manipulation

The central part of the user interface as shown in Fig. 9 allows to display the generated network and the changes in its communities at each time step. Each graph in the sequence can be viewed separately in the Graph tab. The sequence of graphs can also be visualized through the timestamp scrollbar at the right side of the panel.

For each graph plotted, in the Graph View tab, we can set different options (see Fig. 32) allowing, for example, to hide or display the edges and vertices through the Graph View section at the right side panel. The graph can then be displayed with different layout options (kamada-kawai, fruchtman-reynolds or self-organizing map) where the sizes of the plotted vertices are chosen according to their degree, age or community membership. Moreover, we can select or filter the displayed vertices according to their different events, as described in the micro-dynamic operations, from the Select Vertices panel.

Fig. 32

Graph options panel

In the plotted graph, vertices of the same color are member of the same community. The user can then interactively select or manipulate a vertex (respectively a group of vertices) using the cursor. The informations for each node (id, degree, attributes) are displayed when a vertex is pointed.

The community dynamics (see Figs. 24, 25, 26) are available through the Community Dynamics tabs, in the central part of the user interface. It displays the size and the evolution of the different communities in the sequence of graphs according to the macro-dynamic operations (split, merge and migrate).

1.3 Graph measures

Several measures, listed in Table 3, such as modularity or homophily are computed on each graph of the dynamic network to describe its properties, notably P1, P2, P3, P4 and P5 detailed in Sect. 2. The changes in these different measures on the sequence of graphs is presented at the bottom of the interface as Fig. 9 shows.

1.3.1 Attribute measures

• Observed homophily : Ratio of edges connecting similar vertices w.r.t. their attribute values;

• Expected homophily : Ratio of pair of similar vertices among all possible pairs of vertices;

  The difference between the expected and observed homophily allows to measure if similar vertices according to the attributes tend to be more connected than dissimilar vertices (cf. P5);

• Within inertia : Measure of the dispersion of the attribute values inside the communities (cf. P4). A low within inertia indicates that the communities are highly homogeneous with regard to the attribute values;

1.3.2 Structural measures

• Modularity : gives the partition modularity measure as defined by [28] (cf. P3);

• Average clustering coefficient : is given as an indication of the transitivity of connections in the network [32];

• Random clustering coefficient : gives the clustering coefficient in a Erdös–Renyi random graph having the same number of vertices and edges;

  The network average clustering coefficient is a measure of the clustering tendency of the network (cf. P3). This observed value can be compared with the expected value computed on a random graph having the same vertex set: An observed value higher than the expected value confirms the community structure;

• Average degree : the average number of neighbors of the vertices (cf. P1);

• Average shortest path length : the average minimum number of hops required to reach two arbitrary vertices (cf. P2). It is not computed when the graph is formed by several disconnected components (i.e., $E^\mathrm{max}_{btw}{}=0$);

• Diameter : length of the longest shortest path between any pair of vertices (cf. P2);

• Nb. edges between : number of edges connecting two vertices belonging to different communities;

• Nb. edges within : number of edges connecting two vertices belonging to the same community (cf. P3);

• Nb. edges : total number of edges in the graph, i.e., $\mathcal {E}$.

1.3.3 Degree distribution

The bottom of the user interface includes also a panel displaying the distribution of vertex degrees on each graph of the sequence as shown in Fig. 33.

Fig. 33

Degree distribution panel

Table 5 Predefined benchmark profiles

1.4 Output files

The generated dynamic network can be saved as a collection of files, one for each time step, under the out directory located in the same working directory as the generator. For each graph of the sequence, the file with the extension “.graph” indicates the composition of the graph (vertices and edges), and the “parameters” file enumerates all the parameters used by the generator.

• Parameters : The parameters are output in a separated file. Each line starts by the parameter name and its value.

• Vertices : In the graph file, the vertices section starts with the line # Vertices. Each consecutive line describes a vertex. A line consists of an integer corresponding to the vertex id, the list of its attribute values separated by “; ” and an integer corresponding to the vertex community id.

• Edges : This section starts with the line # Edges. Each consecutive line corresponds to an edge. A line is composed of two vertex ids separated by a “; ”.

• Measures : the measures are saved in a separated file. Each line gives the measure name and its consecutive values at each time step.

Appendix 3: Benchmark profiles

Table 5 presents a first network (Configuration 1 obtained with parameters given in Table 6), having a good community structure according to the relationships and the attributes and then three other networks in which the link-based structure or the attribute-based structure or the both are weaken. Table 7 presents modifications related to the dynamicity of the first network. The parameter setting is given for each network as well as its characteristics (modularity and within inertia).

Table 6 Default parameters

Table 7 Benchmark profiles derived from configuration 1 by changing dynamicity